introduction to string field theory - sorbonne-universiteerbin/files/reviews/string_theory.pdf ·...

String theory: a field theory perspective

Harold Erbin1*

*Dipartimento di Fisica, Università di Torino Infn Sezione di Torino and Arnold–Regge Center,Via Pietro Giuria 1, I-10125 Torino, Italy

*Ludwig–Maximilians–Universität, Theresienstraße 37, 80333 München, Germany

31st March 2020

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Abstract

This review describe different topics related to string field theory (SFT) and its applic-ations. The aim is to cover all the prerequisite materials to make the review self-contained.

It is a work in continuous progress. As such, the sections have various levels of comple-tion and sometimes incoherent normalizations. Major and consistent milestones should bepublished by Springer. Any comment or correction is most welcome.

The text is organized on three levels: the main content (augmented with examples),computations, and remarks. The latter two levels can be omitted in a first lecture: thecomputations and remarks are clearly separated from the text (respectively by a verticalline on the left and by italics) such that the reader can more directly navigate the text tofind the formula and concepts she is interested in.

This review grew up from lecture notes for a course given at the Ludwig-Maximilians-Universität LMU (winter semesters 2017–2018 and 2018–2019).

Contents

Preface 10

1 Introduction 131.1 Strings, a distinguished theory . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Worldvolume and spacetime descriptions . . . . . . . . . . . . . . . . . . . . . 17

1.2.1 Classical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.2 Quantum aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3 String theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.3.2 Classification of superstring theories . . . . . . . . . . . . . . . . . . . 341.3.3 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.4 String field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.4.1 String field action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.4.2 Expression with spacetime fields . . . . . . . . . . . . . . . . . . . . . 461.4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

1.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

I Point-particle 50

2 Classical relativistic point-particle 512.1 Proper-time action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.2 Parametrization and non-linear action . . . . . . . . . . . . . . . . . . . . . . 52

2.2.1 Induced metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.2.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3 Polynomial action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.3.1 Dirac procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

II Bosonic string theory 56

3 Worldsheet path integral: vacuum amplitudes 573.1 Worldsheet action and symmetries . . . . . . . . . . . . . . . . . . . . . . . . 573.2 Path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3 Faddeev–Popov gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3.1 Metrics on Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . 65

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3.3.2 Reparametrizations and analysis of P1 . . . . . . . . . . . . . . . . . . 703.3.3 Weyl transformations and quantum anomalies . . . . . . . . . . . . . . 753.3.4 Ambiguities, ultralocality and cosmological constant . . . . . . . . . . 773.3.5 Gauge-fixed path integral . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4 Ghost action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.4.1 Actions and equations of motion . . . . . . . . . . . . . . . . . . . . . 793.4.2 Weyl ghost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.4.3 Zero-modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.5 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4 Worldsheet path integral: scattering amplitudes 864.1 Scattering amplitudes on moduli space . . . . . . . . . . . . . . . . . . . . . . 86

4.1.1 Vertex operators and path integral . . . . . . . . . . . . . . . . . . . . 864.1.2 Gauge fixing: general case . . . . . . . . . . . . . . . . . . . . . . . . . 894.1.3 Gauge fixing: 2-point amplitude . . . . . . . . . . . . . . . . . . . . . 92

4.2 BRST quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.1 BRST symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.2 BRST cohomology and physical states . . . . . . . . . . . . . . . . . . 97

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5 Worldsheet path integral: complex coordinates 1005.1 Geometry of complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2 Complex representation of path integral . . . . . . . . . . . . . . . . . . . . . 1035.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Introduction to off-shell string theory 1066.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.1.1 3-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.1.2 4-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2 Off-shell states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.2.1 Open string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2.2 Closed string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3 Off-shell amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.3.1 Amplitudes from the marked moduli space . . . . . . . . . . . . . . . 1136.3.2 Local coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7 Geometry of moduli spaces and Riemann surfaces 1177.1 Parametrization of Pg,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.2 Tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207.3 Plumbing fixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.3.1 Separating case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227.3.2 Non-separating case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.3.3 Decomposition of moduli spaces and degeneration limit . . . . . . . . 1267.3.4 Stubs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129


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8 Off-shell amplitudes 1328.1 Cotangent spaces and amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 132

8.1.1 Construction of forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328.1.2 Amplitudes and surface states . . . . . . . . . . . . . . . . . . . . . . . 134

8.2 Properties of forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368.2.1 Vanishing of forms with trivial vectors . . . . . . . . . . . . . . . . . . 1378.2.2 BRST identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8.3 Properties of amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.3.1 Restriction to Pg,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.3.2 Consequences of the BRST identity . . . . . . . . . . . . . . . . . . . 141

8.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9 Amplitude factorization and Feynman diagrams 1439.1 Amplitude factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.1.1 Separating case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439.1.2 Non-separating case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9.2 Feynman diagrams and Feynman rules . . . . . . . . . . . . . . . . . . . . . . 1489.2.1 Feynman graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1499.2.2 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1509.2.3 Fundamental vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1529.2.4 Stubs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.2.5 1PI vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

9.3 Properties of fundamental vertices . . . . . . . . . . . . . . . . . . . . . . . . 1589.3.1 String product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1589.3.2 Feynman graph interpretation . . . . . . . . . . . . . . . . . . . . . . . 159

9.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

10 Off-shell scattering amplitudes and Feynman diagrams – examples 16110.1 Feynman diagrams – closed string . . . . . . . . . . . . . . . . . . . . . . . . 161

10.1.1 3-point graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16110.1.2 4-point graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

10.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

III Bosonic string field theory 165

11 String field 16611.1 Field functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16611.2 Field expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16711.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16811.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

12 Light-cone string field theory 17012.1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

13 Free BRST string field theory 17113.1 Classical action for the open string . . . . . . . . . . . . . . . . . . . . . . . . 171

13.1.1 Warm-up: point-particle . . . . . . . . . . . . . . . . . . . . . . . . . . 17113.1.2 Open string action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17213.1.3 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17413.1.4 Siegel gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

13.2 Open string field expansion, parity and ghost number . . . . . . . . . . . . . 177

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13.3 Path integral quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17913.3.1 Tentative Faddeev–Popov gauge fixing . . . . . . . . . . . . . . . . . . 17913.3.2 Tower of ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

13.4 Spacetime action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18213.4.1 Classical action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

13.5 Closed string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18513.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18613.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

14 Closed string field theory 18814.1 Closed string field expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 18814.2 Gauge fixed theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

14.2.1 Kinetic term and propagator . . . . . . . . . . . . . . . . . . . . . . . 18914.2.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19114.2.3 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

14.3 Classical gauge invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . 19414.4 BV theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19614.5 1PI theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19814.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

15 Witten’s cubic string field theory 20015.1 Axiomatic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20015.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

16 Homotopy algebras in string field theory 20216.1 Closed string field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

17 Background independence 20417.1 The concept of background independence . . . . . . . . . . . . . . . . . . . . 20417.2 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20517.3 Deformation of the CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20617.4 Expansion of the action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20817.5 Relating the equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 20817.6 Idea of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20917.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

IV Superstring theory 211

18 Superstring 21218.1 Worldsheet superstring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

18.1.1 Heterotic worldsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21218.1.2 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

18.2 Off-shell superstring amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 21618.2.1 Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21718.2.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21818.2.3 Spurious poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

18.3 Superstring field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22218.3.1 String field and propagator . . . . . . . . . . . . . . . . . . . . . . . . 22318.3.2 Constraint approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22318.3.3 Auxiliary field approach . . . . . . . . . . . . . . . . . . . . . . . . . . 22418.3.4 Large Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

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18.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

V Spacetime string field theory 227

19 Momentum-space SFT 22819.1 General form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22819.2 Generalized Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23019.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

VI Classical solutions 234

20 Classical solutions 23520.1 Tachyon condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23520.2 Analytic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

20.2.1 Algebraic tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23620.2.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

20.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

VII Conformal field theory 241

21 Conformal field theory in D dimensions 24221.1 CFT on a general manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

21.1.1 Conformal group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24221.1.2 Conformal field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

21.2 CFT on Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24421.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

22 Conformal field theory on the plane 24622.1 The Riemann sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

22.1.1 Map to the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . 24622.1.2 Relation to the cylinder – string theory . . . . . . . . . . . . . . . . . 248

22.2 Classical CFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24922.2.1 Witt conformal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 24922.2.2 PSL(2,C) conformal group . . . . . . . . . . . . . . . . . . . . . . . . 25122.2.3 Definition of a CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

22.3 Quantum CFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25522.3.1 Virasoro algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25522.3.2 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

22.4 Operator formalism and radial quantization . . . . . . . . . . . . . . . . . . . 25722.4.1 Radial ordering and commutators . . . . . . . . . . . . . . . . . . . . 25722.4.2 Operator product expansions . . . . . . . . . . . . . . . . . . . . . . . 25922.4.3 Hermitian and BPZ conjugation . . . . . . . . . . . . . . . . . . . . . 26122.4.4 Mode expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26222.4.5 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26522.4.6 CFT on the cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

22.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

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23 CFT systems 27223.1 Free scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

23.1.1 Covariant action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27223.1.2 Action on the complex plane . . . . . . . . . . . . . . . . . . . . . . . 27423.1.3 OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27623.1.4 Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27923.1.5 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28123.1.6 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28223.1.7 Euclidean and BPZ conjugates . . . . . . . . . . . . . . . . . . . . . . 284

23.2 First-order bc ghost system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28423.2.1 Covariant action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28523.2.2 Action on the complex plane . . . . . . . . . . . . . . . . . . . . . . . 28523.2.3 OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28723.2.4 Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28923.2.5 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29123.2.6 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29223.2.7 Euclidean and BPZ conjugates . . . . . . . . . . . . . . . . . . . . . . 29823.2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

23.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

24 BRST quantization 30024.1 BRST for reparametrization invariance . . . . . . . . . . . . . . . . . . . . . . 30024.2 BRST in the CFT formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

24.2.1 OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30124.2.2 Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30224.2.3 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

24.3 BRST cohomology: two flat directions . . . . . . . . . . . . . . . . . . . . . . 30324.3.1 Conditions on the states . . . . . . . . . . . . . . . . . . . . . . . . . . 30524.3.2 Relative cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30724.3.3 Absolute cohomology, states and no-ghost theorem . . . . . . . . . . . 31224.3.4 Cohomology for holomorphic and anti-holomorphic sectors . . . . . . . 313


VIII Riemann surfaces 315

25 Riemann surfaces 31625.1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

IX Mathematics 317

26 Homotopy algebras 31826.1 A∞ homotopy associative algebra . . . . . . . . . . . . . . . . . . . . . . . . . 31826.2 L∞ homotopy Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

26.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31826.2.2 Coalgebra description . . . . . . . . . . . . . . . . . . . . . . . . . . . 32126.2.3 Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

26.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

8

X Appendices 324

A Conventions 325A.1 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325A.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327A.3 QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327A.4 Curved space and gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328A.5 Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329A.6 List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

B Summary of important formulas 333B.1 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333B.2 QFT, curved spaces and gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 333

B.2.1 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334B.3 Conformal field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

B.3.1 Complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334B.3.2 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335B.3.3 Hermitian and BPZ conjugations . . . . . . . . . . . . . . . . . . . . . 336B.3.4 Scalar field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337B.3.5 Reparametrization ghosts . . . . . . . . . . . . . . . . . . . . . . . . . 337

B.4 Bosonic string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340B.4.1 Closed string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341B.4.2 Open string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

C Quantum field theory 342C.1 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

C.1.1 Integration measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342C.1.2 Field redefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344C.1.3 Zero-modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

C.2 BRST quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347C.3 BV formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

C.3.1 Properties of gauge algebra . . . . . . . . . . . . . . . . . . . . . . . . 350C.3.2 Classical BV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351C.3.3 Quantum BV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

C.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

Bibliography 357

Index 375

9

Preface

This review grew up from lectures delivered within the Elite Master Program Tmp from theLudwig-Maximilians-Universität during the winter semesters 2017–2018 and 2018–2019.

The main focus of this review is the closed bosonic string field theory (SFT). While thereare many resources available for the open bosonic SFT, a single review [84] has been writtensince the final construction of the bosonic closed SFT by Zwiebach [297]. For this reason,it makes sense to provide a modern and extensive study. Moreover, the usual approach toopen SFT focuses on the cubic theory, which is so special that it is difficult to generalize thetechniques to other SFTs. Finally, closed strings are arguably more fundamental than openstrings because they are always present since they describe gravity, which further motivatesmy choice. However, the reader should take this focus as denying the major achievementsand the beauty of the open SFT; reading this book should provide most of the tools neededto feel comfortable also with this theory.

While part of the original goal of SFT was to provide a non-perturbative definitionof string theory and to address important questions such as classifying consistent stringbackgrounds or understanding dualities, no progress on this front has been achieved so far.Hence, there is still much to understand and the recent surge of developments provide anew chance to deepen our understanding of closed SFT. For example, several consistencyproperties of string theory have been proven rigorously using SFT. Moreover, the recentconstruction of the open-closed superstring field theory [186] together with earlier works [55,248, 297, 299] show that all types of string theories can be recast as a SFT. This is why, Ibelieve, it is a good time to provide a complete review on SFT.

The goal of this review is to provide a self-contained description of SFT and all thetools necessary to build it. The emphasis is on describing the concepts behind SFT and tomake the reader build intuitions on what it means. For this reason, there are relatively fewapplications.

The reader is assumed to have some knowledge of QFT, and a basic knowledge of CFTand string theory (classical string, Nambu–Goto action, light-cone and old-covariant quant-izations).

OrganizationThe text is organized on three levels: the main content (augmented with examples), compu-tations, and remarks. The latter two levels can be omitted in a first lecture. The examples,computations and remarks are clearly separated from the text (respectively, by a half-boxon the left and bottom, by a vertical line on the left, and by italics) to help the navigation.

Many computations have been set aside from the main text in order to not break the flowof the argument and to provide the reader with the opportunity to check by himself first.While the derivations contain more details than usual texbooks and may look pedantic tothe expert, I think it is useful for students and newcomers to have complete references whereto check each step. This is even more the case when there are many different conventions in

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the literature. The remarks are not directly relevant to the core of the text but they makeconnections with other parts or with other topics. The goal is to broaden the perspectivesof the main text.

Text in dark gray indicates draft material which I started to check and organize but which stillneed further checking. This can also indicates more conceptual reflections which I did not (or couldnot) verify.

General references can be found at the end of each chapter to avoid overloading thetext. In-text references are reserved for specific points or explicit quotations (of a formula, adiscussion, a proof, etc.). I did not try to be exhaustive in the citations and I have certainlymissed important references: this should be imputed to my lack of familiarity with themand not to their values.

I tried to make the different chapters relatively independent. This implies repetitions,but this allows to present the same concept according to different perspectives. This alsoavoids having to always go back and forth.

The thematic organization of the book makes the progression somewhat non-linear. Ac-cordingly, the reader should not feel that she cannot read the rest of the review if she findsa chapter difficult. I would recommend to go on, and come back later.

During the lectures, additional topics have been covered in tutorials, which can be foundonline (together with corrections):

https://www.physik.uni-muenchen.de/lehre/vorlesungen/wise_17_18/sft_ws_17_18/exercises_sft/index.html

My plan is to frequently update this review with new content. You can access the lastversion of the draft on my professional webpage, currently located at:

http://www.lpthe.jussieu.fr/~erbin/

These notes are available in two formats. The first, called the “book” version, is shorter, morestatic and verified more extensively. The second, called the “review” version, is organizedby topics and contains more details but is less thoroughly checked.

AcknowledgementsI have started to learn string field theory in Hri by attending lectures from Ashoke Sen.Since then, I have benefited from collaboration and many insightful discussions with him.Following his lectures have been much helpful in building an intuition that cannot be foundin papers or reviews on the topic. Through this review, I hope being able to make some ofthese insights more accessible.

I am particularly grateful to Ivo Sachs who proposed me to teach this course and toMichael Haack for continuous support and help for the organization, and to both of themfor many interesting discussions during the two years I have spent at Lmu.

Moreover, I have been very lucky to be assigned an excellent tutor for this course,Christoph Chiaffrino. After providing him with the topic and few references, Christophwould prepare all the tutorials and the corrections autonomously. His help brought a lot tothe course.

I am particularly obliged to all the students who have taken this course at Lmu formany interesting discussions and comments: Enrico Andriolo, Hrólfur Ásmundsson, DanielBockisch, Fabrizio Cordonnier, Julian Freigang, Wilfried Kaase, Andriana Makridou, PouriaMazloumi, Daniel Panea, Martin Rojo.

I am also grateful to all the string theory community for many exchanges. For discussionsrelated to the topics of this book, I would like to thank more particularly: Costas Bachas,Adel Bilal, Subhroneel Chakrabarti, Atish Dabholkar, Benoit Douçot, Ted Erler, Dileep

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http://www.lpthe.jussieu.fr/~erbin/

Jatkar, Carlo Maccaferri, Juan Maldacena, Yuji Okawa, Sylvain Ribault, Raoul Santachiara,Martin Schnabl, Dimitri Skliros, Jakub Vošmera.

I am thankful to my colleagues at Lmu for providing a warm and stimulating envir-onment, with special thanks to Livia Ferro for many discussions around coffee. Moreover,the encouragements and advices from Oleg Andreev and Erik Plauschinn have been strongincentives for publishing this review.

I have received a lot of feedback during the different stages of writing this review, and Iam obliged to all the colleagues who sent me feedback.

I would also like to thank Christian Caron and Lisa Scalone for their help and efficiencyduring the publishing process. I am also indebted to Stefan Theisen for having supportedthe publication at Springer.

Finally, writing this review would have been more difficult without the continuous andloving support from Corinne at all levels.

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Chapter 1

Introduction

The goal of this chapter is to motivate in which respect string theory is particularly dis-tinguished and to outline what is required to describe it at a fundamental level. For this,we first put in perspective the different descriptions of quantum relativistic systems with aspatial extension (p-brane). This allows to stress what are the major differences between apoint-particle and a p-brane – an object which extends in p spatial dimensions –, but alsobetween the worldvolume and spacetime approaches (or first- and second-quantization). Thechapter ends with a summary of the main ideas of string theory and why it is desirable toconstruct a field theory of strings. This chapter must be seen as exploratory and couldcontain controversial reasoning. The ideas outlined in this chapter will be deepen in the restof the review.

1.1 Strings, a distinguished theoryThe first and simplest reason for considering theories of fundamental p-branes (fundamentalobjects extended in p spatial dimensions) can be summarized by the following question:“Why would Nature just make use of point-particles?” There is no a priori reason forbiddingthe existence of fundamental extended objects and, according to Gell-Mann’s totalitarianprinciple, “Everything not forbidden is compulsory.” If one cannot build a consistent theory(after a reasonable amount of efforts) or if it contradicts current theories (in their domainsof validity) and experiments, then one can support the claim that only point-particles exist.On the other side, if such a theory can be built, it is of primary interest to understand itdeeper and to see if it can solve the current problems in high-energy theoretical physics.

The simplest case after the point particle is the string, so it makes sense to start withit. It happens that one can construct a consistent theory of strings, and that the latter(in its supersymmetric version) contains all the necessary ingredients for a fully consistenthigh-energy model:1

• quantum gravity (quantization of general relativity plus higher-derivative corrections);

• grand unification (of matter, interactions and gravity);

• no divergences, UV finiteness (finite and renormalizable theory);

• fixed number of dimensions (26 = 25 + 1 for the bosonic string, 10 = 9 + 1 for thesupersymmetric version);

1There are also indications that a theory of membranes (2-branes) in 10+1 dimensions, called M-theory,should exist. No direct and satisfactory description of the latter exists and we will thus focus on stringtheory in this review.

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• existence of all possible branes;

• no dimensionless parameters and one dimensionful parameter (the string length `s).

It can be expected that a theory of fundamental strings (1-branes) occupies a distinguishedplace among fundamental p-branes for the following reasons.

Remark 1.1 (Other applications) Beside its construction as a quantum gravity theory,string theory has lead to many applications in mathematics and in theoretical physics. Inparticular, our understanding of QFT has been greatly challenged and improved by stringtheory. For this reason, string theory will remain an unavoidable theory in the landscapeof theoretical physics, even if it turns out not to be the correct theory for quantum gravity.Another argument is that the construction of a string field theory pushes to its extremity thefield theory formalism.

Interaction non-locality In a QFT of point particles, UV divergences arise becauseinteractions (defined as the place where the number and/or nature of the objects change)are arbitrarily localized at a spacetime point. In Feynman graphs, such divergences can beseen when the momentum of a loop becomes infinite (two vertices collide): this happenswhen trying to concentrate an infinite amount of energy at a single point. On the otherhand, one expects these divergences to be reduced or absent in a field theory of extendedobjects: whereas the interaction between particles is perfectly local in spacetime and agreedupon by all observers (Figure 1.1), the spatial extension of branes makes the interactionsnon-local. This means that two different observers will neither agree on the place of theinteractions (Figure 1.2), nor on the part of the diagram which describes one or two branes.

Due to the non-locality, the number of branes which can interact simultaneously is alsolimited. That several branes seem to interact at the same time (both at the same or differentpoints) can only be an observer-dependent effect. Indeed, a Lorentz transformation willchange the time at which the different interactions take place but in different manners.2

The string lies at the boundary between too much local and too much non-local: in anygiven frame, the interaction is local in space, but not in spacetime. The reason is that a stringis one-dimensional and splits or joins along a point. For p > 1, one needs to break/join thebrane along an extended spatial section, which looks much more non-local. Such a processcould even threaten causality because of the upper limit on the speed of signal propagation.

Another consequence of the non-locality is a drastic reduction of the possible interactions.If the interaction is Lorentz invariant, Lorentz covariant objects can be attached at the vertex(such as momentum or gamma matrices): this gives Lorentz invariants after contracting withindices carried by the field. But, this is impossible if the interaction itself is non-local (andthus not invariant): inserting a covariant object would break the Lorentz invariance.

Remark 1.2 (Ontology of the string) Interestingly, the string and the particle also dif-fer in their existential aspects.3

A fundamental particle is ontologically one and cannot be divided: it can decay in severalother particles but the latter cannot be seen as being part of the first one. As an example,consider a muon decaying into an electron and neutrinos (in the Standard model): it isobvious that the electron and the neutrinos are not “inside” the muon since the latter is afundamental particle. This is because, for an observer, there is a clear notion of a particle,defined as the intersection between its equal-time surfaces and the worldline of the particle.

This is not the case for the string: any set of strings which have interacted together don’thave an independent existence, and only the full worldsheet has an ontological attribute. This

2There is a slight caveat to this statement: in string field theory, the action typically contains interactionsof all orders. The details are technical and will be discussed in Chapter 14. For a discussion of how this iscompatible with the statement of the main text, see [229].

3It should be possible to extend the arguments to branes.

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Figure 1.1: Locality of a particle interaction: two different observers always agree on theinteraction point and which parts of the worldline are 1- and 2-particle states.

is related to the non-locality of the string interactions, implying that two observers cannotagree on the question “when does the string cease to exist?” A string can be cut in two pieces,and these pieces do not acquire some new property making them “other” than just pieces ofthe first string (this idea is illustrated in some string field theories for which interactionsexactly describe this process of gluing or separating two pieces). Nonetheless, one cannot saythat the original string is made of the two strings glued together, since it can be separated atany point (due to the non-locality), and assigning some part to one or the other is not aninvariant statement. The above reasoning would make possible to break the string in points(and thus it would not be fundamental).

Brane degrees of freedom The higher is the number of spatial dimensions of a p-brane, the more possibilities it has to fluctuate. As a consequence, it is expected that newdivergences appear as p increases due to the proliferations of the brane degrees of freedom.From the worldvolume perspective, this is understood from the fact that the worldvolumetheory describes a field theory in (p+ 1) dimensions, and UV divergences become worse asthe number of dimensions increase. The limiting case happens for the string (p = 1) sincetwo-dimensional field theories are well-behaved in this respect (for example, any monomialinteraction for a scalar field is power-counting renormalizable). This can be explained by thelow-dimensionality of the momentum integration and by the enhancement of symmetries intwo dimensions. Hence, strings should display nice properties and are thus of special interest.

Worldvolume theory The point-particle (0-brane) and the string (1-brane) are also re-markable in another aspect: it is possible to construct a simple worldvolume field theory(and the associated functional integral) in terms of a worldvolume metric. All components ofthe latter are fixed by gauge symmetries (diffeomorphisms for the particle, diffeomorphismsand Weyl invariance for the string). This ensures the reparametrization invariance of theworldvolume without having to use a complicated action.

Summary As a conclusion, strings achieve an optimal balance between spacetime andworldsheet divergences, as well as having a simple description with reparametrization in-variance.

Since the construction of a field theory is difficult, it is natural to start with a worldsheettheory and to study it in the first-quantization formalism, which will provide a guideline forwriting the field theory. In particular, this allows to access the physical states in a simpleway and to find other general properties of the theory. When it comes to the interactions

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(a) Observers at rest and boosted. (b) Observers close to the speedof light moving in opposite dir-ections. The interactions arewidely separated in each case.

Figure 1.2: Non-locality of string interaction: two different observers see the interactionhappening at different places (denoted by the filled and empty circles) and they don’t agreeon which parts of the worldsheet are 1- and 2-string states (the litigation is denoted by thegrey zone).

and scattering amplitudes, we will see in Section 1.2.2 that this approach may be hopelessin general since one needs to specify by hand the topology of the worldvolume (describingthe interaction process). In this respect, the case of the string is again exceptional: becauseRiemann surfaces have been classified and are well-understood, the arbitrariness is minimal.Combined with the tools of conformal field theory, many computations can be performed.Moreover, since the modes of vibrations of the strings provide all the necessary ingredientsto describe the Standard model, it is sufficient to consider only one string field (for one typeof strings), instead of the plethora one finds in point-particle field theory (one field for eachparticle). Similarly, non-perturbative information (such as branes and dualities) could befound only due to the specific properties of strings.

Coming back to the question which opened this section, higher-dimensional branes ofall the allowed dimensions naturally appear in string theory as bound states at the per-turbative level. Correspondingly, this hints at the existence of fundamental branes at thenon-perturbative level (think of the case of magnetic monopoles). Hence, even if the simplestworldvolume formulation of branes with p > 1 looks pathological, string theory hints towardsanother definition of these objects.4

In the rest of this introduction, we explain the different possibilities to describe a p-brane.This will help to pinpoint the advantage and difficulties of the field theory formalism. Then,we will focus on strings.

Remark 1.3 (String bit model) While we assume that strings are fundamental, thereare alternative approaches – such as the string bit model [270, 279, 280] – which considerthem as composite.

4The fact that a perturbative description of a single higher-dimensional brane fails is logical if the latterare intrinsically non-perturbative objects and cannot appear independently from each other. However, oneshould still be able to build an effective theory.

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•(a) 0-brane (particle) (b) 1-brane (string)

(c) 2-brane (membrane)

Figure 1.3: Examples of branes.

1.2 Worldvolume and spacetime descriptionsAn overview of the possible descriptions of a relativistic (extended) object is given in thissubsection. They are the worldvolume and spacetime (field) approaches (corresponding re-spectively to first- and second-quantization). In this way, we can highlight how the standardformulation of string theory (worldsheet) differs from the point-particle QFT (spacetime),and how this translates in our ability to study the properties of this system. Many conceptsare explained through heuristic arguments and should not be always taken too literally, butmore as an introduction to familiarise oneself with more precise statements given in the restof this review.

Let’s consider a classical p-dimensional object (relativistic or not) – a p-brane – whichpropagates in a spacetime M with coordinates xµ = (t, xi). Here, p denotes the numberof spatial dimensions of the object, and the shape at a given time is denoted by Σ. Thehistory of a p-brane – the time evolution and the spatial locations of its shape Σ – is calledthe worldvolume W and describes a (p+ 1)-dimensional spacetime hypersurface.

Example 1.1 – Low-dimensional branesA 0-brane is a point –, and thus any particle falls in this category. In this case, theworldvolume is called worldline. 1-branes are strings and gather all objects made froma line, open or closed. Their worldvolume is also called worldsheet. This includes forexample ropes (open) and hoops (closed). A simple example of a 2-brane (also calledmembrane) is a blanket. See Figure 1.3.

Classification First, we assume that branes are compact. Branes can be classified atdifferent levels:

• topology: the geometrical shape (including the existence of boundaries and of anorientation);

• boundary conditions: Dirichlet and Neumann for open domains, periodic and anti-periodic for closed domains;

17

• internal properties: the tension and any other properties distinguishing the differenttypes of branes (generalizations of the particle spin).

Note that the possible boundary conditions depend both of the internal properties and ofthe topology.

Example 1.2 – Shapes for low-dimensional branesThe only 0-brane is the point. There are open and closed strings (1-branes), whichare respectively topologically equivalent to a line and to a circle. These two topologiesrepresent types of strings, which have each distinct properties. Similarly, membranes(2-branes) can be any Riemann surface.

Example 1.3 – String classificationHere is the classification of the string:

• topology: (un)oriented open string, (un)oriented closed string;

• boundary conditions: Dirichlet and Neumann conditions (open string), periodicand anti-periodic (closed string);

• internal property: bosonic string, type I and II superstrings, heterotic superstring.

Open strings are consistent at the classical level but not at the quantum level. Inpractical models, a single family is considered, with the same tension for all its members.The reader is encouraged to go through Section 1.3 with these notions in mind.

Fundamental A p-brane can be fundamental or not. There are two aspects in which abrane can be not fundamental:

1. the brane is an assembly of lower-dimensional branes bound together by some force;

2. it appears to be p-dimensional as an approximation by forgetting about some dimen-sions of a higher-dimensional brane if they are not observable.

On the other hand, a fundamental object cannot be decomposed in terms of lower-dimensionalelements nor can any structure be found inside it. But, as we will see below, this does notprevent the brane from splitting in several other branes (see also Remark 1.2 page 14).

Example 1.4 – Not fundamental objectsA proton is not fundamental since it is composed of quarks and gluons, but it behaveslike a point-particle at a good approximation for many applications.

A guitar string (1-brane) is also not fundamental because it is made of many atoms(0-brane); similarly, a necklace is made of many beads. It makes perfectly sense tobreak the necklace and to say that it has no intrinsic existence without the beads.Likewise, a thin rope appears to be 1-dimensional for a human being (which can onlywalk on one direction on it, for example), but it can appear to be 2-dimensional fora sufficiently small insect (which could walk along the length or circle around). Bothcases are illustrated on Figure 1.4.

Parametrization In order to describe a brane, it is often useful to parametrize its world-volume W in terms of coordinates σa = (σ0, σα) with a = 0, . . . , p. However, since theyare parameters introduced for computational convenience without any physical existence,nothing should depend on them. The simplest way to enforce this constraint is to adopt adescription where reparametrizations of the coordinates σa are a gauge symmetry.

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(a) String made from beads (atoms)

(b) String with an hidden dimension

Figure 1.4: Non-fundamental 1-branes.

1.2.1 Classical aspectsDynamical variables

The dynamics can be described in two different fashions, depending on the variables chosenand the way they relate the object to the spacetime:

1. Worldvolume (or σ-model) approach: the spacetime position of the object Xµ(σa)given as functions of the worldvolume coordinates σa are the dynamical variables.They are maps describing the embedding of the p-brane worldvolume in spacetime.Since they can be viewed as fields on the worldvolume, the description is equivalentto a field theory on a (p + 1)-dimensional space. This description must be invariantunder reparametrizations of the σa.

2. Spacetime (or field) approach: the object position is encoded in a field Ψ[Xµ(Σ)]which is a functional of the object spacetime coordinates. The dependence on Σ of thespatial position denotes that, at a given time, the field is a function of all the points Σoccupied by the brane.5 Even if one can work with function on Σ-space, in practice, itis often convenient to parametrize explicitly the shape Σ in terms of spatial coordinatesσα and to ensure that the description remains invariant. The simplest instance of adynamical field is the density field.

The two approaches correspond to the two possible paths to write a Lorentz-covarianttheory starting from classical mechanics. Indeed, special relativity requires to treat space andtime on an equal footing and thus a mixed description (with time being a label and spatialpositions dynamical variables) is no more adapted. It is clear that there are two possibilities:treat all spacetime coordinates as dynamical variables (worldvolume approach) or treat allspacetime coordinates as labels (spacetime field theory).

Conceptually, both descriptions are fully equivalent, but they differ as computationaltools, insofar as some calculations are simpler (if at all possible) in one of them. Moreover,the first approach quickly displays various limitations when used for relativistic systems, in

5The simplest way to understand that it should depend on the spatial shape Σ and not of the worldvolumeW is by considering the point-particle. The dynamics of the brane, and thus its time evolution, is givenby the equation of motion: as a consequence, the field should not include information about the completeworldvolume, but only about the spatial shape. This is also a direct consequence of second-quantization.

19

particular at the quantum level. We review briefly the main features of each formalism fromvarious points of view.

Remark 1.4 (Mixed description from gauge-fixing) A possible gauge-fixing of the re-parametrization gauge symmetry is to set (some of) the worldvolume coordinates equal tothe spacetime coordinates, σa = xa, which also implies Xa(σa) = σa. This is particularlynatural for the time, σ0 = t, when there is a foliation of the spacetime according to a time-like direction. For a point-particle, this leads to the standard description (of non-relativisticphysics) in terms of Xi(t) by taking X0 = σ0 = t.

Worldvolume approach Since the action is a quantity to be extremized, it is natural toconsider the volume of a geometrical object. Hence, the simplest action for a brane is theworldvolume itself:

Swv = −T∫W

dVol , (1.1)

where T is the tension of the brane which gives the correct mass dimension to the RHS, Wis the worldvolume and Vol its volume.6 For a single particle, one recovers the traditionalintegration of the proper-time over the worldline, with T equal to its mass. Using the mapsXµ(σ), the action is written as

Swv = −T∫

dp+1σ√− det γ (1.2)

where γab is the metricγab = ηµν

∂Xµ

∂σa∂Xν

∂σb(1.3)

induced on the brane from the spacetime metric ηµν (taken to be flat for simplicity). Intro-ducing an independent metric gab(σ) on the worldsheet, the action is rewritten as

S′wv = −T2

∫dp+1σ

√−det g ηµνgab ∂aXµ∂bX

ν + p− 12 T

∫dp+1σ

√−det g . (1.4)

This is usually the starting point for studying the string. The classical equations of motionyield that gab is proportional to γab.

Remark 1.5 (Action (1.4) for p > 1) The action (1.4) is problematic for p > 1 becauseit describes too much degrees of freedom. Indeed, the worldvolume should have no intrinsicdynamics. In particular, the metric gab should have no degrees of freedom. In order toreduce them, one can only impose diffeomorphism and Weyl invariances (local rescalings ofthe metric). This gives a total of p + 2 constraints for an object which has (p + 2)(p +1)/2 degrees of freedom. The worldvolume metric can be totally fixed only for p = 0 (withdiffeomorphisms) and p = 1 (with diffeomorphisms and Weyl invariance).

As already mentioned, nothing should depend on the parametrization of the surface.The coordinates σa are merely internal labels which indicate how to read the spacetimeposition from the dynamical variables (the same apply to the maps Xµ(σa) or the metricgab(σ) which are just tools to describe a physical object). Reparametrization is treatedas a gauge symmetry. For a relativistic system leads to constraints among the canonicalmomenta, which in turn imply the vanishing of the worldvolume energy–momentum tensor.The strongest effect is the vanishing of the worldvolume Hamiltonian:

Hwv = 0 . (1.5)6More complicated actions can be considered: the effect is to change the possible vibrations of the object

(i.e. its spectrum). Examples are the spinning particles and strings.

20

Since the Hamiltonian generates dynamics, the worldvolume is not dynamical. This makessense since the worldvolume, by definition, is fixed and already includes the complete historyof the brane.

The constraints are equivalent to the mass-shell condition for the brane, which is thensaid to be on-shell. In the quantization process, these constraints also have to be imposed(as constraint on the states à la Gupta–Bleuler, by working in the BRST cohomology, etc.).As a consequence, the tools available in QFT (renormalization, analyticity. . . ) which relyon an off-shell formulation and which ensure the consistency of the theory (divergences,gauge invariance, unitarity, causality. . . ) have no direct counterpart in the worldvolumeformulation.

Spacetime approach The spacetime action is found by introducing some fields Ψ[Xµ(Σ)](this is a functional since the position depends on the shape Σ of the object) and the freeaction is quadratic in the fields:7

Sfree = K(Ψ,Ψ) :=∫

dXµ(Σ)dX ′µ(Σ′) Ψ[Xµ(Σ)]K[Xµ(Σ), X ′µ(Σ′)] Ψ[X ′µ(Σ′)]

≈∫

dX ΨKΨ ,

(1.6)

where K is the kinetic operator for the field Ψ, and K[X,X ′] denotes its matrix elementsin position space.8

Example 1.5 – Spin-0 scalar fieldIf Ψ(x) describes a scalar particle of spin 0, then the matrix elements of the kernel Kare

K(x, x′) = δ(D)(x− x′)(−∂2x +m2) (1.7)

such thatS =

∫dDxΨ(x)(−∂2

x +m2)Ψ(x) . (1.8)

Remark 1.6 (Other descriptions) There are other descriptions which do not rely at allon a Lagrangian, but they will not be detailed in this review. Examples include the conformalbootstrap and scattering amplitude programs.

Multi-brane systems

The way both descriptions accommodate systems with N objects (in the Lagrangian form-alism) is different:

1. Worldvolume: one dynamical variable XµA(σa) (A = 1, . . . , N) is introduced for every

brane.

2. Spacetime: different fields are needed only for each different type of objects, but asingle field can describe several identical objects (from one to infinitely many, as for afluid). For example, N identical branes can be described with a density field writtenas a sum of N delta functions.

The worldvolume approach is best adapted to describe a finite and fixed number ofwell-identified objects, which interact with each other and which are acted upon by external(classical) forces. Moreover, the spacetime description would be cumbersome since it requiresto use delta functions. Conversely, it is more suited in two cases:

7The measure dXµ(Σ) denotes a functional integral over Xµ(Σ) for all µ.8In the bra and ket notation, would write K(Ψ,Ψ) =〈Ψ|K |Ψ〉 and |Ψ〉 =

∫dXµ(Σ) Ψ[Xµ(Σ)] |Xµ(Σ)〉.

21

• relativistic system: the number of branes is not conserved in relativistic systems andit is thus natural to look for a framework which allows this number to vary;

• when the number of particles is huge (typically in a fluid) and cannot be followedindividually.

1.2.2 Quantum aspectsQuantization

The canonical quantization (which replaces the dynamical variables by operators) of bothdescriptions leads also to different frameworks:

1. First-quantization (worldvolume): the spacetime positions of each object become op-erators Xµ(σ) in the Schrödinger picture. The evolution is given in terms of a (single-brane) wave function ψk[Xµ(σ), τ ] satisfying a Schrödinger equation with time τ (kdenotes collectively the eigenvalues, which include the centre-of-mass momentum).Multi-brane states are found by solving the Schrödinger equation for the (anti-)sym-metrized tensor product of single-brane wave functions.

2. Second-quantization (spacetime): the field is promoted to an operator Ψ[Xµ(Σ)] whichcan be decomposed in terms of annihilation and creation operators with first-quantizedwave functions as coefficients. These operators build the Fock space (multi-branestates) by acting on the vacuum (no brane state).

In both cases, the coordinates σα of Σ can be Fourier transformed. Since Σ is compact, thisleads to discrete eigenvalues which are simpler to handle.

The second-quantization can also be viewed as follows: the first-quantized wave functionis reinterpreted as a classical spacetime field, which is then quantized (the proper-timedependence is absent due to the reparametrization gauge symmetry).

Even if multi-brane systems can be considered in the first-quantization, the resulting descriptionis much more complicated than with a field: if the wave function of one brane has K degrees offreedom (generically K = ∞ due to the spacetime dependence), then a N -brane wave functionwould have KN components. On the other hand a field always has K components.

Interactions

The definition of scattering amplitudes is also handled very differently in the two formalisms:

1. Worldvolume: amplitudes are computed by a path integral (weighted by the free actionof one brane) with a sum over all the possible processes

〈O1 · · · On〉 ∼∑

W|O1,...,On

λn(W) e−Swv[W]O1 · · · On . (1.9)

The sum is over all the worldvolumes W, with (incoming and outcoming) boundaryconditions given by the external states, which is denoted by W|O1,...,On . The interac-tion vertices with the corresponding coupling constant λ are added by hand and de-termine the possible topologies.9The power of the coupling constant n(W) is obtainedfrom multiplying the coupling constants of all interactions happening in a graph; itthus depends on the worldvolume topology. Moreover, graphs with equivalent topolo-gies should not be counted twice: this principle can help to find the set of fundamentalinteractions as those generating all the possible topologies without over-counting.

9For simplicity, only one coupling constant is introduced, but in principle each vertex comes with itsown coupling.

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2. Spacetime: the interaction between n fields is described by adding a monomial of ordern in the fields in the action:

Sint = Vn(Ψ, . . . ,Ψ︸︷︷︸n times

) ≈∫

dX Ψn , (1.10)

and the scattering amplitudes are built from the Feynman graphs, which are themselvesuniquely fixed by the action

〈O1 · · · On〉 ∼∫

dΨ e−Sfree[Ψ]−Sint[Ψ]O1[Ψ] · · · On[Ψ] . (1.11)

The partition function Z is obtained by considering a single operator equal to the identity.It had to be expected that the worldvolume approach cannot describe interactions since

the path integral is weighted with the action of a single-brane, and its Hilbert space doesnot include multi-brane states. The arbitrariness in the worldvolume approach complicateshighly the models when there are several types of branes which can interact in many differentways. This also implies that the worldvolume formulation is perturbative in nature sincethe sum over the topologies is perturbative. On the other hand, a field theory can probenon-perturbative features of the system since the partition function is not given as a seriesin the coupling constant.

Remark 1.7 (Interactions in worldvolume action) In the worldvolume path integral(1.9), the action is free, and one can ask whether interaction terms can be added. It is in factpossible, but such terms describe the interactions with an external force or a background field.This is what is usually done in classical mechanics, where an object reacts to a spring, orto the electromagnetic field of some source, etc. However, these interactions do not describethe processes of annihilation and creation, and cannot describe scattering amplitudes.

Divergences

Typically, the scattering amplitudes diverge:

• UV divergences: they originate from the high-energy region in loop diagrams. Thedivergent terms are removed by adding counter-terms and by renormalizing.

• IR divergences: they arise when an internal particle is forced to be on-shell, either aspart of a massless particle tadpole (Figure 1.5), of a propagator on an external leg(Figure 1.6) or of an internal propagator at a resonance (Figure 1.7). Physically, theycorrespond to quantum effects which shift the vacuum and the masses. Both problemsare cured by renormalization.

• Spurious divergences: these are non-physical and arise because of a bad parametriza-tion of the amplitudes.

Renormalization is an intrinsically off-shell procedure and thus it can be employed onlywith difficulties in the worldvolume formalism. Moreover, since the masses of the statesare shifted due to quantum effects, which cannot be taken into account, only (quantum)amplitudes for protected states10 can be computed.

10A protected state is a state whose mass is not renormalized due to some symmetry.

23

Figure 1.5: The momentum of the particle connecting the tadpole loop is forced to be zerodue to the conservation of the energy–momentum. If the state is a massless particle, then itis on-shell and the propagator diverges. This means that the classical vacuum is destabilizedand that there is a nearby vacuum with a lower energy.

Figure 1.6: Due to the conservation of energy–momentum, the particle propagating afterthe loop is on-shell and this leads to a divergence of its propagator. This is cured by takinginto account the mass renormalization and resummation of all the 1PR graphs.

Figure 1.7: The amplitude diverges when the external momenta are such that k becomes on-shell. Physically, the internal particle is stable at the classical level, which is in contradictionwith its decay, and hence the divergence. Quantum effects give an imaginary part to themass, indicating that the particle can decay.

24

1.2.3 ComparisonWe can now summarise the disadvantages of the worldsheet approach over the spacetimefield one:

• no natural description of (relativistic) multi-particle states;

• on-shell states:

– lack of renormalization,– presence of infrared divergences,– scattering amplitudes only for protected states;

• interactions added by hand;

• hard to check consistency (unitarity, causality. . . );

• absence of non-perturbative processes.

Some of these problems can be addressed with various prescriptions, but it is desirableto dispose of a unified and systematic procedure, which is to be found in the field theorydescription.

We will come back on the various problems and make them more precise in due course:we are voluntarily vague since appropriate solutions exist only for the particle and the string.These two cases will be more thoroughly studied in the core of this review.

In the rest of this section, we compare briefly both approaches for the point-particle andfor p-branes.

1.2.4 ExamplesPoint-particle

A (point-)particle is an object which has no spatial extension (this can be an approximationor a fundamental property): as a consequence, it corresponds to a point in space, a 0-brane.Hence, its history in spacetime is a curve, called the worldline: since it is 1-dimensional itis parametrized by a single parameter τ (Figure 1.8). Note that τ can be taken to be theproper-time of the particle, but this is not necessary.

The trajectory is denoted by xc(τ): it is parametrized by the interval τ ∈ [τi, τf ] suchthat xc(τ) maps every point of the trajectory to a spacetime point. It begins at the pointxc(τi) = xi and ends at xc(τf ) = xf . Since it is just a label, the precise values are notimportant and one can fix τi and τf to arbitrary values, for example τi = 0 and τf = 1. Theworldline appears as a segment in the τ -space (Figure 1.8): the precise spacetime shape ofthe worldline is encoded in the function xµc (τ) (Figure 1.9).

The next step is to construct a Lagrangian whose equation of motion admits the tra-jectory xc(τ) as a solution. In the worldvolume approach, the dynamical variable is Xµ(τ),and it is possible to fix the gauge τ = t = X0 in order to work with Xi(t). An appropriatefield in the spacetime approach is the density ρ(xµ): its classical solution will be of the formρ(x) ∼ δ(x− xc(t)).

Given a system of N identical particles with trajectories xa(τ) (a = 1, . . . , N), one wouldneed N dynamical variables Xi(τ) in the worldline formulation, whereas the unique fieldρ(x) would be sufficient. The classical field solution describing the above trajectories reads

ρ(x) =N∑a=1

δ(x− xa(t)

). (1.12)

25

−−−−−−−−−−→

Figure 1.8: Trajectory xµc (τ) of a particle in spacetime (worldline). The worldline is describedby the parameter τ ∈ [τi, τf ].

Figure 1.9: Examples of values for xµ(τ) (note that x0 is monotonically increasing).

26

It is obviously very cumbersome to work with the field ρ(x) and one prefers the worldlineapproach in classical mechanics. When one wants to describe inelastic scatterings whereparticles merge (non-relativistic) or are created or destroyed (relativistic) in this approach,the interaction itself is described using conservation laws (kinematics) since the Lagrangian(and its equations of motion) cannot describe a process where the numbers and types ofparticles change, but it is still valid in-between.

For a relativistic quantum theory, one works with a more abstract field φ(x), whoseexcitations represent individual particles. This is equivalent to say that the field can bedecomposed as (in the momentum representation)

φ(x) ∼∑k

akψk(x) (1.13)

where the ak are creation and annihilation operators associated to the 1-particle state foundby solving the worldline Schrödinger equation. Any multi-particle state can then be foundby applying a set of operators ak on the vacuum (no-particle state).

Remark 1.8 (Position dependence of particle) To make contact between the first- andsecond-quantized approaches more easily, it could be convenient to write the field dependenceas φ(Xµ) instead of φ(xµ), because one can view the field as depending not on the spacetimecoordinates directly, but on the locations which can be occupied by a particle. This is ofparticular importance for a p-brane field since the latter depends on the location and theshape, and thus the dependence of the field is not really in terms of spacetime coordinates(but in terms of a collection of coordinates), as we review below.

Branes

A p-brane is an p-dimensional extended object. Its worldvolume is described by a proper-timeτ and by p spatial coordinates (denoted collectively as) σ. It is important to stress that onlythe shape in spacetime is physical, while the parametrization is not.

The trajectory is denoted by xc(τ, σ). It begins and ends respectively at the geometricshapes parametrized by xc(τi, σ) = xi(σ) and by xc(τf , σ) = xf (σ). It is clear that theextension of the object in space leads to complications due to the various topologies andorientations.

For example the spatial section of a string can be topologically closed (circle) or open(line), leading to cylindrical or rectangular worldsheets as illustrated in Figures 1.10 and 1.11.Note that the coordinate system on the worldsheet itself is arbitrary.

The worldvolume Lagrangian is constructed with a fieldXµ(τ, σ). The trajectory xµc (τ, σ)can then be found as a classical solution to the equations of motion.

The description in terms of a spacetime field is more complicated since the object itcreates has a spatial extension: as a consequence, a simple dependence in the spacetimecoordinates is not sufficient, and one needs a way to incorporate the shape. The simplestway to achieve this is by writing the field as a functional of parametrized spacetime shapesXµ(σ), i.e. Ψ[Xµ(σ)] But, this leads to several difficulties since one needs to ensure thatthe functional dependence is reparametrization invariant. Another problem is the non-locality of the interactions: due to the spatial extension of the branes, the question “whereis the interaction localized” has no meaning, and in particular it is not a Lorentz invariantstatement.

Hence, constructing such a field theory is a major endeavour, and the goal of this reviewis to see how this was achieved in the case of strings.

Remark 1.9 The field Xµ(σ) is the restriction of Xµ(τ, σ) on a fixed worldvolume-timeslice, in the same way that the dependence for a particle field is Xµ whereas the worldline

27

−−−−−−−−→

Figure 1.10: Trajectory xµc (τ, σ) of a closed string in spacetime (worldsheet). It begins andends at the circles parametrized by xi(σ) and xf (σ). The worldsheet is topologically acylinder and is parametrized by (τ, σ) ∈ [τi, τf ]× [0, 2π).

−−−−−−−−→

Figure 1.11: Trajectory xµc (τ, σ) of an open string in spacetime (worldsheet). It beginsand ends at the lines parametrized by xi(σ) and xf (σ). The worldsheet is topologically arectangle and is parametrized by (τ, σ) ∈ [τi, τf ]× [0, `].

28

variable was Xµ(τ). This is expected since the worldvolume variable describes the spacetimehistory of the brane while the field creates the brane at a fixed proper-time instead of thefull worldvolume. A precise contact can be made by looking at the worldvolume Schrödingerequation and performing a second-quantization.

1.3 String theory1.3.1 PropertiesThe goal of this section is to give a general idea of string theory by introducing some conceptsand vocabulary. This section is by no mean rigorous and the precise justification of manystatements will be postponed to later. Until Section 1.4, we will focus on the worldsheetformalism since it is more intuitive.

String tension

The string length `s is the only length one can build on dimensional ground from the tensionand the fundamental constants

`s =√

~c2πT =⇒ T = ~c

2π`2s, (1.14)

where the factor of 2π has been introduced for convenience: the string is often parametrizedby σ ∈ [0, 2π). Since the action is proportional to T , the spatial integral is normalised by2π when expressed in terms of `2s.

Next, we will compare how the non-relativistic and relativistic strings differ, and whythe string length can in fact be interpreted as the natural size of the strings (this will bea hand-waving argument, to be taken more as an indication rather than a rigorous proof).Most formulas used in this paragraph can be found in [300].

Remember that a static non-relativistic string is characterized by three parameters: itsmass per unit of length µ, its tension T and its rest length `0 (when it is not stretched).Stretching the string to a length `, the potential energy is given by V = T (` − `0)2/2 andits mass energy is µ`0c2. The energy is obviously minimal for ` = `0.

Remark 1.10 (Independence of the parameters) In order to show that the tension isindependent of the rest length of the object, consider a spring with stiffness k1. The lattergives the force F one must apply to elongate the spring by ∆`1. According to Hooke’s law,the result is F = k1∆`1. Next consider replacing this spring by two springs, of the samematerial but half smaller, attached in series. One finds that the stiffness k2 is twice bigger:exerting the same force F , the total elongation is again ∆`1 and it splits equally betweenthe two springs, so ∆`2 = ∆`1/2. Moreover, the same force applies to each spring and thusF = k2∆`2, yielding k2 = 2k1. Hence, the stiffness is inversely proportional to the length.Considering a string to be a series of many such springs, one obtains the tension as thiscoefficient of proportionality (tension is an energy per unit of length, or equivalently a force).In the simplest model, the tension is given by the Young modulus times the section of thestring. These are not related to the mass. Obviously, all these parameters depend on thematerial.

Consider next a relativistic string with tension T (defined in the rest frame), and attach itsuch that its length is `. Its energy reads E = T` and is entirely equal to the potential energydue to the stretching. Since the energy in special relativity is the sum of the rest mass µ0 andof the potential energy, this implies that the string is massless, µ0 = 0. It can be attributedan effective mass (per unit of length) due to the tension through µc2 = E/` = T . Hence,

29

contrary to the non-relativistic case, mass and tension are not independent parameters. Thesecond point is that the rest length is also zero `0 = 0, since there is no off-set in the energydue to a rest length. This makes sense since the energy is given solely by the potentialenergy, which is interpreted as the energy needed for creating a string of length ` (which canbe viewed as stretching an infinitesimal string). Conversely, this also means that the massis not conserved, the additional length of the string is created thanks to the conversion ofenergy to mass as it is accumulated by stretching (for a non-relativistic mass the stretchingjust rearranges the underlying constituents).

Remark 1.11 (Massive relativistic string) It can be checked that the non-relativisticlimit for the massless relativistic string is singular and does not reproduce the standardnon-relativistic string. However, it is possible to write a massive relativistic string [10, 45].

Taking into account quantum effects, the string cannot shrink without limit due to theuncertainty relation

`p ≥ ~2 , (1.15)

since doing so would require to have an infinite momentum. To find the natural stringlength, one takes p to be the momentum of the centre of mass, which can be related to theenergy

p = E

c= µc` = T`

c, (1.16)

since the centre of mass behaves like a massless particle with pµ = (E,−E/c). Then,requiring the uncertainty relation to be saturated yields

` ∼√

~c2T . (1.17)

One recovers the string length (1.14) up to11 a factor of π.This means that any string which is not attached to something will spontaneously shrink.

In particular, closed strings cannot be attached to anything and will tend to be very tiny,of the order of `s, which is expected to be just few order of magnitude above the Plancklength (since the argument used a quantum argument and since the closed string is relatedto gravity). Note that the size of the string depends of its energy, and thus of its momentumand vibrational mode.

Remark 1.12 (Classical dynamics and shrinking) At the classical level, we have ar-gued that the lowest energy of a string is attained if it shrinks to a point. But, one canconsider classical configurations where the strings is of finite length even if it is not attachedto anything (for example with a rotating string [300]). This is explained by the conservationof energy: the length of the string is fixed as an initial condition and the energy cannotchange because one considers an isolated free string.

The last parameter to be introduced is Regge’s slope α′, defined by the relation

J

~= α′E2 , (1.18)

where J is the angular momentum. Considering a rotating string gives the relations

T = 12π~cα′ , `s = ~c

√α′ . (1.19)

11The precise relation between the energy and the tension depends on the details of the system. Forexample, a rotating string has E =

π

2T`, so the precise factor defining `s is a matter of convention.

30

(a) Open string (b) Closed string

Figure 1.12: Open and closed strings.

Finally, it makes sense to speak about the string tension or the string length because oneconsiders only one type of string.12

Worldsheet CFT

A string is characterized by its worldsheet field theory.13 The simplest description is obtainedby endowing the worldsheet with a metric gab and by adding a set of D scalar fields Xµ(σ)living on the worldsheet. The latter represents the position of the string in theD-dimensionalspacetime. From the classical equations of motion, the metric gab is proportional to themetric induced on the worldsheet from its embedding in spacetime. More generally, oneensures that the worldsheet metric is non-dynamical by imposing that the action is invariantunder (worldsheet) diffeomorphisms and under Weyl transformations (local rescalings of themetric). The consistency of these conditions at the quantum level imposes that D = 26,and this number is called the critical dimension. Gauge fixing the symmetries, and thusthe metric, leads to the conformal invariance of the resulting worldsheet field theory: aconformal field theory (CFT) is a field theory (possibly on a curved background) in whichonly angles and not distances can be measured. This simplifies greatly the analysis sincethe two-dimensional conformal algebra (called the Virasoro algebra) is infinite-dimensional.

One can consider more general CFTs: fields with continuous energy eigenvalues areinterpreted as non-compact dimensions while fields with discrete eigenvalues are interpretedas compact dimensions or some internal structure, such as the spin.14

While the light-cone quantization allows to find quickly the states of the theory, thesimplest covariant method is the BRST quantization. It introduces ghosts (and super-ghosts) associated to the gauge fixing of diffeomorphisms (and local supersymmetry). These(super)ghosts form a CFT which is universal (independent of the matter CFT).

Depending on the boundary conditions, one finds several types of strings, and in partic-ular two topologies (Figure 1.12):

• closed: periodic boundary conditions;

• open: Dirichlet and Neumann boundary conditions.

We will find that, while a closed string theory is consistent by itself, an open string theoryis not and requires closed strings. Thus, a bosonic string theory will contain one type ofclosed string and possibly several types of open strings.

Remark 1.13 (Multi-string models) Nothing but phenomenology prevents us from con-structing a theory with several families of (closed and/or open) strings: since the string

12In fact, there will be two types of strings, closed and open, but unitarity requires them to have thesame elementary properties, only their shapes differ.

13We focus mainly on the bosonic string theory, leaving aside the superstring, except when differencesare important.

14To understand this division, consider the quantum-mechanical problem of a particle on a line or on asegment: wave functions have the form eikx, where k ∈ R in the first case, and k ∈ πZ in the second. Thetwo cases correspond respectively to non-compact and compact dimensions (e.g. consider the Kaluza–Kleinreduction on a circle).

31

spectrum contains a graviton, every different family of string would come with its own grav-iton, and it is not clear that multi-metric theories are consistent. Therefore, it is simplerand sufficient to assume a single type (in particular, it already provides all ingredients forphenomenology).

For a short general discussion, see [11]; applications to multi-metric theories can be foundin [150, 151], while such models are used as a tool in [6, 7, 169] to build higher-spin theories.

Spectrum

In order to gain some intuitions for the states described by a closed string, one can write theFourier expansion of the fields Xµ (in the gauge gab = ηab and after imposing the equationsof motion)

Xµ(τ, σ) ∼ xµ + pµτ + i√2∑n∈Z∗

1n

(αµne−in(τ−σ) + αµne−in(τ+σ)) , (1.20)

where xµ is the centre-of-mass position of the string and pµ its momentum

[xµ, pν ] = iηµν . (1.21)

With respect to a point-particle for which one finds only the first two terms, one gets aninfinite number of oscillators αµn and αµn which satisfy canonical commutation relations forcreation n < 0 and annihilation operators n > 0

[αµm, ανn] = mηµνδm+n,0 . (1.22)

The modes n 6= 0 describe the shape Σ in the momentum space thanks to a Fourier transform.The case of the open string is simply obtained by setting αn = αn. The Hamiltonian forthe closed and open strings read respectively

Hclosed = −m2

2 +N + N − 2 , (1.23a)

Hopen = −m2 +N − 1 (1.23b)

where m2 = −pµpµ is the mass of the state (in Planck units), N and N (level operators)count the numbers Nn and Nn of oscillators αn and αn weighted by their mode index n:

N =∑n∈N

nNn , Nn = 1nα−n · αn ,

N =∑n∈N

nNn , Nn = 1nα−n · αn .

(1.24)

With these elements, one can construct the Hilbert space of the string theory. Invarianceunder reparametrization leads to the on-shell condition, which says that the Hamiltonianvanishes:

H |ψ〉 = 0 (1.25)for any physical state |ψ〉. Another constraint for the closed string is the level-matchingcondition

(N − N) |ψ〉 = 0 . (1.26)It can be understood as fixing an origin on the string.

The ground state |k〉 with momentum k is defined to be the eigenstate of the momentumoperator which does not contain any oscillator

∀n > 0 : αµn |k〉 = 0 . (1.27)

32

A general state can be built by applying successively creation operators

|ψ〉 =∏n>0

d∏µ=0

(αµ−n)Nn,µ |k〉 , (1.28)

where Nn,µ ∈ N counts how many times the oscillator αµ−n appears. In the rest of thissection, we describe the first two levels of states.

The ground state is a tachyon (faster-than-light particle) because the Hamiltonian con-straint shows that it has a negative mass

closed : m2 = −4 , open : m2 = −1 . (1.29)

The first excited state of the open string is found by applying α−1

αµ−1 |k〉 . (1.30)

One finds that it is masslessm2 = 0 (1.31)

and since it transforms as a Lorentz vector (spin 1), it is identified with a U(1) gauge boson.One can write a superposition of such states

|A〉 =∫

dDk Aµ(k)αµ−1 |k〉 (1.32)

and the coefficient Aµ(k) of the Fourier expansion is interpreted as the spacetime field forthe gauge boson. Reparametrization invariance is equivalent to the equation of motion

p2Aµ = 0 . (1.33)

One can prove that the field obeys the Lorentz gauge condition

pµAµ = 0 , (1.34)

which results from gauge fixing the U(1) gauge invariance

Aµ −→ Aµ + pµλ . (1.35)

It can also be checked that the low-energy action reproduces the Maxwell action.The first level of the closed string is obtained by applying both α−1 and α−1 (this is the

only way to match N = N at this level)

αµ−1αν−1 |p〉 (1.36)

and the corresponding states are massless

m2 = 0 . (1.37)

These states can be decomposed into irreducible representations of the Lorentz group(αµ−1α

ν−1 + αν−1α

µ−1 −

1Dηµνα−1 · α−1

)|p〉 ,(

αµ−1αν−1 − αν−1α

µ−1)|p〉 , 1

Dηµνα

µ−1α

ν−1 |p〉

(1.38)

which are respectively associated to the spacetime fields Gµν (metric, spin 2), Bµν (Kalb–Ramond 2-form, spin 1) and Φ (dilaton, spin 0). The appearance of a massless spin 2 particle(with low-energy action being the Einstein–Hilbert action) is a key result and originallyraised interest for string theory.

33

Figure 1.13: A string of length 10−33 cm looks like a point-like particle at higher scales.

Remark 1.14 (Reparametrization constraints) Reparametrization invariance leads toother constraints than H = 0. They imply in particular that the massless fields have thecorrect gauge invariance and hence the correct degrees of freedom.

Note that, after taking into account these constraints, the remaining modes correspondto excitations of the string in the directions transverse to it.

Hence, each vibrational mode of the string corresponds to a spacetime field for a point-particle (and linear superpositions of modes can describe several fields). This is how stringtheory achieves unification since a single type of string (of each topology) is sufficient fordescribing all the possible types of fields one encounters in the standard model and in gravity.They correspond to the lowest excitation modes, the higher massive modes being too heavyto be observed at low energy.

The understanding of the string in terms of spacetime fields can be deepened as follows:a fundamental string is very tiny (10−33 cm) and as such it appears to be point-like whenseen from afar (Figure 1.13). The spin and the other properties of the particles are providedby the internal structure of the string (and in particular its vibrational mode).

Bosonic string theory includes tachyons and is thus unstable. While the instability ofthe open string tachyon is well understood and indicates that open strings are unstable andcondense to closed strings, the status of the closed string tachyon is more worrisome (literallyinterpreted, it indicates a decay of spacetime itself). In order to solve this problem, one canintroduce supersymmetry: in this case, the spectrum does not include the tachyon becauseit cannot be paired with a supersymmetric partner.

Moreover, as its name indicates, the bosonic string possesses only bosons in its spectrum(perturbatively), which is an important obstacle to reproduce the standard model. Byintroducing spacetime fermions, supersymmetry also resolves this problem. The last directadvantage of the superstring is that it reduces the number of dimensions from 26 to 10,which makes the compactification easier.Remark 1.15 (Fermions in bosonic string) In fact, it is possible to show that somebosonic string theories admit fermions as non-perturbative bound states [53] (see also [76,148]).

1.3.2 Classification of superstring theoriesIn this section, we describe the different superstring theories. In order to proceed, we needto introduce some new elements.

The worldsheet field theory of the closed string is made of two sectors, called the left- andright-moving sectors (the αn and αn modes). While they are treated symmetrically in thesimplest models, they are in fact independent (up to the zero-mode) and the correspondingCFT can be chosen to be distinct.

The second ingredient already evoked earlier is supersymmetry. This symmetry associ-ates a fermion to each boson (and conversely) through the action of a supercharge Q

|boson〉 = Q |fermion〉 . (1.39)

34

More generally, one can consider N supercharges which build up a family of several bosonicand fermionic partners. Since each supercharge increases the spin by 1/2 (in D = 4), onefinds an upper limit for the number of supersymmetries – for interacting theories with afinite number of fields15 – in order to keep the spin of a family in the range where consistentactions exist:

• Nmax = 4 without gravity (−1 ≤ spin ≤ 1);

• Nmax = 8 with gravity (−2 ≤ spin ≤ 2).

This counting serves as a basis to determine the maximal number of supersymmetries inother dimensions (by relating them through dimensional reductions).

Let’s turn our attention to the case of the two-dimensional worldsheet theory. Thenumber of supersymmetries of the closed left- and right-moving sectors can be chosen in-dependently and one writes (NL, NR) (the index is omitted when statements are made atthe level of the CFT). The critical dimension (absence of quantum anomaly for the Weylinvariance) depends on the number of supersymmetry

D(N = 0) = 26, D(N = 1) = 10. (1.40)

Type II superstrings have (NL, NR) = (1, 1) and come in two flavours called IIA and IIBaccording to the chiraly of the spacetime gravitini chiralities. A theory is called heteroticif NL > NR; we will mostly be interested in the case NL = 1 and NR = 0.16 In suchtheories, one cannot have open strings since both sectors must be equal in the latter. Sincethe critical dimensions of the two sectors do not match, one needs to get rid of the additionaldimensions of the right-moving sector, and this leads to the next topic – gauge groups.

Gauge groups associated with spacetime gauge bosons appear in two different places. Inheterotic models, the compactification of the unbalanced dimensions of the left sector leadsto the appearance of a gauge symmetry. The possibilities are scarce due to consistencyconditions which ensure a correct gluing with the right-sector. Another possibility is to adddegrees of freedom – known as Chan–Paton indices – at the ends of open strings: one endtransforms in the fundamental representation of a group G, while the other end transforms inthe anti-fundamental. The modes of the open string then reside in the adjoint representation,and the massless spin 1 particles then become the gauge bosons of the non-Abelian gaugesymmetry.

Finally, one can consider oriented or unoriented strings. An oriented string possesses aninternal direction, i.e. there is a distinction between going from the left to the right (for anopen string) or circling in clockwise or anti-clockwise direction (for a closed string). Suchan orientation can be attributed globally to the spacetime history of all strings (interactingor not). The unoriented string is obtained by quotienting the theory by the Z2 worldsheetparity symmetry which exchanges the left- and right-moving sectors. Applying this to thetype IIB gives the type I theory.

The tachyon-free superstring theories together with the bosonic string are summarizedin Table 1.1.

Remark 1.16 (Extended supersymmetries) One can also consider theories with NL =2 and NL = 4 but they are pathological. While the critical dimension of NL = 2 looks morepromising since D(NL = 2) = 4, the theory makes sense only if spacetime has two time dir-ections [179]. Moreover, this theory has no transverse excitations and the S-matrix is almosttrivial. A more interesting model is found in the heterotic case (2, 1) (with D(NR = 1) = 12instead of 10 to compensate for the additional time dimension) which admits transverseexcitations. Finally, the case of NL = 4 is directly ruled out since D(NL = 4) = −2.

15These conditions exclude the cases of free theories and higher-spin theories.16The case NL < NR is identical up to exchange of the left- and right-moving sectors.

35

worldsheetsusy D

spacetimesusy gauge group open string oriented tachyon

bosonic (0, 0) 26 0 any* yes yes / no yestype I (1, 1) 10 (1, 0) SO(32) yes no no

type IIA (1, 1) 10 (1, 1) U(1) (yes)† yes notype IIB (1, 1) 10 (2, 0) none (yes)† yes no

heterotic SO(32) (1, 0) 10 (1, 0) SO(32) no yes noheterotic E8 (1, 0) 10 (1, 0) E8 × E8 no yes no

heterotic SO(16) (1, 0) 10 (0, 0) SO(16)× SO(16) no yes no* UV divergences beyond the tachyon (interpreted as closed string dilaton tadpoles) cancel only for the unorientedopen plus closed strings with gauge group SO(213) = SO(8192).

† The parenthesis indicates that type II theories don’t have open strings in the vacuum: one needs to consider aD-brane background. This is expected since there is no gauge multiplet in d = 10 (1, 1) or (2, 0) supergravities(the D-brane breaks half of the supersymmetry).

Table 1.1: List of the consistent tachyon-free (super)string theories. The bosonic theory isadded for comparison. There are additional heterotic theories without spacetime supersym-metry, but they contain a tachyon and are thus omitted.

1.3.3 InteractionsWorldsheet and Riemann surfaces

After having described the spectrum and the general characteristics of string theory, onecan address the question of interactions. String interactions are equivalent to string joiningand splitting at a point:

1. a string (open or closed) pinches (two parts of the string come close to each other);

2. a string makes a loop and crosses itself;

3. a string breaks at a point.

The last process requires the presence of open strings since the result is necessarily an openstring. If the string is oriented, then the orientations of the resulting strings is inheritedfrom the orientations of the original strings: this limits the number of possible processes.Obviously, any process valid for oriented strings is also available to unoriented strings.

The main question is to find the set of the fundamental interactions, i.e. the interactionsfrom which all processes can be derived. According to the arguments in Section 1.2.2, suchinteractions can involve at most three strings. Indeed, one could imagine trying to addsimultaneous interactions at various points and/or more strings at the same point, but thenotion of simultaneity is frame-dependent (due to the non-locality of the interaction) andanother observer would see the process as a combination of more elementary interactions.Moreover, like in QFT, we exemplify only one process, but the reverse process is valid, alongwith processes where some elements from the LHS (or RHS) are move to the RHS (or LHS).

In order to give some intuition for the interactions, it is simpler to start by consideringonly spatial sections of the interactions (by considering times just before and after, in aspecific Lorentz frame). This simply corresponds to taking a rope and seeing how it can bearranged to reproduce one of the situation listed above. The processes involving at mostthree strings are given in Figures 1.14 and 1.15 for the oriented and unoriented stringsrespectively. Arrows have been indicated on each diagram to show if the orientation ofeach part of each string can be preserved. The strings can break at a point (Figures 1.14cand 1.14d), pinch (Figures 1.14b, 1.14f, 1.15a and 1.15b) or cross (Figures 1.14a and 1.14e).

36

The two unoriented processes show that a string can interact with itself in order to removea twist; however, this cannot be done while preserving a global orientation. To simplify thediscussion, we now focus on the oriented case. In order to make the connection with thenext discussion, graphs for the interactions of four open strings are given in Figure 1.16.

It is easily seen that a theory of open strings also requires closed strings since the twoends of the string can join and form a closed loop: because of causality (and locality) sucha process happens before the two ends “know” that they are part of the same string.

The graphs listed in Figures 1.14 and 1.15 exhaust all the possibilities. However, thereare still redundancies and not all are fundamental interactions: since they are constructedat fixed time, the non-locality is not apparent and some graphs can be equivalent to acombination of other graphs. For example, all four strings of Figure 1.16 look like meetingat a single spacetime point, but this is an artefact of choosing a specific frame for drawingthe picture: performing a Lorentz transformation would show that the interaction happensin two locations (and, as a consequence, it cannot be fundamental).

Describing the interactions requires to look at the full worldsheet by including the timedirection. The propagations of closed and open strings are respectively represented by tubesand strips. Then, the diagrams from Figure 1.14 become the surfaces drawn in Figure 1.17.Remark 1.17 (Drawings) The worldsheet diagrams associated to the diverse spatial dia-grams may not represent them exactly: because the strings are bi-dimensional and evolvein several dimensions, it is hard to draw the diagrams and clearly identify which part ofthe worldsheet corresponds to which part of the spatial section (in the same sense that aFeynman diagram is not an exact depiction of reality). Hence, the correspondence we giveis based at a more intuitive level rather than a rigorous map. This explains why Figure 1.14has more diagrams than Figure 1.17.

To illustrate how a diagram can be equivalent to a series of more elementary processes,consider the interaction of four strings (Figure 1.16). The corresponding worldsheet diagramis given on Figure 1.18a: two strings merge and create two strings. However, this diagramis topologically equivalent to the worldsheet Figure 1.18b: the two initial strings merge intoone string, which propagates, and then decay in two strings (s-channel process). One couldalso consider that one string decays in two strings, among which one merges with the secondinitial string to form the second final string (t- or u-channel, Figures 1.19a and 1.19b).This means that the contact interaction, and the s-, t- and u-channels provide alternativedescriptions of the same process (in QFT one would have to sum over the three differentchannels). This has profound consequences on the properties of string amplitudes; this pointwill be discussed in more details later.

Investigations show also that the processes (shown in Figures 1.14e, 1.14f and 1.17d) ofone open string emitting a closed string are also not fundamental [110].

At the quantum level, strings can also propagate through loops. Such a loop impliesthat a string splits and merges again later: this amounts to introducing holes in the world-sheet. A closed string loop is given by a torus, while one finds an annulus for open strings(Figure 1.21). Higher-loop amplitudes are obtained by adding more holes (of the torus orannulus types) to the surfaces.

The worldsheets one obtains in this way are Riemann surfaces, i.e. 1-dimensional complexmanifolds. They are classified by the numbers of handles (or holes) g (called the genus) andexternal tubes n. In the presence of open strings, one obtains surfaces with boundaries: inaddition to the handles and tubes, they are classified by the numbers of disks b and of stripsm.17 A particularly important number associated to each surface is the Euler characteristics

χ = 2− 2g − b , (1.41)17We ignore unoriented strings in this discussion. They would lead to an additional object called a

cross-cap, which is a place where the surface looses its orientation.

37

(a) closed ↔ closed + closed (b) closed ↔ closed + closed

(c) open ↔ open + open (d) closed ↔ open

(e) open ↔ open + closed (f) open ↔ open + closed

Figure 1.14: Joining and splitting of oriented strings. All these processes are oriented because one canconsistently draw an arrow on each diagram before and after the interactions.

(a) closed ↔ closed (b) open ↔ open

Figure 1.15: Joining and splitting of unoriented strings. These two processes are non-oriented becausepart of the string should flip its orientation during the interaction.

(a) (b)

Figure 1.16: Recombinations of two open strings into two strings.

38

(a) (b)

(c) (d)

Figure 1.17: Worldsheet diagrams for string interactions.

(a) contact vertex

=

(b) s-channel

Figure 1.18: Scattering of four closed strings. Due to the non-locality the process on the leftis equivalent to the one of the right, where an intermediate string propagates. The s-channelis displayed.

(a) t-channel (b) u-channel

Figure 1.19: Scattering of four closed strings through the t- and u-channels.

39

(a)

Figure 1.20: Worldsheet diagrams for string interactions.

(a) Closed strings

(b) Open strings

Figure 1.21: One-loop amplitude for a 4-point scattering of closed strings.

40

(a) Closed strings

(b) Open strings

Figure 1.22: Graphs corresponding to 1-loop 4-point scattering after a conformal mapping.

Figure 1.23: General Riemann surfaces with boundaries and punctures.

which is a topological invariant. It is remarkable that at every loop order there is a singletopology when one considers only closed strings, and just a few more in the presence ofopen strings. The analysis is greatly simplified in contrast to QFT, for which the numberof Feynman graphs increases very rapidly with the number of loops and external particles.

Due to the topological equivalence between surfaces, one can perform a (conformal) mapin order to work with simpler surfaces. In particular, one can collapse the external tubesand strips to points (called punctures or marked points) on the corresponding surfaces orboundaries. For example, the graphs of Figure 1.21 are equivalent to the graphs fromFigure 1.22. A general amplitude will then looks like a sphere from which holes and diskshave been removed and to which marked points have been pierced (Figure 1.23).

Amplitudes

In order to compute an amplitude for a n-string scattering, one must sum over all theinequivalent worldsheets through a path integral weighted by the CFT action chosen todescribe the theory.18 At fixed n, one has to sum over the genus g, such that each term isdescribed by a Riemann surface Σg,n of genus g with n punctures.

The interactions between strings follow from the graph topologies: since the latter are notencoded into the action, one needs to add by hand the dependence in the coupling constant.For closed strings, there is a unique cubic vertex with coupling gs. A direct inspection shows

18For simplicity we focus on closed string amplitudes in this section.

41

that the correct factor is gn−2+2gs :

• for n = 3 one has a factor gs, and every additional external string leads to the additionof one vertex with factor gs, since this process can be obtained from the n− 1 processby splitting one of the external string in two by inserting a vertex;

• each loop comes with two vertices, so g-loops provide a factor g2gs .

Remark 1.18 (Status of gs as a parameter) It was stated earlier that string theory hasno dimensionless parameter, but gs looks to be one. In reality it is determined by the expect-ation value of the dilaton gs = e〈Φ〉. Hence the coupling constant is not a parameter definingthe theory but is rather determined by the dynamics of the theory.

Finally, one needs to specify the external states: this amounts to prescribe boundaryconditions for the path integral or to insert the corresponding wave functions. Under theconformal mapping which brings the external legs to punctures located at zi, the states aremapped to local operators Vi(ki, zi) inserted at the points zi. The latter are built fromthe CFT fields and are called vertex operators: they are characterized by a momentum kµ

which comes from the Fourier transformation of the Xµ fields representing the non-compactdimensions. These operators are inserted inside the path integral and one integrates overthe positions zi in order to describe all possible conformal mappings.

Ultimately, the amplitude (amputated Green function) is computed as

An(k1, . . . , kn) =∑g≥0

gn−2+2gs Ag,n (1.42)

whereAg,n =

∫ n∏i=1

d2zi

∫dgabdΨ e−Scft[gab,Ψ]

n∏i=1

Vi(ki, zi) (1.43)

is the g-loop n-point amplitude (for simplicity we omit the dependence on the states beyondthe momentum). Ψ denotes collectively the CFT fields and gab is the metric on the surface.

The integration over the metrics and over the puncture locations contain a huge redund-ancy due to the invariance under reparametrizations and thus one integrates over manyequivalent surfaces. To avoid this, one must introduce Faddeev–Popov ghosts and to re-strict the integral to only finitely many (real) parameters tλ. They form the moduli spaceMg,n of the Riemann surfaces Σg,n whose dimension is

dimRMg,n = 6g − 6 + 2n. (1.44)

One can then summarize the computation of the amplitude Ag,n by

Ag,n =∫Mg,n

6g−6+2n∏λ=1

dtλ F (t). (1.45)

The function F (t) is a correlation function in the worldsheet CFT defined on the Riemannsurface Σg,n

F (t) =⟨

n∏i=1

Vi × ghosts× super-ghosts⟩

Σg,n

. (1.46)

Note that the (super)ghost part is independent of the choice of matter CFT.

42

Divergences and Feynman graphs

Formally the moduli parameters are equivalent to Schwinger (proper-time) parameters si inusual QFT: these are introduced in order to rewrite propagators as

1k2 +m2 =

∫ ∞0

ds e−s(k2+m2), (1.47)

such that the integration over the momentum k becomes a Gaussian times a polynomial.This form of the propagator is useful to display the three types of divergences which can beencountered:

1. IR: regions si →∞ (for k2 +m2 ≤ 0). These divergences are artificial for k2 +m2 < 0and means that the parametrization is not appropriate. Divergences for k2 +m2 = 0are genuine and translates the fact that quantum effects shift the vacuum and themasses. Taking these effects into account necessitates a field theory framework inwhich one can use renormalization.

2. UV: regions si → 0 (after integrating over k). Such divergences are absent in string the-ories because these regions are excluded from the moduli spaceMg,n (see Figure 1.24for the example of the torus).19

3. Spurious: regions with finite si where the amplitude diverges. This happens typicallyonly in the presence of super-ghosts and it translates a breakdown of the gauge fixingcondition.20 Since these spurious singularities of the amplitudes are not physical, oneneeds to ensure that they can be removed, which is indeed possible to achieve.

Hence, only IR divergences present a real challenge to string theory. As explained pre-viously, dealing with these divergences required renormalizing the amplitudes, but this isnot possible in the standard formulation of worldsheet string theory since the states areon-shell.21 Accordingly, the first step is to find a way to go off-shell: this is made possibleby introducing local coordinates around the punctures of the Riemann surface.

Remark 1.19 (UV divergences and symmetries) A good UV behaviour can also be ex-pected from the huge amount of symmetries (in particular when supersymmetry is included)displayed by the string.

The IR divergences of amplitudes originate from Riemann surfaces close to degeneration,that is, surfaces with long tubes. The latter can be of separating and non-separating types,depending on whether the Riemann surface splits in two pieces if one cuts the tube (Fig-ure 1.25). By exploring the form of the amplitudes in this limit, one finds that the expressionnaturally separates into several pieces, to be interpreted as two amplitudes (of lower n and g)connected by a propagator. The latter can be reinterpreted as a standard (k2 +m2)−1 term,hence solving the divergence problem for k2 +m2 < 0. Taking this decomposition seriously,one is lead to identify each contribution with a Feynman graph. Writing the amplitude interms of Feynman diagrams solves another problem: the amplitude (1.45) computed by asingle diagram yields a real result, which is at odds with unitarity and the expectation of acomplex result for unstable particles.

19There is a caveat to this statement: UV divergences reappear in string field theory in Lorentziansignature due to the way the theory is formulated. The solution requires a generalization of the Wickrotation.Moreover, this does not hold for open strings whose moduli spaces contains those regions: in this case,

one can reinterpret the divergences in terms of closed strings propagating.20Such spurious singularities are also found in supergravity.21The on-shell condition is a consequence of the BRST and conformal invariance. While the first will be

given up, the second will be maintained to facilitate the computations.

43

Figure 1.24: Moduli space of the torus: Re τ ∈ [−1/2, 1/2], Im τ > 0 and |τ | > 1.

(a) Separating.

(b) Non-separating.

Figure 1.25: Degeneration of Riemann surfaces.

44

The last step consists in finding the elementary graphs, i.e. the interaction vertices fromwhich all other graphs (and amplitudes) can be built. With these interactions, one is thenable to build a field theory (the kinetic term being given by the inverse of the propagator).However, this theory will be gauge fixed because, on-shell amplitudes are gauge invariantand only include the physical states. For this reason, one needs to find how to re-establishthe gauge invariance. Due to the complicated structure of string theory, one needs to usethe full-fledged Batalin–Vilkovisky (BV) formalism (it basically amounts to introduce ghostsbefore the gauge fixing).

The procedure we will follow is a kind of reverse-engineering: we know what is the finalresult and we want to study backwards how it is obtained:

on-shell amplitude→ off-shell amplitude→ Feynman graphs→ gauge fixed field theory→ BV field theory

In standard QFT, one follows the opposite process.Remark 1.20 There are some prescriptions (using for example analytic continuation, theoptical theorem, some tricks. . . ) to address the problems mentioned above, but there is nogeneral and universally valid procedure. A field theory is much more satisfactory because itprovides a unique and complete framework.

1.4 String field theory1.4.1 String field actionA string field theory(SFT) for open and closed strings is based on two fields Φ[X(σ)] (openstring field) and Ψ[X(σ)] (closed string field) governed by some action S[Φ,Ψ]. This actionis built from a diagonal kinetic term

S0 = 12 KΨ(Ψ,Ψ) + 1

2 KΦ(Φ,Φ) (1.48)

and from an interaction polynomial in the fields

Sint =∑m,n

Vm,n(Φm,Ψn) (1.49)

where Vm,n is an appropriate product mappingm closed and n open string states to a number(the power is with respect to the tensor product). In particular, it contains the couplingconstant. For example, the terms corresponding to elementary interactions of three closed,three open, and one closed with one open strings (diagrams 1.17a, 1.17b and 1.17c) arerespectively of the forms Ψ3, Φ3 and ΨΦ. But, contrary to the worldsheet approach wherethese interactions were sufficient, one needs typically higher-order elementary interactionssuch as Ψn corresponding to a vertex with n closed strings (without limit on n). A secondspecific feature is that the products also admit a loop (or genus g) expansion: one needsto introduce a fundamental n-point interaction at every loop order g. These terms areinterpreted as (finite) counter-terms needed to restore the gauge invariance of the measure.These two facts come from the decomposition of the moduli spaces in pieces (Section 1.3.3).

As explained in Section 1.2, writing an action for a field Ψ[X(σ)] for which reparamet-rization invariance holds is highly complicated. The most powerful method is to introducea functional dependence in ghost fields Ψ[X(σ), , c(σ)] and to extend the BRST formalismto the string field, leading ultimately to the BV formalism. While the latter formalism isthe most complete and ensures that the theory is consistent at the quantum level, it is diffi-cult to characterize the interactions explicitly. Several constructions which exploit differentproperties of the theory have been proposed:

45

• direct computation by reverse engineering of worldsheet amplitudes;

• specific parametrization of the Riemann surfaces (hyperbolic, minimal area);

• analogy with Chern–Simons and Wess–Zumino–Witten (WZW) theories;

• exploitation of the L∞ and A∞ algebra structures.

It can be shown that these constructions are all equivalent. For the superstring, the simpleststrategy is to dress the bosonic interactions with data from the super-ghost sector, whichmotivates the study of the bosonic SFT by itself. The main difficulty in working with SFT isthat only the first few interactions have benn constructed explicitly. Finally, the advantageof the first formulation is that it provides a general formulation of SFT at the quantumlevel, from which the general structure can be studied.

1.4.2 Expression with spacetime fieldsTo obtain a more intuitive picture and to make contact with the spacetime fields, one canexpand the field in terms of 1-particle states in the momentum representation

|Ψ〉 =∑n

∫ dDk(2π)D ψα(k) |k, α〉 , (1.50)

where α denotes collectively the discrete labels in the CFT eigenstates. The coefficientsψα(k) of the CFT states |k, α〉 are spacetime fields, the first ones being the same as thosefound in the first-quantized picture (Section 1.3.1)

ψα = T,Gµν , Bµν ,Φ, . . .. (1.51)

Then, one can insert this expansion in the action such that one gets an expression likeS[T,Gµν , . . .]. The exact expression of this action is out of reach and one can computeexplicitly only the lowest terms for a given CFT background. Nonetheless, examining thestring field action indicates what is the generic form of the action in terms of the spacetimefields. One can then study the properties of such a general QFT: since it is more generalthan the SFT (expanded) action, any result derived for it will also be valid for SFT. Thisapproach is very fruitful for studying properties related to consistency of QFT (unitarity,soft theorems. . . ) and this can provide helpful phenomenological models.

In conclusion, one can see SFT as a regular QFT with the following properties:

• infinite number of fields;

• non-local interaction (proportional to e−k2#);

• the amplitudes agree with the worldsheet amplitudes (when the latter can be defined);

• genuine (IR) divergences agree but can be handled with the usual QFT tools.

1.4.3 ApplicationsIn Section 1.2, we have given several more pragmatical reasons to motivate the constructionof a string field theory. The first aspect is the possibility to use standard QFT techniques(such as renormalization) to study – and to make sense of – string amplitudes. In this sense,SFT can be viewed as providing recipes for computing quantities in the worldsheet theorywhich are otherwise not defined. This program has been pushed quite far in the last years.

Another reason to use SFT is gauge invariance: it is always easier to describe a sys-tem when its gauge invariance is manifest. We have explained that string theory contains

46

Yang–Mills and graviton fields with the corresponding (spacetime) gauge invariances (non-Abelian gauge symmetry and diffeomorphisms). In fact, these symmetries are enhanced toan enormous gauge invariance when taking into account the higher-spin fields. This invari-ance is hidden in the standard formulation and one cannot exploit it fully. On the otherhand, the full gauge symmetry is manifest in string field theory.

Finally, the worldvolume description of p-brane is difficult because one does not have theanalogue of the Polyakov action. If one could find a first-principle description of SFT whichdoes not rely on CFT and first-quantization, then one may hope to generalize it to build abrane field theory.We can summarize the general motivations for studying SFT:

• field theory (second-quantization);

• more rigorous and constructive formulation;

• make gauge invariance explicit (L∞ algebras et al.);

• use standard QFT techniques (renormalization, analyticity. . . )→ remove IR divergences, prove consistency (Cutkosky rules, unitarity, soft theorems,background independence. . . );

• worldvolume theory ill-defined for (p > 1)-branes.

Beyond these general ideas, SFT has been developed in order to address different questions:

• worldsheet scattering amplitudes;

• effective actions;

• map of the consistent backgrounds (classical solutions, marginal deformations, RRfluxes. . . );

• collective, non-perturbative, thermal, dynamical effects;

• symmetry breaking effects;

• dynamics of compactification;

• proof of dualities;

• proof of the AdS/CFT correspondence.

The last series of points is still out of reach within the current formulation of SFT. However,the last two decades have seen many important develoments developments:

• construction of the open, closed and open-closed superstring field theories:

– 1PI and BV actions and general properties [85, 186, 187, 243, 244, 246, 248, 250,252, 253, 256, 260],

– dressing of bosonic products using the WZW construction and homotopy al-gebra [79–82, 86–88, 92, 93, 126, 153, 156, 162–168],

– light-cone super-SFT [129–132]– supermoduli space [197, 273];

• hyperbolic and minimal area constructions [49, 116, 117, 184, 185, 207];

• open string analytic solutions [90, 91];

47

• level-truncation solutions [157–159];

• field theory properties [43, 54, 172, 211, 251, 254, 255];

• spacetime effective actions [77, 176, 177, 283];

• defining worldsheet scattering amplitudes [208–210, 245, 246, 249, 257–259];

• marginal and RR deformations [44, 257, 283].

Recent reviews are [55, 83, 84].

1.5 LimitationsIn order to give a more balanced view of string theory, one should mentioned some drawbacksthat plague the theory and that have hindered any direct application to the real world.

Even though string theory contains all the ingredients for describing a grand unifiedtheory with quantum gravity, the precise construction is yet unknown. The reason is thatone needs to perform a compactification of the extraneous spatial dimensions (or any otherprocess which “hides” them) to recover a 4-dimensional world. Since no dynamical processstarting from the 10-dimensional string theory leads to such a compactification, one is lead tostudy the possibilities by hand and to introduce many parameters to describe the result. Asa consequence, string theory has made no (practical and testable) experimental prediction.

The bosonic string theory is not consistent due to the presence of a tachyon in its spec-trum. While the superstring does not suffer from that, the presence of supersymmetrycomplicates its relation to the standard model, especially since natural low-energy supersym-metry appears to be more and more excluded experimentally. Studying non-supersymmetriccompactifications is particularly challenging and has mostly been ignored (see however [1]).

All current methods to work with string theory (including string field theory) requiresto fix the background (in the form of a worldsheet CFT) to perform explicit computations.However, one would expect a manifest background independence from a theory of quantumgravity.

Ultimately, these different problems originate from the absence of a (quasi-)axiomaticformulation of string theory. One does not start from first principles and then derive aframework and its consequences, but rather one proceeds through educated guesses andanalogies which have turned into a sets of rules for computing what one is interested in.Nonetheless, the deep meaning of many concepts (such as the branes, the dualities and thedefinition of M-theory as a non-perturbative string theory which encompasses all knownstring theories) is missing. Moreover, most SFT computations aiming at producing explicitnumbers are often reduced to worldsheet computations, with their inherent limitations. Thecomplete understanding of string field theory would help alleviate this point and give a moresystematic framework.

1.6 ReferencesNumerous books have been published on string theory. Useful complements are:

• Zwiebach [300]: the best introductory book, which covers in details the classical aspectof the bosonic string theory and the most important aspects of its quantization. Itdescribes also some advanced aspects and contains important insights on the structureof the theory.

• Lawrie [173, chap. 15]: the best short introduction to the most important concepts.

48

• Blumenhagen, Lüst, Theisen [31]: very complete and pedagogical book, certainly thebest generic introduction and reference.

• Polchinski [218, 219]: an excellent complement to the previous book, it contains addi-tional formal aspects not developed in [31]. The difficulty increases quickly and thisbook is not recommended for a first approach to the topic.

• Kiritsis [149]: very useful as a reference but not suitable as an introduction.

• Deligne et al.: a (huge) book more focused on the mathematical aspects and on stringperturbation theory is [57, 58].

• Kaku [143, 144]: the only books to address SFT in some length. Some parts areoutdated or follow an untraditional approach, which can make them hard to follow.

• Schomerus [234]: short introduction to the main concepts of string theory.

Other books on string theory and related aspects are [66, 110, 111, 125, 138, 204, 287].Good lecture notes include [182, 271, 272, 281, 284, 293].

For references about different aspects in this chapter:

• Differences between the worldvolume and spacetime formalisms – and of the associatedfirst- and second-quantization – for the particle and string [143, chap. 1, 300, chap. 11].

• General properties of relativistic strings [108, 300].

• Divergences in string theory [55, 247, 291, sec. 7.2].

• Motivations for building a string field theory [217, sec. 4].

49

Part I

Point-particle

50

Chapter 2

Classical relativisticpoint-particle

One considers a D-dimensional spacetime with coordinates xµ = (t, xi = x) with signaturemostly plus and metric

ηµν = diag(−1, 1, . . . , 1). (2.1)

The objective of this chapter is to describe the different formulation of a relativistic particlewhich travels in this spacetime along a worldline C.

2.1 Proper-time actionThe invariant line element s is expressed in terms of the coordinates Xµ as

ds2 = ηµνdXµdXν = −dT 2 + dX2. (2.2)

Since a particle cannot go backward in time, it means that T > 0 and going in the rest-frame(where X = 0) shows that a physical particle is timelike (massive) or null (massless)

massive : ds2 < 0, massless : ds2 = 0. (2.3)

Accordingly the proper-time t is defined by

dt2 = −ds2. (2.4)

Since the action is a (Lorentz) scalar, it is natural to postulate that it should be propor-tional to the invariant element (integrated over the worldline) because it is the only availablescalar

S = m

∫C

ds. (2.5)

The problem with this action is that the Lagrangian is constant and thus the equations ofmotion are identically zero. Moreover the previous action cannot describe massless particlessince the limit m → 0 or ds → 0 is singular. The first aspect is remedied by introducingan explicit parametrization of the worldline, while more work is needed to describe masslessparticles.

51

2.2 Parametrization and non-linear actionThe wordline is parametrized by τ (which is not necessarily the proper-time τ)1 which ismonotonically increasing along the wordline. The (spacetime) path followed by the particleis denoted by Xµ(τ), where the uppercase helps to distinguish between the wordline (specificto the particle under concerned) from the coordinates (labels of the spacetime). Derivationwith respect to τ is indicated by a dot

˙ = ddτ . (2.6)

The velocity of the particle corresponds to the changes of its position as τ changes

uµ = dXµ

dτ := Xµ. (2.7)

In terms of these variables the derivative of the invariant reads

s2 =(

dsdτ

)2= ηµνu

µuν . (2.8)

The definition of the proper-time (2.4) is equivalent to

t = 1. (2.9)

This implies that the variation of the proper-time is positive

dt > 0 (2.10)

and thus it can be selected to parametrize the worldline. Integrating the relation gives

τ = t + t0. (2.11)

In this parametrization the velocity is normalized to unity(dXµ

dt

)2= −1. (2.12)

More generally one can consider a constant coefficient of proportionality t = α as the simplesttypes of parametrization

τ = αt + t0. (2.13)In terms of the worldline parameter the action and the associated Lagrangian read

S =∫C

dτ L(X, X), L = −m√−ηµν

dXµ

dτdXν

dτ . (2.14)

This action is obviously invariant under the Lorentz group SO(D, 1).

Computation – Equation (2.14)Starting from (2.5)

S = m

∫ds = −m

∫dτ(−ds

dτ

)= −m

∫dτ

√−(

dsdτ

)2.

Inserting a minus sign is necessary since s < 0 and the argument of the squareroot must

1In (special) relativity textbooks, the worldline parameter is either denoted by another symbol (suchthat λ or s), or is taken to be the proper-time. We don’t follow these conventions in order to match thenotations of string theory in the worldsheet formalism where τ and σ parametrize the worldsheet withoutbeing proper-coordinates. It will be explicitly indicated when τ is taken to be the proper-time.

52

be positive. The result follows by plugging the expression (2.2) of ds2 in terms of Xµ.

The parameter m can be identified with the mass of the particle. First dimensionalanalysis indicates that it has the dimension of a mass [m] = M since [S] = 1 and [τ ] = M−1.This can be further motivated by taking the non-relativistic limit with τ = t = T (staticgauge)

L = −m

√1−

(dXdt

)2≈ −m+ m

2

(dXdt

)2. (2.15)

The second term represents the kinetic energy of a non-relativistic particle of mass m, whilethe first term can be identified with a constant potential (mass energy). One can also putthis in perspective with the definition of the conjugate momenta. Note that the actionsatisfies the bound

S ≤ −m∫

dt (2.16)

since√

1− v2 < 1.The action (2.14) has several characteristics which make it difficult to study:

• it is non-polynomial since the variable appears under a squareroot and as such it isnon-local (infinite number of time derivatives);

• the massless limit m→ 0 is singular;

• the worldline geometry is not parametrized explicitly: while this can be an asset (sinceparametrization independence is ensured) this can also make more difficult to studythe set of possible worldlines.

The last problem can be solved by introducing a metric on the worldline. This will alsoprovide a starting point to address the other two issues.

2.2.1 Induced metricGiven a path Xµ(τ), a metric is induced on the worldline by pulling back the spacetimemetric

gττ = ηµνdXµ

dτdXν

dτ = −(

dtdτ

)2(2.17)

and this implies in particular the equality of the worldline and spacetime invariant intervals

ds2 = ηµν dXµdXν = gττ dτ2. (2.18)Note that the worldline metric has only one component since the worldline C is a one-dimensional manifold (without spatial directions). Moreover it is convenient to work withthe einbein e(τ)

gττ = −e(τ)2, e = dtdτ , (2.19)

where the minus sign follows from ηττ = −1.The induced metric (2.17) is recognized to be the argument of the squareroot in (2.14).The action (2.14) becomes

S = −m∫

dτ√−g = −m

∫dτ e. (2.20)

This action enjoys an invariance under reparametrization of τ

τ = f(τ ′). (2.21)

53

2.2.2 HamiltonianThe conjugate momenta are given by

pµ = ∂L

∂Xµ= mXµ√

−XµXµ

= m

eXµ. (2.22)

and Euler–Lagrange equations say it is conserved

dpµdτ = 0. (2.23)

The Hamiltonian is defined byH = Xµpµ − L. (2.24)

The momentum norm isp2 = pµp

µ = −m2 (2.25)which imposes a constraint

φ(p) = p2 +m2 = 0. (2.26)Equivalently the equation pµ = fµ(Xν) is not invertible. This is another formulation ofthe constraint found between the equations of motion, and by comparing it means that theHamiltonian vanishes

H = e

m(p2 +m2) = 0. (2.27)

2.3 Polynomial action2.3.1 Dirac procedureThe Dirac procedure permits to build an Hamiltonian for constrained systems. Let λ be aLagrange multiplier, then the Dirac hamiltonian is

H ′ = λH = m

eλφ. (2.28)

The Hamilton equation gives the relation between Xµ and pµ

Xµ = ∂H ′

∂pµ= 2λ e

mpµ. (2.29)

The associated (first-order) Lagrangian is

L′ = Xµpµ − λH = Xµpµ −λe

m(p2 +m2). (2.30)

L′ = 2λe m2

(XµXµ

(2λe)2 − 1). (2.31)

For only the combination 2λe appears, it means that both are not independent variablesand one can redefine e to absorb 2λ, which amounts to set

λ = 12 . (2.32)

The final form of the action is obtained by rescaling

e −→ me (2.33)

54

such thatS′ = 1

2

∫dτ e

(XµXµ

e2 −m2). (2.34)

The massless limit m→ 0 is well-defined and yield the action

S′ = 12

∫dτ e−1 XµXµ. (2.35)

The interpretation of S′ can be made easier by replacing the einbein by the metric (2.19)(with gττ = e−2)

S′ = −m2

∫dτ√−g gττ∂τXµ∂τX

µ − m

2

∫dτ√−γττ (2.36)

This is the action forD scalar fieldsXµ(τ) living on the 1-dimensional space C and minimallycoupled to 1-dimensional gravity with a cosmological constant −m/2.

2.4 Bibliography• relativistic particle [12, 281, sec. 1.1, 2, chap. 1].

55

Part II

Bosonic string theory

56

Chapter 3

Worldsheet path integral:vacuum amplitudes

In this chapter, we develop the path integral quantization for a generic closed string theoryin worldsheet Euclidean signature. We focus on the vacuum amplitudes, leaving scatteringamplitudes for the next chapter. This allows to focus on the definition and gauge fixing ofthe path integral measure.

The exposition is distinguishes from most traditional textbooks in three ways: 1) weconsider a general matter CFT, 2) we consider the most general treatment (for any genus)and 3) we don’t use complex coordinates but always a covariant parametrization.

The derivation is technical and the reader is encouraged to not stop at this chapterin case of difficulties and to proceed forward: most concepts will be reintroduced from adifferent point of view later in other chapters of the review.

3.1 Worldsheet action and symmetriesThe string worldsheet is a Riemann surface W = Σg of genus g: the genus counts thenumber of holes or handles. Coordinates on the worldsheet are denoted by σa = (τ, σ).When there is no risk of confusion, σ denotes collectively both coordinates. Since closedstrings are considered, the Riemann surface has locally the topology of a cylinder, with thespatial section being circles S1 with radius taken to be 1, such that

σ ∈ [0, 2π), σ ∼ σ + 2π. (3.1)

The string is embedded in the D-dimensional spacetimeM with metric Gµν through mapsXµ(σa) :M→W with µ = 0, . . . , D − 1.

In Polyakov’s formalism, the worldsheet geometry is endowed with a metric gab(σ) to-gether with a set of matter fields living on it. Starting from the Nambu–Goto action

SNG[Xµ] = 12πα′

∫d2σ

√detGµν(X)∂X

µ

∂σa∂Xν

∂σb, (3.2)

where α′ is the Regge slope (related to the string tension and string length), and translatingit to the Polyakov action

SP[g,Xµ] = 14πα′

∫d2σ√g gabGµν(X)∂X

µ

∂σa∂Xν

∂σb, (3.3)

57

it is clear that the scalar fields Xµ(σ) (µ = 0, . . . D−1) characterizes the string theory underconsideration in two ways. First, by specifying the spacetime in which the string propagates,second, by describing the internal degrees of freedom (vibration modes).1 But, nothing pre-vents to consider a more general matter content in order to describe a different spacetime ordifferent degrees of freedom. The scalar fields Xµ can be described by a general sigma modelwhich encodes the embedding of the string in the D non-compact spacetime dimensions, andother fields can be added, for example to describe compactified dimensions or (spacetime)spin. Different sets of fields (and actions) correspond to different string theories. However,to describe precisely the different possibilities, we first have to understand the constraints onthe worldsheet theories and to introduce conformal field theories (Part VII). In this chapter(and in most of the book), the precise matter content is not important and we will denotethe fields collectively as Ψ(σ).

In order to describe a proper string theory, the metric should not be dynamical. Thismeans that the worldsheet has no intrinsic dynamics and that no supplementary degreesof freedom are introduced when parametrizing the worldsheet with a metric. A solutionto remove these degrees of freedom is to introduce gauge symmetries with as many gaugeparameters as there are of degrees of freedom. The simplest symmetry is invariance underdiffeomorphisms: indeed, the worldsheet is effectively a QFT coupled to gravity and itmakes sense to require this invariance. Physically, this corresponds to the fact that theworldsheet spatial coordinate σ used along the string and worldsheet time are arbitrary.However, diffeomorphisms alone are not sufficient to completely fix the metric. Anothernatural candidate is Weyl invariance (local rescalings of the metric).

A diffeomorphism f ∈ Diff(Σg) acts on the fields as

σ′a = fa(σb), g′(σ′) = f∗g(σ), Ψ′(σ′) = f∗Ψ(σ), (3.4)

where the star denotes the pullback by f : this corresponds simply to the standard coordinatetransformation where each tensor index of the field receives a factor ∂σa/∂σ′b. In particular,the metric and scalar fields transform explicitly as

g′ab(σ′) = ∂σc

∂σ′a∂σd

∂σ′bgcd(σ), X ′µ(σ′) = Xµ(σ). (3.5)

The index µ is inert since it is a target spacetime index: from the worldsheet point of view,it just labels a collection of worldsheet scalar fields. Infinitesimal variations are generatedby vector fields on Σg:

δξσa = ξa, δξΨ = LξΨ, δξgab = Lξgab, (3.6)

where Lξ is the Lie derivative2 with respect to the vector field ξ ∈ diff(Σg) ' TΣg. The Liederivative of the metric is

Lξgab = ξc∂cgab + gac∂bξc + gbc∂aξ

c = ∇aξb +∇bξa. (3.7)

The Lie algebra generates only transformations in the connected component Diff0(Σg) ofthe diffeomorphism group which contains the identity.

Transformations not contained in Diff0(Σg) are called large diffeomorphisms: this in-cludes reflections, for example. The quotient of the two groups is called the modular groupΓg (also mapping class group or MCG):

Γg := π0(Diff(Σg)

)= Diff(Σg)

Diff0(Σg). (3.8)

1Obviously, the vibrational modes are also constrained by the spacetime geometry.2For our purpose here, it is sufficient to accept the definition of the Lie derivative as corresponding to

the infinitesimal variation.

58

It depends only on the genus g (but not on the surface). It is an infinite discrete group forgenus g ≥ 1 surfaces; in particular, Γ1 = SL(2,Z).

Integrating the Riemann curvature R over the surface Σg yields a topological invariant,the Euler characteristics:

χg := χ(Σg) := 2− 2g = 14π

∫Σg

d2σ√g R. (3.9)

Oriented Riemann surfaces without boundaries are completely classified by their Euler char-acteristics, or equivalently by their genus.

A Weyl transformation e2ω ∈ Weyl(Σg) corresponds to a local rescaling of the metricand leaves the other fields unaffected3

g′ab(σ) = e2ω(σ)gab(σ), Ψ′(σ) = Ψ(σ). (3.10)

The exponential parametrization is generally more useful, but one should remember that itis e2ω and not ω which is an element of the group. The infinitesimal variation reads

δωgab = 2ω gab, δωΨ = 0 (3.11)

where ω ∈ weyl(Σ) ' F(Σg) is a function on the manifold. Two metrics related in this wayare said to be conformally equivalent. The conformal structure of the Riemann surface isdefined by

Conf(Σg) := Met(Σg)Weyl(Σg)

, (3.12)

where Met(Σg) denotes the space of all metrics on Σg. Each element is a class of conformallyequivalent metrics.

Diffeomorphisms have two parameters ξa (vector field) and Weyl invariance has one, ω(function). Hence, this is sufficient to fix the three components of the metric (symmetricmatrix) and the total gauge group of the theory is the semi-direct product

G := Diff(Σg) n Weyl(Σg). (3.13)

Similarly, the component connected of the identity is written as

G0 := Diff0(Σg) n Weyl(Σg). (3.14)

The semi-direct product arises because the Weyl parameter is not inert under diffeo-morphisms. Indeed, the combination of two transformations is

g′ = f∗(e2ωg

)= e2f∗ωf∗g, (3.15)

such that the diffeomorphism acts also on the conformal factor.The combination of transformations (3.15) can be chosen to fix the metric in a convenient

gauge. For example, the conformal gauge reads

gab(σ) = e2φ(σ)gab(σ), (3.16)

where gab is some (fixed) background metric and φ(σ) is the conformal factor, also calledthe Liouville field. Fixing only diffeomorphisms amount to keep φ arbitrary: the latter can

3One could also consider global Weyl transformations, but this is not of interest here because one islooking for gauge symmetries in order to fix the metric. In more general contexts, it is also possible toconsider matter fields which transform under the Weyl transformations.

59

then be fixed with a Weyl transformation. For instance, one can adopt the conformally flatgauge

gab = δab, φ arbitrary (3.17)

with a diffeomorphism, and then reach the flat gauge

gab = δab, φ = 0 (3.18)

with a Weyl transformation. Another common choice is the uniformization gauge where gis taken to be the metric of constant curvature on the sphere (g = 0), on the plane (g = 1)or on the hyperbolic space (g > 1). All these gauges are covariant (both in spacetime andworldsheet).

Remark 3.1 (Active and passive transformations) Usually, symmetries are describedby active transformations, which means that the field is seen to be changed by the transform-ation. On the other hand, gauge fixing is seen as a passive transformation, where the fieldis expressed in terms of other fields (i.e. a different parametrization). These are mathem-atically equivalent since both cases correspond to inverse elements, and one can choose themost convenient representation. We will use indifferently the same name for the parametersto avoid introducing minus signs and inverse.

Remark 3.2 (Topology and gauge choices) While it is always possible to adopt locallythe flat gauge (3.18), it may not be possible to extend it globally. The can be seen intuitivelyfrom the fact that the sign of the curvature is given by the one of 1− g, but the curvature ofthe flat metric is zero: curvature must then be localized somewhere and this prevents fromusing a single coordinate patch.

The final step is to write an action Sm[g,Ψ] for the matter fields. According to theprevious discussion, it must have the following properties:

• local in the fields;

• renormalizable;

• non-linear sigma models for the scalar fields;

• periodicity conditions;

• invariant under diffeomorphisms (3.4);

• invariant under Weyl transformations (3.10).

The latter two conditions are summarized by

Sm[f∗g, f∗Ψ] = Sm[g,Ψ], Sm[e2ωg,Ψ] = Sm[g,Ψ]. (3.19)

The invariance under diffeomorphisms is straightforward to enforce by using only covariantobjects. Since the scalar fields represent embedding of the string in spacetime, the non-linear sigma model condition means that spacetime is identified with the target space of thesigma model, of which D dimensions are non-compact, and the spacetime metric appearsin the matter action as in (3.3). The isometries of the target manifold metric becomeglobal symmetries of Sm: while they are not needed in this chapter, they will have theirimportances in other chapters. Finally, to make the action consistent with the topology ofthe worldsheet, the fields must satisfy appropriate boundary conditions. For example, thescalar fields Xµ must be periodic:

Xµ(τ, σ) ∼ Xµ(τ, σ + 2π). (3.20)

60

Remark 3.3 (2d gravity) The setup in two-dimensional gravity is exactly similar, exceptthat the system is, in general, not invariant under Weyl transformations. As a consequence,one component of the metric (usually taken to be the Liouville mode) remains unconstrained:in the conformal gauge, (3.16) only g is fixed.

The symmetries (3.19) of the action have an important consequence: they imply that thematter action is conformally invariant on flat space gab = δab. A two-dimensional conformalfield theory (CFT) is characterized by a central charge cm: roughly, it is a measure of thequantum degrees of freedom. The central charge is additive for decoupled sectors. In partic-ular, the scalar fields Xµ contribute as D, and it is useful to define the perpendicular CFTwith central charge c⊥ as the matter which does not describe the non-compact dimensions:

cm = D + c⊥. (3.21)

This will be discussed in length in Part VII. For this chapter and most of the book, it issufficient to know that the matter is a CFT of central charge cm and includes D scalar fieldsXµ:

matter CFT parameters: D, cm. (3.22)

The energy–momentum is defined by

Tm,ab := − 4π√g

δSmδgab

. (3.23)

The variation of the action under the transformations (3.6) vanishes on-shell if the energy–momentum tensor is conserved

∇aTm,ab = 0 (on-shell). (3.24)

On the other hand, the variation under (3.11) vanishes off-shell (i.e. without using theequations of motion) if the energy–momentum tensor is traceless:

gabTm,ab = 0 (off-shell). (3.25)

The conserved charges associated to the energy–momentum tensor generate worldsheettranslations

P a :=∫

dσ T 0am . (3.26)

The first component is identified with the worldsheet Hamiltonian P 0 = H and generatestime translations, the second component generates spatial translations.

Remark 3.4 (Tracelessness of the energy–momentum tensor) In fact, the trace canalso be proportional to the curvature

gabTm,ab ∝ R. (3.27)

Then, the equations of motion are invariant since the integral of R is topological. The theoryis invariant even if the action is not. Importantly, this happens for fields at the quantumlevel (Weyl anomaly), for the Weyl ghost field (Section 3.4) and for the Liouville theory(two-dimensional gravity coupled to conformal matter).

61

3.2 Path integralThe quantization of the system is achieved by considering the path integral, which yieldsthe genus-g vacuum amplitude (or partition function):

Zg :=∫ dggab

Ωgauge[g] Zm[g], Zm[g] :=∫

dgΨ e−Sm[g,Ψ] (3.28)

at fixed genus g (not to be confused with the metric). The integration over gab is performedover all metrics of the genus-g Riemann surface Σg: gab ∈ Met(Σg). The factor Ωgauge[g]is a normalization inserted in order to make the integral finite: it depends on the metric(but only through the moduli parameters, as we will show later) [63, p. 931], which explainswhy it is included after the integral sign. Its value will be determined in the next section byrequiring the cancellation of the infinities due to the integration over the gauge parameters.This partition function corresponds to the g-loop vacuum amplitude: interactions and theirassociated scattering amplitudes are discussed in Section 4.1.

In order to perform the gauge fixing and to manipulate the path integral (3.28), it isnecessary to define the integration measure over the fields. Because the space is infinite-dimensional, this is a difficult task. One possibility is to define the measure implicitlythrough Gaussian integration over the field tangent space (see also Appendix C.1). A Gaus-sian integral involves a quadratic form, that is, an inner-product (or equivalently a metric)on the field space. The explanation is that a metric also defines a volume form, and thusa measure. To reduce the freedom in the definition of the inner-product, it is useful tointroduce three natural assumptions:

1. ultralocality: the measure is invariant under reparametrizations and defined point-wise,which implies that it can depend on the fields but not on their derivatives;

2. invariant measure: the measure for the matter transforms trivially under any sym-metry of the matter theory by contracting indices with appropriate tensors;

3. free-field measure: for fields other than the worldsheet metric and matter (like ghosts,Killing vectors, etc.), the measure is the one of a free field.

This means that the inner-product is obtained by contracting the worldsheet indices of thefields with a tensor built only from the worldsheet metric, by contracting other indices (likespacetime) with some invariant tensor (like the spacetime metric), and finally by integratingover the worldsheet.

We need to distinguish the matter fields from those appearing in the gauge fixing proced-ure. The matter fields live in some representation and the inner-product must be invariant:this means that it is not possible to define each field measure independently if the ex-ponential of inner-products does not factorize. As an example, on a curved background:dX 6=

∏µ dXµ. However, we will not need to write explicitly the partition function for

performing the gauge fixing: it is sufficient to know that the matter is a CFT. In the gaugefixing procedure, different types of fields (including the metric) appear which don’t carryindices (beyond the worldsheet indices). Below, we focus on defining a measure for each ofthose single fields (and use free-field measures according to the third condition).

Considering the finite elements δΦ1 and δΦ2 of tangent space at the point Φ of the stateof fields, the inner-product (·, ·)g and its associated norm | · |g read

(δΦ1, δΦ2)g :=∫

d2σ√g γg(δΦ1, δΦ2), |δΦ|2g := (δΦ, δΦ)g, (3.29)

where γg is a metric on the δΦ space. It is taken to be flat for all fields except the metric itself,that is, independent of Φ. The dependence in the metric ensures that the inner-product is

62

diffeomorphism invariant, which in turns will lead to a metric-dependent but diffeomorphisminvariant measure. The functional measure is then normalized by a Gaussian integral:∫

dgδΦ e− 12 (δΦ,δΦ)g = 1√

det γg. (3.30)

This, in turn, induces a measure on the field space itself:∫dΦ√

det γg (3.31)

The determinant can be absorbed in the measure, such that∫dgδΦ e− 1

2 (δΦ,δΦ)g = 1. (3.32)

In fact, this normalization and the definition of the inner-product is ambiguous, but theultralocality condition allows to fix uniquely the final result (Section 3.3.4). Moreover, sucha free-field measure is invariant under field translations

Φ(σ) −→ Φ′(σ) = Φ(σ) + ε(σ). (3.33)

The most natural inner-products for single scalar, vector and symmetric tensor fields are

(δf, δf)g :=∫

d2σ√g δf2 (3.34a)

(δV a, δV a)g :=∫

d2σ√g gabδV

aδV b, (3.34b)

(δTab, δTab)g :=∫

d2σ√g GabcdδTabδTcd, (3.34c)

where the (DeWitt) metric for the symmetric tensor is

Gabcd := Gabcd⊥ + u gabgcd, Gabcd⊥ := gacgbd + gadgbc − gabgcd, (3.35)

with u a constant. The first term G⊥ is the projector on the traceless component of thetensor. Indeed, consider a traceless tensor gabTab = 0 and a pure trace tensor Λgab, then wehave:

GabcdTcd = Gabcd⊥ Tcd = 2Tab, Gabcd(Λgcd) = 2u (Λgab). (3.36)

While all measures are invariant under diffeomorphisms, only the vector measure isinvariant under Weyl transformations. This implies the existence of a quantum anomaly(the Weyl or conformal anomaly): the classical symmetry is broken by quantum effectsbecause the path integral measure cannot respect all the classical symmetries. Hence, onecan expect difficulties for imposing it at the quantum level and ensuring that the Liouvillemode in (3.16) remains without dynamics.

The metric variation (symmetric tensor) is decomposed in its trace and traceless parts

δgab = gab δΛ + δg⊥ab, δΛ = 12 g

abδgab, gabδg⊥ab = 0. (3.37)

In this decomposition, both terms are decoupled in the inner-product

|δgab|2g = 4u|δΛ|2g + |δg⊥µν |2g, (3.38)

63

where the norm of δΛ is the one of a scalar field (3.34a). The norm for δg⊥ab is equivalentto (3.34c) with u = 0 (since it is traceless). Requiring positivity of the inner-product for anon-traceless tensor imposes the following constraint on u:

u > 0. (3.39)

One can absorb the coefficient with u in δΛ, which will just contribute as an overall factor:its precise value has no physical meaning. The simple choice u = 1/4 sets the coefficient of|δΛ|2g to 1 in (3.38) (another common choice is u = 1/2). Ultimately, this implies that themeasure factorizes as

dggab = dgΛ dgg⊥ab. (3.40)

Computation – Equation (3.38)

Gabcd δgabδgcd =(Gabcd⊥ + u gabgcd

)(gab δΛ + δg⊥ab

)(gcd δΛ + δg⊥cd

)=(2u gcd δΛ +Gabcd⊥ δg⊥ab

)(gcd δΛ + δg⊥cd

)= 4u (δΛ)2 +Gabcd⊥ δg⊥abδg

⊥cd

= 4u δΛ2 + 2gacgbdδg⊥abδg⊥cd.

Remark 3.5 Another common parametrization is

Gabcd = gacgbd + c gabgcd. (3.41)

It corresponds to (3.35) up to a factor 1/2 and setting u = 1 + 2c.

Remark 3.6 (Matter and curved background measures) As explained previously, mat-ter fields carry a representation and the inner-product must yield an invariant combination.In particular, spacetime indices must be contracted with the spacetime metric Gµν(X) (whichis the non-linear sigma model metric appearing in front of the kinetic term) for a generalcurved background. For example, the inner-product for the scalar fields Xµ is

(δXµ, δXµ)g =∫

d2σ√g Gµν(X)δXµδXν . (3.42)

It is not possible to normalize anymore the measure to set detG(X) = 1 like in (3.32) sinceit depends on the fields. On the other hand, this factor is not important for the manipu-lations performed in this chapter. Any ambiguity in the measure will again corresponds toa renormalization of the cosmological constant [63, p. 923]. Moreover, as explained above,it is not necessary to write explicitly the matter partition function as long as it describes aCFT.

3.3 Faddeev–Popov gauge fixingThe naive integration over the space Met(Σg) of all metrics of Σg (note that the genus isfixed) leads to a divergence of the functional integral since equivalent configurations

(f∗g, f∗Ψ) ∼ (g,Ψ), (e2ωg,Ψ) ∼ (g,Ψ) (3.43)

gives the same contribution to the integral. This infinite redundancy causes the integralto diverge, and since the multiple counting is generated by the gauge group, the infinitecontribution corresponds to the volume of the latter. The Faddeev–Popov procedure is

64

a mean to extract this volume by separating the integration over the gauge and physicaldegrees of freedom

d(fields) = Jacobian× d(gauge)× d(physical). (3.44)

The space of fields (g,Ψ) is divided into equivalence classes and one integrates over only onerepresentative of each class (gauge slice), see Figure 3.1. This change of variables introducesa Jacobian which can be represented by a partition function with ghost fields (fields witha wrong statistics). This program encounters some complications since G is a semi-directproduct and is non-connected.

Example 3.1 – Gauge redundancyA finite-dimensional integral which mimics the problem is

Z =∫R2

dx dy e−(x−y)2. (3.45)

One can perform the change of variables

r = x− y, y = a (3.46)

such thatZ =

∫R

da∫ ∞

0e−r

2=√π

2 Vol(R), (3.47)

and Vol(R) is to be interpreted as the volume of the gauge group (translation by a realnumber a).

Remark 3.7 Mathematically, the Faddeev–Popov procedure consists in identifying the or-bits (class of equivalent metrics) under the gauge group G and to write the integral in termsof G-invariant objects (orbits instead of individual metrics). This can be done by decompos-ing the tangent space into variations generated by G and its complement. Then, one candefine a foliation of the metric space which equips it with a fibre bundle structure: the baseis the push-forward of the complement and the fibre corresponds to the gauge orbits. Theintegral is then defined by selecting a section of this bundle.

3.3.1 Metrics on Riemann surfacesAccording to the above procedure, each metric gab ∈ Met(Σg) has to be expressed in termsof gauge parameters (ξ and ω) and of a metric gab which contains the remaining gauge-independent degrees of freedom. As there are as many gauge parameters as metric com-ponents (Section 3.1), one could expect that there is no remaining physical parameters andthen that g is totally fixed. But, this is not the case and the metric g depends on a finitenumber of parameters ti (moduli). The reason for this is topological: while locally it isalways possible to completely fix the metric, topological obstructions may prevent doing itglobally. This means that not all conformal classes in (3.12) can be (globally) related by adiffeomorphism.

The quotient of the space of metrics by gauge transformations is called the moduli space

Mg := Met(Σg)G

. (3.48)

Accordingly, its coordinates ti with i = 1, . . . ,dimRMg are called moduli parameters. TheTeichmüller space Tg is obtained by taking the quotient of Met(Σg) with the component

65

Figure 3.1: The metric space decomposed in gauge orbits. Two metrics related by a gaugetransformation lie on the same orbit. Choosing a gauge slice amounts to pick one metric ineach orbit, and the projection gives the space of metric classes.

connected to the identityTg := Met(Σg)

G0. (3.49)

The space Tg is the covering space ofMg:

Mg = TgΓg, (3.50)

where Γg is the modular group defined in (3.8). Both spaces can be endowed with a complexstructure and are finite-dimensional [193]:

Mg := dimRMg = dimR Tg =

0 g = 0,2 g = 1,6g − 6 g ≥ 2,

(3.51)

In particular, their volumes are related by∫Mg

dMg t = 1ΩΓg

∫Tg

dMg t (3.52)

where ΩΓg is the volume of Γg.

We will need to extract volumes of different groups, so it is useful to explain how theyare defined. A natural measure on a connected group G is the Haar measure dg, which isthe unique left-invariant measure on G. Integrating the measure gives the volume of thegroup

ΩG :=∫G

dg =∫G

d(hg), (3.53)

for any h ∈ G. Given the Lie algebra g of the group, a general element of the algebra is alinear combinations of the generators Ti with coefficients αi

α = αiTi. (3.54)

66

Group elements can be parametrized in terms of α through the exponential map. Moreover,since a Lie group is a manifold, it is locally isomorphic to Rn: this motivates the use of aflat metric for the Lie algebra, such that

ΩG =∫

dα :=∫ ∏

i

dαi. (3.55)

Finally, it is possible to perform a change of coordinates from the Lie parameters to co-ordinates x on the group: the resulting Jacobian is the Haar measure for the coordinatesx.

Remark 3.8 While Tg is a manifold, this is not the case of Mg for g ≥ 2, which is anorbifold: the quotient by the modular group introduces singularities [195].

Remark 3.9 (Moduli space and fundamental domain) Given a group acting on a space,a fundamental domain for a group is a subspace such that the full space is generated by act-ing with the group on the fundamental domain. Hence, one can view the moduli space Mg

as a fundamental domain (sometimes denoted by Fg) for the group Γg and the space Tg.

In the conformal gauge (3.16), the metric gab can be parametrized by

gab = g(f,φ)ab (t) := e2f∗φf∗gab(t) = f∗

(e2φgab(t)

)(3.56)

where φ := ω and t denotes the dependence in the moduli parameters. To avoid surchargingthe notations, we will continue to write g when there is no ambiguity. In coordinates, thisis equivalent to:

gab(σ) = g(f,φ)ab (σ; t) := e2φ(σ)g′ab(σ; t), g′ab(σ; t) = ∂σ′c

∂σa∂σ′d

∂σbgcd(σ′; t). (3.57)

Remark 3.10 Strictly speaking, the matter fields also transform and one should write Ψ =Ψ(f) := f∗Ψ and include them in the change of integration measures of the following sections.But, this does not bring any particular benefits since these changes are trivial because thematter is decoupled from the metric.

Remark 3.11 Although the metric cannot be completely gauge fixed, having just a finite-dimensional integral is much simpler than a functional integral. In higher dimensions, thegauge fixing does not reduce that much the degrees of freedom and a functional integral overg remains (in similarity with Yang–Mills theories).

The corresponding infinitesimal transformations are parametrized by (φ, ξ, δti). Thevariation of the metric (3.56) can be expressed as

δgab = 2φ gab +∇aξb +∇bξa + δti∂igab, (3.58)

which is decomposed in a reparametrization (3.6), a Weyl rescaling (3.11), and a contri-bution from the variations of the moduli parameters. The latter are called Teichmüllerdeformations and describe changes in the metric which cannot be written as a combinationof diffeomorphism and Weyl transformation. Only the last term is written with a deltabecause the parameters ξ and φ are already infinitesimal. There is an implicit sum over iand we have defined

∂i := ∂

∂ti. (3.59)

67

According to the formula (3.55), the volumes ΩDiff0 [g] and ΩWeyl[g] of the diffeomorph-isms connected to the identity and Weyl group are

ΩDiff0 [g] :=∫

dgξ, (3.60a)

ΩWeyl[g] :=∫

dgφ. (3.60b)

The full diffeomorphism group has one connected component for each element of the modulargroup Γg, according to (3.8): the volume ΩDiff[g] of the full group is the volume of thecomponent connected to the identity times the volume ΩΓg

ΩDiff[g] = ΩDiff0 [g] ΩΓg . (3.60c)

We have written that the volume depends on g: but, the metric itself is parametrizedin terms of the integration variables, and thus the LHS of (3.60) cannot depend on thevariable which is integrated over: ΩDiff0 can depend only on φ and ΩWeyl only on ξ. But, allmeasures (3.34b) are invariant under diffeomorphisms, and thus the result cannot depend onξ. Moreover, the measure for vector is invariant under Weyl transformation, which meansthat ΩDiff0 does not depend on φ. This implies that the volumes depend only on the moduliparameters

ΩDiff0 [g] := ΩDiff0 [e2φg] = ΩDiff0 [g], ΩWeyl[g] := ΩWeyl[Lξ g] = ΩWeyl[g]. (3.61a)

For this reason, it is also sufficient to take the normalization factor Ωgauge to have the samedependence:

Ωgauge[g] := Ωgauge[g]. (3.61b)These volumes are also discussed in Section 3.3.4.


ΩDiff0 [e2φg] =∫

de2φLξ gξ =∫

de2φgξ =∫

dgξ = ΩDiff0 [g],

ΩWeyl[Lξ g] =∫

de2φLξ gφ =∫

de2φgφ = ΩWeyl[g].

Remark 3.12 (Free-field measure for the Liouville mode) The explicit measure (3.60b)of the Liouville mode is complicated since the inner-product contains an exponential of thefield:

|δφ|2 =∫

d2σ√g δφ2 =

∫d2σ

√g e2φδφ2. (3.62)

It has been proposed by David–Distler–Kawai [52, 67], and later checked explicitly [60, 61,181], how to rewrite the measure in terms of a free measure weighted by an effective action.The latter is identified with the Liouville action (Section 3.3.3).

In principle, we could follow the standard Faddeev–Popov procedure by inserting a deltafunction for the gauge fixing condition

Fab := gab − g(f,φ)ab (t), (3.63)

with g(f,φ)ab (t) defined in (3.56). However, we will take a detour to take the opportunity to

study in details manipulations of path integrals and to understand several aspects of the

68

geometry of Riemann surfaces. In any case, several points are necessary even when goingthe short way, but less apparent.

In order to make use of the factorization (3.40) of the integration measure, the variation(3.58) is decomposed into its trace (first term) and traceless parts (last two terms) (3.37)

δgab = 2Λ gab + (P1ξ)ab + δti µiab, (3.64)

where4

(P1ξ)ab = ∇aξb +∇bξa − gab∇cξc, (3.65a)

µiab = ∂igab −12 gab g

cd∂igcd, (3.65b)

Λ = Λ + 12 δti g

ab∂igab, Λ = φ+ 12 ∇cξ

c. (3.65c)

The objects µi are called Beltrami differentials and correspond to traceless Teichmüllerdeformations (the factor of 1/2 comes from the symmetrization of the metric indices). Thedecomposition emphasizes which variations are independent from each other. In particular,changes to the trace of the metric due to a diffeomorphism generated by ξ or a modificationof the moduli parameters can be compensated by a Weyl rescaling.

One can use (3.40) to replace the integration over gab by one over the gauge parametersξ and φ and over the moduli ti since they contain all the information about the metric:

Zg =∫

dMg tdgΛ dg(P1ξ) Ωgauge[g]−1 Zm[g]. (3.66)

It is tempting to perform the change of variables

(P1ξ, Λ) −→ (ξ, φ) (3.67)

such thatdg(P1ξ) dgΛ

?= dgξ dgφ∆FP[g] (3.68)

where ∆FP[g] is the Jacobian of the transformation

∆FP[g] = det ∂(P1ξ, Λ)∂(ξ, φ) = det

(P1 0? 1

)= detP1. (3.69)

But, one needs to be more careful:

1. The variations involving P1ξ and δti are not orthogonal and, as a consequence, themeasure does not factorize.

2. P1 has zero-modes, i.e. vectors such that P1ξ = 0, which causes the determinant tovanish, detP1 = 0.

A rigorous analysis will be performed in Section 3.3.2 and will lead to additional factors inthe path integral.

Next, if the actions and measures were invariant under diffeomorphisms and Weyl trans-formations (which amounts to replace g by g everywhere), it would be possible to factor outthe integrations over the gauge parameters and to cancel the corresponding infinite factorsthanks to the normalization Ωgauge[g]. A new problem arises because the measures are notWeyl invariant as explained above and one should be careful when replacing the metric(Section 3.3.3).

4For comparison, Polchinski [218] defines P1 with an overall factor 1/2.

69

3.3.2 Reparametrizations and analysis of P1

The properties of the operator P1 are responsible for both problems preventing a directfactorization of the measure; for this reason, it is useful to study it in more details.

The operator P1 is an object which takes a vector v to a symmetric traceless 2-tensor T ,see (3.65a). Conversely, its adjoint P †1 can be defined from the scalar product (3.34c)

(T, P1v)g = (P †1T, v)g, (3.70)

and takes symmetric traceless tensors to vectors. In components, one finds

(P †1T )a = −2∇bTab. (3.71)

The Riemann–Roch theorem relates the dimension of the kernels of both operators [193]:

dim kerP †1 − dim kerP1 = −3χg = 6g − 6. (3.72)

Teichmüller deformations

We first need to characterize Teichmüller deformations, the variations of moduli parameterswhich lead to transformations of the metric independent from diffeomorphisms and Weylrescalings. This means that the different variations must be orthogonal for the inner-product(3.34).

First, the deformations must be traceless, otherwise they can be compensated by a Weyltransformation. The traceless metric variations δg which cannot be generated by a vectorfield ξ are perpendicular to P1ξ (otherwise, the former would a linear combination of thelatter):

(δg, P1ξ)g = 0 =⇒ (P †1 δg, ξ)g = 0. (3.73)

Since ξ is arbitrary, this means that the first argument vanishes

P †1 δg = 0. (3.74)

Metric variations induced by a change in the moduli ti are in the kernel of P †1

δg ∈ kerP †1 . (3.75)

Elements of kerP †1 are called quadratic differentials and a basis (not necessarily orthonor-mal) of kerP †1 is denoted as:

kerP †1 = Spanφi, i = 1, . . . ,dim kerP †1 (3.76)

(these should not be confused with the Liouville field). The dimension of kerP †1 is in factequal to the dimension of the moduli space (3.51):

dimR kerP †1 = Mg =

0 g = 0,2 g = 1,6g − 6 g > 1.

(3.77)

The last two terms in the variation (3.64) of δgab are not orthogonal. Let’s introducethe projector on the complement space of kerP †1

Π := P11

P †1P1P †1 . (3.78)

70

The moduli variations can then be rewritten as

δti µi = δti (1−Π)µi + δti Πµi = δti (1−Π)µi + δti P1ζi. (3.79)

The ζi exist because Πµi ∈ ImP1, and they read

ζi := 1P †1P1

P †1µi. (3.80)

The first term can be decomposed on the quadratic differential basis (3.76)

(1−Π)µi = φj(M−1)jk(φk, µi)g (3.81)

whereMij := (φi, φj)g. (3.82)

Ultimately, the variation (3.64) becomes

δgab = (P1ξ)ab + 2Λ gab +Qiab δti. (3.83)

whereξ = ξ + ζiδti, Qiab = φjab (M−1)jk(φk, µi)g. (3.84)

Correspondingly, the norm of the variation splits in three terms since each variation isorthogonal to the others:

|δg|2g = |δΛ|2g + |P1ξ|2g + |Qiδti|2g. (3.85)

Since the norm is decomposed as a sum, the measure factorizes:

dggab = dgΛ dg(P1ξ) dg(Qiδti). (3.86)

One can then perform a change of coordinates

(ξ, Λ, Qiδti) −→ (ξ,Λ, δti), (3.87)

where Λ was defined in (3.65c). The goal of this transformation is to remove the dependencein the moduli from the measures on the Weyl factor and vector fields, and to recover a finite-dimensional integral over the moduli:

dgΛ dg(P1ξ) dg(Qiδti) = dMg tdgΛ dg(P1ξ)det(φi, µj)g√det(φi, φj)g

, (3.88)

where the determinants correspond to the Jacobian. The role of the determinant in thedenominator is to ensure a correct normalization when the basis is not orthonormal (inparticular, it ensures that the Jacobian is independent of the basis). Plugging this result in(3.28) gives the partition function as

Zg =∫Tg

dMg t1

Ωgauge[g]

∫dgΛ dg(P1ξ)

det(φi, µj)g√det(φi, φj)g

Zm[g]. (3.89)

The ti are integrated over the Teichmüller space Tg defined by (3.49) because the vectors ξgenerate only reparametrizations connected to the identity, and thus the remaining freedomlies in Met(Σg)/G0. Next, we study how to perform the changes of variables to remove P1from the measure.

71

Conformal Killing vectors

In this section, we focus on the dgΛ dg(P1ξ) part of the measure and we make contact withthe rest at the end.

Infinitesimal reparametrizations generated by a vector field ξa produce only transform-ations close to the identity. For this reason, integrating over all possible vector fields yieldsthe volume (3.60a) of the component of the diffeomorphism group connected to the identity:∫

dgξ = ΩDiff0 [g]. (3.90)

Remember that the volume depends only on the moduli, but obviously not on ξ (integratedover) nor φ (the inner-product (3.34b) is invariant). But, due to the existence of zero-modes,one gets an integration over a subset of all vector fields, and this complicates the program,as we discuss now.

Zero-modes ξ(0) of P1 are called conformal Killing vectors (CKV)

ξ(0) ∈ Kg := kerP1 (3.91)

and satisfy the conformal Killing equation (see also Section 21.1):

(P1ξ(0))ab = ∇aξ(0)

b +∇bξ(0)a − gab∇cξ(0)c = 0. (3.92)

CKVs correspond to reparametrizations which can be absorbed by a change of the con-formal factor. They should be removed from the ξ integration in order to not double-countthe corresponding metrics. The dimension of the zero-modes CKV space depends on thegenus [193]:

Kg := dimRKg = dimR kerP1 =

6 g = 0,2 g = 1,0 g > 1.

(3.93)

The associated transformations will be interpreted later (Chapter 21). The groups generatedby the CKVs are

g = 0 : K0 = SL(2,C), g = 1 : K1 = U(1)×U(1). (3.94)

Note that the first group is non-compact while the second is compact.A general vector ξ can be separated into a zero-mode part and its orthogonal complement

ξ′:ξ = ξ(0) + ξ′, (3.95)

such that(ξ(0), ξ′)g = 0 (3.96)

for the inner-product (3.34b). Because zero-modes are annihilated by P1, the correct changeof variables in the partition function (3.66) maps to ξ′ only:

(P1ξ,Λ) −→ (ξ′, φ). (3.97)

Integrating over ξ at this stage would double count the CKV (since they are already describedby the φ integration). The appropriate Jacobian reads

dgΛ dg(P1ξ) = dgφdgξ′∆FP[g], (3.98)

where the Faddeev–Popov determinant is

∆FP[g] = det′ ∂(P1ξ,Λ)∂(ξ′, φ) = det′ P1 =

√det′ P1P

†1 , (3.99)

72

the prime on the determinant indicating that the zero-modes are excluded. This brings thepartition function (3.89) to the form

Zg =∫Tg

dMg tΩgauge[g]−1∫

dgφdgξ′det(φi, µj)g√det(φi, φj)g

∆FP[g]Zm[g]. (3.100)

Computation – Equation (3.98)The Jacobian can be evaluated directly:

∆FP[g] = det′ ∂(P1ξ,Λ)∂(ξ′, φ) = det′

(P1 012∇ 1

)= det′ P1. (3.101)

As a consequence of det′ P †1 = det′ P1, the Jacobian can be rewritten as:√det′ P †1P1 = det′ P1. (3.102)

It is instructive to derive this result also by manipulating the path integral. Con-sidering small variations of the fields, one has:

1 =∫

dgδΛ dg(P1δξ) e−|δΛ|2g−|P1δξ

′|2g

= ∆FP[g]∫

dgδφdgδξ′ e−|δφ+ 12∇cδξ

c|2g−|P1δξ

′|2g

= ∆FP[g]∫

dgδφdgδξ′ e−|δφ|2g−(δξ′,P †1P1δξ

′)g

= ∆FP[g](

det′ P †1P1

)−1/2.

That the expression is equal to 1 follows from the normalization of symmetric tensorsand scalars (3.34) (the measures appearing in the path integral (3.89) arises without anyfactor). The third equality holds because the measure is invariant under translationsof the fields, and we used the definition of the adjoint.

The volume of the group generated by the vectors orthogonal to the CKV is denoted as

Ω′Diff0[g] := Ω′Diff0

[g] =∫

dgξ′. (3.103)

As explained in the beginning of this section, one should extract the volume of the full Diff0group, not only the volume Ω′Diff0

[g]. Since the two sets of vectors are orthogonal, we canexpect the measures, and thus the volumes, to factorize. However, a Jacobian can and doesarise: its role it to take into account the normalization of the zero-modes. Denoting by ψia basis (not necessarily orthonormal) for the zero-modes

kerP1 = Spanψi, i = 1, . . . ,Kg, (3.104)

the change of variablesξ′ −→ ξ (3.105)

readsdgξ′ = 1√

det(ψi, ψj)gdgξ

Ωckv[g] , (3.106)

73

where Ωckv[g] is the volume of the CKV group. The determinant is necessary when the basisis not orthonormal. The relation between the gauge volumes is then

ΩDiff0 [g] =√

det(ψi, ψj)g Ωckv[g] Ω′Diff0[g]. (3.107)

Note that the CKV volume is given in (3.111) and depends only on the topology but not onthe metric. By using arguments similar to the ones which lead to (3.61), one can expect thateach term is independently invariant under Weyl rescaling: this is indeed true (Section 3.3.3).

Computation – Equation (3.106)Let’s expand ξ(0) on the zero-mode basis

ξ(0) = αiψi, (3.108)

where the αi are real numbers, such that one can write the changes of variables

ξ −→ (ξ′, αi). (3.109)

The Jacobian is computed from

1 =∫

dξ e−|ξ|2g = J

∫dξ(0) dξ′ e−|ξ

′|2g−|ξ(0)|2g

= J

∫ ∏i

dαi e−αiαj(ψi,ψj)g∫

dξ′ e−|ξ′|2g

= J (det(ψi, ψj)g)−1/2.

Note that the integration over the αi is a standard finite-dimensional integral. Thisgives

dξ =√

det(ψi, ψj)g dξ′∏i

dαi. (3.110)

Since nothing depends on the αi, they can be integrated over as in (3.53), giving thevolume of the CKV group

Ωckv[g] =∫ ∏

i

dαi. (3.111)

Replacing the integration over ξ′ thanks to (3.106), the path integral becomes

Zg =∫Tg

dMg tΩgauge[g]−1∫

dgφdgξdet(φi, µj)g√det(φi, φj)g

Ωckv[g]−1√det(ψi, ψj)g

∆FP[g]Zm[g]. (3.112)

Since the matter action and measure, and the Liouville measure are invariant underreparametrizations, one can perform a change of variables

(f∗g, f∗φ, f∗Ψ) −→ (g, φ,Ψ) (3.113)

such that everything becomes independent of f (or equivalently ξ). Since the measure for ξis Weyl invariant, it is possible to separate it from the rest of the expression, which yieldsan overall factor of ΩDiff0 [g]. This brings the partition function to the form

Zg =∫Tg

dMg tΩDiff0 [g]Ωgauge[g]

∫dgφ



∆FP[g]Zm[g] (3.114)

74

where the same symbol is used for the metric

gab := g(φ)ab = e2φgab. (3.115)

Since the expression is invariant under the full diffeomorphism group Diff(Σg) and notjust under its component Diff0(Σg), one needs to extract the volume of the full diffeomorph-ism group before cancelling it with the normalization factor. Otherwise, there is still anover-counting the configurations. Using the relation (3.60c) leads to:

Zg = 1ΩΓg

∫Tg

dMg tΩDiff[g]

Ωgauge[g]

∫dgφ



∆FP[g]Zm[g]. (3.116)

The volume ΩΓg can be factorized outside the integral because it depends only on the genusand not on the metric. Finally, using the relation (3.52), one can replace the integrationover the Teichmüller space by an integration over the moduli space

Zg =∫Mg

dMg tΩDiff[g]

Ωgauge[g]

∫dgφ



∆FP[g]Zm[g]. (3.117)

3.3.3 Weyl transformations and quantum anomaliesThe next question is whether the integrand depends on the Liouville mode φ such thatthe Weyl volume can be factorized out. While the matter action has been chosen to beWeyl invariant – see the condition (3.19) – the measures cannot be defined to be Weylinvariant. This means that there is a Weyl (or conformal) anomaly, i.e. a violation of theWeyl invariance due to quantum effects. Since the techniques needed to derive the resultsof this section are outside the scope of this review, we simply state the results.

It is possible to show that the Weyl anomaly reads [63, p. 929]5

∆FP[e2φg]√det(φi, φj)e2φg

= ecgh6 SL[g,φ] ∆FP[g]√

det(φi, φj)g(3.118a)

Zm[e2φg] = ecm6 SL[g,φ]Zm[g], (3.118b)

where SL is the Liouville action

SL[g, φ] := 14π

∫d2σ

√g(gab∂aφ∂bφ+ Rφ

), (3.119)

where R is the Ricci scalar of the metric gab. These relations require to introduce counter-terms, discussed further in Section 3.3.4. The coefficients cm and cgh are the central chargesrespectively of the matter and ghost systems, with:

cgh = −26. (3.120)

This value will be derived in Section 23.2.The inner-products between φi and µj , and between the ψi, and the CKV volume are

independent of φ [193, sec. 14.2.2, 63, p. 931]

det(φi, µj)e2φg = det(φi, µj)g, det(ψi, ψj)e2φg = det(ψi, ψj)g,Ωckv[e2φg] = Ωckv[g].

(3.121)

5The relation is written for Zm since the action is invariant and is not affected by the anomaly.

75

Remark 3.13 (Weyl and gravitational anomalies) The Weyl anomaly translates intoa non-zero trace of the quantum energy–momentum tensor

〈gµνTµν〉 = c

12 R, (3.122)

where c is the central charge of the theory. The Weyl anomaly can be traded for a gravita-tional anomaly, which means that diffeomorphisms are broken at the quantum level [137].

Inserting (3.118) in (3.117) yields

Zg =∫Mg

dMg tΩDiff[g]

Ωgauge[g]det(φi, µj)g√det(φi, φj)g


∆FP[g]Zm[g]∫

dgφ e−cL6 SL[g,φ],

(3.123)with the Liouville central charge

cL := 26− cm. (3.124)

The critical “dimension” is defined to be the value of the matter central charge cm such thatthe Liouville central charge cancels

cL = 0 =⇒ cm = 26. (3.125)

If the number of non-compact dimensions is D, it means that the central charge (3.21) ofthe transverse CFT satisfies

c⊥ = 26−D. (3.126)

In this case, the integrand does not depend on the Liouville mode (because ΩDiff isinvariant under Weyl transformations) and the integration over φ can be factored out andyields the volume of the Weyl group (3.60b)∫

dgφ = ΩWeyl[g]. (3.127)

Then, takingΩgauge[g] = ΩDiff[g]× ΩWeyl[g] (3.128)

removes the infinite gauge contributions and gives the partition function

Zg =∫Mg

dMg tdet(φi, µj)g√det(φi, φj)g


∆FP[g]Zm[g]. (3.129)

Exercise 3.1 – Faddeev–Popov in particular casesIn order to better understand the Faddeev–Popov procedures, redo the computation ofthe full section by focusing on the following specific cases:

1. ignore the CKV and the Teichmüller parameters

2. ignore the Teichmüller parameters (case of the sphere, g = 0)

3. ignore the CKV (case of a surface with g > 1)

To simplify even more the problem, you can consider Sm to be the Polyakov action.

76

3.3.4 Ambiguities, ultralocality and cosmological constantDifferent ambiguities remain in the previous computations, starting with the definitions ofthe measures (3.32) and (3.34), then in obtaining the volume of the diffeomorphism (3.60a)and Weyl (3.60b) groups, and finally in deriving the conformal anomaly (3.118).

These different ambiguities can be removed by renormalizing the worldsheet cosmologicalconstant. This implies that the action

Sµ[g] =∫

d2σ√g (3.130)

must be added to the classical Lagrangian, where µ0 is the bare cosmological constant. Thismeans that Weyl invariance is explicitly broken at the classical level. After performing allthe manipulations, µ0 is determined by removing all ambiguities and enforcing invarianceunder the Weyl symmetry at the quantum level. This amounts to set the renormalizedcosmological constant to zero (since it breaks the Weyl symmetry). The possibility tointroduce a counter-term violating a classical symmetry arises because the symmetry itselfis broken by a quantum anomaly, so there is no reason to enforce it in the classical action.

We now review each issue separately. First, consider the inner-product of a single tensor(3.32): the determinant det γg depends on the metric and one should be more careful whenfixing the gauge or integrating over all metrics. However, ultralocality implies that thedeterminant can only be of the form [63, pp. 923]√

det γg = e−µγ Sµ[g], (3.131)

for some µγ ∈ R, since Sµ is the only renormalizable covariant functional depending on themetric but not on its derivatives. The effect is just to redefine the cosmological constant.

Second, the volume of the field space can be defined as the limit λ → 0 of a Gaussianintegral [63, pp. 931]:

ΩΦ = limλ→0

∫dgΦ e−λ (Φ,Φ)g . (3.132)

Due to ultralocality, the Gaussian integral should again be of the form∫dgΦ e−λ (Φ,Φ)g = e−µ(λ)Sµ[g], (3.133)

for some constant µ(λ). Hence, the limit λ→ 0 gives

ΩΦ =∫

dgΦ = e−µ(0)Sµ[g], (3.134)

which can be absorbed in the cosmological constant. However, the situation is more com-plicated if Φ = ξ, φ since the integration variables also appear in the measure, as it wasalso discussed before (3.61). But, in that case, it cannot appear in the expression of thevolume in the LHS. Moreover, invariances under diffeomorphisms for both measures, andunder Weyl rescalings for the vector measure, imply that the LHS can only depend on themoduli through the background metric g. The diffeomorphism and Weyl volumes can bewritten in terms of e−µ Sµ[g]: since there is no counter-term left (the cosmological constantcounter-term is already fixed to cancel the coefficient of Sµ[g]), it is necessary to divide byΩgauge to cancel the volumes.

Finally, the computation of the Weyl anomaly (3.118) yields divergent terms of the form

limε→0

1ε

∫d2σ√g. (3.135)

These divergences are cancelled by the cosmological constant counter-term. See [64, app. 5.A]for more details.

77

3.3.5 Gauge-fixed path integralAs a conclusion of this section, we found that the partition function (3.28) can be writtenas

Zg =∫Mg



∆FP[g]Zm[g], (3.136a)

=∫Mg

dMg t

√det(φi, µj)2

g

det(φi, φj)gdet′ P †1 P1

det(ψi, ψj)gZm[g]Ωckv[g] . (3.136b)

after gauge fixing of the worldsheet diffeomorphisms and Weyl rescalings. It is implicit thatthe factors for the CKV and moduli are respectively absent for g > 1 and g < 1. For g = 0the CKV group is non-compact and its volume is infinite. It looks like the partition vanishes,but there are subtleties which will be discussed in Section 4.1.3.

Remark 3.14 (Weil–Petersson metric) When the metric is chosen to be of constantcurvature R = −1, the moduli measure together with the determinants form the Weil–Petersson measure

d(WP) =∫Mg


. (3.137)

In (3.136), the background metric gab is fixed. However, the derivation holds for anychoice of gab: as a consequence, it makes sense to relax the gauge fixing and allow it to varywhile adding gauge symmetries. The first symmetry is background diffeomorphisms:

σ′a = fa(σb), g′(σ′) = f∗g(σ), φ′(σ′) = f∗φ(σ), Ψ′(σ′) = f∗Ψ(σ). (3.138)

This symmetry is automatic for Sm[g,Ψ] since Sm[g,Ψ] was invariant under (3.4). Similarly,the integration measures are also invariant. A second symmetry is found by inspecting thedecomposition (3.56)

gab = f∗(e2φgab(t)

), (3.139)

which is left invariant under a background Weyl symmetry (also called emergent):

g′ab(σ) = e2ω(σ)gab(σ), φ′(σ) = φ(σ)− ω(σ), Ψ′(σ) = Ψ(σ). (3.140)

Let us stress that it is not related to the Weyl rescaling (3.10) of the metric gab. Thebackground Weyl rescaling (3.140) is a symmetry even when the physical Weyl rescaling(3.10) is not. Together, the background diffeomorphisms and Weyl symmetry have threegauge parameters, which is sufficient to completely fix the background metric g up to moduli.

In fact, the combination of both symmetries is equivalent to invariance under the physicaldiffeomorphisms. To prove this statement, consider two metrics g and g′ related by adiffeomorphism F and both gauge fixed to pairs (f, φ, g) and (f ′, φ′, g′):

g′ab = F ∗gab, g′ab = f ′∗(e2φ′ g′ab

), gab = f∗

(e2φgab

). (3.141)

Then, the gauge fixing parametrizations are related by background symmetries (F , ω) as

F = f ′−1 F f, φ′ = F ∗(φ− ω), g′ab = F ∗(e2ω gab). (3.142)

Moreover, this also implies that there is a diffeomorphism f = F f such that g′ is gaugefixed in terms of (φ, g):

g′ab = f∗(e2φgab

). (3.143)

78

Computation – Equation (3.142)The functions F , f , f ′, φ, φ′ and the metrics gab, g′ab, gab and g′ab are all fixed and onemust find F and ω such that the relations (3.141) are compatible. First, one rewritesg′ab in terms of gab and compare with the expression with g′ab:

g′ab = F ∗gab = F ∗(f∗(e2φgab

))= F ∗

(f∗(e2(φ−ω)e2ω gab

))= f ′∗

(e2φ′ g′ab

).

In the third equality, we have introduced ω because g′ab = F ∗gab is not true in generalsince there are 3 independent components but F has only 2 parameters, so we cannotjust define f ′ = F f and φ′ = φ. This explains the importance of the emergent Weylsymmetry.

Remark 3.15 (Gauge fixing and field redefinition) Although it looks like we are un-doing the gauge fixing, this is not exactly the case since the original metric is not used any-more. One can understand the procedure of this section as a field redefinition: the degrees offreedom in gab are repackaged into two fields (φ, gab) adapted to make some properties of thesystem more salient. A new gauge symmetry is introduced to maintain the number of degreesof freedom. The latter helps to understand the structure of the action on the background.Finally, in this context, the Liouville action is understood as a Wess–Zumino action, whichis defined as the difference between the effective actions evaluated in each metric. Anothertypical use of this point of view is to rewrite a massive vector field as a massless gauge fieldtogether with an axion [220].

Remark 3.16 (Two-dimensional gravity) In 2d gravity, one does not work in the crit-ical dimension (3.125) and cL 6= 0. Thus, the Liouville mode does not decouple: the con-formal anomaly breaks the Weyl symmetry at the quantum level which gives dynamics togravity, even if it has no degree of freedom classically. As a consequence, one choosesΩgauge = ΩDiff .

Since the role of the classical Weyl symmetry is not as important as for string theory, it iseven not necessary to impose it classically. This leads to consider non-conformal matter [23–25, 38, 97, 98]. Following the arguments from Section 3.1, the existence of the emergentWeyl symmetry (3.140) implies that the total action Sgrav[g, φ] + Sm[g,Ψ] must be a CFTfor a flat background g = δ, even if the two actions are not independently CFTs.

3.4 Ghost action3.4.1 Actions and equations of motionIt is well-known that a determinant can be represented with two anticommuting fields, calledghosts. The fields carry indices dictated by the map induced by the operator of the Faddeev–Popov determinant: one needs a symmetric and traceless anti-ghost bab and a vector ghostca fields:

∆FP[g] =∫

d′gbd′gc e−Sgh[g,b,c], (3.144)

where the prime indicates that the ghost zero-modes are omitted. The ghost action is

Sgh[g, b, c] = 14π

∫d2σ√g gabgcdbac(P1c)bd (3.145a)

= 14π

∫d2σ√g gab

(bac∇bcc + bbc∇acc − bab∇ccc

). (3.145b)

79

The ghosts ca and anti-ghosts bab are associated respectively to the variations due to thediffeomorphisms ξa and to the variations perpendicular to the gauge slice. The normalizationof 1/4π is conventional. In Minkowski signature, the action is multiplied by a factor i.

Since bab is traceless, the last term of the action vanishes and could be removed. However,this implies to consider traceless variations of the bab when varying the action (to computethe equations of motion, the energy–momentum tensor, etc.). On the other hand, one cankeep the term and consider unconstrained variation of bab (since the structure of the actionwill force the variation to have the correct symmetry), which is simpler. A last possibility isto introduce a Lagrange multiplier. These aspects are related to the question of introducinga ghost for the Weyl symmetry, which is described in Section 3.4.2.

The equations of motion are

(P1c)ab = ∇acb +∇bca − gab∇ccc = 0, (P †1 b)a = −2∇bbab = 0. (3.146)

Hence, the classical solutions of b and c are respectively mapped to the zero-modes of theoperators P †1 and P1, and they are thus associated to the CKV and Teichmüller parameters.

The energy–momentum tensor is

T ghab = −bac∇bcc − bbc∇acc + cc∇cbab + gabbcd∇ccd. (3.147)

Its trace vanishes off-shell (i.e. without using the b and c equations of motion)

gabT ghab = 0, (3.148)

which shows that the action (3.145) is invariant under Weyl transformations

Sgh[e2ωg, b, c] = Sgh[g, b, c]. (3.149)

Exercise 3.2Derive (3.146) and (3.147) by varying the action (3.145).

The action (3.145) also has a U(1) global symmetry. The associated conserved charge iscalled the ghost number and counts the number of c ghosts minus the number of b ghosts,i.e.

Ngh(b) = −1, Ngh(c) = 1. (3.150a)

The matter fields are inert under this symmetry:

Ngh(Ψ) = 0. (3.150b)

In terms of actions, the path integral (3.136) can be rewritten as

Zg =∫Mg



∫dgΨ d′gbd′gc e−Sm[g,Ψ]−Sgh[g,b,c]. (3.151)

One can use (3.136) or (3.151) indifferently: the first is more appropriate when using spectralanalysis to compute the determinant explicitly, while the second is more natural in thecontext of CFTs.

3.4.2 Weyl ghostGhosts have been introduced for the reparametrizations (generated by ξa) and the tracelesspart of the metric (the gauge field associated to the transformation): one may wonder whythere is not a ghost cw associated to the Weyl symmetry along with an antighost for the trace

80

of the metric (i.e. the conformal factor). This can be understood from several viewpoints.First, the relation between a metric and its transformation – and the corresponding gaugefixing condition – does not involve any derivative: as such, the Jacobian is trivial. Second,one could choose

F⊥ab = √ggab −√ggab = 0 (3.152)

as a gauge fixing condition instead of (3.63), and the trace component does not appearanywhere. Finally, a local Weyl symmetry is not independent from the diffeomorphisms.

Remark 3.17 (Local Weyl symmetry) The topic of obtaining a local Weyl symmetry bygauging a global Weyl symmetry (dilatation) is very interesting [101, chap. 15, 128]. Undergeneral conditions, one can express the new action in terms of the Ricci tensor (or of thecurvature): this means that the Weyl gauge field and its curvature are composite fields.

Moreover, one finds that local Weyl invariance leads to an off-shell condition while diffeo-morphisms give on-shell conditions. This explains why one imposes only Virasoro constraints(associated to reparametrizations) and no constraints for the Weyl symmetry in the covariantquantization.

However, it can be useful to introduce a ghost field cw for the Weyl symmetry nonetheless.In view of the previous discussion, this field should appear as a Lagrange multiplier whichensures that bab is traceless. Starting from the action (3.145), one finds

S′gh[g, b, c, cw] = 14π

∫d2σ√g gab

(bac∇bcc + bbc∇acc + 2babcw

), (3.153)

where bab is not traceless anymore. The ghost cw is not dynamical since the action does notcontain derivatives of it, and it can be integrated out of the path integral to recover (3.145).

The equations of motion for this modified action are

∇acb +∇bca + 2gabcw = 0, ∇abab = 0, gabbab = 0. (3.154)

Contracting the first equation with the metric gives

cw = −12 ∇ac

a, (3.155)

and thus cw is nothing else than the divergence of the ca field: the Weyl ghost is a compositefield (this makes connection with Remark 3.17) – see also (3.65c). The energy–momentumtensor of the ghosts with action (3.153) is

T ′ghab =−(bac∇bcc + bbc∇acc + 2babcw

)−∇c(babcc)

+ 12 gabg

cd(bce∇dce + bde∇cce + 2bcdcw

).

(3.156)

The trace of this tensorgabT ′ghab = −gab∇c(babcc) (3.157)

does not vanish off-shell, but it does on-shell since gabbab = 0. This implies that the theoryis Weyl invariant even if the action is not. It is interesting to contrast this with the trace(3.148) when the Weyl ghost has been integrated out.

The equations of motion (3.146) and energy–momentum tensor (3.147) for the action(3.145) can be easily derived by replacing cw by its solution in the previous formulas.

Computation – Equation (3.156)The first parenthesis comes from varying gab, the second from the covariant derivatives,

81

the last from the √g. The second term comes from

gab(bacδ∇bcc + bbcδ∇acc

)= 2gabbacδ∇bcc = 2gabbacδΓcbdcd

= gabbacgce(∇bδgde +∇dδgbe −∇eδgbd

)cd

= bab(∇aδgbc +∇cδgab −∇bδgac

)cc

= bab∇cδgabcc,

where the two terms have cancelled due to the symmetry of bab. Integrating by partgives the term in the previous equation.

Note that the integration on the Weyl ghost yields a delta function∫dgcw e−(cw,gabbab)g = δ

(gabbab

). (3.158)

3.4.3 Zero-modesThe path integral (3.151) excludes the zero-modes of the ghosts. One can expect them tobe related to the determinants of elements of kerP1 and kerP †1 with Grassmann coefficients.They can be included after few simple manipulations (see also Appendix C.1.3).

It is simpler to first focus on the b ghost (to avoid the problems related to the CKV).The path integral (3.151) can be rewritten as

Zg =∫Mg

dMg tΩckv[g]−1√det(ψi, ψj)g

∫dgΨ dgbd′gc

Mg∏i=1

(b, µi)g e−Sm[g,Ψ]−Sgh[g,b,c]. (3.159)

In this expression, c zero-modes are not integrated over, only the b zero-modes are. This isthe standard starting point on Riemann surfaces with genus g ≥ 1. The inner-product readsexplicitly

(b, µi)g =∫

d2σ√g Gabcd⊥ babµi,cd =

∫d2σ

√g gacgbdbabµi,cd. (3.160)

Computation – Equation (3.159)Since the zero-modes of b are in the kernel of P †1 , it means that the quadratic differentials(3.76) also provide a suitable basis:

b = b0 + b′, b0 = b0iφi,

where the b0i are Grassmann-odd coefficients. The first step is to find the Jacobian forthe changes of variables b→ (b′, b0i):

1 =∫

dgb e−|b|2g = J

∫dgb′

∏i

db0i e−|b′|2g−|b0iφi|

2= J

√det(φi, φj).

Next, (3.151) has no zero-modes, so one must insert Mg of them at arbitrary positionsσ0j to get a non-vanishing result when integrating over dMgb0i. The result of the integral

is: ∫dMgb0i

∏j

b0(σ0j ) =

∫dMgb0i

∏j

[b0iφi(σ0

j )]

= detφi(σ0j ).

The only combination of the φi which does not vanish is the determinant due to the

82

anti-symmetry of the Grassmann numbers. Combining both results leads to:

dgb′√det(φi, φj)g

= dgbdetφi(σ0

j )

Mg∏j=1

b(σ0j ). (3.161)

The locations positions σ0j are arbitrary (in particular, the RHS does not depend on

them since the LHS does not either). Note that more details are provided in Ap-pendix C.1.3.

An even simpler result can be obtained by combining the previous formula with thefactor det(φi, µj)g:

dgb′det(φi, µj)g√det(φi, φj)g

= dgbMg∏j=1

(b, µj)g. (3.162)

This follows fromMg∏j=1

b(σ0j ) =

Mg∏j=1

[b0iφi(σ0

j )]

= detφi(σ0j )

Mg∏j=1

b0i,

det(φi, µj)gMg∏j=1

b0i =Mg∏j=1

[b0i(φi, µj)g

]=

Mg∏j=1

(b0iφi, µj)g =Mg∏j=1

(b, µj)g.

Note that the previous manipulations are slightly formal: the symmetric traceless fieldsbab and φi,ab carry indices and there should be a product over the (two) independentcomponents. This is a trivial extension and would just make the notations heavier.

Similar manipulations lead to a new expression which includes also the c zero-mode (butwhich is not very illuminating):

Zg =∫Mg

dMg tΩckv[g]−1

detψi(σ0j )

∫dgΨ dgbdgc

Kcg∏j=1

εab2 ca(σ0

j )cb(σ0j )

×Mg∏i=1

(µi, b)g e−Sm[g,Ψ]−Sgh[g,b,c].

(3.163)

The σ0aj are Kcg = Kg/2 fixed positions and the integral does not depend on their values.

Note that only Kcg positions are needed because the coordinate is 2-dimensional: fixing 3points with 2 components correctly gives 6 constraints. Then, ψi(σ0a

j is a 6-dimensionalmatrix, with the rows indexed by i and the columns by the pair (a, j).

The expression cannot be simplified further because the CKV factor is infinite for g = 0.This is connected to a fact mentioned previously: there is a remaining gauge symmetrywhich is not taken into account

c −→ c+ c0, P1c0 = 0. (3.164)

A proper account requires to gauge fix this symmetry: the simplest possibility is to insertthree or more vertex operators – this topic is discussed in Section 4.1.

Finally, note that the same question arises for the b-ghost since one has the symmetry

b −→ b+ b0, P †1 b0 = 0. (3.165)

That there is no problem in this case is related to the presence of the moduli.

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3.5 NormalizationIn the previous sections, the closed string coupling constant gs did not appear in the ex-pressions. Loops in vacuum amplitudes are generated by splitting of closed strings. Byinspecting the amplitudes, it seems that there are 2g such splittings (Figure 3.2), whichwould lead to a factor g2g

s . However, this is not quite correct: this result holds for a 2-pointfunction. Gluing the two external legs to get a partition function (that is, taking the trace)leads to an additional factor g−2

s (to be determined later), such that the overall factor isg2g−2s . The fact that it is the appropriate power of the coupling constant can be more easilyunderstood by considering n-point amplitudes (Section 4.1). The normalization of the pathintegral can be completely fixed by unitarity [218].

The above factor has a nice geometrical interpretation. Defining

Φ0 = ln gs (3.166)

and remembering the expression (3.9) of the Euler characteristics χg = 2− 2g, the couplingfactor can be rewritten as

g2g−2s = e−Φ0χg = exp

(−Φ0

4π

∫d2σ√gR

)= e−Φ0SEH[g], (3.167)

where SEH is the Einstein–Hilbert action. This action is topological in two dimensions.Hence, the coupling constant can be inserted in the path integral simply by shifting theaction by the above term. This shows that string theory on a flat target spacetime iscompletely equivalent to matter minimally coupled to Einstein–Hilbert gravity with a cos-mological constant (tuned to impose Weyl invariance at the quantum level). The advantageof describing the coupling power in this fashion is that it directly generalizes to scatteringamplitudes and to open strings. The parameter Φ0 is interpreted as the expectation value ofthe dilaton. Replacing it by a general field Φ(Xµ) is a generalization of the matter non-linearsigma model, but this topic is beyond the scope of this review.

Figure 3.2: g-loop partition function.

3.6 SummaryIn this chapter, we started with a fairly general matter CFT – containing at least D scalarfields Xµ – and explained under which condition it describes a string theory. The mostimportant consequence is that the matter 2d QFT must in fact be a 2d CFT. We thencontinued by describing how to gauge fix the integration over the surfaces and we identifiedthe remaining degrees of freedom – the moduli spaceMg – up to some residual redundancy– the conformal Killing vector (CKV). Then, we showed how to rewrite the result in termsof ghosts and proved that they are also a CFT. This means that a string theory can be com-pletely described by two decoupled CFTs: a universal ghost CFT and a theory-dependentmatter CFT describing the string spacetime embedding and the internal structure. The

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advantage is that one can forget the path integral formalism altogether and employ onlyCFT techniques to perform the computations. This point of view will be developed for off-shell amplitudes (Chapter 6) in order to provide an alternative description of how to buildamplitudes. It is particularly fruitful because one can also consider matter CFTs which donot have a Lagrangian description. In the next chapter, we describe scattering amplitudes.

3.7 References• The definition of a field measure from a Gaussian integral and manipulations thereof

can be found in [115, sec. 15.1, 22.1, 193, chap. 14, 215, 63].

• The most complete explanations of the gauge fixing procedure are to be found in [115,sec. 15.1, 22.1, 31, sec. 3.4, 6.2, 218, chap. 5, 58, 143, chap. 5]. The original derivationcan be found in [62, 183].

• For the geometry of the moduli space, see [193, 195].

• Ultralocality and its consequences are described in [63, 215] (see also [113, sec. 2.4]).

• The use of a Weyl ghost is shown in [272, sec. 8, 293, sec. 9.2].

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Chapter 4

Worldsheet path integral:scattering amplitudes

In this chapter, we generalize the worldsheet path integral to compute scattering amplitudes,which corresponds to insert vertex operators. The gauge fixing from the previous chapteris generalized to this case. In particular, we discuss the 2-point amplitude on the sphere.Finally, we introduce the BRST symmetry and motivate some properties of the BRSTquantization, which will be performed in details later. The formulas in this chapter are allcovariant: they will be rewritten in complex coordinates in the next chapter.

4.1 Scattering amplitudes on moduli spaceIn this section, we describe the scattering of n strings. The momentum representation ismore natural for describing interactions, especially in string theory. Therefore, each string ischaracterized by a state Vαi(ki) with momentum ki and some additional quantum numbersαi (i = 1, . . . , n). We start from the worldsheet path integral (3.28) before gauge fixing:

Zg =∫ dggab

Ωgauge[g] Zm[g], Zm[g] =∫

dgΨ e−Sm[g,Ψ]. (4.1)

4.1.1 Vertex operators and path integralThe external states are represented by infinite semi-tubes attached to the surfaces. Under aconformal mapping, the tubes can be mapped to points called punctures on the worldsheet.At g loops, the resulting space is a Riemann surface Σg,n of genus g with n punctures (ormarked points). The external states are represented by integrated vertex operators

Vα(ki) :=∫

d2σ√g(σ)Vα(k;σ). (4.2)

The vertex operators Vα(k;σ) are built from the matter CFT operators and from the world-sheet metric gab. The functional dependence is omitted to not overload the notation, but oneshould read Vα(k;σ) := Vα[g,Ψ](k;σ). The integration over the state positions is necessarybecause the mapping of the tube to a point is arbitrary. Another viewpoint is that it isneeded to obtain an expression invariant under worldsheet diffeomorphisms.

Following Section 3.3.5, the Einstein–Hilbert action is inserted in the path integral. Ona surface with punctures, the Euler characteristics is shifted by the number of punctures

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(which are equivalent to boundaries or disks):

χg,n := χ(Σg,n) = 2− 2g − n. (4.3)

This gives the normalization factor:

g−χg,ns = e−Φ0SEH[g], Φ0 := ln gs. (4.4)

The correctness factor can be verified by inspection of the Riemann surface for the scat-tering of n string at g loops. In particular, the string coupling constant is by definitionthe interaction strength for the scattering of 3 strings at tree-level. Moreover, the tree-level2-point amplitude contains no interaction and should have no power of gs. This factor canalso be obtained by unitarity [218].

By inserting these factors in (3.28), the g-loop n-point scattering amplitude is describedby:

Ag,n(ki)αi :=∫ dggab

Ωgauge[g] dgΨ e−Sm[g,Ψ]−Φ0SEH[g]n∏i=1

(∫d2σi

√g(σi)Vαi(ki;σi)

).

(4.5)The σi dependence of each √g will be omitted from now on since no confusion is possible.The following equivalent notations will be used:

Ag,n(ki)αi := Ag,n(k1, . . . , kn)α1,...,αn := Ag,n(Vα1(k1), . . . , Vαn(kn)

). (4.6)

The complete (perturbative) amplitude is found by summing over all genus:

An(k1, . . . , kn)α1,...,αn =∞∑g=0

Ag,n(k1, . . . , kn)α1,...,αn . (4.7)

Sometimes, it is convenient to extract the factor e−Φ0χg,n of the amplitude Ag,n to displayexplicitly the genus expansion, but we will not follow this convention here. Since each term ofthe sum scales as Ag,n ∝ g2g+n−2

s , this expression clearly shows that worldsheet amplitudesare perturbative by definition: this motivates the construction of a string field theory fromwhich the full non-perturbative S-matrix can theoretically be computed.

Finally, the amplitude (4.5) can be rewritten in terms of correlation functions of thematter QFT integrated over worldsheet metrics:

Ag,n(ki)αi =∫ dggab

Ωgauge[g] e−Φ0SEH[g]∫ n∏

i=1d2σi√g

⟨n∏i=1

Vαi(ki;σi)⟩m,g

. (4.8)

The correlation function plays the same role as the partition function in (3.28). This showsthat string expression are integrals of CFT expressions over the space of worldsheet metrics(to be reduced to the moduli space).

We address a last question before performing the gauge fixing: what does (4.5) computesexactly: on-shell or off-shell? Green functions or amplitudes? if amplitudes, the S-matrixor just the interacting part T (amputated Green functions)? The first point is that a pathintegral over connected worldsheets will compute connected processes. We will prove later,when discussing the BRST quantization, that string states must be on-shell (Sections 4.2and 4.2.2) and that it corresponds to setting the Hamiltonian (3.26) to zero:

H = 0. (4.9)

From this fact, it follows that (3.28) must compute amplitudes since non-amputated Greenfunctions diverge on-shell (due to external propagators). Finally, the question of whether it

87

computes the S-matrix S = 1 + iT , or just the interacting part T is subtler. At tree-level,they agree for n ≥ 3, while T = 0 for n = 2 and S reduces to the identity. This difficulty(discussed further in Section 4.1.2) is thus related to the question of gauge-fixing tree-level2-point amplitude (Section 4.1.3). It has long been believed that (3.28) computes only theinteracting part (amputated Green functions), but it has been understood recently that thisis not correct and that (3.28) computes the S-matrix.Remark 4.1 (Scattering amplitudes in QFT) Remember that the S-matrix is separ-ated as:

S = 1 + iT, (4.10)where 1 denotes the contribution where all particles propagate without interaction. Theconnected components of S and T are denoted by Sc and T c. The n-point (connected)scattering amplitudes An for n ≥ 3 can be computed from the Green functions Gn throughthe LSZ prescription (amputation of the external propagators):

An(k1, . . . , kn) = Gn(k1, . . . , kn)n∏i=1

(k2i +m2

i ). (4.11)

The path integral computes the Green functions Gn; perturbatively, they are obtained fromthe Feynman rules. They include a D-dimensional delta function

Gn(k1, . . . , kn) ∝ δ(D)(k1 + · · ·+ kn). (4.12)

The 2-point amputated Green function T2 computed from the LSZ prescription vanisheson-shell. For example, considering a scalar field at tree-level, one finds:

T2 = G2(k, k′) (k2 +m2)2 ∼ (k2 +m2) δ(D)(k + k′) −−−−−−→k2→−m2

0 (4.13)

sinceG2(k, k′) = δ(D)(k + k′)

k2 +m2 . (4.14)

Hence, T2 = 0 and the S-matrix (4.10) reduces to the identity component Sc2 = 12 (which isa connected process). There are several way to understand this result:

1. The recursive definition of the connected S-matrix Sc from the cluster decompositionprinciple requires a non-vanishing 2-point amplitude [136, sec. 5.1.5, 286, sec. 4.3, 74,sec. 6.1].

2. The 2-point amplitude corresponds to the normalization of the 1-particle states (overlapof a particle state with itself, which is non-trivial) [285, eq. 4.1.4, 269, chap. 5].

3. A single particle in the far past propagating to the far future without interacting is aconnected and physical process [74, p. 133].

4. It is required by the unitarity of the 2-point amplitude [78].These points indicate that the 2-point amplitude is proportional to the identity in the mo-mentum representation [136, p. 212, 285, eq. 4.3.3 and 4.1.5]

A2(k, k′) = 2k0 (2π)D−1δ(D−1)(k − k′). (4.15)

The absence of interactions implies that the spatial momentum does not change (the on-shell condition implies that the energy is also conserved). This relation is consistent withthe commutation relation of the operators with the Lorentz invariant measure1

[a(k), a†(k′)] = 2k0 (2π)D−1δ(D−1)(k − k′). (4.16)1If the modes are defined as a(k) = a(k)/

√2k0 such that [a(k), a†(k′)] = (2π)D−1δ(D−1)(k − k′), then

one finds A2(k, k′) = (2π)D−1δ(D−1)(k − k′).

88

That this holds for all particles at all loops can be proven using the Källen–Lehman repres-entation [136, p. 212].

On the other hand, the identity part in (4.10) is absent for n ≥ 3 for connected amp-litudes: Scn = T cn for n ≥ 3. This shows that the Feynman rules and the LSZ prescriptioncompute only the interacting part T of the on-shell scattering amplitudes. The reason is thatthe derivation of the LSZ formula assumes that the incoming and outgoing states have nooverlap, which is not the case for the 2-point function. A complete derivation of the S-matrixfrom the path integral is more involved [136, sec. 5.1.5, 295, sec. 6.7, 95] (see also [48]).The main idea is to consider a superposition of momentum states (here, in the holomorphicrepresentation [295, sec. 5.1, 6.4])

φ(α) =∫

dD−1kα(k)∗a†(k). (4.17)

They contribute a quadratic piece to the connected S-matrix and, setting them to delta func-tions, one recovers the above result.

4.1.2 Gauge fixing: general caseThe Faddeev–Popov gauge fixing of the worldsheet diffeomorphisms and Weyl rescaling(3.15) goes through also in this case if the integrated vertex operators are diffeomorphismand Weyl invariant:

δξVαi(ki) = δξ

∫d2σ√g Vαi(ki;σ) = 0, (4.18a)

δωVαi(ki) = δω

∫d2σ√g Vαi(ki;σ) = 0, (4.18b)

with the variations defined in (3.6) and (3.11). Diffeomorphism invariance is straightforwardif the states are integrated worldsheet scalars. However, if the states are classically Weylinvariant, they are not necessary so at the quantum level: vertex operators are compositeoperators, which need to be renormalized to be well-defined at the quantum level. Renor-malization introduces a scale which breaks Weyl invariance. Enforcing it to be a symmetryof the vertex operators leads to constraints on the latter. We will not enter in the detailssince it depends on the matter CFT and we will assume that the operators Vαi(ki) are indeedWeyl invariant (see [218, sec. 3.6] for more details). In the rest of this book, we will useCFT techniques developed in Chapter 22. The Einstein–Hilbert action is clearly invariantunder both symmetries since it is a topological quantity.

Following the computations from Section 3.3 leads to a generalization of (3.136) withthe vertex operators inserted for the amplitude (4.5):

Ag,n(ki)αi = g−χg,ns

∫Mg



×∫ n∏

i=1d2σi

√g

⟨n∏i=1

Vαi(ki;σi)⟩m,g

.

(4.19)The hat on the vertex operators indicates that they are evaluated in the background metricg.

The next step is to introduce the ghosts: following Section 3.4, the generalization of

89

(3.159) is

Ag,n(ki)αi = g−χg,ns

∫Mg

dMg tΩckv[g]−1√det(ψi, ψj)g

∫dgbd′gc

Mg∏i=1

(b, µi)g e−Sgh[g,b,c]

×∫ n∏

i=1d2σi

√g

⟨n∏i=1

Vαi(ki;σi)⟩m,g

.

(4.20)

For the moment, only the b ghosts come with zero-modes. Then, c zero-modes can beintroduced in (4.20)

Ag,n =∫Mg

dMg tΩckv[g]−1

detψi(σ0j )

∫dgbdgc

Kcg∏j=1

εab2 ca(σ0

j )cb(σ0j )

Mg∏i=1

(µi, b)g e−Sgh[g,b,c]

×∫ n∏

i=1d2σi

√g

⟨n∏i=1

Vαi(ki;σi)⟩m,g

,

(4.21)

by following the same derivation as (3.163). The formulas (4.20) and (4.21) are the correctstarting point for all g and n. In particular, the c ghosts are not paired with any vertex(a condition often assumed or presented as mandatory). This fact will help resolve somedifficulties for the 2-point function on the sphere.

Remember that there is no CKV and no c zero-mode for g ≥ 2. For the sphere g = 0and the torus g = 1, there are CKVs, indicating that there is a residual symmetry in (4.20)and (4.21), which is the global conformal group of the worldsheet. It can be gauge fixed byimposing conditions on the vertex operators.2 The simplest gauge fixing condition amountsto fix the positions of Kcg vertex operators through the Faddeev–Popov trick:

1 = ∆(σ0j )∫

dξKcg∏j=1

δ(2)(σj − σ0(ξ)j ), σ

0(ξ)j = σ0

j + δξσ0j , δξσ

0j = ξ(σ0

j ), (4.22)

where ξ is a conformal Killing vector, and the variation of σ was given in (3.6). We findthat

∆(σ0j ) = detψi(σ0

j ). (4.23)A priori, the positions σ0

j are not the same as the one appearing in (3.163) (since both setsare arbitrary): however, considering the same positions allows to cancel the factor (4.23)with the same one in (3.163).

Computation – Equation (4.23)The first step is to compute ∆ in (4.22). For this, we decompose the CKV ξ on thebasis (3.104)

ξ(σ0j ) = αiψi(σ0

j )

and write the Gaussian integral:

1 =∫ Kcg∏

j=1d2δσj e−

∑j(δσj ,δσj) = ∆

∫ Kg∏j=1

dαi e−∑

j,i,i′(αiψi(σj),αi′ψi′ (σj))

= ∆(

detψi(σj))−1

.

2In fact, it is only important to gauge fix for the sphere because the volume of the group is infinite. Onthe other hand, the volume of the CKV group for the torus is finite-dimensional such that dividing by Ωckvis not ambiguous.

90

Again, we have reduced rigour in order to simplify the manipulations.

After inserting the identity (4.22) into (4.21), one can integrate over Kcg vertex operatorpositions to remove the delta functions – at the condition that there are at least Kcg operators.As a consequence, we learn that the proposed gauge fixing works only for n ≥ 1 if g = 1 orn ≥ 3 if g = 0. This condition is equivalent to

χg,n = 2− 2g − n < 0. (4.24)

In this case, the factors detψi(σ0j ) cancel and (4.20) becomes

Ag,n(ki)αi =∫Mg

dMg t

∫dgbdgc

Kcg∏j=1

εab2 ca(σ0

j )cb(σ0j )

Mg∏i=1

(µi, b)g e−Sgh[g,b,c]

×∫ n∏i=Kcg+1

d2σi√g

⟨ Kcg∏j=1

Vαi(ki;σ0j )

n∏i=Kcg+1

Vαi(ki;σi)⟩m,g

.

(4.25)

The result may be divided by a symmetry factor if the delta functions have solutions forseveral points [218, sec. 5.3]. Performing the gauge fixing for the other cases (in particular,g = 0, n = 2 and g = 1, n = 0) is more subtle (Section 4.1.3 and [218]).

The amplitude can be rewritten in two different ways. First, the ghost insertions can berewritten in terms of a ghost correlation functions

Ag,n(ki)αi =∫Mg

dMg t

∫ n∏i=Kcg+1

d2σi√g

⟨ Kcg∏j=1

εab2 ca(σ0

j )cb(σ0j )

Mg∏i=1

(µi, b)g

⟩gh,g

×

⟨ Kcg∏j=1

Vαi(ki;σ0j )

n∏i=Kcg+1

Vαi(ki;σi)⟩m,g

.

(4.26)

This form is particularly interesting because it shows that, before integration over the mod-uli, the amplitudes factorize. This is one of the main advantage of the conformal gauge,since the original complicated amplitude (4.5) for a QFT on a dynamical spacetime re-duces to the product of two correlation functions of QFTs on a fixed curved background.In fact, the situation is even simpler when taking a flat background g = δ since both theghost and matter sectors are CFTs and one can employ all the tools from two-dimensionalCFT (Part VII) to perform the computations and mostly forget about the path integralorigin of these formulas. This approach is particularly fruitful for off-shell (Chapter 6) andsuperstring amplitudes (Part IV).

Remark 4.2 (Amplitudes in 2d gravity) The derivation of amplitudes for 2d gravityfollows the same procedure, up to two differences: 1) there is an additional decoupled (beforemoduli and position integrations) gravitational sector described by the Liouville field, 2) thematter and gravitational action are not CFTs if the original matter was not.

A second formula can be obtained by bringing the c-ghost on top of the matter vertexoperators which are at the same positions

Ag,n(ki)αi =∫Mg

dMg t

∫ n∏i=Kcg+1

d2σi√g

⟨Mg∏i=1

Bi

Kcg∏j=1

Vαi(ki;σ0j )

n∏i=Kcg+1

Vαi(ki;σi)⟩g

, (4.27)

91

and where

Vαi(ki;σ0j ) := εab

2 ca(σ0j )cb(σ0

j ) Vαi(ki;σ0j ), Bi := (µi, b)g. (4.28)

The operators Vαi(ki;σ0j ) are called unintegrated operators, by opposition to the integrated

operators Vαi(ki). We will see that both are natural elements of the BRST cohomology.To stress that the Bi insertions are really an element of the measure, it is finally possible

to rewrite the previous expression as

Ag,n(ki)αi =∫Mg×Cn

⟨Mg∧i=1

Bi dtiKcg∏j=1

Vαi(ki;σ0j )

n∏i=Kcg+1

Vαi(ki;σi) d2σi√g

⟩g

. (4.29)

The result (4.27) suggests a last possibility for improving the expression of the amplitude.Indeed, the different vertex operators don’t appear symmetrically: some are integrated overand other come with c ghosts. Similarly, the two types of integrals have different roles: themoduli are related to geometry while the positions look like external data (vertex operators).However, punctures can obviously be interpreted as part of the geometry, and one maywonder if it is possible to unify the moduli and positions integrals. It is, in fact, possible toput all vertex operators and integrals on the same footing by considering the amplitude tobe defined on the moduli spaceMg,n of genus-g Riemann surfaces with n punctures insteadof justMg [218] (see also Section 6.3.1).

4.1.3 Gauge fixing: 2-point amplitudeAs discussed at the end of Section 4.1.1, it has long been believed that the tree-level 2-pointamplitude vanishes. There were two main arguments: there are not sufficiently many vertexoperators 1) to fix completely the SL(2,C) invariance or 2) to saturate the number of c-ghost zero-modes. Let’s review both points and then explain why they are incorrect. Wewill provide the simplest arguments, referring the reader to the literature [78, 237] for moregeneral approaches.

For simplicity, we consider the flat metric g = δ and an orthonormal basis of CKV.The two weight-(1, 1) matter vertex operators are denoted as Vk(z, z) and Vk′(z′, z′) suchthat the 2-point correlation function on the sphere reads (see Chapters 22 and 23 for moredetails):

〈Vk(z, z)Vk′(z′, z′)〉S2 = i (2π)Dδ(D)(k + k′)|z − z′|4

. (4.30)

The numerator comes from the zero-modes ei(k+k′)·x for a target spacetime with a Lorentziansignature [58, p. 866, 218] (required to make use of the on-shell condition).

Review of the problem

The tree-level amplitude (4.19) for n = 2 reads:

A0,2(k, k′) = CS2

VolK0,0

∫d2zd2z′ 〈Vk(z, z)Vk′(z′, z′)〉S2 , (4.31)

where K0,n is the CKV group of Σ0,n, the sphere with n punctures. In particular, the groupof the sphere without puncture is K0,0 = PSL(2,C). The normalization of the amplitude isCS2 = 8πα′−1 for gs = 1 [218, 284]. Since there are two insertions, the symmetry can be

92

partially gauge fixed by fixing the positions of the two punctures to z = 0 and z′ = ∞. Inthis case, the amplitude (4.31) becomes:

A0,2(k, k′) = CS2

VolK0,2〈Vk(∞,∞)Vk′(0, 0)〉S2 , (4.32)

where K0,2 = Dil× Rot is the CKV group of the 2-punctured sphere – dilatation R∗+ timesrotation U(1). Since the volume of this group is infinite VolK0,2 =∞, it looks like A0,2 = 0.However, this forgets that the 2-point correlation function (4.30) contains a D-dimensionaldelta function. The on-shell condition implies that the conservation of the momentumk + k′ = 0 is automatic for one component, such that the numerator in (4.32) contains adivergent factor δ(0):

A0,2(k, k′) = (2π)D−1δ(D−1)(k + k′) CS2 2πi δ(0)VolK0,2

. (4.33)

Hence, (4.32) is of the form A0,2 =∞/∞ and one should be careful when evaluating it.The second argument relies on a loophole in the understanding of the gauge fixed amp-

litude (4.27). The result (4.27) is often summarized by saying that one can go from (4.19)to (4.27) by replacing Kcg integrated vertices

∫V by unintegrated vertices ccV in order to

saturate the ghost zero-modes and to obtain a non-zero result. For g = 0, this requires 3unintegrated vertices. But, since there are only two operators in (4.31), this is impossibleand the result must be zero. However, this is also incorrect because it is always possibleto insert 6 c zero-modes, as show the formulas (3.163) and (4.26). Indeed, they are partof how the path integral measure is defined and do not care of the matter operators. Thequestion is whether they can be attached to vertex operators (for aesthetic reasons or morepragmatically to get natural states of the BRST cohomology). To find the correct resultwith ghosts requires to start with (4.26) and to see how this can be simplified when thereare only two operators.

Computation of the amplitude

In this section, we compute the 2-point amplitude from (4.32):

A0,2(k, k′) = CS2

VolK0,2

⟨Vk(∞,∞)tVk′(0, 0)

⟩S2 . (4.34)

The volume of CKV2 reads:

VolK0,2 =∫ d2z

|z|2= 2

∫ 2π

0dθ∫ ∞

0

drr, (4.35)

by doing the change of variables z = reiθ. Since the volume is infinite, it must be regu-larized. A first possibility is to cut-off a small circle of radius ε around r = 0 and r = ∞(corresponding to removing the two punctures at z = 0,∞). A second possibility consistsin performing the change of variables r = eτ and to add an imaginary exponential:

VolK0,2 = 4π∫ ∞

0

drr

= 4π∫ ∞−∞

dτ = 4π limε→0

∫ ∞−∞

dτ eiετ = 4π × 2π limε→0

δ(ε), (4.36)

such that the regularized volume reads

VolεK0,2 = 8π2 δ(ε). (4.37)

In fact, τ can be interpreted as the Euclidean worldsheet time on the cylinder since rcorresponds to the radial direction of the complex plane.

93

Since the worldsheet is an embedding into the target spacetime, both must have thesame signature. As a consequence, for the worldsheet to be also Lorentzian, the formula(4.36) must be analytically continued as ε = −iE and τ = it such that

VolM,E K0,2 = 8π2i δ(E), (4.38)

where the subscript M reminds that one considers the Lorentzian signature. Inserting thisexpression in (4.33) and taking the limit E → 0, it looks like the two δ(0) will cancel.However, we need to be careful about the dimensions. Indeed, the worldsheet time τ andenergy E are dimensionless, while the spacetime time and energy are not. Thus, it is notquite correct to cancel directly both δ(0) since they don’t have the same dimensions. In orderto find the correct relation between the integrals in (4.36) and of the zero-mode in (4.30),we can look at the mode expansion for the scalar field (removing the useless oscillators):

X0(z, z) = x0 + ik0 ln |z|2 = x0 + 2iα′k0τ, (4.39)

where the second equality follows by setting z = eτ . After analytic continuation k0 = −ik0M ,

X0 = iX0M , x0 = ix0

M and τ = it, we find [300, p. 186]:

X0M = x0

M + 2α′k0M t. (4.40)

This indicates that the measure of the worldsheet time in (4.38) must be rescaled by 1/2α′k0M

such that:VolM K0,2 −→

8π2i δ(0)2α′k0 = CS2 2πi δ(0)

2k0 . (4.41)

This is equivalent to rescale E by 2α′k0 and to use δ(ax) = a−1δ(x).Ultimately, the 2-point amplitude becomes

A0,2(k, k′) = 2k0(2π)D−1δ(D−1)(k + k′) (4.42)

and matches the QFT formula (4.15). We see that taking into account the scale of thecoordinates is important to reproduce this result.

The computation displayed here presents some ambiguities because of the regularization.However, this ambiguity can be fixed from unitarity of the scattering amplitudes. A moregeneral version of the Faddeev–Popov gauge fixing has been introduced in [78] to avoiddealing altogether with infinities. It is an interesting question whether these techniquescan be extended to the compute the tree-level 1- and 0-point amplitudes on the sphere. Inmost cases, the 1-point amplitude is expected to vanish since 1-point correlation functionsof primary operators other than the identity vanish in unitary CFTs.3 The 0-point functioncorresponds to the sphere partition function: the saddle point approximation to leadingorder allows to relate it to the spacetime action evaluated on the classical solution φ0,Z0 ∼ e−S[φ0]/~. Since the normalization is not known and because S[φ0] is expected tobe infinite, only comparison between two spacetimes should be meaningful (à la Gibbons–Hawking–York [214, sec. 4.1]). In particular, for Minkowski spacetime we find naively

Z0 ∼δ(D)(0)VolK0

, (4.43)

which is not well-defined. This question has no yet been investigated.3The integral over the zero-mode gives a factor δ(D)(k) which implies k = 0. At zero momentum,

the time scalar X0 is effectively described by unitary CFT. However, there can be some subtleties whenconsidering marginal operator.

94

Expression with ghosts

There are different ways to rewrite the 2-point amplitude in terms of ghosts. In all cases, onecorrectly finds the 6 insertions necessary to get a non-vanishing result since, by definition,it is always possible to rewrite the Faddeev–Popov determinant in terms of ghosts. A firstapproach is to insert 1 =

∫d2z δ(2)(z) inside (4.31) to mimic the presence of a third operator.

This is equivalent to use the identity

〈0| c−1c−1c0c0c1c1 |0〉 = 1 (4.44)

inside (4.32), leading to:

A0,2(k, k′) = CS2

VolK0,2〈Vk(∞,∞)c0c0 Vk′(0, 0)〉S2 , (4.45)

where Vk(z, z) = ccVk(z, z). This shows that (4.15) can also be recovered using the correctinsertions of ghosts. The presence of c0c0 can be expected from string field theory since theyappear in the kinetic term (13.115).

The disadvantage of this formula is to still contain the infinite volume of the dilatationgroup. It is also possible to introduce ghosts for the more general gauge fixing presentedin [78]. An alternative approach has been proposed in [237].

4.2 BRST quantizationThe symmetries of a Lagrangian dictate the possible terms which can be considered. Thiscontinues to hold at the quantum level and the counter-terms introduced by renormalizationare constrained by the symmetries. However, if the path integral is gauge fixed, the originalsymmetry is no more available for this purpose. Fortunately, one can show that there isa global symmetry (with anticommuting parameters) remnant of the local symmetry: theBRST symmetry. It ensures consistency of the quantum theory. It also provides a directaccess to the physical spectrum.

The goal of this section is to provide a general idea of the BRST quantization for theworldsheet path integral. A more detailed CFT analysis and the consequence for stringtheory are given in Chapter 24. The reader is assumed to have some familiarity with theBRST quantization in field theory – a summary is given in Appendix C.2.

4.2.1 BRST symmetryThe partition function (3.159) is not the most suitable to display the BRST symmetry.The first step is to restore the dependence in the original metric gab by introducing a deltafunction

Zg =∫Mg

dMg t

Ωckv[g]

∫dggab dgΨ dgbd′gc δ

(√ggab −

√ggab

) Mg∏i=1

(φi, b)g e−Sm[g,Ψ]−Sgh[g,b,c].

(4.46)Note that it is necessary to use the traceless gauge fixing condition (3.152) as it will becomeclear. The delta function is Fourier transformed in an exponential thanks to an auxiliarybosonic field:

Zg =∫Mg

dMg t

Ωckv[g]

∫dggab dgBab dgΨ dgbd′gc

Mg∏i=1

(φi, b)g e−Sm[g,Ψ]−Sgh[g,g,B]−Sgh[g,b,c]

(4.47)

95

where the gauge-fixing action reads:

Sgf[g, g, B] = − i4π

∫d2σ Bab

(√ggab −

√ggab

). (4.48)

Varying the action with respect to the auxiliary field Bab, called the Nakanish–Lautrup field,produces the gauge-fixing condition.

The BRST transformations areδεgab = iεLcgab, δεΨ = iεLcΨ,

δεca = iεLcca, δεbab = εBab, δεBab = 0,

(4.49)

where ε is a Grassmann parameter (anticommuting number) independent of the position.If the traceless gauge fixing (3.152) is not used, then Bab is not traceless: in that case, thevariation δεbab will generate a trace, which is not consistent. Since the transformations acton the matter action Sm as a diffeomorphism with vector εca, it is obvious that it is invariantby itself. It is easy to show that the transformations (4.49) leave the total action invariantin (4.47). The invariance of the measure is given in [218].

Remark 4.3 (BRST transformations with Weyl ghost) One can also consider the ac-tion (3.153) with the Weyl ghost. In this case, the transformation law of the metric ismodified and the Weyl ghost transforms as a scalar:

δεgab = iεLcgab + iε gabcw, δεcw = iεLccw. (4.50)

The second term in δεgab is a Weyl transformation with parameter εcw. Moreover, bab andBab are not symmetric traceless.

The equation of motion for the auxiliary field is

Bab = iTab := i(Tmab + T gh

ab

), (4.51)

where the RHS is the total energy–momentum tensor (matter plus ghosts). Integrating itout imposes the gauge condition gab = gab and yields the modified BRST transformations

δεΨ = iεLcΨ, δεca = iεLcca, δεbab = iε Tab. (4.52)

Without starting with the path integral (4.47) with auxiliary field, it would have beendifficult to guess the transformation of the b ghost. Since ca is a vector, one can also write

δεca = ε cb∂bc

a. (4.53)

Associated to this symmetry is the BRST current jaB and the associated conserved BRSTcharge QB

QB =∫

dσ j0B . (4.54)

The charge is nilpotentQ2B = 0. (4.55)

and, through the presence of the c-ghost in the BRST transformation, the BRST charge hasghost number one

Ngh(QB) = 1. (4.56)Variations of the matter fields can be written as

δεΨ = i [εQB ,Ψ]±. (4.57)

Note that the energy–momentum tensor is BRST exact

Tab = [QB , bab]. (4.58)

96

4.2.2 BRST cohomology and physical statesPhysical state |ψ〉 are elements of the absolute cohomology of the BRST operator:

|ψ〉 ∈ H(QB) := kerQBImQB

, (4.59)

or, more explicitly, closed but non-exact states:

QB |ψ〉 = 0, @ |χ〉 : |ψ〉 = QB |χ〉 . (4.60)

The adjective “absolute” is used to distinguish it from two other cohomologies (relativeand semi-relative) defined below. Two states of the cohomology differing by an exact staterepresent identical physical states:

|ψ〉 ∼ |ψ〉+QB |Λ〉 . (4.61)

This equivalence relation, translated in terms of spacetime fields, correspond to spacetimegauge transformations. In particular, it contains the (linearized) reparametrization invari-ance of the spacetime metric in the closed string sector, and, for the open string sector, itcontains Yang–Mills symmetries. We will find that it corresponds to the gauge invarianceof free string field theory (Chapter 13).

However, physical states satisfy two additional constraints (remember that bab is tracelesssymmetric): ∫

dσ bab |ψ〉 = 0. (4.62)

These conditions are central to string (field) theory, so they will appear regularly in thisreview. For this reason, it is useful to provide first some general motivations, and to refinethe analysis later since the CFT language will be more appropriate. Moreover, these twoconditions will naturally emerge in string field theory.

In order to introduce some additional terminology, let’s define the following quantities:4

b+ :=∫

dσ b00, b− :=∫

dσ b01. (4.63)

The semi-relative and relative cohomologies H−(QB) and H0(QB) are defined as5

H−(QB) = H(QB) ∩ ker b−, H0(QB) = H−(QB) ∩ ker b+. (4.64)

The first constraint arises as a consequence of the topology of the closed string worldsheet:the spatial direction is a circle, which implies that the theory must be invariant undertranslations along the σ direction (the circle is invariant under rotation). However, choosinga parametrization implies to fix an origin for the spatial direction: this is equivalent toa gauge fixing condition. As usual, this implies that the corresponding generator Pσ ofworldsheet spatial translations (3.26) must annihilate the states:

Pσ |ψ〉 = 0. (4.65)

This is called the level-matching condition. Using (4.58), this can be rewritten as

Pσ |ψ〉 =∫

dσ T01 |ψ〉 =∫

dσ QB , b01 |ψ〉 = QB

∫dσ b01 |ψ〉 , (4.66)

4The objects b± are zero-modes of the b ghost fields. They correspond (up to a possible irrelevant factor)to the modes b±0 in the CFT formulation of the ghost system (23.132).

5The BRST cohomologies described in this section are slightly different from the ones used in the restof this review. To distinguish them, indices are written as superscripts in this section, and as subscriptsotherwise.

97

since QB |ψ〉 = 0 for a state |ψ〉 in the cohomology. The simplest way to enforce thiscondition is to set the state on which QB acts to zero:6

b− |ψ〉 = 0, (4.67)

which is equivalent to one of the conditions in (4.62).The second condition does not follow as simply. The Hilbert space can be decomposed

according to b+ as

H− := H↓ ⊕H↑, H↓ := H0 := H− ∩ ker b+. (4.68)

Indeed, b+ is a Grassmann variable and generates a 2-state system. In the ghost sector, thetwo Hilbert spaces are generated from the ghost vacua | ↓〉 and | ↑〉 obeying

b+ | ↓〉 = 0, b+ | ↑〉 = | ↓〉 . (4.69)

The action of the BRST charge on states |ψ↓〉 ∈ H↓ and |ψ↑〉 ∈ H↑ follow from these relationsand from the commutation relation (4.58):

QB |ψ↓〉 = H |ψ↑〉 , QB |ψ↑〉 = 0, (4.70)

where H is the worldsheet Hamiltonian defined in (3.26). To prove this relation, start firstwith H |ψ↑〉, then use (4.58)) to get the LHS of the first condition; then apply QB to getthe second condition (using that QB commutes with H, and b+ with any other operatorsbuilding the states). For H 6= 0, the state |ψ↓〉 is not in the cohomology and |ψ↑〉 is exact.Thus, the exact and closed states are

ImQB =|ψ↑〉 ∈ H↑ | H |ψ↑〉 6= 0

, (4.71a)

kerQB =|ψ↑〉 ∈ H↑

∪|ψ↓〉 ∈ H↓ | H |ψ↓〉 = 0

. (4.71b)

This implies that eigenstates of H in the cohomology satisfy the on-shell condition:

H |ψ〉 = 0. (4.72)

This is consistent with the fact that scattering amplitudes involve on-shell states. In thiscase, |ψ↑〉 is not exact and is thus a member of the cohomology H(QB), as well as |ψ↓〉 sinceit becomes close. But, the Hilbert space H↑ must be rejected for two reasons: there wouldbe an apparent doubling of states and scattering amplitudes would behave badly. The firstproblem arises because one can show that the cohomological subspaces of each space areisomorphic: H↓(QB) ' H↑(QB). Hence, keeping both subspaces would lead to a doublingof the physical states. For the second problem, consider an amplitude where one of theexternal state is built from |ψ↑〉: the amplitude vanishes if the states are off-shell since thestate |ψ↑〉 is exact, but it does not vanish on-shell [218, ch. 4]. This means that it mustbe proportional to δ(H). But, general properties in QFT forbid such dependence in theamplitude (only poles and cuts are allowed, except if D = 2). Projecting out the states inH↑ is equivalent to require

b+ |ψ〉 = 0 (4.73)for physical states, which is the second condition in (4.62).

In fact, this condition can be obtained very similarly as the b− = 0 condition: using theexpression of H (3.26) and the commutation relation (4.58), (4.72) is equivalent to

QB

∫dσ b00 |ψ〉 = 0. (4.74)

6The reverse is not true. We will see in Section 4.2.2 the relation between the two conditions in moredetails.

98

Hence, imposing (4.73) allows to automatically ensure that (4.72) holds.Since the on-shell character (4.72) of the BRST states and of the BRST symmetry are

intimately related to the construction of the worldsheet integral, one can expect difficultyfor going off-shell.

4.3 SummaryIn this chapter, we derived general formulas for string scattering amplitudes. The generalBRST formalism has been summarized. Moreover, we gave general motivations for restrict-ing the absolute cohomology to the smaller relative cohomology. In Chapter 24, a moreprecise derivation of the BRST cohomology is worked out. It includes also a proof of theno-ghost theorem: the ghosts and the negative norm states (in Minkowski signature) areunphysical particles and should not be part of the physical states. This theorem asserts thatit is indeed the case. It will also be the occasion to recover the details of the spectrum invarious cases.

4.4 References• The delta function approach to the gauge fixing is described in [218, sec. 3.3, 173,

sec. 15.3.2], with a more direct computation is in [149].

• The most complete references for scattering amplitudes in the path integral formalismare [63, 218].

• Computation of the tree-level 2-point amplitude [78, 237] (for discussions of 2-pointfunction, see [63, p. 936–7, 236, 72, 73, 58, p. 863–4]).

• The BRST quantization of string theory is discussed in [178, 50, 218, chap. 4]. Fora general discussion see [119, 282, 286]. The use of an auxiliary field is consideredin [287, sec. 3.2].

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Chapter 5

Worldsheet path integral:complex coordinates

In the two previous chapters, the amplitudes computed from the worldsheet path integralshave been written covariantly for a generic curved background metric. In this chapter, westart to use complex coordinates and finally take the background metric to be flat. This isthe usual starting point for computing amplitudes since it allows to make contact with CFTsand to employ tools from complex analysis. We first recall few facts on 2d complex manifoldsbefore briefly describing how to rewrite the scattering amplitudes in complex coordinates.

5.1 Geometry of complex manifoldsChoosing a flat background metric simplifies the computations. However, we have seen inSection 3.3 that there is a topological obstruction to get a globally flat metric. The solutionis to work with coordinate patches (σ0, σ1) = (τ, σ) such that the background metric gab isflat in each patch (conformally flat gauge):

ds2 = gabdσadσb = e2φ(τ,σ)(dτ2 + dσ2), (5.1)

orgab = e2φδab, gab = δab. (5.2)

To simplify the notations, we remove the dependence in the flat metric and the hat forquantities (like the vertex operators) expressed in the background metric when no confusionis possible.

Introducing complex coordinates

z = τ + iσ, z = τ − iσ, (5.3a)

τ = z + z

2 , σ = z − z2i , (5.3b)

the metric reads1ds2 = 2gzzdzdz = e2φ(z,z)|dz|2. (5.4)

1In Section 22.1, we provide more details on the relation between the worldsheet (viewed as a cylinderor a sphere) and the complex plane.

100

The metric and its inverse can also be written in components:

gzz = e2φ

2 , gzz = gzz = 0, (5.5a)

gzz = 2e−2φ, gzz = gzz = 0. (5.5b)

Equivalently, the non-zero components of the background metric are

gzz = 12 , gzz = 2. (5.6)

An oriented two-dimensional manifold is a complex manifold: this means that there existsa complex structure, such that the transition functions and changes of coordinates betweendifferent patches are holomorphic at the intersection of the two patches:

w = w(z), w = w(z). (5.7)

For such a transformation, the Liouville mode transforms as

e2φ(z,z) =∣∣∣∣∂w∂z

∣∣∣∣2e2φ(w,w) (5.8)

such thatds2 = e2φ(w,w)|dw|2. (5.9)

This shows also that a conformal structure (3.12) induces a complex structure since thetransformation law of φ is equivalent to a Weyl rescaling.

The integration measures are related as

d2σ := dτdσ = 12 d2z, d2z := dzdz. (5.10)

Due to the factor of 2 in the expression, the delta function δ(2)(z) also gets a factor of 2with respect to δ(2)(σ)

δ(2)(z) = 12 δ

(2)(σ). (5.11)

Then, one can check that ∫d2z δ(2)(z) =

∫d2σ δ(2)(σ). (5.12)

The basis vectors (derivatives) and one-forms can be found using the chain rule:

∂z = 12 (∂τ − i∂σ), ∂z = 1

2 (∂τ + i∂σ), (5.13a)

dz = dτ + idσ, dz = dτ − idσ. (5.13b)

The Levi–Civita (completely antisymmetric) tensor is normalized by

ε01 = ε01 = 1. (5.14a)

εzz = i2 , εzz = −2i, (5.14b)

remembering that it transforms as a density. Integer indices run over local frame coordinates.The different tensors can be found from the tensor transformation law. For example, the

components of a vector V a in both systems are related by

V z = V 0 + iV 1, V z = V 0 − iV 1 (5.15)

101

such thatV = V 0∂0 + V 1∂1 = V z∂z + V z∂z. (5.16)

For holomorphic coordinate transformations (5.7), the components of the vector do not mix:

V w = ∂w

∂zV z, V w = ∂w

∂zV z. (5.17)

This implies that the tangent space of the Riemann surface is decomposed into holomorphicand anti-holomorphic vectors:2

TΣg ' TΣ+g ⊕ TΣ−g , (5.18a)

V z∂z ∈ TΣ+g , V z∂z ∈ TΣ−g , (5.18b)

as a consequence of the existence of a complex structure. Similarly, the components of a1-form ω – which is the only non-trivial form on Σg – can be written in terms of the realcoordinates as:

ωz = 12 (ω0 − iω1), ωz = 1

2 (ω0 + iω1) (5.19)

such thatω = ω0dσ0 + ω1dσ1 = ωzdz + ωzdz. (5.20)

Hence, a 1-form is decomposed into complex (1, 0)- and (0, 1)-forms:

T ∗Σg ' Ω1,0(Σg)⊕ Ω0,1(Σg), (5.21a)ωzdz ∈ Ω1,0(Σg), ωzdz ∈ Ω0,1(Σg), (5.21b)

since both components will not mixed under holomorphic changes of coordinates (5.7). Fi-nally, the metric provides an isomorphism between TΣ+

g and Ω0,1(Σg), and between TΣ−gand Ω1,0(Σg), since it can be used to lower/raise an index while converting it from holo-morphic to anti-holomorphic, or conversely:

Vz = gzzVz, Vz = gzzV

z. (5.22)

This can be generalized further by considering components with more indices: all anti-holomorphic indices can be converted to holomorphic indices thanks to the metric:

T

q++p−︷︸︸︷z···z

z···z︸︷︷︸p++q−

= (gzz)p−(gzz)q−Tq+︷︸︸︷z···z

q−︷︸︸︷z···z

z···z︸︷︷︸p+

z···z︸︷︷︸p−

. (5.23)

Hence, it is sufficient to study (p, q)-tensors with p upper and q lower holomorphic indices.In this case, the transformation rule under (5.7) reads

T

q︷︸︸︷w···w

w···w︸︷︷︸p

=(∂w

∂z

)nT

q︷︸︸︷z···z

z···z︸︷︷︸p

, n := q − p. (5.24)

The number n ∈ Z is called the helicity or rank.3 The set of helicity-n tensors is denotedby T n.

The first example is vectors (or equivalently 1-forms): V z ∈ T 1, Vz ∈ T −1. The secondmost useful case are traceless symmetric tensors, which are elements of T ±2. Consider

2However, at this stage, each component can still depend on both z and z: V z = V z(z, z) and V z =V z(z, z).

3In fact, it is even possible to consider n ∈ Z + 1/2 to describe spinors.

102

a traceless symmetric tensor T ab = T ba and gabTab = 0: this implies T 01 = T 10 and

T 00 = −T 11 in real coordinates. The components in complex coordinates are:

T zz = 2(T 00 + iT 01) ∈ T 2, T zz = 2(T 00 − iT 01) ∈ T −2, T zz = 0. (5.25)

Note thatTzz = gzzgzzT

zz = 12(T 00 − iT 01), (5.26)

and T zz = gzzTzz ∈ T 0 corresponds to the trace.


T zz =(∂z

∂τ

)2T 00 +

(∂z

∂σ

)2T 11 + 2 ∂z

∂τ

∂z

∂σT 01 = T 00 − T 11 + 2iT 01.

Stokes’ theorem in complex coordinates follows directly from (B.10):∫d2z (∂zvz + ∂zv

z) = −i∮ (

dz vz − dzvz)

= −2i∮∂R

(vzdz − vzdz), (5.27)

where the integration contour is anti-clockwise. To obtain this formula, note that d2x = 12d2z

and εzz = i/2, such that the factor 1/2 cancels between both sides.

5.2 Complex representation of path integralIn the previous section, we have found that tensors of a given rank are naturally decomposedinto different subspaces thanks to the complex structure of the manifold. Accordingly,complex coordinates are natural and one can expect most objects in string theory to splitsimilarly into holomorphic and anti-holomorphic sectors (or left- and right-moving). Thiswill be particularly clear using the CFT language (Chapter 22). The main difficulty for thisprogram is due to the matter zero-modes. In this section, we focus on the path integralmeasure and expression of the ghosts.

There is, however, a subtlety in displaying explicitly the factorization: the notion of“holomorphicity” depends on the metric (because the complex structure must be compatiblewith the metric for an hermitian manifold). Since the metric depends on the moduli whichare integrated over in the path integral, it is not clear that there is a consistent holomorphicfactorization. We will not push the question of achieving a global factorization further (butsee Remark 5.1) to focus instead on the integrand. The latter is local (in moduli space) andthere is no ambiguity.

The results of the previous section indicate that the basis of Killing vectors (3.104) andquadratic differentials (3.76) split into holomorphic and anti-holomorphic components:

ψi(z, z) = ψzi ∂z + ψzi ∂z, φi(z, z) = φi,zz(dz)2 + φi,zz(dz)2. (5.28)

Similarly, the operators P1 (3.65a) and P †1 (3.71) also split:

(P1ξ)zz = 2∇zξz = ∂zξz, (P1ξ)zz = 2∇zξz = ∂zξ

z, (5.29a)(P †1T )z = −2∇zTzz = −4 ∂zTzz, (P †1T )z = −2∇zTzz = −4 ∂zTzz (5.29b)

for arbitrary tensor ξ and traceless symmetric tensor T (in the background metric). As aconsequence, the components of Killing vectors and quadratic differentials are holomorphicor anti-holomorphic as a function of z:

ψz = ψz(z), ψz = ψz(z), φzz = φzz(z), φzz = φzz(z), (5.30)

103

such that it makes sense to consider a complex basis instead of the previous real basis:

kerP1 = SpanψK(z) ⊕ SpanψK(z), K = 1, . . . ,Kcg, (5.31a)kerP †1 = SpanφI(z) ⊕ SpanφI(z), I = 1, . . . ,Mc

g. (5.31b)

The last equation can inspire to search for a similar rewriting of the moduli parameters.In fact, the moduli space itself is a complex manifold and can be endowed with complexcoordinates [195, 218]:

mI = t2I−1 + it2I , mI = t2I−1 − it2I , I = 1, . . . ,Mcg (5.32)

with the integration measuredMg t = d2Mc

gm. (5.33)

The last ingredient to rewrite the vacuum amplitudes (3.136) is to obtain the determin-ants. The inner-products of vector and traceless symmetric fields also factorize:

(T1, T2) = 2∫

d2σ√g gacgbdT1,abT2,cd = 4

∫d2z

(T1,zzT2,zz + T1,zzT2,zz

), (5.34a)

(ξ1, ξ2) =∫

d2σ√g gabξ

aξb = 14

∫d2z

(ξz1ξ

z2 + ξz1ξ

z2). (5.34b)

All inner-products are evaluated in the flat background metric. For (anti-)holomorphic fields,only one term survives in each integral: since each field appears twice in the determinants(φi, φj) and (φi, φj), the final expression is a square, which cancels against the squarerootin (3.136). The remaining determinant involves the Beltrami differential (3.65b):

µizz = ∂igzz, µizz = ∂igzz (5.35)

(gzz = 0 in our coordinates system, but its variation under a shift of moduli is not zero).The basis can be changed to a complex basis such that the determinant of inner-productsbetween Beltrami and quadratic differentials is a modulus squared. All together, the differentformulas lead to the following rewriting of the vacuum amplitude :

Zg =∫Mg

d2Mcgm|det(φI , µJ)|2

|det(φI , φJ)|det′ P †1P1

|det(ψI , ψJ)|Zm[δ]Ωckv[δ] , (5.36)

where the absolute values are to be understood with respect to the basis of P1 and P †1 , forexample |f(mI)|2 := f(mI)f(mI).

The same reasoning can be applied to the ghosts. The c and b ghosts are respectivelya vector and a symmetric traceless tensor, both with two independent components: it iscustomary to define

c := cz, c := cz, b := bzz, b := bzz. (5.37)

In that case, the action (3.145) reads

Sgh[g, b, c] = 12π

∫d2z

(b∂zc+ b∂z c

). (5.38)

The action is the sum of two holomorphic and anti-holomorphic contributions and it isindependent of φ(z, z) as expected. In fact, the equations of motion are

∂zc = 0, ∂zb = 0, ∂z c = 0, ∂z b = 0, (5.39)

104

such that b and c (resp. b and c) are holomorphic (anti-holomorphic) functions. Then, theintegration measure is simply

Mg∧i=1

Bi dti =Mcg∧

I=1BIBI dmI ∧ mI , BI := (µI , b). (5.40)

Note that BI does not contain b(z), it is built only from b(z).Finally, the vacuum amplitude (3.163) reads

Zg =∫Mg

d2Mcgm

Ωckv[δ]−1

|detψI(z0j )|2

∫d(b, b) d(c, c)

Kcg∏j=1

c(z0j )c(z0

j )Mcg∏

I=1|(µI , b)|2 e−Sgh[b,c] Zm[δ].

(5.41)The c insertions are separated in holomorphic and anti-holomorphic components because,at the end, only the zero-modes contribute. The measures are written as d(b, b) and d(c, c)because proving that they factorize is difficult (Remark 5.1).

Remark 5.1 (Holomorphic factorization) It was proven in [16, 34, 42] (see [195, sec. 9,63, sec. VII, 267, sec. 3] for reviews) that the ghost and matter path integrals can be globallyfactorized, up to a factor due to zero-modes. Such a result is suggested by the factorizationof the inner-products, which imply a factorization of the measures: the caveat is due to thezero-mode determinants and matter measure. Interestingly, the factorization is possible onlyin the critical dimension (3.125).

5.3 SummaryIn this chapter, we have introduced complex notations for the fields, path integral andmoduli space.

Since the CFT language will play an important role in the rest of the book, the readerwho is not familiar with it is advised to proceed first to Part VII before reading the nextchapter.

5.4 References• Good references for this chapter are [31, 63, 193, 195, 218].

• Geometry of complex manifolds is discussed in [31, sec. 6.2, 193, chap. 14, 63].

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Chapter 6

Introduction to off-shell stringtheory

In this chapter, we introduce a framework to describe off-shell amplitudes in string theory.We first start by motivating various concepts – in particular, local coordinates and factor-ization – by focusing on the 3- and 4-point amplitudes. We then prepare the stage for ageneral description of off-shell amplitudes. We focus again on the closed bosonic string only.

6.1 Motivations6.1.1 3-point functionThe tree-level 3-point amplitude of 3 weight hi vertex operators1 Vi is given by

A0,3 =⟨ 3∏i=1

Vi(zi)⟩S2

∝ (z1 − z2)h3−h1−h2 × perms× c.c. (6.1)

There is no integration since dimM0,3 = 0.The amplitude is independent of the zi only if the matter state is on-shell, hi = 0, for

example if Vi = ccVi with h(Vi) = 1. Indeed, if hi 6= 0, then A0,3 is not invariant underconformal transformations (22.38):

z −→ fg(z) = az + b

cz + d∈ SL(2,C) (6.2)

(it transforms covariantly). This is a consequence of the punctures: the presence of the lattermodifies locally the metric, since they act as sources of negative curvature. When performinga conformal transformation, the metric around the punctures changes in a different way asaway from them. This implies that the final result depends on the metric chosen aroundthe punctures. This looks puzzling because the original path integral derivation (Chapter 4)indicates that the 3-point amplitude should not depend on the locations of the operatorsbecause its moduli space is empty (hence, all choices of zi should be equivalent).

The solution is to introduce local coordinates wi with a flat metric |dwi|2 around eachpuncture conventionally located at wi = 0. The local coordinates are defined by the maps:

z = fi(wi), zi = fi(0). (6.3)1The quantum number (k, j) of the vertex operator is mostly irrelevant for the discussion of the current

and next chapters, and they are omitted. We will distinguish them by a number and reintroduce themomentum k when necessary. We also omit the overall normalization of the amplitudes.

106

This is also useful to characterize in a simpler way the dependence of off-shell amplitudesrather than using the metric around the punctures (computations may be more difficult witha general metric).

The expression of a local operator in the local coordinate system is found by applyingthe corresponding change of coordinates (22.48):

f V (w) = f ′(w)hf ′(w)h V(f(w)

). (6.4)

The amplitude reads then

A0,3 =⟨ 3∏i=1

fi Vi(0)⟩S2

=( 3∏i=1

f ′i(0)hif ′i(0)hi)⟨ 3∏

i=1Vi(fi(0)

)⟩S2

(6.5a)

∝

( 3∏i=1

f ′i(0)hif ′i(0)hi)(

f1(0)− f2(0))h3−h1−h2 × perms× c.c. (6.5b)

The amplitude depends on the local coordinate choice fi, but not on the metric around thepunctures. It is also invariant under SL(2,C): the transformation (6.2) written in terms ofthe local coordinates is

fi −→afi + b

cfi + d(6.6)

from which we get:

f ′i −→f ′i

(cfi + d)2 , fi − fj −→fi − fj

(cfi + d)(cfj + d) . (6.7)

All together, this implies the invariance of the 3-point amplitude since the factors in thedenominator cancel. When the states are on-shell hi = 0, the dependence in the localcoordinate cancels, showing that the latter is non-physical.

One can ask how Feynman graphs can be constructed in string theory. By definition, anamplitude is the sum of Feynman graphs contributing at that order in the loop expansionand for the given number of external legs. The Feynman graphs are themselves built from aset of Feynman rules. These correspond to the data of the fundamental interactions togetherwith the definition of a propagator. Since a tree-level cubic interaction is the interaction ofthe lowest order, it makes sense to promote it to a fundamental cubic vertex2 V0,3:

V0,3(V1,V2,V3) := = A0,3(V1,V2,V3). (6.8)

The index 0 reminds that it is a tree-level interaction.2The notation will become clear later, and should not be confused with the vertex operators.

107

6.1.2 4-point functionThe tree-level 4-point amplitude is expressed as

A0,4 =∫

d2z4

⟨ 3∏i=1

ccVi(zi)V4(z4)⟩S2

. (6.9)

The conformal weights are denoted by h(Vi) = hi. For on-shell states, hi = 1: while thereis no dependence on the positions z1, z2 and z3, there are divergences for

z4 −→ z1, z2, z3, (6.10)

corresponding to collisions of punctures in the integration process. Moreover, the expressiondoes not look symmetric: it would me more satisfactory if all the insertions were accompaniedby ghost insertions and if all the puncture locations were treated on an equal footing.

Example 6.1 – TachyonsGiven tachyon states Vi = eiki·X , the amplitude reads:

A0,4 ∝3∏

i,j=1i<j

|zi − zj |2+ki·kj∫

d2z4

3∏i=1|z4 − zi|ki·k4 . (6.11)

The integral diverges for z4 → zi if ki · k4 ≤ 0. This can happen for physical values ofthe momenta ki.

The idea is to cut out regions around z1, z2 and z3 in the z4-plane and to change theinterpretation of these contributions. First, we consider the case z4 → z3, which correspondsto cutting a region around z3. Writing z4 = qy4 with y4 ∈ C fixed, the contribution of thisregion to the amplitude is denoted by F (s)

0,4 . For simplicity, we take z3 = 0. The contributionreads:

F (s)0,4 =

∫ d2q

|q|2〈ccV1(z1)ccV2(z2)ccV3(0)|qy4|2V4(qy4)〉. (6.12)

The implicit radial ordering pushes V3 to the left of V4 and using the OPE between the band c ghosts gives:

F (s)0,4 = −

∫ d2q

|q|2

⟨ccV1(z1)ccV2(z2)

∮|w|=|q|1/2

dww b(w)∮|w|=|q|1/2

dw w b(w) ccV4(qy4)ccV3(0)⟩.

(6.13)The sign arises by anti-commuting c and b. The integration variable q can be removed fromthe argument of V4 using the L0 and L0 operators:

F (s)0,4 = −

∫ d2q

|q|2

⟨ccV1(z1)ccV2(z2)

∮dww b(w)

∮dw w b(w) qL0 qL0ccV4(y4)ccV3(0)

⟩.

(6.14)This expression is more satisfactory because all vertex operators are accompanied with c-ghost insertions and none of the arguments is integrated over. But, in fact, even better canbe achieved.

Inserting two complete sets of states φr (see Section 6.2) inside this expression gives(restoring a generic z3-dependence):

F (s)0,4 = 〈ccV1(z1)ccV2(z2)φr(0)〉〈ccV3(z3)ccV4(y4)φs(0)〉

∫ d2q

|q|2⟨φcrq

L0 qL0b0b0φcs

⟩, (6.15)

108

where the sum over r and s is implicit. The conjugate states φcr are defined by 〈φcr|φs〉 = δrs.The first two terms are cubic interactions (6.8), and the last term connects both. It is thentempting to identify the latter with a propagator ∆

∆(φcr, φ

cs

):= 〈φcr|∆ |φcs〉 := −

∫ d2q

|q|2⟨φcrq

L0 qL0b0b0φcs

⟩, (6.16)

such that:

F (s)0,4 = V0,3

(ccV1(z1), ccV2(z2), φr(0)

)×∆

(φcr, φ

cs

)× V0,3

(ccV3(z3), ccV4(y4), φs(0)

)

=(6.17)

To make this more precise, change the coordinates as:

q = e−s+iθ, s ∈ R+, θ ∈ [0, 2π), (6.18)

such that the integral becomes:∫ d2q

|q|2qL0 qL0 = 2

∫ ∞0

ds∫ 2π

0dθ e−s(L0+L0)eiθ(L0−L0) = 2

L0 + L0δL0,L0

. (6.19)

This shows that the propagator can be rewritten as

∆ = − 2b0b0L0 + L0

δL0,L0= b+0L+

0b−0 δL−0 ,0

, (6.20)

where L±0 = L0 ± L0 and b±0 = b0 ± b0. The sign is added by anticipating the normalizationto be derived later. Its properties will be studied in details in Section 9.2.2.

Taking the basis states φr := φα(k) to be eigenstates of L0 and L0

L0 |φα(k)〉 = L0 |φα(k)〉 = α′

4 (k2 +m2α) |φα(k)〉 (6.21)

allows to rewrite the last term of F (s)0,4 as

∆αβ(k) =∫ d2q

|q|2⟨φcα(k)qL0 qL0b0b0φ

cβ(−k)

⟩= Mαβ(k)k2 +m2

α

. (6.22)

The finite-dimensional matrix Mαβ gives the overlap of states of identical masses:

Mαβ(k) := 2α′〈φcα(k)| b+0 b−0 |φcβ(−k)〉 . (6.23)

The propagator depends only on one momentum because 〈k|k′〉 ∼ δ(D)(k − k′). This isexactly the standard propagator one finds in QFT and this justifies the above claim. Thecontribution F (s)

0,4 to the amplitude can be seen as a s-channel Feynman graph obtained bygluing two cubic fundamental vertices with a propagator. We will see later the interpretationin terms of Riemann surfaces.

109

The same procedure can be followed by considering z4 ∼ z2 and z4 ∼ z1. This leads tocontributions F (t)

0,4 and F (u)0,4 corresponding to t- and u-channel Feynman graphs:

F (t)0,4 = F (u)

0,4 = (6.24)

In general, the sum of the three contributions F (s,t,u)0,4 does not reproduce the full amp-

litude A0,4. Said differently, the regions cut in the z4-plane does not cover it completely. It isthen natural to interpret the remaining part as a fundamental tree-level quartic interactiondenoted by

V0,4 = (6.25)

such thatA0,4 = F (s)

0,4 + F (t)0,4 + F (u)

0,4 + V0,4. (6.26)Up to (6.14) it was sufficient to consider on-shell states, but the insertion of the complete

basis requires to consider also off-shell states since on-shell states do not form a basis of theHilbert space. As discussed for the 3-point functions, it is necessary to introduce localcoordinates to describe off-shell states properly.

With the 3- and 4-point functions, we motivated the use of off-shell states and introducedthe two important ideas of local coordinates and amplitude factorization. We also indicatedthat amplitudes can be written in a more symmetric way (see also the discussion at theend of Section 4.1.2). In the rest of this chapter, we give additional ideas on off-shell stringtheory.

Remark 6.1 (Riemann surface interpretation) The interpretation of the insertion ofa propagator in terms of Riemann surface consists in gluing two of them thanks to theplumbing fixture procedure (Section 7.3).

6.2 Off-shell statesAn off-shell state is a generic state of the CFT Hilbert space

H = Hm ⊗Hgh. (6.27)

without any constraint. A basis for the off-shell states is denoted by

H = Span|φr〉. (6.28)

110

Since there is no constraint on the states, the ghost number of φr are arbitrary and denotedas:

nr := Ngh(φr) ∈ Z (6.29)(the ghost number is restricted for states in the cohomology of QB). The Grassmann parityof a state φr is denoted as |φr|. When there is no fermions in the matter sector (usually thecase for the bosonic string), only ghosts are odd. Then, the Grassmann parity of a state isodd (resp. even) if its ghost number is odd (resp. even):

|φr| := Ngh(φr) mod 2 =

0 Ngh(φr) even,1 Ngh(φr) odd.

(6.30)

The dual basis |φcr〉 is defined from the BPZ inner product:〈φcr|φs〉 = δrs. (6.31)

Denoting the ghost numbers of the dual states byncr := Ngh(φcr), (6.32)

the product is non-vanishing if

ncr + nr =

6 closed,3 open,

(6.33)

due to the ghost number anomaly on the sphere. This condition cannot be satisfied if thedual state φcr is simply taken to be the BPZ conjugate φtr since the BPZ conjugation doesnot change the ghost number. This implies that

〈φr|φs〉 = 0 (6.34)from the ghost anomaly for every state, except for the closed string states with Ngh = 3 (infact, the inner product of these states is also zero as can be seen after investigation). Onecan show that

〈φr|φcs〉 = (−1)|φr| δrs. (6.35)Hence, the resolution of the identity can be written in the two equivalent ways

1 =∑r

|φr〉〈φcr| =∑r

(−1)|φr| |φcr〉〈φr| . (6.36)

6.2.1 Open stringThe Hilbert space H can be separated according to the ghost zero-modes (Section 23.2.6):

H ∼ H0 ⊕ c0H0, (6.37)

where the relative Hilbert space is defined as:

H0 := H ∩ ker b0 =|φ〉 ∈ H | b0 |φ〉 = 0

∼ b0H. (6.38)

Given a state ψ ∈ c0H, it is mapped to another state ψ ∈ H0 under the isomorphism

|ψ〉 = c0 |ψ〉 , |ψ〉 = b0 |ψ〉 . (6.39)

The basis states of the open string Hilbert space are decomposed as

φr = φ↓,r + φ↑,r, b0 |φ↓,r〉 = 0, c0 |φ↑,r〉 = 0. (6.40)

Each state φ↑,r ∈ c0H0 can be associated to a state φ↓,r ∈ H0 (the arrow is changed to indicateproperly in which subspace the state lies):

|φ↑,r〉 = c0 |φ↓,r〉 , |φ↓,r〉 = b0 |φ↑,r〉 ,

b0 |φ↓,r〉 = 0, Ngh(φ↑,r) = Ngh(φ↓,r) + 1.(6.41)

111

6.2.2 Closed stringFollowing (23.176), the Hilbert space can be decomposed as:

H = H± ⊕ c±0 H±, (6.42)

whereH± := H ∩ ker b±0 = H0 ⊕ c∓0 H0, H0 := H ∩ ker b−0 ∩ ker b+0 . (6.43)

In fact, we will find that a consistent description of the off-shell amplitudes for the closedstring requires to impose some conditions on the states even at the off-shell level. The off-shell states will have to satisfy the level-matching condition and to be annihilated by b−0 :

L−0 |φ〉 = 0, b−0 |φ〉 = 0. (6.44)

This implies that the off-shell states will be elements of H− ∩ kerL−0 . This will appear asconsistency conditions on the geometry of the moduli space and by studying the propagator.In general, we shall work with H and indicate when necessary the restriction to H− (keepingthe condition kerL−0 implicit to avoid new notations).

The Hilbert space H can be separated according to the ghost zero-modes

H ∼ H↓↓ ⊕H↓↑ ⊕H↑↓ ⊕H↑↑, (6.45)

with the following definitions:

H↓↑ ∼ H0, H↓↑ ∼ c0H↓↓ H↑↓ ∼ c0H↓↓ H↑↑ ∼ c0c0H↓↓. (6.46)

Accordingly, every basis state can be split as

φr = φ↓↓,r + φ↓↑,r + φ↑↓,r + φ↑↑,r (6.47)

such thatb0 |φ↓↓,r〉 = b0 |φ↓↓,r〉 = 0, b0 |φ↓↑,r〉 = c0 |φ↓↑,r〉 = 0,c0 |φ↑↓,r〉 = b0 |φ↑↓,r〉 = 0, c0 |φ↑↑,r〉 = c0 |φ↑↑,r〉 = 0.

(6.48)

Moreover, the basis can be indexed such that

|φ↓↑,r〉 = c0 |φ↓↓,r〉 |φ↑↓,r〉 = c0 |φ↓↓,r〉 |φ↑↑,r〉 = c0c0 |φ↓↓,r〉 . (6.49)

A dual state φcr is also expanded:

φcr = φc↓↓,r + φc↓↑,r + φc↑↓,r + φc↑↑,r (6.50)

and the components satisfy

〈φc↓↓,r| c0 = 〈φc↓↓,r| c0 = 0, 〈φc↓↑,r| c0 = 〈φc↓↑,r| b0 = 0,〈φc↑↓,r| b0 = 〈φc↑↓,r| c0 = 0, 〈φc↑↑,r| b0 = 〈φc↑↑,r| b0 = 0.

(6.51)

The indexing of the basis is chosen such that:

〈φc↓↑,r| = 〈φc↓↓,r| b0 〈φc↑↓,r| = 〈φc↓↓,r| b0 〈φc↑↑,r| = 〈φc↓↓,r| b0b0. (6.52)

such that〈φcx,r|φy,s〉 = δxyδrs, (6.53)

where x, y =↓↓, ↑↓, ↓↑, ↑↑.

112

Consider the Hilbert space H−, then a basis state must satisfy

b−0 |φr〉 = 0 =⇒ b0 |φr〉 = b0 |φr〉 . (6.54)

The expansion (6.47) gives the relation

φ↑↓,r + φ↑↑,r = φ↓↓,r + φ↓↑,r (6.55)

such thatφr = 2(φ↓↓,r + φ↓↑,r). (6.56)

For convenience, the factor of 2 can be omitted (which amounts to rescaling the basis states).

6.3 Off-shell amplitudesIn this section, we provide a guideline of what we need to look for in order to write anoff-shell amplitude. The geometrical tools will be described in the next chapter, and theconstruction of off-shell amplitudes in the following one.

6.3.1 Amplitudes from the marked moduli spaceIn Chapter 4, the scattering amplitudes were written as an integral over the moduli spaceMg of the Riemann surface Σg. As a consequence, the moduli ofMg and the position of thevertex operators are not treated on an equal footing. Moreover, the insertions of operatorsis not symmetric since some are integrated, and others have factors of c. These problemscan be solved by reinterpreting the scattering amplitudes in a more geometrical way.

The key is to consider the punctures where vertex operators are inserted as part of thegeometry and not as external data added on top of the Riemann surface Σg.

Then, the worldsheet with the external states is described as a punctured (or marked)Riemann surface Σg,n, which is a Riemann surface Σg with n punctures (marked points) zi.The Euler number of such a surface was given in (4.3):

χg,n := χ(Σg,n) = 2− 2g − n. (6.57)

This makes sense since punctures can be interpreted as disks (boundaries). Note that thepunctures are labeled and are thus distinguishable.

Since the marked points are distinguished, marked Riemann surfaces with identical gand n but with punctures located at different points are seen as different (this statementrequires some care for g = 0 and g = 1 due to the presence of CKV). The correspondingmoduli space is denoted by Mg,n, and it can be viewed as a fibre bundle with Mg as thebase and the puncture positions as the fibre. The dimension ofMg,n is

Mg,n := dimRMg,n = 6g − 6 + 2n, for

g ≥ 2,g = 1, n ≥ 1,g = 0, n ≥ 3.

(6.58)

These cases are equivalent to χg,n < 0, that is, when the surfaces have a negative curvature.The corresponding coordinates are denoted by tλ, λ = 1, . . . ,Mg,n. Comparing with (3.51)and (3.93), this corresponds to the situation where Σg,n has no CKV left unfixed.

Example 6.2 – 4-punctured sphere Σ0,4The positions of the punctures are denoted by zi with i = 1, . . . , 4. Since there are threeCKV, the positions of three punctures (say z1, z2 and z3) can be fixed, leaving onlyone position which characterizes Σ0,4. Hence, the moduli space has dimension M0,4 = 2

113

andM0,4 is parametrized by z4.

Example 6.3 – 2-punctured torus Σ1,2The positions of the punctures are denoted by zi with i = 1, 2. One puncture can befixed using the single CKV of the surface, which leaves one position. Together with themoduli parameter τ of the torus, this gives M1,2 = 4 and the coordinates ofM1,2 arez2, τ.

The g-loop n-point scattering amplitude with external states Vi can be written as anintegral overMg,n of some Mg,n-form ω

(g,n)Mg,n

:

Ag,n(V1, . . . ,Vn) =∫Mg,n

ωg,nMg,n(V1, . . . ,Vn). (6.59)

The integration over Mg,n has the correct dimension to reproduce the formulas from Sec-tion 4.1.

While it is possible to derive this amplitude from the path integral (see the commentsat the end of Section 4.1.2), we will make only use of the properties of CFT on Riemannsurfaces in the next chapter. This provides an alternative point of view on the computationof scattering amplitudes and how to derive the formulas, which can be helpful when themanipulation of the path integral is more complicated (for example, with the superstring).

The expression of the form ωg,nMg,nmust 1) provide a measure on the moduli space and 2)

extract a function of the moduli from the states Vi. It is natural to achieve the second pointby computing a correlation function on the Riemann surfaces Σg,n. Moreover, Chapters 3and 4 indicate that the ghosts are part of the definition of the measure. Hence, one canexpect the ωg,nMg,n

to have the form:

ωg,nMg,n(V1, . . . ,Vn) =

⟨ghosts×

n∏i=1

Vi

⟩Σg,n

×Mg,n∧λ=1

dtλ. (6.60)

We will motivate an expression in Chapter 9 before checking that it has the correct prop-erties. The ghost insertions are necessary to saturate the number of zero-modes to obtaina non-vanishing result. By convention, the ghosts are inserted on the left: while this doesnot make difference for on-shell closed states, this will for off-shell states and for the othertypes of strings (open and supersymmetric) since the operators can be Grassmann odd.

6.3.2 Local coordinatesThe next step is to consider off-shell states Vi ∈ H. As motivated previously, one needs tointroduce local coordinates defined by the maps:

z = fi(wi), zi = fi(0). (6.61)

There is one local coordinate for each operator, which is inserted at the origin. Local coordin-ates on the surfaces can be seen in two different fashions (Figure 6.1). Either as describingpatches on the surface, in which case the maps fi correspond to transition functions. Or,one can interpret them by cutting disks centred at the punctures and whose interiors aremapped to complex planes, and the maps fi tell how to insert the plane inside the disk.

When the amplitude Ag,n is defined in terms of local coordinates, it will depend onthe maps fi and one needs to ensure that this cancels when the Ai are on-shell. But, thechoice of the maps fi is arbitrary: selecting a specific set hides that all choices are physically

114

(a) Original surface with three punctures.

(b) Disks delimiting the local coordinate patches.

(c) Complex plane mapped to disks centred around the previouspuncture location.

Figure 6.1: Usage of local coordinates for a Riemann surface.

115

equivalent and should lead to the same results on-shell. For this reason, the geometry canbe enriched with the local coordinates, in the same way that the puncture locations wereadded as a fibre to the moduli space Mg to get the marked moduli space Mg,n. Hence,the fundamental geometrical object is the fibre bundle Pg,n withMg,n being the base andthe local coordinates the fibre. Since there is an infinite number of functions, the fibre isinfinite-dimensional, and so is the space Pg,n.

Every point of Pg,n corresponds to a genus-g Riemann surface with n punctures togetherwith a choice of local coordinates around the punctures. The form ωg,nMg,n

is defined in thisbigger space and the integration giving the off-shell amplitude (6.59) is performed over aMg,n-dimensional section Sg,n ⊂ Pg,n (Figure 6.2):

Ag,n(V1, . . . ,Vn)Sg,n =∫Sg,n

ωg,nMg,n(V1, . . . ,Vn)

∣∣Sg,n

. (6.62)

The subscript in the LHS indicates that the amplitudes depend on Sg,n through the choiceof local coordinates. The on-shell independence of Ag,n on the local coordinates translateinto the independence on the choice of the section:

∀Sg,n : Ag,n(V1, . . . ,Vn)Sg,n = Ag,n(V1, . . . ,Vn) (on-shell). (6.63)

The section is taken to be continuous, which means that two neighbouring surfaces of themoduli space must have close local coordinates.

Figure 6.2: Section Sg,n of the fibre bundle Pg,n, the latter havingMg,n as a base the localcoordinates as a fibre.

In order to define the amplitude, one needs to find the expression of the Mg,n-form ωg,nMg,n

on Pg,n. It is in fact simpler to define general p-forms ωg,np on Pg,n, in particular, for provinggeneral properties about the forms and the amplitudes. Given a manifold, a p-form is anelement of the cotangent space and it can be defined through its contraction with vectors(tangent space). Since vectors correspond to small variation of the manifold coordinates, itis necessary to find a parametrization of Pg,n. The geometry of Pg,n – and of its relevantsubspaces and tangent space – is studied in the next chapter. Then, we will come back onthe construction of the amplitudes in Chapter 8.

6.4 References• General references on off-shell string theory include [297, sec. 7, 245, sec. 2, 55, 218].

• Interpretation of local coordinates [218, sec. 5.2].

116

Chapter 7

Geometry of moduli spaces andRiemann surfaces

In this chapter, we describe how to parametrize the moduli space Mg,n and the local co-ordinates which together form the fibre bundle Pg,n introduced in the previous chapter.Then, we can characterize the tangent space which we will need in the next chapter to writethe p-forms on Pg,n necessary to write the amplitudes. Finally, we introduce the notionof plumbing fixture, an operation which glue together punctures located on the same ordifferent surfaces.

7.1 Parametrization of Pg,nThe first step is to find a parametrization of the Riemann surfaces. As we have seen(Chapter 6), the dependence of the surface on the punctures can be described by localcoordinates, that is, transition functions. The patch is defined by cutting a disk aroundeach puncture and the transition functions are defined on the circle given by the intersectionof the disk with the rest of the surface. The number of disks is simply:

#disks = n. (7.1)

It makes sense to look for a similar description of the other moduli (associated to thegenus) by introducing additional coordinate patches. One can imagine that all the depend-ence of the moduli and punctures will reside in the transition functions between patchesif the different patches are isomorphic to a surface without any moduli: the 3-puncturedsphere Σ0,3. Hence, one can look for a decomposition of the surface by cutting disks suchthat one is left with 3-punctured spheres only, and transition functions are defined on thecircles at the intersection of the spheres.

Next, we need to find the number of spheres with 3 holes (or punctures). We start firstwith Σ0,n: in this case, it is straightforward to find that there will be n− 2 spheres. Indeed,for each additional puncture beyond n = 3, an additional sphere is created by cutting acircle. For g ≥ 1, natural places to split the surface are handles: two circles can be cut foreach of them. By inspection, one finds that it leads to 2 spheres for each handle (one on theright and one on the left).1 This shows that the number of spheres is:

#spheres = 2g − 2 + n. (7.2)1The simplest way to find this result is to consider Σg,2 and to write one puncture at each side of the

surface (as in Figure 7.2). To generalize further, one can consider a generic n and put all punctures but oneon one side of the surfaces.

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The number of circles corresponds to the number of boundaries divided by two since theboundaries are glued pairwise: each disk has one boundary and each sphere has 3, whichleads to:

#circles = n+ 3(2g − 2 + n)2 = 3g − 3 + 2n. (7.3)

The idea of the construction is to split the surface into elementary objects (spheres anddisks) such that the full surface is seen as the union of all of them (gluing along circles),and no information is left in the individual geometries. This parametrization is particularlyuseful because there are simple coordinate systems on spheres and disks and these surfacesare easy to visualize and to work with. For example, they can be easily mapped to thecomplex plane.

To conclude, a genus-g Riemann surface Σg,n with n punctures can be seen as thecollection of:

• 2g − 2 + n three-punctured spheres Sa with coordinates za

• n disks Di with coordinates wi around each puncture

• 3g − 3 + 2n circles Cα at the intersection of the spheres and disks

Examples for Σ0,4 and Σ2,2 are given in Figures 7.1 and 7.2.

Figure 7.1: Parametrization of Σ0,4.

Figure 7.2: Parametrization of Σ2,2.

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There are two types of circles: respectively, the ones at the overlap between two spheres,and between a disk and a 3-sphere:

CΛ(ab) := Sa ∩ Sb, Ci(a) := Sa ∩Di, Cα = CΛ, Ci, (7.4)

where Λ counts in fact the number of moduli:

Λ = 1, . . . ,Mcg,n, Mc

g,n = 3g − 3 + n. (7.5)

On the overlap circles, the coordinate systems are related by transitions functions:

on CΛ(ab): za = Fab(zb),on Ci(a): za = fai(wi).

(7.6)

Then, the set of functions Fab, fai completely specifies the Riemann surface Σg,n togetherwith the choice of the local coordinate systems around the punctures. The transition func-tions can thus be used to parametrize the moduli spaceMg,n and the fibre bundle Pg,n, butit is highly redundant because many different functions lead to the same Riemann surface.A unique characterization of the different spaces is obtained by making identifications up tosymmetries.

In the previous chapter, we have seen that the metric in the local coordinate system isflat, ds2 = |dw|2. This means that two systems differing by a global phase rotation

wi −→ wi = eiαiwi (7.7)

lead to surfaces with local coordinates which cannot be distinguished. Correspondingly, thetwo maps fi and fi which relate the local coordinates wi and wi to the coordinate z

z = fi(wi), z = fi(wi) (7.8)

are related as:fi(wi) = fi(eiαiwi). (7.9)

Hence, this motivates to consider the smaller space

Pg,n = Pg,n/U(1)n, (7.10)

where the action of each U(1) is defined by the equivalence (7.9). The necessity to considerthis subspace will be strengthen further later, and will correspond to the level-matchingcondition. Below, global phase rotations are also interpreted in terms of the plumbingfixture, see (7.57).

The different spaces which we need are parametrized by the transition functions up tothe following identifications:

• Pg,n = Fab, fai modulo reparametrizations of za

• Pg,n = Fab, fai modulo reparametrizations of za and phase rotations of wi

• Mg,n = Fab, faimodulo reparametrizations of za and of wi keeping the points wi = 0fixed

• Mg = Fab, fai modulo reparametrizations of za and wi

At each step, the dimension of the space is reduced because one divides by bigger and biggergroups. The highest reduction occurs when dividing by the reparametrizations of wi whichform an infinite-dimensional group (phase rotations form a finite-dimensional subgroup ofthem).

119

For concreteness, it is useful to introduce explicit coordinates xs on Pg,n (s ∈ N since thespace is infinite-dimensional). The transition functions on the Riemann surface depend onthe xs which explains why they can be used to parametrize the moduli spaces. Describingthe spheres Sa by complex planes with punctures located at za,1, za,2 and za,3, the transitionfunctions on the Mg,n circles CΛ(ab) = Sa ∩ Sb for all a < b can be taken to be:

on CΛ(ab): za − za,m = qΛzb − zb,n

, (7.11)

where za,m and zb,n denote the punctures of Sa and Sb lying in CΛ. Then, the complexparameters qΛ with Λ = 1, . . . ,Mc

g,n are coordinates on the moduli space Mg,n. On theremaining n circles Ci(a) = Sa ∩Di, the transition functions can be expanded in series

on Ci(a): za − za,m = wi +∞∑N=1

pi,NwNi , (7.12)

where za,m is the puncture of Sa lying in Ci. There is no negative index in the series becausethe RHS must vanish for wi = 0 which maps to za = za,m (puncture location). The complexcoefficients of the series pi,N (i = 1, . . . , n and N ∈ N∗) provide coordinates for the fibre.Thus, coordinates for Pg,n are:

xs = qΛ, pi,N. (7.13)

As usual, derivatives with respect to xs are abbreviated by ∂s.When the dependence in the xs must be stressed, the transition functions (7.6) are

denoted by:za = Fab(zb;xs), za = fai(wi;xs). (7.14)

In this coordinate system, each parameter appears in only one transition function and it lookslike one can separate the fibre from the basis. But, this is not an invariant statement as thiswould not hold in other coordinate systems. For example, one can rescale the coordinatesto lump all dependence on qΛ in a single circle.

Since both cases are formally identical, it is convenient to fix the orientation of each Cαand to denote by σα (resp. τα) the coordinate on the left (resp. right) of the contour, suchthat the transition functions reads:

on Cα: σα = Fα(τα;xs). (7.15)

Now that we have coordinates on Pg,n, it is possible to construct tangent vectors.

7.2 Tangent spaceA tangent vector Vs ∈ TPg,n corresponds to an infinitesimal variation of the coordinates onthe manifold

δxs = ε Vs, (7.16)

where ε is a small parameter, such that functions of xs vary as:

ε Vs ∂sf = f(xs + ε Vs)− f(xs). (7.17)

The transition functions Fα provide an equivalent (but redundant) set of coordinates forPg,n. Hence, vectors in TPg,n can also be obtained by considering small variations of thetransition functions Fα:

Fα −→ Fα + ε δFα. (7.18)

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Considering an overlap circle Cα, a deformation of the transition function

σα = Fα(τα) (7.19)

for fixed τα can be interpreted as a change of the coordinate σα:

σ′α = Fα(τα) + ε δFα(τα) = σα + ε δFα(τα) = σα + ε δFα(F−1α (σα)

). (7.20)

This transformation is generated by a vector field v(α) on the Riemann surface Σg,n:

σ′α = σα + εv(α)(σα), v(α) = δFα F−1α . (7.21)

The situation is symmetrical and one can obviously fix σα and vary τα. The vector field isregular around the circle Cα (to have a well-defined changes of coordinates) but it can havesingularities away from the circle Cα. Hence, the vector field v(α) together with the circleCα define a vector of Pg,n:

V (α) ∼(v(α), C(α)

). (7.22)

This provides a basis of TPg,n. This is sufficient when using the coordinate system (7.13),but, in more general situations, one needs to consider linear combinations. For example, ifa modulus appears in several transitions functions, then the associated vector field will bedefined on the corresponding circles. A general vector V is described by a vector field v withsupport on a subset C of the circles Cα:

V ∼(v, C

), C ⊆

⋃α

Cα, (7.23)

and the restriction of v on the various circles is written as:

v|Cα = v(α). (7.24)

Note that the vector field v(α) and its complex conjugate v(α) are independent and areassociated to different tangent vectors. This construction is called the Schiffer variation.

The simplest tangent vectors ∂s are given by varying one coordinate of Pg,n while keepingthe other fixed:

xs −→ xs + ε δxs. (7.25)

On each circle Cα, this gives a deformation of the transition functions

Cα : Fα −→ Fα + ε δFα, δFα = ∂Fα∂xs

δxs (7.26)

(no sum over s), such that the change of coordinates reads

σ′α = σα + ε v(α)s (σα) δxs, v(α)

s (σα) = ∂Fα∂xs

(F−1α (σα)

). (7.27)

If the xs are given by (7.13), the vectors have support in only one circle.There is, however, a redundancy in these vectors. Not all of them leads to a motion in

Pg,n because some modifications can be absorbed with a reparametrization of the za. Forexample, if a given v(α) can be extended holomorphically outside the circle Cα in the neigh-bour sphere, then its effect can be undone by reparametrizing the corresponding coordinate.A similar discussion holds for the other spaces and relations can be found by restricting thevector on subspaces. A non-trivial vector (v(i), Ci) of Pg,n becomes trivial onMg,n if it canbe cancelled with a reparametrization of wi which leaves the origin fixed.

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7.3 Plumbing fixtureThe plumbing fixture is a way to glue together two Riemann surfaces (separating case) ortwo parts of the same surface (non-separating case) together, in order to build a surface witha higher number of holes and punctures. This geometric operation will correspond preciselyto the concept of gluing two Feynman graphs with a propagator in Siegel gauge.

The plumbing fixture depends on a (complex) one-parameter, which leads to a family ofsurfaces. This provides the correct number of moduli for the surface obtained after gluing.This brings to the question of describing the moduli spaces Mg,n in terms of the modulispaces with lower genus and number of punctures.

7.3.1 Separating caseConsider two Riemann surfaces Σg1,n1 and Σg2,n2 with local coordinates w(1)

1 , . . . , w(1)n1 and

w(2)1 , . . . , w(2)

n2 .The first step is to cut two disks D(1)

q and D(2)q of radius |q|1/2 around a puncture on

each surface, taken to be the n1-th and n2-th for definiteness:

D(1)q =

|w(1)n1| ≤ |q|1/2

, D(2)

q =|w(2)n2| ≤ |q|1/2

, (7.28)

where q ∈ C is fixed (Figure 7.4).2 Then, both surfaces can be glued (indicated by thebinary operation #) together into a new surface

Σg,n = Σg1,n1#Σg2,n2 ,

g = g1 + g2,

n = n1 + n2 − 2(7.29)

by removing the disks D(1)q and D(2)

q and by identifying the circles ∂D(1)q and ∂D(2)

q . At thelevel of the coordinates, this is achieved by the plumbing fixture operation:

w(1)n1w(2)n2

= q, |q| ≤ 1, (7.30)

The restriction on q arises because we have |w(1)n1 |, |w

(2)n2 | ≤ 1 (for a discussion, see [84]). This

case is called separating because cutting the new tube splits the surface in two components.Locally, the new surface looks like Figure 7.3. It is also convenient to parametrize q as

q = e−s+iθ, s ∈ R+, θ ∈ [0, 2π). (7.31)

The parameters s and θ are interpreted below as moduli of the Riemann surface.

Figure 7.3: Integration contour on the circle between the two local coordinates which areglued together.

The geometry of the new surface can be viewed in three different ways:2The disks D(i)

q should be equal or smaller than the disks D(1)n1 and D(2)

n2 .

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Figure 7.4: Disks around one puncture of the surfaces Σ1,1 and Σ0,3. The disks appear as acap because it is on top of the surface, which is curved.

1. both surfaces Σg1,n1 and Σg2,n2 (with the disks removed) are connected directly attheir boundaries (Figure 7.5a);

2. both surfaces Σg1,n1 and Σg2,n2 (with the disks removed) are connected by a cylinderof finite size (Figure 7.5b);

3. the surface Σg2,n2 is inserted inside the disk D(1)q , or conversely Σg1,n1 inside D(2)

q

(Figures 7.5c and 7.5d).

The first interpretation is the most direct one: the disks are simply removed and theboundaries are glued together by a small tube of radius |w(1)

1 | < |q|1/2. The connection

between both surfaces can be smoothed (and figures are often drawn in this way – forexample Figure 7.6), but this is not necessary (the smoothing is achieved by cutting disksof size |q|1/2 − ε and gluing the boundaries to the disks of radius |q|1/2).

In the second interpretation, one rescales the local coordinates in order to bring theradius of the disk to 1 instead of |q|1/2. In terms of these new coordinates, the surfaces areconnected by a tube of length s = − ln |q| after a Weyl transformation (see [218, sec. 9.3]for a longer discussion).

The last interpretation is obtained by performing a conformal mapping of the secondcase: the region |w(1)

n1 | < |q|1/2 is mapped to the region

|w(2)n2| = |q||w(1)n1 |

> |q|1/2, (7.32)

and conversely. The idea is that the disk D(1)q of Σg1,n1 is removed and replaced by the

complement of D(2)q in Σg2,n2 , i.e. the full surface Σg2,n2−D

(2)q is glued inside D(1)

q . While itis clear geometrically, this statement may look confusing from the coordinate point of viewbecause the local coordinates w(1)

n1 and w(2)n2 do not cover completely the Riemann surfaces,

but their relation still encode informations about the complete surface. The reason is thatone can always use transition functions to relate the coordinates on the two surfaces.

Example 7.1Denote by S(1)

a and S(2)b the spheres sharing a boundary with D(1)

n1 and D(2)n2 , and write

the corresponding coordinates by z(1)a and z(2)

b such that the transition functions are

z(1)a = f (1)

an1(w(1)

n1), z

(2)b = f

(2)bn2

(w(2)n2

). (7.33)

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(a) Direct gluing of circles.

(b) Connection by a long tube.

(c) Insertion of the second surface into the first one. (d) Insertion of the first surfaceinto the second one.

Figure 7.5: Different representations of the surface Σ1,2 obtained after gluing Σ1,1 and Σ0,3through the plumbing fixture.

Figure 7.6: Smoothed connection between both surfaces.

124

Then the coordinates za and zb are related by

z(1)a = f (1)

an1

(w(1)n1

)= f (1)

an1

(q

w(2)n2

)= f (1)

an1

(q

f(2)−1bn2

(z

(2)b

)) (7.34)

such that the new transition function reads

za = Fab(zb), Fab = f (1)an1 (q · I) f (2)−1

bn2, (7.35)

where I is the inversion (the superscript on the coordinates za and zb has been removedto indicate that they are now seen as coordinates on the same surface Σg,n).

The Riemann surface Σg,n is a point of Mg,n. By varying the moduli parameters ofΣg1,n1 and Σg2,n2 , one obtains other surfaces in Mg,n. But the number of parametersfurnished by Σg1,n1 and Σg2,n2 does not match the dimension (6.58) ofMg,n:

Mg1,n1 + Mg2,n2 = 6g1 − 6 + 2n1 + 6g2 − 6 + 2n2 = Mg,n − 2. (7.36)

This means that the subspace ofMg,n obtained by gluing all the possible surfaces inMg1,n1

and Mg2,n2 is of codimension 2. The missing complex parameter is q: in writing theplumbing fixture, it was taken to be fixed, but it can be varied to generate a 2-parameterfamily of Riemann surfaces inMg,n, with the moduli of the original surfaces held fixed.

The surface Σg,n is equipped with local coordinates inherited from the original surfacesΣg1,n1 and Σg2,n2 . Hence, the plumbing fixture of points in Pg1,n1 and Pg2,n2 automaticallyleads to a point of Pg,n. The fact that the local coordinates are inherited from lower-ordersurfaces is called gluing compatibility. It is also not necessary to add parameters to describethe fibre direction.

7.3.2 Non-separating caseIn the previous section, the plumbing fixture was used to glue punctures on two differentsurfaces. In fact, one can also glue two punctures on the same surface to get a new surfacewith an additional handle:

Σg,n = #Σg1,n1 ,

g = g1 + 1,n = n1 − 2,

(7.37)

defining # as a unary operator. This gluing is called non-separating because there is a singlesurface before the identification of the disks.

In terms of the local coordinates, the gluing relation reads

w(1)n1−1w

(1)n1

= q, (7.38)

where we consider the last two punctures for definiteness.The dimensions of both moduli spaces are related by

Mg1,n1 = Mg,n − 2. (7.39)

Again, the two missing parameters are provided by varying q and we obtained a Mg,n-dimensional subspace ofMg,n.

Example 7.2Here are some examples of surfaces obtained by gluing:

125

• Σ0,4 = Σ0,3#Σ0,3

• Σ0,5 = Σ0,3#Σ0,3#Σ0,3,Σ0,3#Σ0,4

• Σ1,1 = #Σ0,3

• Σ1,2 = #Σ0,4,Σ1,1#Σ0,3

Note that the moduli on the LHS and RHS are fixed (we will see later that not allsurfaces can be obtained by gluing).

7.3.3 Decomposition of moduli spaces and degeneration limitWe have seen that the separating and non-separating plumbing fixtures yield a family ofsurfaces in Mg,n described in terms of lower-dimensional moduli spaces. The question iswhether all points inMg,n can be obtained in this way by looking at all the possible gluing(varying g1, n1, g2 and n2). It turns out that this is not possible, which is at the core of thedifficulties to construct a string field theory.

Which surfaces are obtained from this construction? In order to interpret the regionsof Mg,n covered by the plumbing fixture, the parametrization (7.31) is the most useful.Previously, we explained that s gives the size of the tube connecting the two surfaces. Sincethe latter is like a sphere with two punctures, it corresponds to a cylinder (interpreted as anintermediate closed string propagating). The angle θ in (7.31) is the twist of the cylinderconnecting both components. This amounts to start with θ = 0, then to cut the cylinder,to twist it by an angle θ and to glue again.

The limit s → ∞ (|q| → 0) is called the degeneration limit: the degenerate surfaceΣg,n reduces to Σg1,n1 and Σg2,n2 connected by a very long tube attached to two punctures(separating case), or to Σg−1,n+2 with a very long handle (non-separating case). So it meansthat the family of surfaces described by the plumbing fixture are “close” to degeneration.Another characterization (for the separating case) is that the punctures on Σg1,n1 are closer(according to some distance, possibly after a conformal transformation) to each other thanto the punctures on Σg2,n2 .

Conversely, there are surfaces which cannot be described in this way: the plumbingfixture does not cover all the possible values of the moduli. For a given Mg,n, we denotethe surfaces which cannot be obtained by the plumbing fixture by Vg,n. This space doesnot contain any surface arbitrarily close to degeneration (i.e. with long handles or tubes).In terms of punctures, it also means that there is no conformal frame where the puncturessplit in two sets.

In the previous subsection, we considered two specific punctures, but any other puncturescould be chosen. Hence, there are many ways to split Σg,n in two surfaces Σg1,n1 and Σg2,n2

(with fixed g1, g2, n1 and n2): every partition of the punctures and holes in two sets lead todifferent degeneration limits (because they are associated to different moduli – Figure 7.7).Since each puncture is described by a modulus, choosing different punctures for gluing givedifferent set of moduli for Σg,n, such that each possibility covers a different subspace ofMg,n. The part of the moduli space Mg,n covered by the plumbing fixture of all surfacesΣg1,n1 and Σg2,n2 (with fixed g1, g2, n1, n2) is denoted byMg1,n1#Mg2,n2 :

Mg1,n1#Mg2,n2 ⊂Mg,n, (7.40)

where the operation # includes the plumbing fixture for all values of q and all pairs ofpunctures. Similarly, the part covered by the non-separating plumbing fixture is written as#Mg1,n1 :

#Mg1,n1 ⊂Mg,n. (7.41)

Importantly, the regions covered by the plumbing fixture depends on the choice of thelocal coordinates because (7.30) is written in terms of local coordinates. The subspacesMg1,n1#Mg2,n2 and #Mg1,n1 are not necessarily connected (in the topological sense).

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(a) Degeneration 12→ 34 (b) Degeneration 13→ 24

(c) Degeneration 14→ 23

Figure 7.7: Permutations of punctures while gluing two spheres: they correspond to different(disconnected) parts ofM0,4.

The moduli spaceMg,n cannot be completely covered by the plumbing fixture of lower-dimensional surfaces. We define the propagator and fundamental vertex regions Fg,n andVg,n as the subspaces which can and cannot be described by the plumbing fixture:

Fg,n := #Mg−1,n+2⋃ ( ⋃

n1+n2=n+2g1+g2=g

Mg1,n1#Mg2,n2

), (7.42a)

Vg,n :=Mg,n −Fg,n, (7.42b)

In the RHS, it is not necessary to consider multiple non-separating plumbing fixture for thefirst term because #Mg−2,n+4 ⊂ Mg−1,n+2, etc. For the same reason, it is sufficient toconsider a single separating plumbing fixture. Note that Vg,n and Fg,n are in general notconnected subspaces. A simple illustration is given in Figure 7.9. The actual decompositionofM0,4 is given in Figure 7.8. Importantly, Fg,n and Vg,n depend on the choice of the localcoordinates for all Vg′,n′ appearing in the RHS.

It is also useful to define the subspaces F1PRg,n and V1PIg,n of Mg,n which can and cannot

be described with the separating plumbing fixture only:

F1PRg,n :=

⋃n1+n2=n+2g1+g2=g

Mg1,n1#Mg2,n2 , (7.43a)

V1PIg,n :=Mg,n −F1PRg,n . (7.43b)

1PR (1PI) stands for 1-particle (ir)reducible, a terminology which will become clear later.Note the relation:

V1PIg,n = Vg,n⋃ (⋃

g′

#Mg−g′,n+g′

). (7.44)

The two plumbing fixtures behave as follow:

• separating: increases both n and g (if both surfaces have a non-vanishing g);

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Figure 7.8: In white are the subspaces of the moduli space M0,4 covered by the plumb-ing fixture. The three different regions correspond to the three different ways to pair thepunctures (see Figure 7.7). In grey is the fundamental vertex region V0,4.

Figure 7.9: Schematic illustration of the covering ofMg,n from the plumbing fixture of lower-dimensional spaces. The fundamental region Vg,n (usually disconnected) is not covered bythe plumbing fixture.

128

• non-separating plumbing: increases g but decreases n.

The construction is obviously recursive: starting from the lowest-dimensional moduli space,which isM0,3 (no moduli), one has:

V0,3 =M0,3, F0,3 = ∅. (7.45)

Next, the subspace ofM0,4 obtained from the plumbing fixture is:

F0,4 = V0,3#V0,3, (7.46)

and V0,4 is characterized as the remaining region. Then, one has:

F0,5 =M0,4#M0,3

= F0,4#V0,3 + V0,4#V0,3 = V0,3#V0,3#V0,3 + V0,4#V0,3,(7.47)

and V0,5 is what remains ofM0,5. The pattern continues for g = 0. The same story holdsfor g ≥ 1: the first such space is

F1,1 = #V0,3, (7.48)and V1,1 = M1,1 − F1,1. The gluing of a 3-punctured sphere and the addition of a handleare the two most elementary operations.

To keep track of which moduli spaces can contribute, it is useful to find a function ofΣg,n, called the index, which increases by 1 for each of the two elementary operations:

r(Σg1,n1#Σ0,3) = r(Σg1,n1) + 1, r(#Σg1,n1) = r(Σg1,n1) + 1. (7.49)

An appropriate function isr(Σg,n) = 3g + n− 2 ∈ N∗. (7.50)

which is normalized such that:r(Σ0,3) = 1. (7.51)

For a generic separating plumbing fixture, we find:

r(Σg1,n1#Σg2,n2) = r(Σg1,n1) + r(Σg2,n2). (7.52)

Since the index increases, surfaces with a given r can be obtained by considering all thegluings of surfaces with r′ < r.

7.3.4 StubsTo conclude this chapter, we introduce the concept of stubs. Previously in (7.31), the rangeof the parameter s was the complete line of positive numbers, s ∈ R+. This means thattube of all lengths were considered to glue surfaces. But, we could also introduce a minimallength s0 > 0, called the stub parameter, for the tube. In this case, the plumbing fixtureparameter is generalized to:

q = e−s+iθ, s ∈ [s0,∞), θ ∈ [0, 2π), s0 ≥ 0. (7.53)

What is the effect on the subspaces Fg,n(s0) and Vg,n(s0)? Obviously, less surfaces can bedescribed by the plumbing fixture if s0 > 0 than if s0 = 0, since the plumbing fixture cannotdescribe anymore surfaces which contain a tube of length less than s0. Equivalently, thevalues of the moduli described by the plumbing fixture is more restricted when s0 > 0. Moregenerally, one has:

s0 < s′0 : Fg,n(s′0) ⊂ Fg,n(s0) Vg,n(s0) ⊂ Vg,n(s′0). (7.54)

129

Figure 7.10: In light grey is the subspace covered by the V0,4(s0) as in Figure 7.8. In darkgrey is the difference δV0,4 = V0,4(s0 + δs0)− V0,4(s0) with δs0 > 0.

This is illustrated on Figure 7.10. Even if s0 is very large, Vg,n still does not include surfacesarbitrarily close to degeneracy. In general, we omit the dependence in s0 except when it isnecessary.

To interpret the stub parameter, consider two local coordinates w1 and w2 and rescalethem by λ ∈ C with Reλ > 0:

w1 = λ w1, w2 = λ w2. (7.55)

Then, the plumbing fixture (7.30) becomes

w1w2 = e−s+iθ. (7.56)

withs = s+ 2 ln |λ|, θ = θ + i ln λ

λ. (7.57)

If s ∈ R+, the corresponding range of s is

s ∈ [s0,∞), s0 := 2 ln |λ|. (7.58)

This shows that rescaling the local coordinates by a constant parameter is equivalent tochange the stub parameter.

Note also how performing a global phase rotation in (7.57) is equivalent to shift the twistparameter. Working in Pg,n forces to take λ ∈ R+.

7.4 SummaryIn this chapter, we have explained how to parametrize the fibre bundle Pg,n, that is, appro-priate coordinates for the moduli space and the local coordinate systems. This was realized

130

by introducing different coordinate patches and encoding all the informations of Pg,n inthe transition functions. Then, this description lead to a simple description of the tangentvectors through the Schiffer variation.

In the next chapter, we will continue the program by building the p-forms required todescribe off-shell amplitudes.

7.5 References• Plumbing fixture [218, sec. 9.3].

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Chapter 8

Off-shell amplitudes

While the previous chapter was purely geometrical, this one makes contact with stringtheory through the worldsheet CFT. We continue the description of Pg,n by constructingp-forms. The reason why we need to consider the CFT is that ghosts are necessary to buildthe p-forms: this can be understood from Chapter 3, where we found that the ghosts mustbe interpreted as part of the measure on the moduli space. Then, we build the off-shellamplitudes and discuss some properties.

8.1 Cotangent spaces and amplitudesIn this section, we construct the p-forms on Pg,n which are needed for the amplitudes. Wefirst motivate the expressions from general ideas, and check later that they have the correctproperties.

8.1.1 Construction of formsA p-form ω

(g,n)p ∈

∧pT ∗Pg,n is a multilinear antisymmetric map from

∧pTPg,n to a function

of the moduli parameters. The superscript on the form is omitted when there is no ambiguityabout the space considered. The components ωi1···ip of the p-form are defined by insertingp basis vectors ∂s1 , . . . , ∂sp

ωi1···ip := ωp(∂s1 , . . . , ∂sp), (8.1)

where ∂s = ∂∂xs

and xs are the coordinates (7.13). It is antisymmetric in any pair of twoindices

ωi1i2···ip = −ωi2i1···ip , (8.2)and multilinearity implies that

ωp(V (1), . . . , V (p)) = ωp

(V (1)s1 ∂s1 , . . . , V

(p)sp ∂sp

)= ωi1···ipV

(1)s1 · · ·V

(p)sp , (8.3)

given vectors V (α) = V(α)s ∂s.

The p-forms which are needed to define off-shell amplitudes depend on the external statesVi (i = 1, . . . , n) inserted at the punctures zi. They are maps from

∧pTPg,n × Hn to a

function on Pg,n. The dependence on the states is denoted equivalently as

ωp(V1, . . . ,Vn) := ωp(⊗iVi). (8.4)

The simplest way to get a function on Pg,n from the states Vi is to compute a CFT correlationfunction of the operators inserted at the points zi = fi(0) on the surface Σg,n described bythe point inMg,n.

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The 0-form is just a function and is defined by:

ω0 = (2πi)−Mcg,n

⟨n∏i=1

fi Vi(0)⟩

Σg,n

. (8.5)

For simplicity, the dependence in the local coordinates fi is kept implicit in the rest of thechapter.

A natural approach for constructing p-form is to build them from elementary 1-formsand to use ghosts to enforce the antisymmetry. Remembering the Beltrami differentialsfound in Chapter 3, the contour integral of ghosts b(z) weighted by some vector field is agood starting point. In the current language, it is defined by its contraction with a vectorV = (v, C) ∈ TPg,n defined in (7.23):

B(V ) :=∮C

dz2πi b(z)v(z) +

∮C

dz2πi b(z)v(z), (8.6)

where b(z) and b(z) are the b-ghost components, and v is the vector field on Σg,n definingV . The contours run anti-clockwise. If the contour C include several circles (C = ∪αCα),B(V ) is defined as the sum of the contour integral on each circle:

B(V ) :=∑α

∮Cα

dz2πi b(z)v(z) + c.c. (8.7)

It is also useful to define another object built from the energy–momentum tensor:

T (V ) :=∮C

dz2πi T (z)v(z) +

∮C

dz2πi T (z)v(z), (8.8)

where T and T are the components of the energy–momentum tensor. It is defined such that

T (V ) = QB , B(V ). (8.9)

Considering the coordinate system (7.13), the Beltrami form can be decomposed as:

B = Bsdxs, Bs := B(∂s), (8.10a)

Bs =∑α

∮Cα

dσα2πi b(σα) ∂Fα

∂xs

(F−1α (σα)

)+∑α

∮Cα


∂xs

(F−1α (σα)

), (8.10b)

where the contour orientations are defined by having the σα coordinate system on the left.We define the p-form contracted with a set of vectors V (1), . . . , V (p) by

ωp(V (1), . . . , V (p))(V1, . . . ,Vn) := (2πi)−Mc

g,n

⟨B(V (1)) · · ·B(V (p))

n∏i=1

Vi

⟩Σg,n

, (8.11)

and the corresponding p-form reads

ωp = ωp,s1···sp dxs1 ∧ · · · ∧ dxsp (8.12a)

= (2πi)−Mcg,n

⟨Bs1dxs1 ∧ · · · ∧Bspdxsp

n∏i=1

Vi

⟩Σg,n

. (8.12b)

In this expression, the form contains an infinite numbers of components ωp,s1···sp since thereis an infinite number of coordinates. Note that the normalization is independent of p.

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In practice, one is not interested in Pg,n, but rather in a subspace of it. Given a q-dimensional subspace S of Pg,n parametrized by q real coordinates t1, . . . , tq

xs = xs(t1, . . . , tq), (8.13)

the restriction of a p-form to this subspace is obtained by the chain rule:

∀p ≤ q : ωp|S = (2πi)−Mcg,n

⟨Br1

∂xs1∂tr1

dtr1 ∧ · · · ∧Brp∂xsp∂trp

dtrpn∏i=1

Vi

⟩Σg,n

,

∀p > q : ωp|S = 0.

(8.14)

We will often write the expression directly in terms of the coordinates of S and abbreviatethe notation as:

Br := ∂xs∂tr

Bs. (8.15)

8.1.2 Amplitudes and surface statesIt is now possible to write the amplitude more explicitly. An on-shell amplitude is definedas an integral over Mg,n. Off-shell, one needs to consider local coordinates around eachpuncture, that is, a point of the fibre for each point of the baseMg,n. This defines a Mg,n-dimensional section Sg,n of Pg,n (Figure 6.2). The g-loop n-point off-shell amplitude of thestates V1, . . . ,Vn reads:

Ag,n(V1, . . . ,Vn)Sg,n :=∫Sg,n

ωg,nMg,n(V1, . . . ,Vn)

∣∣Sg,n

, (8.16a)

ωg,nMg,n(V1, . . . ,Vn)

∣∣Sg,n

= (2πi)−Mcg,n

⟨Mg,n∧λ=1

Bs∂xs∂tλ

dtλn∏i=1

fi Vi(0)⟩

Σg,n

, (8.16b)

where the choice of the fi is dictated by the section Sg,n. From now on, we stop to write therestriction of the form to the section. We also restrict to the cases where χg,n = 2−2g−n < 0.

The complete (perturbative) n-point amplitude is the sum of contributions from all loops:

An(V1, . . . ,Vn) :=∑g≥0

Ag,n(V1, . . . ,Vn). (8.17)

More generally, we define the integral over a section Rg,n which projection on the baseis a subspace ofMg,n (and not the full space as for the amplitude) as:

Rg,n(V1, . . . ,Vn) :=∫Rg,n

ωg,nMg,n(V1, . . . ,Vn), (8.18)

For simplicity, we will sometimes use the same notation for the section of Pg,n and itsprojection on the baseMg,n. For this reason, the reader should assume that some choice oflocal coordinates around the punctures is made except otherwise stated.

Given sections Rg,n, the sum over all genus contribution is written formally as

Rn :=∑g≥0Rg,n, (8.19)

such that

Rn(V1, . . . ,Vn) :=∑g≥0Rg,n(V1, . . . ,Vn) =

∑g≥0

∫Rg,n

ωg,nMg,n(V1, . . . ,Vn). (8.20)

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A surface state is defined as a n-fold bra which reproduces the expression of a givenfunction when contracted with n states Ai. The surface 〈Σg,n|, form 〈ωg,n|, section 〈Rg,n|and amplitude 〈Ag,n| n-fold states are defined by the following expressions:

〈Σg,n|Bs1 · · ·Bsp | ⊗i Vi〉 := ωs1···sp(V1, . . . ,Vn), (8.21a)〈ωg,np | ⊗i Vi〉 := ωp(V1, . . . ,Vn), (8.21b)〈Ag,n| ⊗i Vi〉 := Ag,n(V1, . . . ,Vn). (8.21c)

The last relation is generalized to any section Rg,n:

〈Rg,n| ⊗i Vi〉 := Rg,n(V1, . . . ,Vn). (8.21d)

The reason for introducing these objects is that the form (8.12) is a linear map from H⊗nto a form on Mg,n – see (8.4). Thus, there is always a state 〈Σg,n| such that its BPZproduct with the states reproduces the form. In particular, the state 〈Σg,n| contains all theinformation about the local coordinates and the moduli (the dependence is kept implicit).The definition of the other states follow similarly. These states are defined as bras, but theycan be mapped to kets.

One finds the obvious relations:

〈ωg,np | = 〈Σg,n|Bs1dxs1 · · ·Bspdxsp , 〈Ag,n| =∫Mg,n

〈ωg,np | . (8.22)

The surface states don’t contain information about the matter CFT: they collect theuniversal data (like local coordinates) needed to describe amplitudes. Hence, it is an im-portant step in the description of off-shell string theory to characterize this data. However,note that the relation between a surface state and the corresponding form does depend onthe CFT.

Example 8.1 – On-shell amplitude A0,4The transition functions are given by (see Figure 7.1):

C1 : w1 = z1 − y1, C3 : w3 = z2 − y3, C5 : z1 = z2,

C2 : w2 = z1 − y2, C4 : w4 = z2 − y4.(8.23)

Three of the parameters (y1, y2 and y3) are fixed while the single complex modulusof M0,4 is taken to be y4. Since we are interested in the on-shell amplitude, it is notnecessary to introduce local coordinates and the associated parameters.

A variation of the modulus

y4 −→ y4 + δy4, y4 −→ y4 + δy4 (8.24)

is equivalent to a change in the transition function of C4. This translates in turn intoa transformation of z2:

z′2 = z2 + δy4, z′2 = z2 + δy4. (8.25)

Then, the tangent vector V = ∂y4 is associated to the vector field

v = 1, v = 0, (8.26)

with support on C4. For V = ∂y4 , one finds

v = 0, v = 1. (8.27)

135

The Beltrami 1-form for the unit vectors are

B(∂y4) =∮C4

dz2 b(z2)(+1), B(∂y4) =∮C4

dz2 b(z2)(+1), (8.28)

with both contours running anti-clockwise.The components of the 2-form reads

ω2(∂y4 , ∂y4) = 12πi

⟨B(∂y4)B(∂y4)

4∏i=1

Vi

⟩Σ0,4

= 12πi

⟨∮C4

dz2 b(z2)∮C4

dz2 b(z2)4∏i=1

Vi

⟩Σ0,4

.

For on-shell states Vi = ccVi(yi, yi), this becomes

ω2(∂y4 , ∂y4) = 12πi

⟨ 3∏i=1

ccVi(yi, yi)∮C4

dz2 b(z2)∮C4

dz2 b(z2)c(y4)c(y4)V4(y4, y4)⟩

Σ0,4

.

The first three operators could be moved to the left because they are not encircled bythe integration contour. Note the difference with the example discussed in Section 6.1.2:here, the contour encircles z3, while it was encircling y3 for the s-channel.

Using the OPE ∮C4

dz2 b(z2)c(y4) ∼∮C4

dz21

z2 − y4(8.29)

to simplify the product of b and c gives the amplitude

A0,4 = 12πi

∫dy4 ∧ dy4

⟨ 3∏i=1

ccVi(yi)V4(y4)⟩

Σ0,4

. (8.30)

This is the standard formula for the 4-point function derived from the Polyakov pathintegral.

8.2 Properties of formsIn this section, we check that the form (8.12) has the correct properties:

• antisymmetry under exchange of two vectors;

• given a trivial vector of (a subspace of) Pg,n (Section 7.1), its contraction with theform vanishes: ωp(V (1), . . . , V (p)) = 0 if any of the V (i) generates:

– reparametrizations of za for V (i) ∈ TPg,n,– rotation wi → (1 + iαi)wi for V (i) ∈ T Pg,n,– reparametrizations of wi keeping wi = 0 if the states are on-shell for V (i) ∈TMg,n;

• BRST identity, which is necessary to prove several properties of the amplitudes.

The first property is obvious. Indeed, the form is correctly antisymmetric under theexchange of two vectors V (i) and V (j) due to the ghost insertions.

136

8.2.1 Vanishing of forms with trivial vectorsReparametrization of za Consider the sphere Sa with coordinate za, and denote by C1,C2 and C3 the three boundaries. Then, a reparametrization

za −→ za + φ(za) (8.31)

is generated by a vector field φ(z) which is regular on Sa. This transformation modifiesthe transition functions on the three circles and is thus associated to a tangent vector Vdescribed by a vector field v with support on the three circles:

Ci : v(i) = φ|Ci . (8.32)

The Beltrami form then reads

B(V ) =3∑i=1

∮Ci

dza b(za)φ(za) + c.c. (8.33)

where the orientations of the contours are such that Sa is on the left. Since the vectorfield φ is regular in Sa, two of the contours can be deformed until they merge together. Theresulting orientation is opposite to the one of the last contour (Figure 8.1). As a consequence,both cancel and the integral vanishes.

Figure 8.1: Deformation of the contour of integration defining the Beltrami form for areparametrization of za. The figure is drawn for two circles at a hole, but the proof isidentical for other types of circles.

Rotation of wi Consider an infinitesimal phase rotation of the local coordinate wi in thedisk Di:

wi −→ (1 + iαi)wi, wi −→ (1− iαi)wi, (8.34)with αi ∈ R. The tangent vector is defined by the circle Ci and the vector field by

v = iwi, v = −iwi. (8.35)

The Beltrami form for this vector is

B(V ) = i∮Ci

dwi wi b(wi)− i∮Ci

dwi wi b(wi), (8.36)

where Di is kept to the left.In the p-form (8.11), the ith operator Vi is inserted in Di and encircled by Ci. Because

there is no other operator inside Di, the contribution of this disk to the form is

B(V )Vi(0) = i∮Ci

dwi wi b(wi)Vi(0)− i∮Ci

dwi wi b(wi)Vi(0), (8.37)

137

The state–operator correspondence allows to rewrite this result as

i(b0 − b0) |Vi〉 , (8.38)

since the contour integral picks the zero-modes of b and of b. Requiring that the formvanishes implies the ghost counter-part of the level-matching condition:

b−0 |Vi〉 = 0. (8.39)

Hence, consistency of off-shell amplitudes imply that

Vi ∈ H−, (8.40)

where H− is defined in (6.43).

Reparametrization of wi A reparametrization of the local coordinate wi keeping theorigin of Di fixed reads:

wi −→ f(wi), f(0) = 0. (8.41)

The function can be expanded in series:

f(wi) =∑m≥0

pmwm+1i . (8.42)

Because the transformation is holomorphic, it can be extended on Ci. Each parameter pmprovides a coordinate of Pg,n and whose deformation corresponds to a vector field:

vm = wm+1i , vm = 0. (8.43)

The corresponding Beltrami differential is

B(∂pm) =∮Ci

dwi b(wi)wm+1i . (8.44)

Since only the operator Vi is inserted in the disk, the state–operator correspondence givesbm |Vi〉. Requiring that the form vanishes onMg,n for allm and also for the anti-holomorphicvectors gives the conditions:

∀m ≥ 0 : bm |Vi〉 = 0, bm |Vi〉 = 0. (8.45)

This holds automatically for on-shell states Vi = ccVi.

8.2.2 BRST identityThe BRST identity for the p-form (8.12) reads

ωp

(∑i

Q(i)B ⊗i Vi

)= (−1)pdωp−1(⊗Vi), (8.46)

using the notation (8.4). The BRST operator acting on the ith Hilbert space is written as

Q(i)B = 1i−1 ⊗QB ⊗ 1n−i (8.47)

and acts asQBVi(z, z) = 1

2πi

∮dw jB(w)Vi(z, z) + c.c. (8.48)

138

More explicitly, the LHS corresponds to

ωp

(∑i

Q(i)B ⊗i Vi

)= ωp(QBV1,V2, . . . ,Vn) + (−1)|V1|ωp(V1, QBV2, . . . ,Vn)

+ · · ·+ (−1)|V1|+···+|Vn−1|ωp(V1,V2, . . . , QBVn).(8.49)

We give just an hint of this identity, the complete proof can be found in [297, pp. 85–89,245, sec. 2.5].

The contour of the BRST current around each puncture can be deformed, picking singu-larities due to the presence of the Beltrami forms. Using (8.9), we find that anti-commutingthe BRST charge with the Beltrami form Bs leads to an insertion of

Ts = QB , Bs. (8.50)

The energy–momentum tensor generates changes of coordinates. Hence, Ts = T∂s is preciselythe generator associated to an infinitesimal change of the coordinate xs on Pg,n. The latteris given by the vector ∂s. For this reason, one can write:

dxs QB , Bs = dxs Ts = dxs ∂s = d, (8.51)

where d is the exterior derivative on Pg,n. The minus signs arise if the states Vi are Grass-mann odd.

8.3 Properties of amplitudesIn order for the p-form (8.12) to be non-vanishing, its total ghost number should match theghost number anomaly:

Ngh(ωp(V1, . . . ,Vn)

)=

n∑i=1

Ngh(Vi)− p = 6− 6g, (8.52)

using Ngh(B) = −1. For an amplitude, one has p = Mg,n = 6g − 6 + 2n and thus:

Ngh(ωMg,n) = 6− 6g =⇒

n∑i=1

Ngh(Vi) = 2n. (8.53)

This condition holds automatically for on-shell states since Ngh(ccVi) = 2.

8.3.1 Restriction to Pg,nThe goal of this section is to explain why amplitudes must be described in terms of a sectionof Pg,n (7.10) instead of Pg,n. This means that one should identify local coordinates differingby a global phase rotation.

The off-shell amplitudes (8.16) are multi-valued on Pg,n. Indeed, the amplitude dependson the local coordinates1 and changes by a factor under a global phase rotation of any localcoordinate wi → eiαwi. However, such global rotation leaves the surface unchanged, sincethe flat metric |dwi|2 is invariant. This means that the same surface leads to different valuesfor the amplitude. To prevent this multi-valuedness of the amplitudes, it is necessary toidentify local coordinates differing by a constant phase.

A second way to obtain this condition is to require that the section Sg,n is globallydefined: every point of the section should correspond to a single point of the moduli space

1The current argument does not apply for on-shell amplitudes.

139

Mg,n. However, there is a topological obstruction which prevents finding a global section inPg,n in general. One hint [68, sec. 2, 84, sec. 3] is to exhibit a nowhere vanishing 1-form ifSg,n is globally defined: this leads to a contradiction since such a 1-form does not generallyexist (see for example [71, 6.3.2, ch. 7]). Then, consider a closed curve in the moduli space(such curves exist since Mg,n is compact). Starting at a given point Σ of the curve, onefinds that the local coordinates typically change by a global phase when coming back to thepoint Σ (Figure 8.2), since this describes the same surface and there is no reason to expectthe phase to be invariant. Up to this identification, it is possible to find a global section.The latter corresponds to a section of Pg,n.

(a) Closed curve inMg,n. (b) Change in the phase of wi.

Figure 8.2: Schematic plot of the change in the phase of the local coordinate wi as onefollows a closed curve in Mg,n. If the original phase at Σ is α0 and if the phase variescontinuously along the path, then α1 6= α0 when returning back to Σ by continuity.

Remark 8.1 (Degeneracy of the antibracket) It is possible to define a BV structure onRiemann surfaces [263, 264]. The antibracket is degenerate in Pg,n but not in Pg,n [264].

Global phase rotations of the local coordinates are generated by L−0 . Hence, identifyingthe local coordinates wi → eiαiwi amounts to require that the amplitude is invariant underL−0 . This is equivalent to imposing the level-matching condition

L−0 |Vi〉 = 0 (8.54)

on the off-shell states. This condition was interpreted in Section 4.2.2 as a gauge-fixingcondition for translations along the S1 of the string. This shows, in agreement with earliercomments, that the level-matching condition should also be imposed off-shell because nogauge symmetry is introduced for the corresponding transformation.

If the generator L−0 is trivial, this means that the ghost associated to the correspondingtangent vector must be decoupled. According to Section 8.2.1, this corresponds to theconstraint:

b−0 |Vi〉 = 0. (8.55)This can be interpreted as a gauge fixing condition (Section 13.5), which could in principlebe relaxed. However, the decoupling of physical states (equivalent to gauge invariance inSFT) happens only after integrating over the moduli space. This requires having a globallydefined section.

As a consequence, off-shell states are elements of the semi-relative Hilbert space

Vi ∈ H− ∩ kerL−0 , (8.56)

and the amplitudes are defined by integrating the form ωMg,n over a section Sg,n ⊂ Pg,n.

140

Computation – Equation (8.54)The operator associated to the state through |Ai〉 = Ai(0) |0〉 transforms as

Vi(0) −→ (eiαi)h(e−iαi)hVi(0) (8.57)

which translates into|Vi〉 −→ eiαi(L0−L0) |Vi〉 (8.58)

for the state, using the fact that the vacuum is invariant under L0 and L0. Then,requiring the invariance of the state leads to (8.54).

8.3.2 Consequences of the BRST identityTwo important properties of the on-shell amplitudes can be deduced from the BRST identity(8.46): the independence of physical results on the choice of local coordinates and thedecoupling of pure gauge states.

Given BRST closed states, the LHS of (8.46) vanishes identically

∀i : QB |Vi〉 = 0 =⇒ dωp−1(V1, . . . ,Vn) = 0. (8.59)

Using this result, one can compare the on-shell amplitudes computed for two different sec-tions S and S ′: ∫

SωMg,n

−∫S′ωMg,n

=∫∂T

ωMg,n−1 =∫T

dωMg,n−1 = 0, (8.60)

using Stokes’ theorem and where T is the surface delimited by the two sections (Figure 8.3).This implies that on-shell amplitudes do not depend on the section, and thus on the localcoordinates. In obtaining the result, one needs to assume that the vertical segments donot contribute. The latter correspond to boundary contributions of the moduli space. Ingeneral, many statements hold up to this condition, which we will not comment more in thisbook.

Figure 8.3: Two sections S and S ′ of Pg,n delimiting a surface T .

Next, we consider a pure gauge state together with BRST closed states:

|V1〉 = QB |Λ〉 , QB |Vi〉 = 0. (8.61)

The BRST identity (8.46) reads:

ωMg,n(QBΛ,V2, . . . ,Vn) = dωMg,n−1(Λ,V2, . . . ,Vn), (8.62)

141

which gives the amplitude∫SωMg,n

(QBΛ,V2, . . . ,Vn) =∫S

dωMg,n−1(Λ,V2, . . . ,Vn) =∫∂SωMg,n−1(Λ,V2, . . . ,Vn)

(8.63)where the last equality follows from Stokes’ theorem. Assuming again that there is noboundary contribution, this vanishes:∫

SωMg,n

(QBΛ,V2, . . . ,Vn) = 0. (8.64)

This implies that pure gauge states decouple from the physical states.

8.4 References• Definition of the forms [84, 85, 245, 297].

• Global phase rotation of local coordinates [68, sec. 2, 84, sec. 3, 196, 297, p. 54].

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Chapter 9

Amplitude factorization andFeynman diagrams

In the previous chapter, we built the off-shell amplitudes by integrating forms on sectionsof Pg,n. Studying their factorizations lead to rewrite them in terms of Feynman diagrams,which allows to identify the fundamental interactions vertices. We will then be able to writethe SFT action in the next chapter.

9.1 Amplitude factorizationWe have seen how to write off-shell amplitudes. The next step is to rewrite them as a sumof Feynman diagrams through factorization of amplitudes.

Factorization consists in writing a g-loop n-point amplitude in terms of lower-orderamplitudes in both g and n connected by propagators. Since an amplitude correspondsto a sum over all possible processes, which corresponds to integrating over the moduli space,it is natural to associate Feynman diagrams to different subspaces of the moduli space.One can expect that the plumbing fixture (Section 7.3) is the appropriate translation of thefactorization at the level of Riemann surfaces. We will assume that it is the case and checkthat it is correct a posteriori.

To proceed, we consider the contribution to the amplitude Ag,n of the family of surfacesobtained by the plumbing fixture of two surfaces (separating case) or a surface with itself(non-separating case).

9.1.1 Separating caseIn this section, we consider the separating plumbing fixture where part of the moduli spaceMg,n is covered byMg1,n1#Mg2,n2 with g = g1 + g2 and n = n1 + n2 − 2 (Section 7.3.1).The local coordinates read w(1)

i and w(2)j for i = 1, . . . , n1 and j = 1, . . . , n2. By convention,

the last coordinate of each set is used for the plumbing fixture:

w(1)n1w(2)n2

= q. (9.1)

The g-loop n-point amplitude with external states V (1)1 , . . . ,V

(1)n1−1,V

(2)1 , . . . ,V

(2)n2−1 is

denoted as:Ag,n =

∫Sg,n

ωg,nMg,n

(V

(1)1 , . . . ,V

(1)n1−1,V

(2)1 , . . . ,V

(2)n2−1

). (9.2)

143

We need to study the form ωg,nMg,nonMg1,n1#Mg2,n2 , which means to rewrite it in terms

of the data fromMg1,n1 and fromMg2,n2 . This corresponds to the degeneration limit wherethe two groups of punctures denoted by V

(1)i and V

(2)j (i = 1, . . . , n1 − 1, j = 1, . . . , n2 − 1)

together with g1 and g2 holes move apart from each other (Figure 9.1).

Figure 9.1: Degeneration limit of Σg,n where the punctures V(1)i and V

(2)j move apart from

each other.

Since q is a coordinate of Pg,n, its variation is associated with a tangent vector and aBeltrami 1-form. The latter has to be inserted inside ωg,nMg,n

. A change q → q+ δq translatesinto a change of coordinate

w′(1)n1

= w(1)n1

+ w(1)n1

qδq, (9.3)

where w(2)n2 is kept fixed (obviously, this choice is conventional as explained in Section 7.2).

Thus, the vector field and the Beltrami form are

vq = w(1)n1

q, Bq = 1

q

∮Cq

dw(1)n1b(w(1)n1

)w(1)n1. (9.4)

Computation – Equation (9.3)Starting from (9.1), vary q → q + δq while keeping w(2)

n2 fixed:

w′(1)n1

w(2)n2

= q + δq

w′(1)n1

= w(1)n1

q(q + δq) = w(1)

n1+ q

w(1)n1

δq.

The second line follows by replacing w(2)n2 using (9.1).

The Mg,n-form for the moduli described by the plumbing fixture can be expressed as:

ωMg,n

(V

(1)1 , . . . , V

(1)Mg1,n1

, ∂q, ∂q, V(2)1 , . . . , V

(2)Mg2,n2

)(9.5)

= (2πi)−Mcg,n

⟨Mg1,n1∏λ=1

B(V

(1)λ

)B(∂q)B(∂q)

Mg2,n2∏κ=1

B(V (2)κ

)n1−1∏i=1

V(1)i

n2−1∏j=1

V(2)j

⟩Σg,n

.

We introduce the surface states Σn1 and Σn2 such that the BPZ inner product with the

144

new states V(1)n1 and V

(2)n2 reproduce the Mg1,n1 - and Mg2,n2-forms:

〈Σn1 |V (1)n1〉 := ωMg1,n1

(V (1)1 , . . . ,V (1)

n1) = (2πi)−Mc

g1,n1

⟨Mg1,n1∏λ=1

B(V

(1)λ

) n1−1∏i=1

V(1)i

⟩Σg1,n1

, (9.6a)

〈Σn2 |V (2)n2〉 := ωMg2,n2

(V (2)1 , . . . ,V (2)

n2) = (2πi)−Mc

g2,n2

⟨Mg1,n1∏λ=1

B(V

(2)λ

) n2−1∏j=1

V(2)j

⟩Σg2,n2

. (9.6b)

As described in Section 8.1.2, these states exist since the p-form are linear in each of theexternal state and the BPZ inner-product is non-degenerate. Each of the surface statescorresponds to an operator

〈Σ1| = 〈0| I Σn1(0), 〈Σ2| = 〈0| I Σn2(0), (9.7)

defined from (22.135). Then, the forms can be interpreted as 2-point functions on thecomplex plane:

〈Σn1 |V (1)n1〉 = 〈I Σn1(0)Vn1(0)〉

w(1)n1, 〈Σn2 |V (2)

n2〉 = 〈I Σn2(0)Vn2(0)〉

w(2)n2. (9.8)

All the complexity of the amplitudes has been lumped into the definitions of the surfacestates which contain information about the surface moduli (including the ghost insertions)and about the n1 − 1 remaining states (including the local coordinate systems). The localcoordinates around V

(1)n1 and V

(2)n2 are denoted respectively as w(1)

n1 and w(2)n2 . Correspond-

ingly, the surface operators are inserted in the local coordinates w1 and w2 which are relatedto w(1)

n1 and w(2)n2 through the inversion:

w1 = I(w(1)n1

), w2 = I

(w(2)n2

). (9.9)

In order to rewrite (9.5) in terms of Σ1 and Σ2, it is first necessary to express all operatorsin one coordinate system, for example w(1)

n1 . Hence, we need to find its relation to w2. Usingthe plumbing fixture (9.1), the relation between w(1)

n1 and w2 is:

w(1)n1

= q

w(2)n2

= q

I(w2) = qw2 := f(w2). (9.10)

Then, the form (9.5) becomes

ωMg,n = 12πi 〈I Σn1(0)BqBq f Σn2(0)〉

w(1)n1

= 12πi 〈Σn1 |BqBq qL0 qL0 |Σ2〉 , (9.11)

using that Σ2 has a well-defined scaling dimension. The factor of 2πi arises by comparingthe contribution from Σn1 and Σn2 with the factor in (9.5). The expression can be simplifiedby using the relation

〈Σn1 |BqBq |V (1)n1〉 = 1

qq〈Σn1 | b0b0 |V (1)

n1〉 (9.12)

using the expression (9.4) for Bq and the state–operator correspondence:

BqV(1)n1

(z, z) = 1q

∮Cq

dw(1)n1b(w(1)n1

)w(1)n1

V (1)n1

(z, z) −→ 1qqb0b0 |V (1)

n1〉 . (9.13)

Ultimately, the form (9.5) reads

ωMg,n = 12πi

1qq〈Σn1 | b0b0 qL0 qL0 |Σn2〉 . (9.14)

145

It is important to remember that the plumbing fixture describes only a patch of themoduli space, and the form defined in this way is valid only locally. As a consequence, theintegration over all moduli of Mg1,n1#Mg2,n2 does not describe Mg,n, but only a part ofit (Section 7.3.3). Every degeneration limit with a different puncture distribution in twodifferent groups contributes to a different part of the amplitude.

We denote the contribution to the total amplitude (9.2) from the region of the modulispace connected to this degeneration limit as:

Fg,n(V

(1)i |V

(2)j

):= 1

2πi

∫ Mg1,n1∧λ=1

dt(1)λ

Mg2,n2∧κ=1

dt(2)κ ∧

dqq∧ dqq〈Σn1 | b0b0 qL0 qL0 |Σn2〉 . (9.15)

To proceed, we introduce a basis φα(k) of eigenstates of L0 and L0, where kµ is the D-dimensional momentum and α denotes the remaining quantum number. Then, introducingtwice the resolution of the identity (6.36) gives:

Fg,n(V

(1)i |V

(2)j

)= 1

2πi

∫ dDk(2π)D

dDk′(2π)D (−1)|φα|

×∫ dq

q∧ dqq〈φα(k)c| b0b0 qL0 qL0 |φβ(k′)c〉 (9.16)

×∫ Mg1,n1∧

λ=1dt(1)λ 〈Σn1 |φα(k)〉

∫ Mg2,n2∧κ=1

dt(2)κ 〈φβ(k′)|Σn2〉

(with implicit sums over α and β). In the last line, one recognizes the expressions of theg1-loop n1-point amplitude with external states V (1)

1 , . . . ,V(1)n1−1, φα and of the g2-loop

and n2-point amplitudes with external states V (2)1 , . . . ,V

(2)n2−1, φβ:

Ag1,n1

(V

(1)1 , . . . ,V

(1)n1−1, φα(k)

)=∫Sg1,n1

ωMg1,n1

(V

(1)1 , . . . ,V

(1)n1−1, φα(k)

)=∫Sg1,n1

Mg1,n1∧λ=1

dt(1)λ 〈Σn1 |φα(k)〉 ,

(9.17a)

Ag2,n2

(V

(2)1 , . . . ,V

(2)n2−1, φβ(k′)

)=∫Sg2,n2

ωMg2,n2

(V

(2)1 , . . . ,V

(2)n2−2, φβ(k′)

)=∫Sg2,n2

Mg2,n2∧λ=1

dt(2)λ 〈Σn2 |φβ(k′)〉 .

(9.17b)

The property (B.30) has been used to reverse the order of the BPZ product for the secondRiemann surface, and this cancels the factor (−1)|φα|.

Defining the second line of (9.16) as

∆αβ(k, k′) := ∆(φα(k)c, φβ(k′)c

):= 1

2πi

∫ dqq∧ dqq〈φα(k)c| b0b0 qL0 qL0 |φβ(k′)c〉 , (9.18)

one has:

Fg,n(V

(1)i |V

(2)j

)=∫ dDk

(2π)DdDk′(2π)D Ag1,n1

(V

(1)1 , . . . ,V

(1)n1−1, φα(k)

)∆αβ(k, k′)

×Ag2,n2

(V

(2)1 , . . . ,V

(2)n2−1, φβ(k′)

).

(9.19)

146

We recover the expressions from Section 6.1.2, but for a more general amplitude. We hadfound that ∆ corresponds to the propagator: its properties are studied further in Sec-tion 9.2.2. Hence, the object (9.16) corresponds to the product of two amplitudes connectedby a propagator (Figure 9.2).

There are several points to mention about this amplitude:

• We will find that the propagator depends only on one momentum because 〈k|k′〉 ∼δ(D)(k + k′), which removes one of the integral. Then, both amplitudes Ag1,n1 andAg2,n2 contain a delta function for the momenta:

Ag1,n1 ∼ δ(D)(k(1)1 +· · ·+k(1)

n1−1+k), Ag2,n2 ∼ δ(D)(k(2)

1 +· · ·+k(2)n2−1+k′

). (9.20)

As a consequence, the second momentum integral can be performed and yields a deltafunction:

Fg,n ∼ δ(D)(k(1)1 + · · ·+ k

(1)n1−1 + k

(2)1 + · · ·+ k

(2)n2−1

). (9.21)

Hence, the momentum flowing in the internal line is fixed and this ensures the overallmomentum conservation as expected.

• The ghost numbers of the states φα and φβ are also fixed (in terms of the externalstates). Indeed, because of the ghost number anomaly, the amplitudes onMg1,n1 andMg2,n2 are non-vanishing only if the ghost numbers of these states satisfy:

Ngh(φα) = 2n1 −n1−1∑i=1

Ngh(V

(1)i

), Ngh(φβ) = 2n2 −

n2−1∑j=1

Ngh(V

(2)j

). (9.22)

The non-vanishing of Fg,n also gives another relation:

Ngh(φα) +Ngh(φβ) = 4. (9.23)

In particular, if the external states are on-shell with Ngh = 2, we find:

Ngh(φα) = Ngh(φβ) = 2. (9.24)

As indicated in Chapter 13, such states are appropriate at the classical level since theydo not contain spacetime ghosts.

• The sum over α and β is over an infinite number of states and could diverge. In fact,the sum can be made convergent by tuning the stub parameter (Section 9.2.4).

Properties of Feynman graphs and amplitudes in the momentum space will be discussedfurther in Chapter 19.

9.1.2 Non-separating caseNext, we consider the non-separating plumbing fixture (Section 7.3.2). The computationsare almost identical to the separating case, thus we outline only the general steps.

Part of the moduli spaceMg,n is covered by #Mg1,n1 , with g = g1 + 1 and n = n1 − 2.The local coordinates are denoted as wi for i = 1, . . . , n1 and the plumbing fixture reads:

wn1−1wn1 = q. (9.25)

The g-loop n-point amplitude with external states V (1)1 , . . . ,V

(1)n1−2 is denoted as:

Ag,n =∫Sg,n

ωg,nMg,n

(V

(1)1 , . . . ,V

(1)n1−2

). (9.26)

147

Figure 9.2: Factorization of the amplitude into two sub-amplitudes connected by a propag-ator (dashed line).

When the n1 − 2 punctures and g1 = g − 1 holes move lose to each other, the form canbe written as:

ωMg,n

(V

(1)1 , . . . , V

(1)Mg1,n1

, ∂q, ∂q)

= (2πi)−Mcg,n

⟨Mg1,n1∏λ=1

B(V

(1)λ

)B(∂q)B(∂q)

n1−2∏i=1

V(1)i

⟩Σg,n

.

(9.27)To proceed, one needs to introduce the surface state Σn1−1,n1 :

〈Σn1−1,n1 |V(1)n1−1 ⊗ V (1)

n1〉 := ωMg1,n1

(V (1)1 , . . . ,V (1)

n1). (9.28)

Following the same step as in the previous section leads to:

Fg,n(V

(1)i |

)=∫ dDk

(2π)DdDk′(2π)D Ag1,n1

(V

(1)1 , . . . ,V

(1)n1−2, φα(k), φβ(k′)

)∆αβ(k, k′), (9.29)

where the propagator is given in (9.18). This is equivalent to an amplitude for which twoexternal legs are glued together with a propagator, giving a loop (Figure 9.3).

Since both states φα and φβ are inserted on the same surface, their ghost numbers arenot fixed, even if the external states are physical. The non-vanishing of Fg,n only leads tothe constraint:

Ngh(φα) +Ngh(φβ) = 2n1 −n1−2∑i=1

Ngh(V

(1)i

)= 4. (9.30)

As a consequence, loop diagrams force to introduce states of every ghost number. Internalstates with Ngh 6= 2 correspond to spacetime ghosts.

Since the propagator contains a delta function δ(D)(k − k′), the integral over k′ can beremoved by setting k′ = −k. However, the integral over k remains since

Ag1,n1

(V

(1)1 , . . . ,V

(1)n1−2, φα(k), φβ(−k)

)∼ δ(D)(k(1)

1 + · · ·+ k(1)n1−2

). (9.31)

Hence, the loop momentum k is not fixed, as expected in QFT.

Remark 9.1 Not all values of the moduli associated to the holes can be associated to loopsin Feynman diagrams. Only the values close to the degeneration limit can be interpreted inthis way, the other being just standard (quantum) vertices.

9.2 Feynman diagrams and Feynman rulesIn the standard QFT approach, Feynman graphs compute Green functions, and scatteringamplitudes are obtained by amputating the external propagators through the LSZ prescrip-tion. For connected tree-level processes, this requires n ≥ 3 (corresponding to χ0,n < 0).

148

Figure 9.3: Factorization of the amplitude into two sub-amplitudes connected by a propag-ator (dashed line). The propagator connects two punctures of the same surface, which isequivalent to a loop.

Given a theory, there is a minimal set of Feynman diagrams – the Feynman rules – fromwhich every other diagram can be constructed. These rules include the definitions of thefundamental vertices – the fundamental interactions – and of the propagator – how statespropagate between two interactions (or, how to glue vertices together). In this section, wedescribe these different elements.

9.2.1 Feynman graphsThe amplitude factorization described in Section 9.1 gives a natural separation of amplitudesinto several contributions. Considering all the possible degeneration limits lead to a set ofdiagrams with amplitudes of lower order connected by propagators (Figure 9.2, Figure 9.3).This corresponds exactly to the idea behind Feynman graphs. Then, the goal is to findthe Feynman rules of the theory: since the propagator has already been identified (furtherstudied in Section 9.2.2), it is sufficient to find the interaction vertices.

Let’s make this more precise by considering an amplitude Ag,n(V1, . . . ,Vn). The indexof an amplitude is defined to be the index (7.50) of the corresponding Riemann surfaces

r(Ag,n) := r(Σg,n) = 3g + n− 2. (9.32)

Contributions to an amplitude with a given r(Ag,n) can be described in terms of amp-litudes Ag′,n′ with r(Ag′,n′) < r(Ag,n). But, the moduli spaceMg,n cannot (generically) becompletely covered with the plumbing fixture of lower-dimensional moduli spaces, i.e. withr(Mg′,n′) < r(Mg,n) (Section 7.3.3). Then, the same must be true for the amplitudes, suchthat Ag,n cannot be uniquely expressed in terms of amplitudes Ag′,n′ .

The g-loop n-point fundamental vertex is defined by:

Vg,n(V1, . . . ,Vn) := :=∫Rg,n

ωg,nMg,n(V1, . . . ,Vn), (9.33)

The form defined in (8.16) is integrated over a sub-sectionRg,n ⊂ Sg,n of Pg,n. Its projectionon the base is the region Vg,n ⊂ Mg,n which cannot be described by the plumbing fixture,

149

see (7.42b). In general, we will keep the choice of local coordinates implicit and always writeVg,n to avoid surcharging the notations.

It corresponds to the remaining contribution of the amplitude once all graphs containingpropagators have been taken into account:

=∑

0≤h≤g0≤m<n−1

+ + perms +

(9.34)

where the permutations are taken over all legs exiting the amplitudes in the first two terms(this includes the two legs glued together in the second term), including if necessary a weightto avoid overcounting. In the RHS, the amplitudes Ag,1 are tadpoles and have no externalvertices Vi (from Ag,n); this corresponds to the terms for m = 0 and m = n− 2.

In general, the fundamental vertex is non-vanishing for every value g, n ∈ N such thatχg,n < 0. For this reason, the index g helps to distinguish between graphs with identicalvalues of n. It may look strange that one needs vertices at every loop: the interpretation willbe made clearer when translating this in the language of string field theory (Chapter 14). Westress again that the definition of the fundamental vertex (and the region covered) depends onthe choice of local coordinates for all lower-order vertices Vg′,n′ such that r(Vg′,n′) < r(Vg,n).Remark 9.2 There are different alternative notations for (9.33):

Vg,n(V1, . . . ,Vn) := Vg,n(⊗iVi) := V1, . . . ,Vng. (9.35)

Example 9.1 – Scalar QFTConsider a scalar field theory with a cubic and a quartic interaction. The 4-point amp-litude contains four contributions, three from gluing 3-point vertices with a propagator,and one from the fundamental quartic vertex. The mismatch between the amplitudeand the three graphs with a propagator hints at the existence of the quartic interac-tions. This example gives an idea of how one can identify the fundamental interactionsrecursively.

The definition (9.34) of the vertex shows that it can also be interpreted as an ampu-tated Green function without internal propagator (i.e. there is no propagator at all). Thisdefinition is expected from the definition of the interactions vertices from an action, as willbe exemplified in Chapter 14. Before describing (and generalizing) the vertices, we describefirst the properties of the propagator.

9.2.2 PropagatorThe propagator has been defined in (9.18):

∆ = 12πi

∫ dqq∧ dqqb0b0 q

L0 qL0 . (9.36)

150

The plumbing moduli q is parametrized by (7.31)

q = e−s+iθ, s ∈ R+, θ ∈ [0, 2π), (9.37)

such that the integration measure becomes

dqq∧ dqq

= −2i ds ∧ dθ. (9.38)

Using the variables L±0 = L0 ± L0 and b±0 = b0 ± b0, the propagator can be recast as:

∆ = 12π b

+0 b−0

∫ ∞0

ds e−sL+0

∫ 2π

0dθ eiθL−0 . (9.39)

The form of the first integral is recognized as the Schwinger parametrization of the propag-ator, while the second is the Fourier transformation of the discrete delta function:∫ ∞

0ds e−sL

+0 = 1

L+0,

∫ 2π

0dθ eiθL−0 = 2π δL−0 ,0. (9.40)

In fact, the first integral converges only if L+0 > 0. As argued in the introduction, divergences

for L+0 ≤ 0 are either non-physical or IR divergences which can be cured by renormalization.

For this reason, we take the RHS as a definition of the integral, which would be the correctresult if one starts with a field theory action instead of a first-quantized formalism.

In this case, the propagator becomes

∆ = b+0L+

0b−0 δL−0 ,0

. (9.41)

This is the standard expression for the propagator. For completeness, the form in terms ofthe holomorphic and anti-holomorphic components is:

∆ = −2b0b01

L0 + L0δL0,L0

. (9.42)

The delta function restricts the amplitude to states satisfying the level-matching condition,that is, annihilated by L−0 .

Considering a basis φα(k) of eigenstates of both L0 and L0:

L+0 |φα(k)〉 = α′

2 (k2 +m2α) |φα(k)〉 , L−0 |φα(k)〉 = 0 (9.43)

leads to the following momentum-space kernel for the propagator:

∆αβ(k, k′) := 〈φα(k)c|∆ |φβ(k′)c〉 := (2π)Dδ(D)(k + k′) ∆αβ(k), (9.44a)

∆αβ(k) := Mαβ(k)k2 +m2

α

, Mαβ(k) := 2α′〈φcα(k)| b+0 b−0 |φcβ(−k)〉 , (9.44b)

withMαβ a finite-dimensional matrix giving the overlap of states of identical masses (becausethe number of states at a given level is finite).

For the propagator to be well-defined, it must be invertible (in particular, to define akinetic term). The propagator (9.41) is non-vanishing if the states it acts on satisfy:

b+0 |φcα〉 6= 0, b−0 |φcα〉 6= 0. (9.45)

151

Necessary and sufficient conditions for this to be true are

c+0 |φcα〉 = 0, c−0 |φcα〉 = 0. (9.46)

Indeed, decomposing the state on the ghost zero-modes

|φcα〉 = |φ1〉+ b±0 |φ2〉 , c±0 |φ1〉 = c±0 |φ2〉 = 0 (9.47)

givesc±0 |φcα〉 = 0 =⇒ |φ2〉 = 0, (9.48)

and one has correctly b±0 |φ1〉 6= 0.These conditions are given for the dual states: translating them on the normal states

reverses the roles of b0 and c0. Hence, the states must satisfy the conditions:

b+0 |φα〉 = 0, b−0 |φα〉 = 0. (9.49)

The second condition is satisfied automatically because the Hilbert space is H− when work-ing with Pg,n (Section 8.3.1). However, the first condition further restricts the states whichpropagate in internal lines. This leads to postulate that the external states should also betaken to satisfy this condition

b+0 |Vi〉 = 0, (9.50)

since external states are usually a subset of the internal states. This provides another mo-tivation of the statement in Section 4.2.2 that scattering amplitudes for the states not anni-hilated by b+0 must be trivial. A field interpretation of this condition is given in Chapters 13and 14.

Under these constraints on the states, the propagator can be inverted:

∆−1 = c+0 c−0 L

+0 δL−0 ,0

. (9.51)

9.2.3 Fundamental verticesThe vertices (9.33) can be constructed recursively assuming that all amplitudes are known.The starting point is the tree-level cubic amplitude A0,3: since it does not contain anyinternal propagator, it is equal to the fundamental vertex V0,3.

The fist thing to extract from the recursion relations are the background independentdata. This amounts to find local coordinates and a characterization of the subspaces Vg,n ⊂Mg,n, starting with P0,3 and iterating.

In the rest of this section, we show how this works schematically.

Recursive definition: tree-level vertices

The description of tree-level amplitudes A0,n is the simplest since only the separating plumb-ing fixture is used and Feynman graphs are trees. The possible factorizations of the amp-litude correspond basically to all the partitions of the set Vi into subsets.

Tree-level cubic vertex Since M0,3 = 0, the moduli space of the 3-punctured sphereΣ0,3 reduces to a point, and so does the section S0,3 of P0,3 (Figure 9.4a):

V0,3(V1,V2,V3) := A0,3(V1,V2,V3) = ω0,30 (V1,V2,V3). (9.52)

The corresponding graph is indicated in Figure 9.4b.

152

(a) A section S0,3 over P0,3 reduces toa point.

(b) Fundamental cubic vertex.

Figure 9.4: Section of P0,3 and cubic vertex.

Tree-level quartic vertex Part of the contributions to the 4-point amplitude A0,4 withexternal states Vi (i = 1, . . . , 4) comes from gluing two cubic vertices. Because there are fourexternal states, there are three different partitions 2→ 2 which are described in Figure 9.5(see also Figure 7.7). The sum of these three diagrams does not reproduce A0,4: the modulispaceM0,4 is not completely covered by the three amplitudes. Equivalently, the projectionof the section over P0,4 does not cover all ofM0,4. The missing contribution is defined bythe quartic vertex (Figure 9.7)

V0,4(V1,V2,V3,V4) :=∫R0,4

ω0,42 (V1, . . . ,V4), (9.53)

and the corresponding section is denoted by R0,4 (Figure 9.6). Denoting by F (s,t,u)0,4 the

graphs 9.5 in the s-, t- and u-channels, one has the relation

A0,4 = F (s)0,4 + F (t)

0,4 + F (u)0,4 + V0,4. (9.54)

Tree-level quintic vertex The amplitude A0,5 can be factorized in a greater numberof channels, the two types being 2 → 2|1 and 2 → 3. The possible Feynman graphs arebuilt either from three cubic vertices and two propagators (Figure 9.8a and permutations),or from one cubic and one quartic vertices together with one propagator (Figure 9.8b andpermutations). The remaining contribution is the fundamental vertex (Figure 9.8c):

V0,5(V1, . . . ,V5) :=∫R0,5

ω0,54 (V1, . . . ,V5). (9.55)

The construction to higher-order follows exactly this scheme.

Recursive definition: general vertices

Next, one needs to consider Feynman diagrams with loops. The first amplitude which canbe considered is the one-loop tadpole A1,1(V1). The factorization region corresponds to thegraph obtained by gluing two legs of the cubic vertex (Section 9.2.3)Section 9.2.3. Theremaining contribution is the fundamental tadpole vertex V1,1(V1) (Section 9.2.3) – note theindex g = 1 on the vertex, indicating that it is a 1-loop effect.

Next, the 1-loop 2-point amplitude can be obtained using the cubic and quartic tree-levelvertices V0,3 and V0,4, but also the one-loop tadpole V1,1. Iterating, the number of loops

153

(a) s-channel.

(b) t-channel. (c) u-channel.

Figure 9.5: Factorization of the quartic amplitude A0,4 in the s-, t- and u-channels.

Figure 9.6: A section S0,4 over P0,4, the contribution from the s-, t- and u-channels (Fig-ure 9.5) are indicated by the corresponding indices. The fundamental vertex is defined bythe section V0,4.

154

Figure 9.7: Fundamental quartic vertex.

(a) Factorization 12→ 3|45.

(b) Factorization 12→ 345. (c) Fundamental vertex.

Figure 9.8: Factorization of the amplitude G0,5 in channels and fundamental quintic vertex.Only the cases where V1 and V2 factorize on one side is indicated, the other cases follow bypermutations of the external states.

155

can be increased either by gluing together two external legs of a graph, or by gluing twodifferent graphs with loops together.

For g ≥ 2, the recursion implies the existence of vertices with no external states Vg,0:they should be interpreted as loop corrections to the vacuum energy density.

It is important to realize that, in this language, a handle in the Riemann surface isnot necessarily mapped to a loop in the Feynman graph: only handles described by theregion Fg,n = Mg,n − Vg,n do. The higher-order vertices – corresponding to surfaces withsmall handles only and described by Vg,n – should be regarded as quantum fundamentalinteractions. In Chapter 14, it will be explained that they really correspond to (finite)counter-terms: the measure is not invariant under the gauge symmetry of the theory andthese terms must be introduced to restore it.

(a) Internal loop. (b) Fundamental vertex.

Figure 9.9: Factorization of the amplitude G1,1 and fundamental tadpole at 1-loop.

Other vertices

The definition given at the end of (9.2.1) suggests to introduce additional vertices. Theprevious recursive definition gives only vertices with χg,n = 2− 2g − n < 0, but, in fact, itmakes sense to consider the additional cases: g = 0 and n = 0, 1, 2, and g = 1, n = 0.

The definition of the vertices as amputated Green function without internal propagatorsprovides a hint for the tree-level quadratic vertex V0,2. We define the latter as the amputatedtree-level 2-point Green function:

V0,2 := ∆−1∆∆−1 = ∆−1. (9.56)

Hence, we haveV0,2(V1,V2) := 〈V1| c+0 c

−0 L

+0 δL−0 ,0

|V2〉 . (9.57)Note that V0,2 is not the 2-point scattering amplitude.

We denote the tree-level 1-point and 0-point vertices as V0,1(V1) and V0,0. The first canbe interpreted as a classical source in the action, while the second is a classical vacuumenergy. They are set to zero in most applications and can be safely ignored. However, theyappear when formulating the theory on a background which does not solve the equation ofmotion [298].

Finally, the 1-loop vacuum energy V1,0 can also be defined as the partition function ofthe worldsheet CFT integrated over the torus modulus.

This allows to define the vertices Vg,n for all g, n ∈ N. We define the sum of all loopcontributions for a fixed n as:

Vn(V1, . . . ,Vn) :=∑g≥0

(~g2s)g Vg,n(V1, . . . ,Vn). (9.58)

156

9.2.4 StubsIn Section 7.3.4, we have indicated that the plumbing fixture can be modified by addingstubs or, equivalently, by rescaling the local coordinates. This amounts to introduce acut-off (7.53) on the variable s such that

q = e−s+iθ, s ∈ [s0,∞), θ ∈ [0, 2π). (9.59)

instead of (9.37). In this case, the s-integral in the propagator (9.36) is modified to∫ ∞s0

ds e−sL+0 = e−s0L+

0

L+0

. (9.60)

This leads to a new expression for the propagator:

∆(s0) = b+0e−s0L+

0

L+0

b−0 δL−0 ,0. (9.61)

In momentum space, this reads

∆αβ(k) := e−α′s0

2 (k2+m2α)

k2 +m2α

Mαβ(k). (9.62)

It is more convenient to work with the canonical propagator (9.41). This can be achievedby absorbing e−

s02 L

+0 in the interaction vertex: a n-point interaction will get n such factors.1

Since s0 changes the local coordinates, this means that it also changes the region Vg,n(Figure 7.10). The freedom in the choice of s0 translates into a freedom to choose which partof the amplitude is described by propagator graphs Fg,n(s0), and which part is described bya fundamental vertex Vg,n(s0). The amplitude Ag,n is independent of s0 since it is describedin terms of the complete moduli spaceMg,n. This also means that the parameter s0 mustdisappear when summing over the contributions from Vg,n(s0) and Fg,n(s0). This indicatesthat the value of s0 is not relevant, even off-shell: it can be taken to any convenient value.

The possibility of adding stubs solves the problem that the sum over all states coulddiverge (see Section 9.1.1). Indeed, the expression (9.62) in momentum space shows thatthe propagator includes an exponential suppression for very massive particle propagating asintermediate states. Since the mass of a particle increases with the level, this shows that thesum converges for a sufficiently large value of s0 thanks to the factor e−α′s0m2 . A secondinteresting aspect is the exponential momentum suppression e−α′s0k2 : this is responsible forthe nice UV behaviour of string theory. Since the value of s0 is not physical, this meansthat all Feynman graphs must share these properties. These two points will be made moreprecise in Chapter 19.

9.2.5 1PI verticesWe can follow the same procedure as before, but considering only the separating plumb-ing fixture. In this case, the Feynman diagrams are all 1PR (1-particle reducible): if thepropagator line is cut, then the graphs split in two disconnected components. The regionof the moduli space covered by these graphs is written as F1PR

g,n (7.43a). The complement1To make this identification precise for vertices involving external states, one has to consider the non-

amputated Green functions.

157

defines the 1PI region V1PIg,n (7.43b). Then, the 1PI g-loop n-point fundamental vertices aredefined as:

V1PIg,n (V1, . . . ,Vn) := :=∫R1PIg,n

ωg,nMg,n(V1, . . . ,Vn), (9.63)

where R1PIg,n is a section of Pg,n which projection on the base is V1PIg,n .

9.3 Properties of fundamental vertices9.3.1 String productFollowing the definition of surfaces states (Section 8.1.2), the vertex state is defined as:

〈Vg,n| ⊗i Vi〉 := Vg,n(⊗iVi). (9.64)

The vertex is a map Vg,n : H⊗n → C where C ' H⊗0. We will find very useful tointroduce the string products `g,n : H⊗n → H through the closed string inner product:

Vg,n+1(V0,V1, . . . ,Vn) := 〈V0| c−0 |`g,n(V1, . . . ,Vn)〉 . (9.65)

An alternative notation is:

`g,n(V1, . . . ,Vn) := [V1, . . . ,Vn]g (9.66)

The advantage of the second notation is to show that the products with g ≥ 3 are directgeneralization of the 2-product, which is very similar to a super-Lie bracket. These productsplay a central role in SFT – in fact, the description of SFT is more natural using the `g,nrather than the Vg,n.

Note that the products with n = 0 are maps C→ H, which means that they correspondto a particular fixed state.

`g,0 := [·]g ∈ H. (9.67)The ghost number of the product (9.65) is

Ngh(`g,n(V1, . . . ,Vn)

)= 3− 2n+

n∑i=1

Ngh(Vi) = 3 +n∑i=1

(Ngh(Vi)− 2

), (9.68)

and it is independent of the genus g. As a consequence, the parity of the product is

|`g,n(V1, . . . ,Vn)| = 1 +n∑i=1|Vi| mod 2, (9.69)

and the string product itself is always odd.The vertices satisfy the following identity for g ≥ 0 and n ≥ 1 [297, pp. 41–42]

0 =∑

g1,g2≥0g1+g2=g

∑n1,n2≥0n1+n2=n

n!n1!n2!Vg1,n1+1

(Ψn1 , `g2,n2(Ψn2)

)+ (−1)|φs|Vg−1,n+2

(φs, b

−0 φ

cs,Ψn

).

(9.70)The last term is absent for g = 0. It is a consequence of the definition of the vertices as themissing region from gluing lower-order vertices.

158

9.3.2 Feynman graph interpretationThe vertices must satisfy a certain number of conditions to be interpreted as Feynmandiagrams. The first is that they must be symmetric under permutations of the states. Notevery choice of local coordinates satisfies this requirement: this can be solved by definingthe vertex over a generalized section. In this case, the vertex is defined as the average ofthe integrals over N sections S(a)

g,n of Pg,n:

Vg,n(V1, . . . ,Vn) = 1N

N∑a=1

∫R(a)g,n

ωg,nMg,n(V1, . . . ,Vn). (9.71)

Example 9.2 – 3-point vertexThe cubic vertex must be symmetric under permutations

V0,3(V1,V2,V3) = V0,3(V3,V1,V2) + · · · (9.72)

Taking the vertex to be given by a section S0,3 with local coordinates fi

V0,3(V1,V2,V3) = ω0,30 (V1,V2,V3)|S0,3 = 〈f1 V1(0)f2 V2(0)f3 V3(0)〉, (9.73)

one finds that a permutation looks different

V0,3(V3,V1,V2) = 〈f1 V3(0)f2 V1(0)f3 V2(0)〉 6= V0,3(V1,V2,V3), (9.74)

unless the local coordinates satisfy special properties (remember that the local co-ordinates are specified by the vertex state V and not by the external states Vi, so apermutation of them does not permute the local maps). Obviously, both amplitudesagree on-shell since the dependence in the local coordinates cancel (equivalently onecan rotate the punctures using SL(2,C)).

Writing zi = fi(0), there is a SL(2,C) transformation g(z) such that

g(z1) = z2, g(z2) = z3, g(z3) = z1 (9.75)

such that

V0,3(V3,V1,V2) = 〈g f1 V3(0)g f2 V1(0)g f3 V2(0)〉. (9.76)

While the state Vi is correctly inserted at the puncture zi in this expression, this is notsufficient to guarantee the equality of the amplitudes. Indeed the fibre is defined by thecomplete functions fi(w) and not only by their values at w = 0. For this reason theamplitudes can be equal only if

g f1 = f2, g f2 = f3, g f3 = f1. (9.77)

This provides constraints on the functions fi, but it is often not possible to solve them.If the constraints cannot be solved, then one must introduce a general section. In

this case a generalized section will be made of 6 sections S(a) (a = 1, . . . , 6) becausethere are 6 permutations. Then the amplitude reads

V0,3(V1,V2,V3) = 16

6∑a=1

ω0,30 (V1,V2,V3)|S(a)

0,3. (9.78)

159

Figure 9.10: A generalized section S(a)0,3 (a = 1, . . . , 6) of P0,3 for the 3-point vertex. This

is to be compared with Figure 9.4a.

When computing the Feynman graphs by gluing lower-dimensional amplitudes, it ispossible that parts of the section overlap, meaning that several graphs cover the same partof the moduli space. In this case, the fundamental vertex should be defined as a negativecontribution in the overlap region. This procedure is perfectly well-defined since all graphsare finite and there is no ambiguity. In practice, it is always simpler to work with non-overlapping sections (i.e. a single covering of the moduli space). A simple way to preventoverlaps is to tune the stub parameter s0 to a large value.

By construction, the integral over Vg,n should be finite. If this is not the case, it meansthat the propagator graphs also diverge and that the parametrization is not good. This canalso be solved by considering a sufficiently large value of the stub parameter s0.

9.4 References• Plumbing fixture and amplitude factorization [218, sec. 9.3, 9.4, 291, sec. 6].

160

Chapter 10

Off-shell scattering amplitudesand Feynman diagrams –examples

In this chapter we describe explicitly several applications of the off-shell formalism.

10.1 Feynman diagrams – closed string10.1.1 3-point graphsIn this section we look for parametrizations of the 3-point vertex

A1, A2, A30 =⟨ 3∏i=1

fi Ai(0)⟩S2

. (10.1)

The punctures are taken to be located at

f1(0) = 0, f2(0) = 1, f3(0) =∞ (10.2)

using a SL(2,C) transformation.

First parametrization

Consider the local mapf1(w) = λw, (10.3)

where λ ∈ C is a fixed parameter.The SL(2,C) transformation (22.43)

g(z) = 11− z (10.4)

permutes the three punctures

g(0) = 1, g(1) =∞, g(∞) = 0. (10.5)

Then the cyclic symmetry of the vertex imposes

f1(w) = λw, f2(w) = g f1(w) = 11− λw , f3(w) = g f2(w) = 1− 1

λw. (10.6)

161

Obviously one has f1 = g f3.The permutation of the punctures 0 and 1 is achieved by the SL(2,C) transformation

h(z) = 1− z, (10.7)

such thath(0) = 1, h(1) = 0, h(∞) =∞. (10.8)

For the vertex to be invariant the functions must satisfy

h f1 = f2, h f2 = f1, h f3 = f3 (10.9)

(this exhausts all possibilities since they can be found by composing with g), but one findsthat these relations does not hold. Let’s introduce a second set of functions defined by

f1(w) = h f2(w) = λw

1− λw , f2(w) = h f1(w) = 1− λw,

f3(w) = h f3(w) = 1λw

.

(10.10)

Then any permutation of the punctures brings any function fi or fi into another functionof this set.

Then the tree-level cubic vertex can be defined as the weighted sum of the amplitudesevaluated in the two sets of local coordinates

A1, A2, A30 = 12

[⟨ 3∏i=1

fi Ai(0)⟩S2

+⟨ 3∏i=1

fi Ai(0)⟩S2

]. (10.11)

This corresponds to a generalized section, see (9.71).

10.1.2 4-point graphsThe punctures of Σ0,4 are located at Z = Za with a = 1, . . . , 4 (Z is the global coordinateof the sphere) and the local coordinates Wa and maps Fa are such that

Z = Fa(Wa), Za = Fa(0). (10.12)

In order to identify the quartic fundamental graph, one needs first to compute the threegraphs resulting from gluing the cubic vertex from Section 10.1.1.

First parametrization

In this section we compute the quartic graphs using the parametrization (10.11) of the cubicvertex, with the local maps given by (10.6) and (10.10). The gluing of two cubic verticeswith local coordinates

w(1,2)i = f

(1,2)i (0) (10.13)

is performed by identifying the punctures w(1,2)1 through the plumbing fixture

w(1)1 w

(2)1 = q = e−s+iθ. (10.14)

The global coordinates of the spheres are denoted by z1 and z2 and the global coordinate Zof Σ0,4 is identified with z1

Z = z1. (10.15)

162

In total four different gluing are possibles due to the presence of both the fi and fi:(fff, fff), (fff, f f f), (f f f , fff), (f f f , f f f). (10.16)

Consider first the (fff, fff) gluing for the s-channel, where 12 and 34 come close andwhich translates into the following identification of the local coordinates

W1 = w(1)2 , W2 = w

(1)3 , W3 = w

(2)2 , W4 = w

(2)3 (10.17)

(remember that local coordinates are inherited from the lowest-dimensional surfaces).The local map F1 and F2 are directly found to be

F1(W ) = 11− λW , F2(W ) = 1− 1

λW. (10.18a)

The two other functions are found after relating the z2 coordinate to z1

F3(W ) = λ2q(1− λW ) F4(W ) = − λ3qW

1− λW . (10.18b)

The locations of the punctures are found by setting W = 0F1(0) = 1, F2(0) =∞, F3(0) = λ2q, F4(0) = 0. (10.19)

The part of the moduli space covered by this diagram is found by varying q with |q| < 1: theregion covered by Z3 is a circle of radius λ2 centred at Z4 (Figure 10.1). This was expectedsince the graph under concerned corresponds to the degeneration limit where Z3 and Z4 areclose.Computation – Equation (10.18)Using the identification z1 = Z and of W1 = w

(1)2 , and the expression (10.6) of the local

map f2 one findsz1 = f

(1)2 (w(1)

2 ) = 11− λw(1)

2

= Z = F1(W1) = F1(w(1)2 ).

(10.20)

A similar computation gives F2.For the two other functions one needs to relate the z1 and z2 coordinate systems

z1 = λw(1)1 , z2 = λw

(2)1 , =⇒ z1z2 = λ2q. (10.21)

Then one findsF3(w(2)

2 ) = z1 = λ2q

z2= λ2q(1− λw(2)

2 ). (10.22)

The last function F4 is obtained similarly.

The u- and t-channel graphs can be constructed by exchanging two punctures beforeperforming a SL(2,C) transformation to put them in their original places. This moves thedisks and one finds that they are centred respectively around Z1 = 1 and Z2 =∞.

Note that the parameter λ is arbitrary, but it is convenient to choose it such that thedisks do not overlap

|λ| ≤ 1√2. (10.23)

10.2 Bibliography• Off-shell amplitudes [21, 39, 40, 175].

• Feynman diagrams [17, 27, 28, 51, 65, 89, 174, 230].

163

Figure 10.1: Part of the moduli space M0,4 = C (parametrized by Z3) covered by thes-channel graph.

164

Part III

Bosonic string field theory

165

Chapter 11

String field

In this chapter, we introduce general concepts about the string field. The goal is to give anidea of which type of object it is and of the different possibilities for describing it. We willsee that the string field is a functional and, for this reason, it is more convenient to workwith the associated ket field, which can itself be represented in momentum space. We focuson what to expect from a free field, taking inspiration from the worldsheet theory. Theinterpretation becomes more difficult when taking into account the interactions.

11.1 Field functionalA string field, after quantization, is an operator which creates or destroys a string at agiven time. Since a string is a 1-dimension extended object, the string field Ψ must dependon the spatial positions of each point of the string Xµ(shape). Hence, the string field is afunctional Ψ[Xµ(shape)]. The fact that it is a functional rather than a function makes theconstruction of a field theory much more challenging: it asks for revisiting all concepts weknow in point-particle QFT without any prior experience with a simple model.1

It is important that the dependence is only on the shape and not on the parametrization.However, it is simpler to first work with a specific parametrization X(σ) and make sure thatnothing depends on it at the end (equivalent to imposing the invariance under reparamet-rization of the worldsheet). This leads to work with a functional Ψ[X(σ)] of fields on theworldsheet (at fixed time). To proceed, one should first determine the degrees of freedom ofthe string, and then to find the interactions. The simplest way to achieve the first step is toperform a second-quantization of the string wave-functional: the string field is written as alinear combination of first-quantized states with spacetime wave functionals as coefficients.2This provides a free Hamiltonian; trying to add interactions perturbatively does not workwell.

It is not possible to go very far with this approach and one is lead to choose a specificgauge, breaking the manifest invariance under reparametrizations. The simplest is the light-cone gauge since one works only with the physical degrees of freedom of the string. Whilethis approach is interesting to gain some intuitions and to show that, in principle, it ispossible to build a string field theory, it requires making various assumptions and ends upwith problems (especially for superstrings).3

1In fact, generalizing further to functionals of extended p-dimensional objects – branes – shows that SFTis the simplest of such field theories.

2The description of the first-quantized states depends on the CFT used to describe the theory. Thisexplains the lack of manifest background independence of SFT. Unfortunately, no better approach has beenfound until now.

3While this approach has been mostly abandoned, recent results show that it can still be used when

166

Since worldsheet reparametrization invariance is just a kind of gauge symmetry – maybeless intuitive than the non-Abelian gauge symmetries in Yang–Mills, but still a gauge sym-metry –, one may surmise that it should be possible to gauge fix this symmetry and tointroduce a BRST symmetry in its place. This is the program of the BRST (or covariant)string field theory in which the string field depends not only of the worldsheet (at fixedtime), but also on the ghosts: Ψ[X(σ), c(σ)]. There is no dependence on the b ghost becausethe latter is the conjugate momentum of the c ghost: in the operator language, b(σ) ∼ δ

δc(σ) .The BRST formalism has the major advantage to allow to move easily from D = 26

dimensions – described by Xµ scalars (µ = 0, . . . , 25) – to a (possibly curved) D-dimensionalspacetime and a string with some internal structure – described by a more general CFT,in which D scalars Xµ represent the non-compact dimensions and the remaining systemwith central charge 26 −D describes the compactification and structure. It is sufficient toconsider the string field as a general functional of all the worldsheet fields. For simplicity,we will continue to write X in the functional dependence, keeping the other matter fieldsimplicit.

It is complicated to find an explicit expression for the string field as a functional of X(σ)and c(σ). In fact, the field written in this way is in the position representation and, as usualin quantum mechanics, one can choose to work with the representation independent ket |Ψ〉:

Ψ[X(σ), c(σ)] := 〈X(σ), c(σ)|Ψ〉 . (11.1)It is often more convenient to work with |Ψ〉 (which we will also denote simply as Ψ, notdistinguishing between states and operators). The latter will be the basic object of SFT inmost of this review.

Writing a field theory in terms of |Ψ〉 may not be intuitive since in point-particle QFT,one is used to work with the position or momentum representation. In fact, there is a verysimple way to recover a formulation in terms of spacetime point-particle fields, which can beused almost whenever there is a doubt about what is going on. Indeed, as is well-known fromstandard worldsheet string theory, the string states behave like a collection of particles. Thisis because the modes of the CFT fields (like αµn) carry spacetime indices (Lorentz, grouprepresentation. . . ) such that the states themselves carries indices. Indeed, these quantumnumbers classify eigenstates of the operators L0 and L0. On the other hand, positions andshapes are not eigenstates of any simple CFT operator.

11.2 Field expansionIt follows that the second-quantized string field can be written as a linear combinationof first-quantized off-shell states |φα(k)〉 = Vα(k; 0, 0) |0〉 (which form a basis of the CFTHilbert space H):

|Ψ〉 =∑α

∫ dDk(2π)D ψα(k) |φα(k)〉 , (11.2)

where k is the D-dimensional momentum of the string (conjugated to the position of thecentre-of-mass) and α is a collection of discrete quantum numbers (Lorentz indices, grouprepresentation. . . ). When inserting this expansion inside the action, we find that it reducesto a standard field theory with an infinite number of particles described by the spacetimefields ψα(k) (momentum representation). The fields can also be written in the positionrepresentation by Fourier transforming only the momentum k to the centre-of-mass x:

ψα(x) =∫ dDk

(2π)D eik·xψα(k). (11.3)

defined with a proper regularization [9, 129–132].

167

However, we will see that it is often not convenient because the action is non-local in positionspace (including for example exponentials of derivatives).

The physical intuition is that the string is a non-local object in spacetime. It can beexpressed in momentum space through a Fourier transformation: variables dual to non-compact (resp. compact) dimensions are continuous (discrete). As a consequence, the mo-mentum is continuous since the centre-of-mass move in the non-compact spacetime, whilethe string itself has a finite extension and the associated modes are discrete but still notbounded (and similarly for compact dimensions). This indicates that the spectrum is thecollection of a set of continuous and discrete modes. Hence, the non-locality of the string(due to the spatial extension) is traded for an infinite number of modes which behave likestandard particles. In this description, the non-locality arises: 1) in the infinite number offields, 2) in the coupling between the modes, 3) as a complicated momentum-dependence ofthe action.

When we are not interested in the spacetime properties, we will write a generic basis ofthe Hilbert space H as φr:

|Ψ〉 =∑r

ψr |φr〉 . (11.4)

The sum over r includes discrete and continuous labels.Remark 11.1 (Classical fields and quantum states) The states |φα(k)〉, being eigenstates ofthe CFT Hamiltonian, are (first-quantized) quantum states. A classical field Ψ can then be writtenas a linear combination of such states with coefficients ψα(k) corresponding to wave functions. Aspecific choice of these functions gives a profile to the classical field. As long as the coefficients arefunctions, the field Ψ is classical: (second-)quantization replaces the functions by operators, and,at this point, the field Ψ is promoted to a quantum operator. The fact that one uses first-quantizedstates to describe the field configuration does not mean that the field itself is quantized: this is justa particularly useful description since one knows already a basis of possible configurations.

Example 11.1 – Scalar fieldIn order to illustrate the notations for a point-particle, consider a scalar field φ(x). Itcan be expanded in Fourier modes as:

φ(x) =∫ dDk

(2π)D φ(k)eik·x. (11.5)

The corresponding ket |φ〉 is found by expanding on a basis |k〉:

|φ〉 =∫ dDk

(2π)D φ(k) |k〉 , φ(k) = 〈k|φ〉 . (11.6)

Similarly, the position space field is defined from the basis |x〉 such that:

φ(x) = 〈x|φ〉 =∫ dDk

(2π)D 〈x|k〉〈k|φ〉 , 〈x|k〉 = eik·x. (11.7)

11.3 SummaryIn this chapter, we introduced general ideas about what is a string field. We now need towrite an action. In general, one proceeds in two steps:

1. build the kinetic term (free theory):

(a) equations of motion → physical states

168

(b) equivalence relation → gauge symmetry

2. add interactions and deform the gauge transformation

We consider the first point in the next chapter, but we will have to introduce more machineryin order to discuss interactions.

11.4 References• General discussions of the string field and of the ideas of string field theory can be

found in [217, sec. 4, 296].

• Light-cone SFT is reviewed in [278, 143, chap. 6, 144, chap. 9].

169

Chapter 12

Light-cone string field theory

12.1 BibliographyLight-cone string field theory is discussed in [278, 143, chap. 6, 144, chap. 9].

170

Chapter 13

Free BRST string field theory

In this chapter, we construct the BRST (or covariant) free bosonic string field theories. It isuseful to first ignore the interactions in order to introduce some general tools and structuresin a simpler setting. Moreover, the free SFT is easily constructed and does not require asmuch inputs as the interactions. In this chapter, we discuss mostly the open string, keepingthe closed string for the last section. We start by describing the classical theory: equationsof motion, action, gauge invariance and gauge fixing. Then, we perform the path integralquantization and compute the action in terms of spacetime fields for the first two levels(tachyon and gauge field).

13.1 Classical action for the open stringContrary to most of this review, we will exemplify the discussion with the open string. Thereason is that most computations are the same in both the open and closed string theories,but the latter requires twice more writing. There are also a few subtleties which can be moreeasily explained once the general structure is understood. Everything needed for the openstring for this chapter can be found in Chapter 24: in fact, describing the open string (atthis level) is equivalent to consider only the holomorphic sector of the CFT and to set pL = p(instead of p/2). We consider a generic matter CFT in addition to the ghost system and wedenote as H the space of states. The open and closed string fields are denoted respectivelyby Φ and Ψ, such that it is clear which theory is studied.

An action can be either constructed from first principles, or it can be derived from theequations of motion. Since the fundamental structure of string field theory is not (really)known, one needs to rely on the second approach. But do we already know the (free)equations of motion for the string field? The answer is yes. But, before showing how thesecan be found from the worldsheet formalism, we will study the case of the point-particle tofix ideas and notations.

13.1.1 Warm-up: point-particleThe free (or linearized) equation of motion for a scalar particle reads:

(−∆ +m2)φ(x) = 0. (13.1)

Solutions to this equation provides one-particle state of the free theory: a convenient basisis eikx, where each state satisfies the on-shell condition

k2 = −m2. (13.2)

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The field φ(x) is decomposed on the basis as

φ(x) =∫

dk φ(k)eikx, (13.3)

where φ(k) are the coefficients of the expansion. Since the field is off-shell, the conditionk2 = −m2 is not imposed. Following Chapter 11, the field can also be represented as a ket:

φ(x) = 〈x|φ〉 , φ(k) = 〈k|φ〉 , (13.4)

or, conversely:|φ〉 =

∫dxφ(x) |x〉 =

∫dk φ(k) |k〉 . (13.5)

Writing the kinetic operator as a kernel:

K(x, x′) := 〈x|K |x′〉 = δ(x− x′) (−∆x +m2), (13.6)

the equations of motion reads∫dx′K(x, x′)φ(x′) = 0 ⇐⇒ K |φ〉 = 0. (13.7)

An action can easily be found from the equation of motion by multiplying with φ(x) andintegrating:

S = 12

∫dxφ(x)(−∆ +m2)φ(x) = 1

2

∫dxdx′ φ(x)K(x, x′)φ(x′). (13.8)

It is straightforward to write the action in terms of the ket:

S = 12 〈φ|K |φ〉 . (13.9)

There is one hidden assumption in the previous lines: the definition of a scalar product.A natural inner product is provided in the usual quantum mechanics by the pairing of braand ket. Similarly, integration provides another definition of the inner product when workingwith functions. We will find that the definition of the inner product requires more care inclosed SFT. To summarize, to write the kinetic term of the action, one needs the linearizedequation of motion and an appropriate inner product on the space of states.

13.1.2 Open string actionThe worldsheet equation which yields precisely all the string physical states |ψ〉 is the BRSTcondition:

QB |ψ〉 = 0. (13.10)

Considering the open string field Φ to be a linear combination of all possible one-stringstates |ψ〉

Φ ∈ H, (13.11)

the equation of motion is:QB |Φ〉 = 0. (13.12)

Moving away from the physical state condition, the string field Φ is off-shell and is expandedon a general basis φr of H. This presents a first difficulty because the worldsheet approach– and the description of amplitudes – looks ill-defined for off-shell states: extending the usual

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formalism will be the topic of Chapter 6. However, this is not necessary for the free theoryand we can directly proceed.

Next, we need to find an inner product 〈·, ·〉 on the Hilbert space H. A natural candidateis the BPZ inner product since it is not degenerate

〈A,B〉 := 〈A|B〉 , (13.13)

where 〈A| = |A〉t is the BPZ conjugate (22.98) of |A〉, using I−. This leads to the action:

S = 12 〈Φ, QBΦ〉 = 1

2 〈Φ|QB |Φ〉 . (13.14)

Due to the definition of the BPZ product, the action is equivalent to a 2-point correlationfunction on the disk.

The inner product satisfies the following identities:

〈A,B〉 = (−1)|A||B|〈B,A〉, 〈QBA,B〉 = −(−1)|A|〈A,QBB〉. (13.15)

A first consistency check is to verify that the ghost number of the string can be definedsuch that the action is not vanishing. Indeed, the ghost number anomaly on the disk impliesthat the total ghost number must be Ngh = 3. Since physical states have Ngh = 1, it islogical to take the string field to satisfy the same condition, even off-shell:

Ngh(Φ) = 1. (13.16)

This condition means that there is no ghost at the classical level beyond the one of theenergy vacuum | ↓〉, which has Ngh = 1. Moreover, the BRST charge has Ngh(QB) = 1,such that the action has ghost number 3.

One needs to find the Grassmann parity of the string field. Using the properties of theBPZ inner product, the string field should be Grassmann odd

|Φ| = 1 (13.17)

for the action to be even. This is in agreement with the fact that the string field has ghostnumber 1 and that the ghosts are Grassmann odd. One must impose a reality condition onthe string field (a complex field would behave like two real fields and have too much states).The appropriate reality condition identifies the Euclidean and BPZ conjugates:

|Φ〉‡ = |Φ〉t . (13.18)

That this relation is correct will be checked a posteriori for the tachyon field in Section 13.4.


〈Φ〉QBΦ = (−1)|Φ|(|QBΦ|)〈QBΦ,Φ〉 = (−1)|Φ|(1+|Φ|)〈QBΦ,Φ〉= 〈QBΦ,Φ〉 = −(−1)|Φ|〈Φ, QBΦ〉,

where both properties (13.15), together with the fact that |Φ|(1 + |Φ|) is necessarilyeven. In order for the bracket to be non-zero, one must have |Φ| = 1.

Since the Hilbert space splits as H = H0 ⊕ c0H0 with H0 = H ∩ ker b0, see (24.31), it isnatural to split the field as (this is discussed further in Section 13.2):

|Φ〉 = |Φ↓〉+ c0 |Φ↓〉 , (13.19)

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whereΦ↓, Φ↓ ∈ H0 =⇒ b0 |Φ↓〉 = b0 |Φ↓〉 = 0. (13.20)

The ghost number of each component is

Ngh(Φ↓) = 1, Ngh(Φ↓) = 0. (13.21)

Remembering the decomposition (24.14a) of the BRST operator

QB = c0L0 − b0M + QB , (13.22)

inserting the decomposition (13.19) in the action (13.14) gives:

S = 12 〈Φ↓| c0L0 |Φ↓〉+ 1

2 〈Φ↓| c0M |Φ↓〉+〈Φ↓| c0QB |Φ↓〉 . (13.23)

The equations of motion are obtained by varying the different fields:

0 = −M |Φ↓〉+ QB |Φ↓〉 , 0 = c0L0 |Φ↓〉+ c0QB |Φ↓〉 . (13.24)

Computation – Equation (13.23)Let’s introduce the projector Πs = b0c0 on the space H0 = H∩ker b0 and the orthogonalprojector Πs = c0b0 such that

|Φ〉 = |Φ↓〉+ |Φ↑〉 , |Φ↓〉 = Πs |Φ〉 , |Φ↑〉 = Πs |Φ〉 . (13.25)

We then have:

ΠsQB |Φ〉 = −b0M |Φ↑〉+ QB |Φ↓〉 , ΠsQB |Φ〉 = c0L0 |Φ↓〉+ QB |Φ↑〉 , (13.26)

using[Πs, QB ] = [Πs,M ] = [Πs, L0] = 0. (13.27)

Then, we need the fact that Π†s = Πs. to compute the action:

S = 12 〈Φ, QBΦ〉

= 12 〈ΠsΦ + ΠsΦ, QBΦ〉

= 12 〈ΠsΦ, ΠsQBΦ〉+ 1

2 〈ΠsΦ,ΠsQBΦ〉

= 12 〈Φ↓, c0L0Φ↓ + QBΦ↑〉+ 1

2 〈Φ↑,−b0MΦ↑ + QBΦ↓〉

= 12 〈Φ↓, c0L0Φ↓〉+ 1

2 〈Φ↓, QBΦ↑〉 −12 〈Φ↑, b0MΦ↑〉+ 1

2 〈Φ↑, QBΦ↓〉.

The result follows by setting |Φ↑〉 = c0 |Φ〉, using (13.15) and that the BPZ conjugateof c0 is −c0.

13.1.3 Gauge invarianceIn writing the action, only the condition that the states are BRST closed has been used. Oneneeds to interpret the condition that the state are not BRST-exact, or phrased differentlythat two states differing by a BRST exact state are equivalent:

|φ〉 ∼ |ψ〉+QB |λ〉 . (13.28)

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Uplifting this condition to the string field, the most direct interpretation is that it corres-ponds to a gauge invariance:

|Φ〉 −→ |Φ′〉 = |Φ〉+ δΛ |Φ〉 , δΛ |Φ〉 = QB |Λ〉 Ngh(Λ) = 0. (13.29)

In order for the ghost numbers to match, the gauge parameter has vanishing ghost number.The action (13.14) is obviously invariant since the BRST charge is nilpotent.

13.1.4 Siegel gaugeIn writing the action (13.14), the condition b0 |ψ〉 = 0 has not been imposed on the stringfield. In Section 4.2.2, this condition was found by restricting the BRST cohomology, pro-jecting out states built on the ghost vacuum | ↑〉, as required by the behaviour of the on-shellscattering amplitudes. In Chapter 24, we obtained it by finding that the absolute cohomo-logy contains twice more states as necessary. This was also understood as a way to work witha specific representative of the BRST cohomology. Since the field is off-shell and since theaction computes off-shell Green functions, these arguments cannot be used, which explainswhy we did not use this condition earlier.

On the other hand, the condition

b0 |Φ〉 = 0 (13.30)

can be interpreted as a gauge fixing condition, called Siegel gauge. It can be reached fromany field through a gauge transformation (13.29) with

|Λ〉 = −∆ |Φ〉 , ∆ = b0L0, (13.31)

where ∆ was defined in (24.38) and will be identified with the propagator. Note that b0 = 0does not imply L0 = 0 since the string field is not BRST closed.

This gauge choice is well-defined and completely fixes the gauge symmetry off-shell,meaning that no solution of the equation of motion is pure gauge after the gauge fixing.This is shown as follows: assume that |ψ〉 = QB |χ〉 is an off-shell pure-gauge state withL0 6= 0, then, because it is also annihilated by b0, one finds:

0 = QB , b0 |ψ〉 = L0 |ψ〉 (13.32)

which yields a contradiction.The gauge fixing condition breaks down for L0 = 0, but this does not pose any problem

when working with Feynman diagrams since they are not physical by themselves (nor arethe off-shell and on-shell Green functions). Only the sum giving the scattering amplitudes(truncated on-shell Green functions) is physical: in this case, the singularity L0 = 0 cor-responds to the on-shell condition and it is well-known how such infrared divergences forintermediate states are removed (through the LSZ prescription, mass renormalization andtadpole cancellation).

Computation – Equation (13.31)Performing a gauge transformation gives

b0 |Φ′〉 = b0 |Φ〉+ b0QB |Λ〉 = 0. (13.33)

Then, one writesb0 |Φ〉 = b0[QB ,∆] |Φ〉 = b0QB∆ |Φ〉 , (13.34)

using the relation (24.36), the expression (24.38) and the fact that b20 = 0. Plugging

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this back in the first equation gives:

b0QB (∆ |Φ〉+ |Λ〉) = 0. (13.35)

The factor of b0 can be removed by multiplying with c0, and the parenthesis shouldvanish (since it is not identically closed), which means that (13.31) holds up to a BRSTexact state.

Example 13.1 – Gauge fixing and singularityIn Maxwell theory, the gauge transformation

A′µ = Aµ + ∂µλ, (13.36)

is used to impose the Lorentz condition

∂µA′µ = 0 =⇒ ∆λ = −∂µAµ. (13.37)

In momentum space, the parameter reads

λ = −kµ

k2 Aµ. (13.38)

It is singular when k is on-shell, k2 = 0. However, this does not prevent from computingFeynman diagrams.

To understand the effect of the gauge fixing on the string field components, decomposethe field as (13.19) |Φ〉 = |Φ↓〉+ c0 |Φ↓〉. Then, imposing the condition (13.30) yields

|Φ↓〉 = 0 =⇒ |Φ〉 = |Φ↓〉 . (13.39)

This has the expected effect of dividing by two the number of states and show that they arenot physical.

Plugging this condition in the action (13.23) leads to gauge fixed action

S = 12 〈Φ| c0L0 |Φ〉 , (13.40)

for which the equation of motion is

L0 |Φ〉 = 0. (13.41)

But, note that this equation contains much less information than the original (13.12): as |Φ↓〉is truncated from (13.40), a part of the equations of motion is lost. The missing equationcan be found by setting |Φ〉 = 0 in (13.24) and must be imposed on top of the action:

QB |Φ〉 = 0. (13.42)

It is called out-of-Siegel gauge constraint and is equivalent to the Gauss constraint in elec-tromagnetism: the equations of motion for pure gauge states contain also the physical fields,thus, when one fixes a gauge, these relations are lost and must be imposed on the side of theaction. This procedure mimics what happens in the old covariant theory, where the Virasoroconstraints are imposed after choosing the flat gauge (if Φ contains no ghost on top of | ↓〉,then QB = 0 implies Ln = 0, see Section 24.3.3). Moreover, the states which do not satisfythe condition b0 = 0 do not propagate: this restricts the external states to be considered inamplitudes.

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Remark 13.1 Another way to derive (13.40) is to insert b0, c0 = 1 in the action:

S = 12 〈Φ|QBc0, b0 |Φ〉 = 1

2 〈Φ|QBb0c0 |Φ〉

= 12 〈Φ| b0, QBc0 |Φ〉 −

12 〈Φ| b0QBc0 |Φ〉

= 12 〈Φ| c0L0 |Φ〉 .

The drawback of this computation is that it does not show directly how the constraints (13.42)arise.

Remark 13.2 (Generalized gauge fixing) It is possible to generalize the Siegel gauge,in the same way that the Feynman gauge generalizes the Lorentz gauge. This has been studiedin [4, 5].

In this section, we have motivated different properties and adopted some normalizations.The simplest way to check that they are consistent is to derive the action in terms of thespacetime fields and to check that it has the expected properties from standard QFT. Thiswill be the topic of Section 13.4.

13.2 Open string field expansion, parity and ghost num-ber

A basis for the off-shell Hilbert space H is denoted by φr, where the ghost numbers andparity of the states are written as:

nr := Ngh(φr), |φr| = nr mod 2. (13.43)

The corresponding basis of dual (or conjugate) states φcr is defined by (22.144):

〈φcr|φs〉 = δrs. (13.44)

The basis states can be decomposed according to the ghost zero-modes

|φr〉 = |φ↓,r〉+ |φ↑,r〉 , b0 |φ↓,r〉 = c0 |φ↑,r〉 = 0. (13.45)

Finally, each state ψ↑ ∈ c0H can be associated to a state ψ:

|ψ↑〉 = c0 |ψ↓〉 , b0 |ψ↓〉 = 0, Ngh(ψ↑) = Ngh(ψ↓) + 1. (13.46)

More details can be found in Section 6.2.Any field Φ can be expanded as

|Φ〉 =∑r

ψr |φr〉 , (13.47)

where the ψr are spacetime fields (remembering that r denotes collectively the continuousand discrete quantum numbers).1

1The notation is slightly ambiguous: from (13.45), it looks like both components of φr have the samecoefficient ψr. But, in fact, one sums over all linearly independent states: in terms of the components of φr,different basis can be considered; for example φ↓,r, φ↑,r, or φ↓,r ± φ↑,r. A more precise expression canbe found in (13.54) and (13.56).

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Obviously, the coefficients do not carry a ghost number since they are not worldsheetoperators. However, they can be Grassmann even or odd such that each term of the sumhas the same parity, so that the field has a definite parity:

∀r : |Φ| = |ψr| |φr|. (13.48)

If the field is Grassmann odd (resp. even) then the coefficients ψr and the basis states musthave opposite (resp. identical) parities, such that |Φ| = 1.

Since the parity results from worldsheet ghosts and since there would be Grassmann oddstates even in a purely bosonic theory, it suggests that the parity of the coefficients ψr isalso related to a spacetime ghost number G defined as:

G(ψr) = 1− nr. (13.49)

The normalization is chosen such that the component of a classical string field (Ngh = 1)are classical spacetime fields with G = 0 (no ghost). We will see later that this definitionmakes sense.

A quantum string field Φ generally contains components Φn of all worldsheet ghostnumbers n:

Φ =∑n∈Z

Φn, Ngh(Φn) = n. (13.50)

The projection on the positive and negative (cylinder) ghost numbers are denoted by Φ±:

Φ = Φ+ + Φ−, Φ+ =∑n>1

Φn, Φ− =∑n≤1

Φn. (13.51)

The shift in the indices is explained by the relation (B.59) between the cylinder and planeghost numbers.

For a field Φn of fixed ghost number, coefficients of the expansion vanish whenever theghost number of the basis state does not match the one of the field:

∀nr 6= n : ψr = 0. (13.52)

Another possibility to define the field Φn is to insert a delta function:

|Φn〉 = δ(Ngh − n) |Ψ〉 =∑r

δ(nr − n)ψr |φr〉 . (13.53)

According to (13.45), a string field Φ can also be separated in terms of the ghost zero-modes:

|Φ〉 = |Φ↓〉+ |Φ↑〉 = |Φ↓〉+ c0 |Φ↓〉 , (13.54a)|Φ↑〉 = c0 |Φ↓〉 , |Φ↓〉 = b0 |Φ↑〉 , (13.54b)

where the components satisfy the constraints

b0 |Φ↓〉 = 0, c0 |Φ↑〉 = 0, b0 |Φ↓〉 = 0. (13.55)

The fields |Φ↓〉 and |Φ↑〉 (or |Φ↓〉) are called the down and top components and they can beexpanded as:

|Φ↓〉 =∑r

ψ↓,r |φ↓,r〉 , |Φ↑〉 =∑r

ψ↑,r |φ↑,r〉 . (13.56)

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13.3 Path integral quantizationThe string field theory can be quantized with a path integral:

Z =∫

dΦcl e−S[Φcl] =∫

dΦcl e−12〈Φcl|QB |Φcl〉. (13.57)

An index has been added to the field to emphasize that it is the classical field (no spacetimeghosts). The simplest way to define the measure is to use the expansion (11.4) such that

Z =∫ ∏

s

dψs e−S[ψr]. (13.58)

13.3.1 Tentative Faddeev–Popov gauge fixingThe action can be gauge fixed using the Faddeev–Popov formalism. The gauge fixing con-dition is

F (Φcl) := b0 |Φcl〉 = 0. (13.59)Its variation under a gauge transformation (13.29) reads

δF = b0QB |Λcl〉 , (13.60)

which implies that the Faddeev–Popov determinant is

det δF

δΛcl= det b0QB . (13.61)

This determinant is rewritten as a path integral by introducing a ghost C and an antighostB′ string fields (the prime on B′ will become clear below):

det b0QB =∫

dB′dC e−SFP , SFP = −〈B′| b0QB |C〉 . (13.62)

The ghost numbers are attributed by selecting the same ghost number for the C ghost and forthe gauge parameter, and then requiring that the Faddeev–Popov action is non-vanishing:

Ngh(B′) = 3, Ngh(C) = 0. (13.63)

The ghosts can be expanded as

|B′〉 = δ(Ngh − 3)∑r

b′r |φr〉 , |C〉 = δ(Ngh)∑r

cr |φr〉 , (13.64)

where the coefficients br and cr are Grassmann odd in order for the determinant formula tomake sense:

|br| = |cr| = 1. (13.65)Then, since the basis states appearing in B′ and C are respectively odd and even, thisimplies

|B′| = 0, |C| = 1. (13.66)However, there is a redundancy in the gauge fixing because the Faddeev–Popov action

is itself invariant under two independent transformations:

δ |C〉 = QB |Λ−1〉 , Ngh(Λ−1) = −1, (13.67a)δ |B′〉 = b0 |Λ′〉 , Ngh(Λ′) = 4. (13.67b)

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This residual invariance arises because not all |Λcl〉 generate a gauge transformation. Indeed,if

|Λ〉 = |Λ0〉+QB |Λ−1〉 , (13.68)

the field transforms as|Φ′cl〉 −→ |Φcl〉+QB |Λ0〉 (13.69)

and there is no trace left of |Λ−1〉, so it should not be counted.The second invariance (13.67b) is not problematic because b0 is an algebraic operator (the

Faddeev–Popov action associated to the determinant has no dynamics). The decompositionsof the gauge parameter Λ′ and the B′ field into components (13.54) read:

|B′〉 = |B′↓〉+ c0 |B〉 , |B〉 := |B′↓〉 , (13.70a)

|Λ′〉 = |Λ′↓〉+ c0 |Λ′↓〉 . (13.70b)

The gauge transformations act on the components as:

δ |B′↓〉 = |Λ′↓〉 , δ |B〉 = 0. (13.71)

This shows that B is gauge invariant and B′↓ can be completely removed by the gaugetransformation. This makes sense because B′↓ does not appear in the action (13.62). Thegauge transformation (13.67b) can be used to fix the gauge:

|F ′〉 = c0 |B′〉 = 0 =⇒ |B′↓〉 = 0. (13.72)

This fixes completely the gauge invariance since the field B is restricted to satisfy b0 |B〉 = 0,and the component form (13.71) of the gauge transformation shows that no transformationis allowed. Moreover, there is no need to introduce a Faddeev–Popov determinant for thisgauge fixing because the corresponding ghosts would not couple to the other fields (and thiswould continue to hold even in the presence of interactions, see Remark 13.4). Indeed, fromthe absence of derivatives in the gauge transformation, one finds that the determinant isfinite-dimensional and thus a ghost-representation is not necessary:

det δF′

δΛ′ = det c0b0 = det c0 det b0 = 12 detb0, c0 = 1

2 . (13.73)

Then, redefining the measure, the partition function and action reduce to

∆FP =∫

dB dC e−SFP[B,C], SFP = 〈B|QB |C〉 . (13.74)

Note that the field B satisfies

b0 |B〉 = 0, Ngh(B) = 2, |B| = 1. (13.75)

Since both fields are Grassmann odd, the action can be rewritten in a symmetric way:

SFP = 12

(〈B|QB |C〉+〈C|QB |B〉

). (13.76)

Remark 13.3 (Ghost and anti-ghost definitions) The definition of the anti-ghost Band ghost C is appropriate because the worldsheet and spacetime ghost numbers are relatedby a minus sign (and a shift of one unit). In the BV formalism, we will see that the fieldscontain the matter and ghost fields, while the antifields contain the anti-ghosts. These twosets are respectively defined with Ngh ≤ 1 and Ngh > 1.

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The constraint b0 |B〉 = 0 can be lifted by adding a top component:

|B〉 = |B↓〉+ c0 |B↓〉 (13.77)

together with the gauge invariance

δ |B〉 = QB |Λ1〉 . (13.78)

Note the difference with (13.70): while B = B′↓ was the top component of the B′ field, here,it is defined to be the down component, such that |B↓〉 = |B′↓〉. However, for the moment,we keep B to satisfy b0 |B〉 = 0.

Remark 13.4 (Decoupling of the ghosts) Since the theory is free the Faddeev–Popovaction (13.74) could be ignored and absorbed in the normalization because it does not coupleto the field. On the other hand, when interactions are included, the gauge transformationis modified and the ghosts couple to the matter fields. But this is true only for the Ctransformation (13.67a), not for (13.67b). Then it means that ghosts introduced for gaugefixing (13.67b) will never couple to the matter and other ghosts.

The invariance (13.67a) is a gauge invariance for C and must be treated in the sameway as (13.29). Then, following the Faddeev–Popov procedure, one is lead to introduce newghosts for the ghosts. But, the same structure appears again. This leads to a residual gaugeinvariance, which has the same form. This process continues recursively and one finds aninfinite tower of ghosts.

13.3.2 Tower of ghostsIn order to simplify the notations, all the fields are denoted by Φn where n gives the ghostnumber:

• Φ1 := Φcl is the original physical field

• Φ0 := C and, more generally, Φn with n < 1 are ghosts

• Φ2 := B and, more generally, Φn with n > 1 are anti-ghosts

The recipe is that each pair of ghost fields (Φn+2,Φ−n) is associated to a gauge parameterΛ−n−1 with n ≥ 0. It is then natural to gather all the fields in a single field

|Φ〉 =∑n

|Φn〉 (13.79)

satisfying the gauge fixing constraint:

b0 |Φ〉 = 0 =⇒ b0 |Φn〉 = 0. (13.80)

For n ≤ 1, these constraints are gauge fixing conditions for the invariance δ |Φn〉 = QBΛn.For n > 1, they arise by considering only the top component of the B field.

Finally, the gauge fixing condition can be incorporated inside the action by using aLagrange multiplier β, which is an auxiliary string field containing also components of allghost numbers:

|β〉 =∑n∈Z|βn〉 . (13.81)

The path integral then readsZ =

∫dΦdβ e−S[Φ,β], (13.82)

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where

S[Φ, β] = 12 〈Φ|QB |Φ〉+〈β| b0 |Φ〉 (13.83a)

=∑n∈Z

(12 〈Φ2−n|QB |Φn〉+〈β4−n| b0 |Φn〉

). (13.83b)

The first term of the action has the same form as the classical action (13.14), but now includesfields at every ghost number. The complete BV analysis is relegated to the interacting theory.

Removing the auxiliary field β = 0, one finds that the action is invariant under theextended gauge transformation

δ |Φ〉 = QB |Λ〉 , (13.84)where the gauge parameter has also components of all ghost numbers:

|Λ〉 =∑n∈Z|Λn〉 . (13.85)

13.4 Spacetime actionIn order to make the string field action more concrete, and as emphasized in Chapter 11,it is useful to expand the string field in spacetime fields and to write the action for thelowest modes. This also helps to check that the normalization chosen until here correctlyreproduces the standard QFT normalizations. For simplicity we focus on the open string inD = 26.

13.4.1 Classical actionWe build the string from the vacuum |k, ↓〉 (Chapter 24) by acting with the ghost positive-frequency modes b−n and c−n, the zero-mode c0, and from the scalar oscillators iαµ−n. Thefactor of i arises from the state–operator correspondence, which maps a products of eikX

with derivatives of ∂X, b and c acting on the conformal vacuum to a state without anyfactor of i.

Up to level ` = 1, the classical open string field can be expanded as

|Φ〉 = 1√α′

∫ dDk(2π)D

(T (k) +Aµ(k)αµ−1 + i

√α′

2 B(k)b−1c0 + · · ·)|k, ↓〉 (13.86)

before gauge fixing. The spacetime fields are T (k), Aµ(k) and B(k): their roles will beinterpreted below. The first two terms are part of the |Φ↓〉 component, while the last termis part of the |Φ↑〉 component. All terms are correctly Grassmann even and they havevanishing spacetime ghost numbers. The normalizations are chosen in order to retrieve thecanonical normalization in QFT.

The equation (13.12) leads to the following equations of motion of the spacetime fields:

(α′k2 − 1)T (k) = 0, k2Aµ(k) + ikµB(k) = 0,kµAµ(k) + iB(k) = 0.

(13.87)

Moreover, plugging the last equation into the second one gives

k2Aµ(k)− kµk ·Aν(k) = 0. (13.88)

After Fourier transformation, the equations in position space read:

(∆ + 1)T = 0, B = ∂µAµ, ∆Aµ = ∂µB. (13.89)

182

This shows that T (k) is a tachyon with mass m2 = −1/α′ and Aµ(k) is a massless gaugefield. The field B(k) is the Nakanishi–Lautrup auxiliary field: it is completely fixed onceAµ is known since its equation has no derivative. Siegel gauge imposes B = 0 which showsthat it generalizes the Feynman gauge to the string field.

Computation – Equation (13.87)Keeping only the levels 0 and 1 terms in the string field, it is sufficient to truncate theBRST operator as

QB = c0L0 − b0M + QB ,

M ∼ 2c−1c1, QB ∼ c1Lm−1 + c−1Lm1 ,

Lm1 ∼ α0 · α1, Lm−1 ∼ α0 · α−1.

(13.90)

Acting on the string field gives

QB |Φ〉 = 1√α′

∫ dDk(2π)D

(T (k)c0L0 |k, ↓〉+Aµ(k)

(c0L0 + ηνρc−1α

ν1α

ρ0

)αµ−1 |k, ↓〉

+ i√α′

2 B(k)(− 2b0c−1c1 + ηνρc1α

ν−1α

ρ0

)b−1c0 |k, ↓〉

)= 1√

α′

∫ dDk(2π)D

(T (k)(α′k2 − 1)c0 |k, ↓〉

+Aµ(k)(α′k2c0α

µ−1 +

√2α′ηνρηµν kρ c−1

)|k, ↓〉

+ i√α′

2 B(k)(

2c−1 +√

2α′ηνρkραν−1c0

)|k, ↓〉

)= 1√

α′

∫ dDk(2π)D

(T (k)(α′k2 − 1)c0 |k, ↓〉

+ α′(Aµ(k)k2 + ikµB(k

)c0α

µ−1 |k, ↓〉

+√

2α′(kµAµ(k) + iB(k)

)c−1 |k, ↓〉

).

One needs to be careful when anticommuting the ghosts and we used that pL = k andα0 =

√2α′k for the open string. It remains to require that the coefficient of each state

vanishes.In order to confirm that Aµ is indeed a gauge field, we must study the gauge transform-

ation. The gauge parameter is expanded at the first level:

|Λ〉 = i√2α′

∫ dDk(2π)D

(λ(k) b−1 |k, ↓〉+ · · ·

). (13.91)

Note that b−1 | ↓〉 is the SL(2,C) ghost vacuum. Since

QB |Λ〉 = i√α′

∫ dDk(2π)D λ(k)

(−√α′

2 k2 b−1c0 + kµαµ−1

)|k, ↓〉 , (13.92)

matching the coefficients in (13.29) gives

δAµ = −ikµλ, δB = k2λ. (13.93)

This is the appropriate transformation for a U(1) gauge field.

183

Finally, one can derive the action; for simplicity, we work in the Siegel gauge. We consideronly the tachyon component:

|T 〉 =∫ dDk

(2π)D T (k)c1 |k, 0〉 , (13.94)

with c1 |0〉 = | ↓〉. The BPZ conjugate and Hermitian conjugates are respectively:

〈T | =∫ dDk

(2π)D T (k)〈−k, 0| c−1, (13.95a)

〈T ‡| =∫ dDk

(2π)D T (k)∗〈k, 0| c−1. (13.95b)

since ct1 = c−1 when using the operator I− in (22.111). Imposing equality of both leads tothe reality condition

T (k)∗ = T (−k), (13.96)

which agrees with the fact that the tachyon is real (the integration measure changes asdDk → −dDk, but the contour is reversed).

Then, the action reads:

S[T ] = 12

∫ dDk(2π)D T (−k)

(k2 − 1

α′

)T (k). (13.97)

This shows that the action is canonically normalized as it should for a real scalar field.Similarly, one can compute the action for the gauge field:

S[A] = 12

∫ dDk(2π)D Aµ(−k)k2Aµ(k). (13.98)

The correct normalization of the tachyon (real scalar field of negative mass) gives a jus-tification a posteriori for the normalization of the action (13.14). Typically, string fieldactions are normalized in this way, by requiring that the first physical spacetime fields hasthe correct normalization. Note how this implies the correct normalization for all the othersphysical fields. Generalizing this computation for higher-levels, one always find the kineticterm to be:

L+02 = 1

2(k2 +m2), (13.99)

which is the canonical normalization.


〈T | c0L0 |T 〉 = 1α′

∫ dDk(2π)D

dDk′(2π)D T (k)T (k′)〈−k′, 0| c−1c0L0c1 |k, 0〉

= 1α′

∫ dDk(2π)D

dDk′(2π)D T (k)T (k′)(α′k2 − 1)〈−k′, 0| c−1c0c1 |k, 0〉

= 1α′

∫ dDk(2π)D dDk′ T (k)T (k′)(α′k2 − 1) δ(D)(k + k′),

where we used 〈0| c−1c0c1 |0〉 = 1 and 〈k′|k〉 = (2π)Dδ(D)(k + k′).

184

13.5 Closed stringThe derivation of the BRST free action for the closed string is very similar. The startingpoint is the equation of motion

QB |Ψ〉 = 0 (13.100)for the closed string field |Ψ〉. The difference with (13.12) is that the BRST charge QB nowincludes both the left- and right-moving sectors. In the case of the open string, the field Φwas free of any constraint: we will see shortly that this is not the case for the closed string.

The next step is to find an inner product 〈·, ·〉 to write the action:

S = 12 〈Ψ, QBΨ〉. (13.101)

Following the open string, it seems logical to give the string field Ψ the same ghost numberas the states in the cohomology:

Ngh(Ψ) = 2. (13.102)In this case, the ghost number of the arguments of 〈·, ·〉 in (13.101) is Ngh = 5. The ghostnumber anomaly requires the total ghost number to be 6, that is:

Ngh(〈·, ·〉) = 1. (13.103)

There is no other choice because Ngh(Ψ) must be integer. The simplest solution is to insertone c zero-mode c0 or c0, or a linear combination. The BRST operator QB contains bothL±0 (see the decomposition (24.88)): the natural expectation (and by analogy with theopen string) is that the gauge fixed equation of motion (to be discussed below) should beequivalent to the on-shell equation L+

0 = 0 (see also Section 24.3.4). This is possible only ifthe insertion is c−0 . With this insertion, 〈·, ·〉 can be formed from the BPZ product:

〈A,B〉 = 〈A| c−0 |B〉 . (13.104)

Then, the action reads:S = 1

2 〈Ψ| c−0 QB |Ψ〉 . (13.105)

However, the presence of c−0 has a drastic effect because it annihilates part of the stringfield. Decomposing the Hilbert space as in (23.175)

H = H− ⊕ c−0 H−, H− := H ∩ ker b−0 , (13.106)

the string field reads:

|Ψ〉 = |Ψ−〉+ c−0 |Ψ−〉 , Ψ−, Ψ− ∈ H−, (13.107)

such thatc−0 |Ψ〉 = c−0 |Ψ−〉 . (13.108)

The problem in such cases is that the kinetic term may become non-invertible. This motiv-ates to project out the component Ψ− by imposing the following constraint on the stringfield:

b−0 |Ψ〉 = 0. (13.109)The constraint (13.109) is stronger than the constraint L−0 = 0 for states in the cohomo-

logy (Section 24.3.1), so there is no information lost on-shell by imposing it. For this reason,we will also impose the level-matching condition:

L−0 |Ψ〉 = 0, (13.110)

185

such thatΨ ∈ H− ∩ kerL−0 . (13.111)

This will later be motivated by studying the propagator and the off-shell scattering amp-litudes. To avoid introducing more notations, we will not use a new symbol for this spaceand keep implicit that Ψ ∈ kerL−0 .

The necessity of this condition can be understood differently. We had found that it isnecessary to ensure that the closed string parametrization is invariant under translationsalong the string (Section 4.2.2). Since there is no BRST symmetry associated to this sym-metry, one needs to keep the constraint.2 This suggests that one may enlarge further thegauge symmetry and interpret (13.109) as a gauge fixing condition. This would be quitedesirable: one could argue that a fundamental field should be completely described by theLagrangian (if such a description exists) and that it should not be necessary to supplementit with constraints imposed by hand. While this can be achieved at the free level, this idearuns into problems in the presence of interactions (Section 8.3.1) and the interpretation isnot clear.3

The action (13.105) is gauge invariant under:

|Ψ〉 −→ |Ψ′〉 = |Ψ〉+ δΛ |Ψ〉 , δΛ |Ψ〉 = QB |Λ〉 , (13.112)

where the gauge parameter has ghost number 1 and also lives in H− ∩ kerL−0 :

Ngh(Λ) = 1, L−0 |Λ〉 = 0, b−0 |Λ〉 = 0. (13.113)

As for the open string, the gauge invariance (13.112) can be gauge fixed in the Siegelgauge:

b+0 |Ψ〉 = 0. (13.114)

Then, the action reduces to:

S = 12 〈Ψ| c

−0 c

+0 L

+0 |Ψ〉 = 1

4 〈Ψ| c0c0L+0 |Ψ〉 . (13.115)

The equation of motion is to the on-shell condition as expected:

L+0 |Ψ〉 = 0. (13.116)

Additional constraints must be imposed to ensure that only the physical degrees of freedompropagate.


c−0 QB = (c0 − c0)(c0L0 + c0L0) = c0c0(L0 + L0).

13.6 SummaryIn this chapter, we have shown how the BRST conditions defining the cohomology can beinterpreted as an equation of motion for a string field together with a gauge invariance. We

2Yet another reason can be found in Section 4.2.2 (see also Section 24.3.4): to motivate the need of theb+0 condition, we could take the on-shell limit from off-shell states because L+

0 is continuous. However, theL−0 operator is discrete and there is no such limit we can consider [278]. So we must always impose thiscondition, both off- and on-shell.

3A recent proposal can be found in [202].

186

found a subtlety for the closed string due to the ghost number anomaly and because of thelevel-matching condition. Then, we studied several basic properties in order to prove thatthe free action has the expected properties.

The next step is to add the interactions to the action, but we don’t know first principlesto write them. For this reason, we need to take a detour and to consider off-shell amplitudes.By introducing a factorization of the amplitudes, it is possible to rewrite them as Feynmandiagrams, where fundamental interactions are connected by propagators (which we will findto match the one in the Siegel gauge). This can be used to extract the interacting terms ofthe action.

13.7 References• The free BRST string field theory is discussed in details in [278] (see also [143, chap. 7,

144, chap. 9, 265, chap. 11]). Shorter discussions can be found in [296, 217, sec. 4,288, 4, 277].

• Spacetime fields and actions are discussed in [275, 217, sec. 4].

• Gauge fixing [169, 275, sec. 6.5, 7.2, 7.4, 4, 156, sec. 2.1, 5, 33].

• General properties of string field (reality, parity, etc.) [4, 297].

187

Chapter 14

Closed string field theory

We bring together the elements from the previous chapters in order to write the closed stringfield action. We first study the gauge fixed theory before reintroducing the gauge invariance.We then prove that the action satisfies the BV master equation meaning that closed SFT iscompletely consistent at the quantum level. Finally, we describe the 1PI effective action.

14.1 Closed string field expansionIn Chapters 6, 8 and 9, constraints on the external and internal states were found to benecessary. But, to provide another perspective and decouples the properties of the field fromthe ones of the state, we assume that the string field does not obey any constraint. Theywill be derived later in order to reproduce the scattering amplitudes from the action and tomake the latter well-defined.

The string field is expanded on a basis φr of the CFT Hilbert space H (see Section 6.2for more details)

|Ψ〉 =∑r

ψr |φr〉 . (14.1)

Using the decomposition (6.45) of the Hilbert space according to the ghost zero-modes, thestring field can also be expanded as

|Ψ〉 =∑r

(ψ↓↓,r |φ↓↓,r〉+ ψ↓↑,r |φ↓↑,r〉+ ψ↑↓,r |φ↑↓,r〉+ ψ↑↑,r |φ↑↑,r〉

), (14.2)

where we recall that the basis states satisfy

b0 |φ↓↓,r〉 = b0 |φ↓↓,r〉 = 0, b0 |φ↓↑,r〉 = c0 |φ↓↑,r〉 = 0,c0 |φ↑↓,r〉 = b0 |φ↑↓,r〉 = 0, c0 |φ↑↑,r〉 = c0 |φ↑↑,r〉 = 0.

(14.3)

We recall the definition of the dual basis φcr through the BPZ inner product

〈φcr|φs〉 = δrs. (14.4)

In terms of the ghost decomposition, the components of the dual states satisfy:

〈φc↓↓,r| c0 = 〈φc↓↓,r| c0 = 0, 〈φc↓↑,r| c0 = 〈φc↓↑,r| b0 = 0,〈φc↑↓,r| b0 = 〈φc↑↓,r| c0 = 0, 〈φc↑↑,r| b0 = 〈φc↑↑,r| b0 = 0,

〈φcx,r|φy,s〉 = δxyδrs,

(14.5)

188

where x, y =↓↓, ↑↓, ↓↑, ↑↑. The spacetime ghost number of the fields ψr is defined by

G(ψr) = 2− nr. (14.6)

Remember that the ghost number of the basis states are denoted by

nr = Ngh(φr), ncr = Ngh(φcr) = 6− nr. (14.7)

14.2 Gauge fixed theoryHaving built the kinetic term (Chapter 11), one needs to construct the interactions. Forthe same reason – our ignorance of SFT first principles – that forced us to start with thefree equation of motion to derive the quadratic action (Chapter 13), we also need to inferthe interactions from the scattering amplitudes. Preparing the stage for this analysis wasthe goal of Chapter 9, where we introduced the factorization of amplitudes to derive thefundamental interactions.

Scattering amplitudes are expressed in terms of gauge fixed states since only them arephysical. This allows to give an alternative derivation of the kinetic term by defining it asthe inverse of the propagator, which is well-defined for gauge fixed states.1 The price topay by constructing interactions in this way is that the SFT action itself is gauge fixed. Toundercover its deeper structure it is necessary to release the gauge fixing condition. In viewof the analysis of the quadratic action in Chapter 13, we can expect that the BV formalismis required. Another possibility is to consider directly the 1PI action.

In this section, we first derive the kinetic term by inverting the propagator. For this tobe possible, the string field must obey some constraints: we will find that they correspondto the level-matching and Siegel gauge conditions. Then, we introduce the interactions intothe action.

14.2.1 Kinetic term and propagatorIn Chapter 9, it was found that the propagator reads (9.41):

∆ = b+0 b−0

1L+

0δL−0 ,0

, ∆rs = 〈φcr| b+0 b−01L+

0δL−0 ,0

|φcs〉 . (14.8)

The most natural guess for the kinetic term is

S0,2 = 12 〈Ψ|K |Ψ〉 = 1

2 ψrKrsψs (14.9)

whereK = c−0 c

+0 L

+0 δL−0 ,0

Krs = 〈φr| c−0 c+0 L+0 δL−0 ,0

|φs〉 . (14.10)

Indeed, it looks like K∆ = 1 using the identities c±0 b±0 ∼ 1 and it matches (13.115). Interms of the holomorphic and anti-holomorphic modes, we have

K = 12 c0c0L

+0 δL−0 ,0

. (14.11)

But, when writing c±0 b±0 ∼ 1, the second part of the anti-commutator b±0 , c±0 = 1 ismissing. The relation c±0 b±0 ∼ 1 is correct only when acting on basis dual states annihilated

1This step is not necessary because the propagator corresponding to the plumbing fixture (Section 9.2.2)matches the one found in Section 13.5 by considering the simplest gauge fixing. However, this would havebeen necessary if the factorization had given another propagator, or if the structure of the theory was morecomplicated, for example for the superstring.

189

by c±0 . The problem stems from the fact that Ψ is not yet subject to any constraint.Moreover, some of the string field components will not appear in the expression since theyare annihilated by the ghost zero-mode. As a consequence, the kinetic operator in (14.10)(or equivalently the propagator) is not invertible in the Hilbert space H because its kernelis not empty:

kerK|H 6= ∅. (14.12)This can be seen by writing φr as a 4-vector and Krs as a 4× 4-matrix:

Krs = 12

〈φ↓↓,r|〈φ↓↑,r|〈φ↑↓,r|〈φ↑↑,r|

t

c0c0L+0 0 0 0

0 0 0 00 0 0 00 0 0 0

|φ↓↓,s〉|φ↓↑,s〉|φ↑↓,s〉|φ↑↑,s〉

. (14.13)

The matrix is mostly empty because the states φx,r with different x =↓↓, ↑↓, ↓↑, ↑↑ areorthogonal (no non-diagonal terms) and the states with x 6=↓↓ are annihilated by c0 or c0.The same consideration applies for the delta-function: if the field does not satisfy L−0 = 0,then the kinetic operator is non-invertible.

To summarize the string field must satisfy three conditions in order to have an invertiblekinetic term

L−0 |Ψ〉 = 0, b−0 |Ψ〉 = 0, b+0 |Ψ〉 = 0. (14.14)This means that the string field is expanded on the H0 ∩ kerL−0 Hilbert space:

|Ψ〉 =∑r

ψ↓↓,r |φ↓↓,r〉 . (14.15)

Ill-defined kinetic terms are expected in the presence of a gauge symmetry: this was alreadydiscussed in Sections 13.1.4 and 13.5 for the free theory, and this will be discussed furtherlater in this chapter for the interacting case.

ComputationLet’s check that Krs is correctly the inverse of ∆rs when Ψ is restricted to H0:

Krs∆st = 〈φr| c−0 c+0 L+0 δL−0 ,0

|φs〉〈φcs| b+0 b−0

1L+

0δL−0 ,0

|φct〉

= 〈φr| c−0 c+0 L+0 δL−0 ,0

b+0 b−0

1L+

0δL−0 ,0

|φct〉

= 〈φr| c−0 , b−0 c+0 , b+0 |φct〉= 〈φr|φct〉 = δrt.

The second equality follows from the resolution of the identity (6.36): due to the zero-mode insertions, the resolution of the identity collapses to a sum over the ↓↓ states

1 =∑r

|φr〉〈φcr| =∑r

|φ↓↓,r〉〈φc↓↓,r| . (14.16)

The third equality uses that L+0 commutes with the ghost modes, that φr is annihilated

by b±0 , and that (δL−0 ,0)2 = δL−0 ,0= 1 on states with L−0 = 0.

Finally, we find that the kinetic term matches the classical quadratic vertex V0,2 definedin (9.56) such that

S0,2 = 12 V0,2(Ψ2) = 1

2 〈Ψ| c−0 c

+0 L

+0 δL−0 ,0

|Ψ〉 . (14.17)

190

14.2.2 InteractionsThe second step to build the action is to write the interaction terms from the Feynman rules.Before proceeding to SFT, it is useful to remember how this works for a standard QFT.

Example 14.1 – Feynman rules for a scalar fieldConsider a scalar field with a standard kinetic term and a n-point interaction:

S =∫

dDx(

12 φ(x)(−∂2 +m2)φ(x) + λ

n! φ(x)n). (14.18)

First, one needs to find the physical states, which correspond to solutions of the lin-earised equation of motion. In the current case, they are plane-waves (in momentumrepresentation):

φk(x) = eik·x. (14.19)

Then, the vertex (in momentum representation) Vn(k1, . . . , kn) is found by replacingin the interaction each occurrence of the field by a different state, and summing overall the different contributions. Here, this means that one considers states φki(x) withdifferent momenta:

Vn(k1, . . . , kn) = λ

n!

∫dDxn!

n∏i=1

φki(x) = λ

∫dDx ei(k1+···+kn)x

= λ(2π)D δ(D)(k1 + · · ·+ kn).(14.20)

The factor n! comes from all the permutations of the n states in the monomial of ordern. Reversing the argument, one sees how to move from the vertex Vn(k1, . . . , kn) writtenin terms of states to the interaction in the action in terms of the field.

Obviously, if the field has more states (for example if it has a spin or if it is in arepresentation of a group), then one needs to consider all the different possibilities. Theabove prescription also yields directly the insertion of the momentum necessary if theinteraction contains derivatives.

In Section 9.2, the Feynman rule for a g-loop n-point fundamental vertex of states(V1, . . . ,Vn) was found to be given by (9.33):

Vg,n(V1, . . . ,Vn) =∫Rg,n

ωg,nMg,n(V1, . . . ,Vn) = (14.21)

where Rg,n is a section over the fundamental region Vg,n ⊂ Mg,n (7.42b) which cannot becovered from the plumbing fixture of lower-dimensional surfaces.

From the example Example 14.1, it should be clear that the g-loop n-point contributionto the action can be obtained simply by replacing every state with a string field in Vg,n:

Sg,n = ~gg2g−2+ns

n! Vg,n(Ψn). (14.22)

where Ψn := Ψ⊗n. The power of the coupling constant has been reinstated: it can bemotivated by the fact that it should have the same power as the corresponding amplitude(Section 4.1.1). Note that the interactions are defined only when the power of gs is positive:

191

χg,n = 2 − 2g − n < 0. We have also written explicitly the power of ~, which counts thenumber of loops.

Before closing this section, we need to comment on the effect of the constraints (14.14)on the interactions. Building a Feynman graph by gluing two m- and n-point interactionswith a propagator, one finds that the states proportional to φx,r for x 6=↓↓ do not propagateinside internal legs

Vg,m(V1, . . . ,Vm−1, φr)〈φcr| b+0 b−01L+

0|φcs〉 Vg′,n(W1, . . . ,Wn−1, φs)

= Vg,m(V1, . . . ,Vm−1, φ↓↓,r)〈φc↓↓,r| b+0 b−01L+

0|φc↓↓,s〉 Vg′,n(W1, . . . ,Wn−1, φ↓↓,s). (14.23)

Thus, they do not contribute to the final result even if the interactions contain them. Whilethe conditions L−0 = b−0 = 0 were found to be necessary for defining off-shell amplitudes, thecondition b+0 = 0 does not arise from any consistency requirement. But, it is also consistentwith the interactions, since only fundamental vertices have a chance to give a non-vanishingresult for states which do not satisfy (14.14). Hence, the interactions (14.22) are compatiblewith the definition of the kinetic term and the restriction of the string field.

14.2.3 ActionThe interacting gauge-fixed action is built from the kinetic term V0,2 (14.17) and from theinteractions Vg,n (14.22) with χg,n < 0. However, this is not sufficient: we have seen inSection 9.3 that it makes sense to consider the vertices with χg,n ≥ 0. First, we shouldconsider the 1-loop cosmological constant V1,0. Then, we can also add the classical sourceV0,1 and the tree-level cosmological constant V0,0. With all the terms together, the actionreads:

S =∑g,n≥0

~gg2g−2+ns

n! Vg,n(Ψn)

:= 12 〈Ψ| c

−0 c

+0 L

+0 δL−0 ,0

|Ψ〉+∑′

g,n≥0~gg2g−2+ns

n! Vg,n(Ψn).(14.24)

where Vn was defined in (9.58). A prime on the sum indicates that the term g = 0, n = 2is removed, such that one can single out the kinetic term. We will often drop the deltafunction imposing L−0 = 0 because the field are taken to satisfy this constraint.

Rewriting the vertices in terms of the products `g,n defined in (9.65)

Vg,n(Ψn) := 〈Ψ| c−0∣∣`g,n−1(Ψn−1)

⟩(14.25)

leads to the alternative form

S =∑g,n≥0

~gg2g−2+ns

n! 〈Ψ| c−0∣∣`g,n−1(Ψn−1)

⟩. (14.26)

The definition (9.56) leads to the following explicit expression for `0,1:

`0,1(Ψcl) = c+0 L+0 |Ψcl〉 . (14.27)

In most cases, the terms g = 0, n = 0, 1 vanish such that the action reads:

S =∑g,n≥0χg,n≤0

~gg2g−2+ns

n! Vg,n(Ψn). (14.28)

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However, we will often omit the condition χg,n ≤ 0 to simplify the notation, except when thedistinction is important, and the reader can safely assumes V0,0 = V0,1 = 0 if not otherwisestated. The classical action is obtained by setting ~ = 0:

Scl = 12 〈Ψcl| c−0 c

+0 L

+0 |Ψcl〉+

∑n≥3

gnsn! V0,n(Ψn

cl). (14.29)

Rescaling the string field by g−1s gives the more canonical form of the action (using the

same symbol):

S =∑g,n≥0

~gg2g−2s

1n! Vg,n(Ψn)

:= 12g2s

〈Ψ| c−0 c+0 L+0 δL−0 ,0

|Ψ〉+ 1g2s

∑′

g,n≥0

(~g2s)gn! Vg,n(Ψn).

(14.30)

In the path integral, the action is divided by ~ such that

S

~=∑g,n≥0

(~g2s)g−1 1

n! Vg,n(Ψn). (14.31)

This shows that there is a single coupling constant ~g2s , instead of two (~ and gs separately)

as it looks at the first sight. This makes sense because gs is in fact the expectation valueof the dilaton field (3.166) and its value can be changed by deforming the background withdilatons [18, 19, 222].

The previous remark also allows to easily change the normalization of the action, forexample, to perform a Wick rotation, to normalize canonically the action in terms of space-time fields, or reintroduce ~. Rescaling the action by α is equivalent to rescale g2

s by α−1:

S → αS =⇒ g2s →

g2s

α. (14.32)

The linearized equation of motion is:

L+0 |Ψ〉 = 0, (14.33)

which corresponds to the Siegel gauge equation of motion of the free theory (13.116). Hence,this equation is not sufficient to determine the physical states (cohomology of the BRSToperator, Chapter 24), as discussed in Chapter 13, and additional constraints must beimposed. One can interpret this by saying that the action (14.24) provides only the Feynmanrules, not the physical states. Removing the gauge fixing will be done in Sections 14.3and 14.4.

The action (14.24) looks overly more complicated than a typical QFT theory: insteadof few interaction terms for low n (n ≤ 4 in d = 4 renormalizable theories), it has contactinteractions of all orders n ∈ N. The terms with g ≥ 1 are associated to quantum correctionsas indicate the power of ~, which means that they can be interpreted as counter-terms. But,how is it that one needs counter-terms despite the claim that every Feynman graphs (includ-ing the fundamental vertices) in SFT are finite? The role of renormalization is not only tocure UV divergences, but also IR divergences (due to vacuum shift and mass renormaliza-tion). Equivalently, this can be understood by the necessity to correct the asymptotic statesof the theory, or to consider renormalized instead of bare quantities. Indeed, the asymptoticstates obtained from the linearized classical equations of motion are idealization: turningon interactions modify the states. In typical QFTs, these corrections are infinite and renor-malization is crucial to extract a number; however, even if the effect is finite, it is needed to

193

describe correctly the physical quantities [285, p. 411]. There is a second reason for theseadditional terms: when relaxing the gauge fixing condition, the path integral is anomalousunder the gauge symmetry, and the terms with g > 0 are necessary to cancel the anomaly(this will be discussed more precisely in Section 14.4). It may thus seem that SFT cannotbe predictive because of the infinite number of counter-terms. Fortunately, this is not thecase: the main reason for the loss of predictability in non-renormalizable theory is thatthe renormalization procedure introduces an infinite number2 of arbitrary parameters (andthus making a prediction would require to have already made an infinite number of observa-tions to determine all the parameters). These parameters come from the subtraction of twoinfinities: there is no unique way to perform it and thus one needs to introduce a new para-meter. The case of SFT is different: since every quantity is finite, the renormalization hasno ambiguity because one subtracts two finite numbers, and the result is unambiguous. Asa consequence, renormalization does not introduce any new parameter and there is a uniquecoupling constant gs in the theory, which is determined by the tree-level cubic interaction.The coupling constants of higher-order and higher-loop interactions are all determined bypowers of gs, and thus a unique measurement is sufficient to make predictions.

Another important point is that the action (14.24) is not uniquely defined. The definitionof the vertices depends on the choice of the local coordinates and of the stub parameter s0.Changing them modifies the vertices, and thus the action. But, one can show that thedifferent theories are related by field redefinitions and are thus equivalent.

14.3 Classical gauge invariant theoryIn the previous section, we have found the gauge fixed action (14.24). Since the completegauge invariant quantum action has a complicated structure, it is instructive to first focuson the classical action (14.29). The full action is discussed in Section 14.4.

The gauge fixing is removed by relaxing the b+0 = 0 constraint on the field (the otherconstraints must be kept in order to have well-defined the interactions). The classical fieldΨcl is then defined by:

Ψcl ∈ H− ∩ kerL−0 , Ngh(Ψcl) = 2. (14.34)

The restriction on the ghost number translates the condition that the field is classical, i.e.that there are no spacetime ghosts at the classical level. The relation (14.6) implies that allcomponents have vanishing spacetime ghost number.

In the free limit, the gauge invariant action should match (13.105)

S0,2 = 12 〈Ψ| c

−0 QB |Ψ〉 . (14.35)

and lead to the results from Section 13.5. A natural guess is that the form of the interactionsis not affected by the gauge fixing (the latter usually modifies the propagator but not theinteractions). This leads to the gauge invariant classical action:

Scl = 12 〈Ψcl| c−0 QB |Ψcl〉+ 1

g2s

∑n≥3

gnsn! V0,n(Ψn

cl), (14.36)

where the vertices V0,n with n ≥ 3 are the ones defined in (9.33) (we consider the case whereV0,0 = V0,1 = 0). It is natural to generalize the definition of V0,2 as:

V0,2(Ψ2cl) := 〈Ψcl| c−0 QB |Ψcl〉 (14.37)

2In practice, this number does not need to be infinite to wreck predictability, it is sufficient that it isvery large.

194

such that

Scl = 1g2s

∑n≥2

gnsn! V0,n(Ψn

cl) = 1g2s

∑n≥2

gnsn! 〈Ψcl| c−0

∣∣`0,n−1(Ψn−1cl )

⟩, (14.38)

where (14.37) implies:`0,1(Ψcl) = QB |Ψcl〉 . (14.39)

The equation of motion is

Fcl(Ψcl) :=∑n≥1

gn−1s

n! `0,n(Ψncl) = QB |Ψcl〉+

∑n≥2

gn−1s

n! `0,n(Ψncl) = 0. (14.40)


δScl = 1g2s

∑n≥2

gnsn! nδΨcl,Ψn−1

cl 0 = 1g2s

∑n≥2

gns(n− 1)! 〈δΨcl| c−0

∣∣`0,n−1(Ψn−1cl⟩. (14.41)

The first equality follows because the vertex is completely symmetric. Simplifying andshifting n, one obtains c−0 |Fcl〉. The factor c−0 is invertible because of the constraintb−0 = 0 imposed on the field.

The action is invariantδΛScl = 0 (14.42)

under the gauge transformation

δΛΨcl =∑n≥0

gnsn! `0,n+1(Ψn

cl,Λ) = QB |Λ〉+∑n≥1

gnsn! `0,n+1(Ψn

cl,Λ). (14.43)

The gauge algebra is [297, sec. 4]:

[δΛ2 , δΛ1 ]Ψcl = δΛ(Λ1,Λ2,Ψcl) |Ψcl〉+∑n≥0

gn+2s

n! `0,n+3(Ψncl,Λ2,Λ1,Fcl(Ψcl)

), (14.44a)

where Fcl is the equation of motion (14.40), and Λ(Λ1,Λ2,Ψcl) is a field-dependent gaugeparameter:

Λ(Λ1,Λ2,Ψcl) =∑n≥0

gn+1s

n! `0,n+2(Λ1,Λ2,Ψncl)

= gs `0,2(Λ1,Λ2) +∑n≥1

gn+1s

n! `0,n+2(Λ1,Λ2,Ψncl).

(14.44b)

The classical gauge algebra is complicated which explains why a direct quantization (forexample through the Faddeev–Popov procedure) cannot work: the second term in (14.44a)indicates that the algebra is open (it closes only on-shell), while the first term is a gaugetransformation with a field-dependent parameter. As reviewed in Appendix C.3, both prop-erties require using the BV formalism for the quantization, and the latter is performed inSection 14.4. An important point is that if the theory had only cubic interactions, i.e. if

∀n ≥ 4 : V0,4(V1, . . . ,Vn) = 0, `g,n−1(V1, . . . ,Vn−1) = 0, (cubic theory), (14.45)

then the algebra closes off-shell and Λ(Λ1,Λ2,Ψcl) becomes field-independent.

195


δΛScl =∑n≥2

gn−2s

n! nV0,n(δΨcl,Ψn−1cl ) =

∑m,n≥0

gm+n−1s

m!n! V0,n+1(`0,m+1(Ψm

cl ,Λ),Ψncl)

=∑m≥0

m∑n=0

gm−1s

(m− n)!n!⟨`m−n+1(Ψm−n

cl ,Λ)∣∣ c−0 |`0,n(Ψn

cl)〉 .

For simplicity we have extended the sum up to n = 0 and m = 0 by using the fact thatlower-order vertices vanish. The bracket can be rewritten as

= 〈`0,n(Ψncl)| c−0

∣∣`0,m−n+1(Ψm−ncl ,Λ)

⟩= V0,m−n+2

(`0,n(Ψn

cl),Ψm−ncl ,Λ

)= −V0,m−n+2

(Λ, `0,n(Ψn

cl),Ψm−ncl

)= 〈Λ| c−0

∣∣`0,m−n+1(`0,n(Ψn

cl),Ψm−ncl

)⟩.

Then, one needs to use the identity (defined for all m ≥ 0)

0 =m∑n=0

m!(m− n)!n! `0,m−n+1

(`0,n(Ψn

cl),Ψm−ncl

), (14.46)

which comes from (9.70). Multiplying this by gm−1s /m! and summing overm ≥ 0 proves

(14.42).

Remark 14.1 (L∞ algebra) The identities satisfied by the products `0,n from the gaugeinvariance of the action implies that they form a L∞ homotopy algebra [86, 190, 297] (formore general references, see [121, 122, 170, 171]). The latter can also be mapped to a BVstructure, which explains why the BV quantization Section 14.4 is straightforward. Thisinterplay between gauge invariance, covering of the moduli space, BV and homotopy algebrais particularly beautiful. It has also been fruitful in constructing super-SFT.

14.4 BV theoryAs indicated in the previous section (Section 14.3), the classical gauge algebra is open andhas field-dependent structure constants. The BV formalism (Appendix C.3) is necessary todefine the theory.

In the BV formalism, the classical action for the physical fields is extended to thequantum master action by solving the quantum master equation (C.120). It is generic-ally difficult to build this action exactly, but the discussion of Section 13.3 can serve as aguide: it was found that the free quantum action (with the tower of ghosts) has exactly thesame form as the free classical action (without ghosts). Hence, this motivates the ansatzthat it should be of the same form as the classical action (14.36) to which are added the

196

counter-terms from (14.24):

S = 1g2s

∑g≥0

~gg2gs

∑n≥0

gnsn! Vg,n(Ψn) (14.47a)

= 12 〈Ψ| c

−0 QB |Ψ〉+

∑′

g,n≥0

~gg2g−2+ns

n! Vg,n(Ψn) (14.47b)

= 1g2s

∑g,n≥0

~gg2g−2+ns

n! 〈Ψ| c−0∣∣`g,n−1(Ψn−1)

⟩, (14.47c)

but without any constraint on the ghost number of Ψ:

Ψ ∈ H− ∩ kerL−0 . (14.48)

In order to show that (14.47) is a consistent quantum master action, it is necessary toshow that it solves the master BV equation (C.120):

(S, S)− 2~∆S = 0. (14.49)

The first step is to introduce the fields and antifields. In fact, because the CFT ghost numberinduces a spacetime ghost number, there is a natural candidate set.

The string field is expanded as (14.1)

|Ψ〉 =∑r

ψr |φr〉 , (14.50)

where the φr forms a basis of H−. The string field can be further separated as:

Ψ = Ψ+ + Ψ−, (14.51)

where Ψ− (Ψ+) contains only states which have negative (positive) cylinder ghost numbers(this gives an offset of 3 when using the plane ghost number):

Ψ− =∑r

∑nr≤2

|φr〉ψr, Ψ+ =∑r

∑ncr>2

b−0 |φcr〉ψ∗r . (14.52)

The order of the basis states and coefficients matter if they anti-commute. The sum in Ψ+can be rewritten as a sum over nr ≤ 2 like the first term since nr+ncr = 6. Correspondingly,the spacetime ghost numbers (14.6) for the coefficients in Ψ− (Ψ+) are positive (negative)

G(ψr) ≥ 0, G(ψ∗r ) < 0. (14.53)

Moreover, one finds that the ghost numbers of ψr and ψ∗r are related as:

G(ψ∗r ) = −1−G(ψr), (14.54)

which also implies that they have opposite parity. Comparing with Appendix C.3, this showsthat the ψr (ψ∗r ) contained in Ψ− (Ψ+) can be identified with the fields (antifields).


G(ψ∗r ) = 2−Ngh(b−0 φcr) = 2 + 1− ncr= 3− (6− nr) = −3 + (2−G(ψr)) = −1−G(ψr).

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In terms of fields and antifields, the master action is

∂RS

∂ψr∂LS

∂ψ∗r+ ~

∂R∂LS

∂ψr∂ψ∗r= 0. (14.55)

Plugging the expression (14.47) of S inside and requiring that the expression vanishes orderby order in g and n give the set of equations:∑

g1,g2≥0g1+g2=g

∑n1,n2≥0n1+n2=n

∂RSg1,n1

∂ψr∂LSg2,n2

∂ψ∗r+ ~

∂R∂LSg−1,n∂ψr∂ψ∗r

= 0, (14.56)

where Sg,n was defined in (14.22). This holds true due to the identity (9.70) (the completeproof can be found in [297, pp. 42–45]). The fact that the second term is not identically zeromeans that the measure is not invariant under the classical gauge symmetry (anomaloussymmetry): corrections need to be introduced to cancel the anomaly. It is a remarkablefact that one can construct directly the quantum master action in SFT and that it takesthe same form as the classical action.

14.5 1PI theoryThe BV action is complicated: instead, it is often simpler and sufficient to work with the1PI effective action. The latter incorporates all the quantum corrections in 1PI verticessuch that scattering amplitudes are expressed only in terms of tree Feynman graphs (thereare no loops in diagrams since they correspond to quantum effects, already included in thedefinitions of the vertices).

A 1PI graph is a Feynman graph which stays connected if one cuts any single internalline. On the other hand, a 1PR graph splits in two disconnected by cutting one of the line.The scattering amplitudes Ag,n are built by summing all the different ways to connect two1PI vertices with a propagator: diagrams connecting two legs of the same 1PI vertex areforbidden by definition.

The g-loop n-point 1PR and 1PI Feynman diagrams are associated to some regions ofthe moduli space Mg,n. Comparing the previous definitions with the gluing of Riemannsurfaces (Section 7.3), 1PR diagrams are obtained by gluing surfaces with the separatingplumbing fixture (Section 9.1.1). Thus, the 1PR and 1PI regions F1PR

g,n and V1PIg,n can beidentified with the regions defined in (7.43a) and (7.43b). In particular, the n-point 1PIinteraction is the sum over g of the g-loop n-point 1PI interactions (9.63):

V1PIn (V1, . . . ,Vn) := :=∑g≥0

(~g2s)g V1PIg,n (V1, . . . ,Vn),

V1PIg,n (V1, . . . ,Vn) :=∫R1PIg,n

ωg,nMg,n(V1, . . . ,Vn),

(14.57)

where R1PIg,n is a section of Pg,n over V1PIg,n .

Given the interactions vertices, it is possible to follow the same reasoning as in Sec-tions 14.2 and 14.3.

198

The gauge fixed 1PI effective action reads:

S1PI = 1g2s

∑n≥0

gnsn! V

1PIn (Ψn) := 1

2 〈Ψ| c−0 c

+0 L

+0 |Ψ〉+ 1

g2s

∑′

n≥0

gnsn! V

1PIn (Ψn). (14.58)

Here, the prime means again that the term g = 0, n = 2 is excluded from the definition ofV1PI2 . The action has the same form as the classical gauge fixed action (14.29), which islogical since it generates only tree-level Feynman graphs. For this reason the vertices V1PIn

have exactly the same properties as the brackets V0,n. This fact can be used to write the1PI gauge invariant action:

S1PI = 12 〈Ψ| c

−0 QB |Ψ〉+ 1

g2s

∑′

n≥0

gnsn! V

1PIn (Ψn), (14.59)

which mirrors the classical gauge invariant action (14.36). Then it is straightforward to seethat it enjoys the same gauge symmetry upon replacing the tree-level vertices by the 1PIvertices. But, since this action incorporates all quantum corrections this also proves thatthe quantum theory is correctly invariant under a quantum gauge symmetry.

Remark 14.2 The 1PI action (14.59) can also be directly constructed from the BV action(14.47).

14.6 References• Gauge fixed and classical gauge invariant closed SFT [297] (see also [160, 161]).

• BV closed SFT [297] (see also [278]).

• Construction of the open–closed BV SFT [299].

• 1PI SFT [243, 244, 55, sec. 4.1, 5.2].

199

Chapter 15

Witten’s cubic string fieldtheory

Witten’s string field theory is an open SFT which arises from the open BV SFT by a specificparametrization of the cubic vertex. It distinguishes itself among all possible SFT by twofacts. First it is cubic (and thus polynomial): all interactions Φn with n ≥ 4 vanish, whichmakes it particularly simple to study. From the Riemann surface perspective this meansthat the moduli space of any surface can be entirely covered from the plumbing fixture of3-punctured spheres. Secondly there is an axiomatic formulation of this theory based on ananalogy with Chern–Simons theory.

In this section we consider an open string theory defined by a given boundary CFT(BCFT).

15.1 Axiomatic formulationThe theory is formulated in terms of a non-commutative ∗-product

A ∗B 6= B ∗A. (15.1)

The product of n identical elements is also written as a power

An := A ∗ · · · ∗A︸︷︷︸n times

. (15.2)

Given this product a notion of integration is defined to obtain a number from an operator.To agree with the previous definition one has∫

A ∗B = 〈A|B〉 (15.3)

where the bracket is the BPZ inner product. This also implies the relations

A ∗B = [A,B],∫A ∗B ∗ C = A,B,C. (15.4)

The action readsS =

∫ (12 Φ ∗QBΦ + 1

3 Φ3). (15.5)

The equations of motion areQBΦ + Φ2 = 0 (15.6)

200

and the gauge transformation is

δΦ = QBΦ + Φ ∗ Λ− Λ ∗ Φ. (15.7)

This transformation can be exponentiated to

Φ −→ Φ′ = e−Λ ∗ (QB + Φ) ∗ eΛ. (15.8)

15.2 Bibliography• Witten’s cubic SFT was constructed in [288].

• Quantization of Witten’s theory [278].

• Most reviews and lecture notes describe Witten’s theory [3, 102, 198, 201, 242, 274,275].

201

Chapter 16

Homotopy algebras in stringfield theory

Classical open and closed string field theories enjoy respectively an homotopy associative A∞and a Lie L∞ algebras (see Chapter 26 for the mathematical definitions). These structuresare useful to derive various properties of the theories, such as background independence.Moreover, while the bosonic theories can be constructed axiomatically from the minimalarea prescription, it is not known how to generalize this approach to the superstring theories;in this case the A∞ and L∞ structures furnish a strong guiding principle.

16.1 Closed string field theoryThe classical closed SFT (??) is written in the L∞ language as

S =∑n≥2

1n! ω

(Ψ, `n−1(Ψn−1)

)(16.1)

Introducing an interpolating field Ψ(t) with t ∈ [0, 1] such that Ψ(0) = 0 and Ψ(1) = Ψ,the action can be rewritten as

S =∫ 1

0dt ω

(Ψ(t), `eΨ(t)). (16.2)


S =∑n≥2

1n! ω

(Ψ, `n−1(Ψn−1)

)=∑n≥1

1(n+ 1)! ω

(Ψ, `n(Ψn)

)=∑n≥1

1(n+ 1)!

∫ 1

0dt d

dt ω(Ψ(t), `n(Ψ(t)n)

)=∑n≥1

1n!

∫ 1

0dt ω

(Ψ(t), `n(Ψ(t)n)

).

The third equality uses that the integral of a total derivative reduces to the boundaryterms, given here by t = 1. The fourth equality follows from the symmetry of the inner

202

product and of the `n under the exchange of the arguments. Next it remains to replace`n by `n and then by ` = `(1) (since only one term will contribute in each case), andto exponentiate the sum of powers of Ψn (the product `0 vanishes).

203

Chapter 17

Background independence

Spacetime background independence is a fundamental property of any candidate quantumgravity theory. In this chapter, we outline the proof of background independence for theclosed SFT by proving that the equations of motion of two background related by a marginaldeformation are equivalent after a field redefinition.

17.1 The concept of background independenceBackground independence means that the formalism does not depend on the background– if any – used to write the theory. A dependence in the background would imply thatthere is a distinguished background among all possibilities, which seems in tension with thedynamics of spacetime and the superposition principle from quantum mechanics. Moreover,one would expect a fundamental theory to tell which backgrounds are consistent and thatthey could be derived instead of postulated. Background independence allows spacetime toemerge as a consequence of the dynamics of the theory and of its defining fundamental laws.

Background independence can be manifest or not. In the second case, one needs to fix abackground to define the theory, but the dynamics on different backgrounds are physicallyequivalent.1 This implies that two theories with different backgrounds can be related, forexample by a field redefinition.

While fields other than the metric can also be expanded around a background, no diffi-culty is expected in this case. Indeed, the topic of background independence is particularlysensible only for the metric because it provides the frame for all other computations – andin particular for the questions of dynamics and quantization. Generally, these questions aresubsumed into the problem of the emergence of time in a generally covariant theory. Inthe previous language, QFTs without gravity are (generically) manifestly background inde-pendent after minimal coupling.2 For example, a classical field theory is defined on a fixedMinkowski background and a well-defined time is necessary to perform its quantization andto obtain a QFT, but it is not needed to choose a background for the other fields. For thisreason, the extension of a QFT on a curved background is generally possible if the space-time is hyperbolic, implying that there is a distinguished time direction. But the couplingto gravity is difficult and restricted to a (semi-)classical description.

What is the status of background independence in string theory? The worldsheet for-mulation requires to fix a background (usually Minkowski) to quantize the theory and tocompute scattering amplitudes. Thus, the quantum theory is at least not manifestly back-

1This does not mean that the physics in all backgrounds are identical, but that the laws are. Hence, acomputation made in one specific background can be translated into another background.

2However, non-minimal coupling terms may be necessary to make the theory physical.

204

ground independent. On the other hand, the worldsheet action can be modified to a genericCFT including a generic non-linear sigma model describing an arbitrary target spacetime.Conformal invariance reproduces (at leading order) Einstein equations coupled to variousmatter and gauge field equations of motion. From this point of view, the classical theory canbe written as a manifestly background independent theory, and this provides hopes that thequantum theory may be also background independent, even if non-manifestly. This idea issupported by other definitions of string theory (e.g. through the AdS/CFT conjecture – andother holographic realizations – or through matrix models) which provide, at least partially,background independent formulations.

Ultimately, the greatest avenue to establish the background independence is string fieldtheory. Indeed, the form of the SFT action and of its properties (gauge invariance, equa-tion of motion. . . ) are identical irrespective of the background [264]. This provides a goodstarting point. The background dependence enters in the precise definition of the stringproducts (BRST operator and vertices). The origin of this dependence lies in the derivationof the action (Chapters 9 and 14): one begins with a particular CFT describing a givenbackground (spacetime compactifications, fluxes, etc.) and defines the vertices from correl-ation functions of vertex operators, and the Hilbert space from the CFT operators. As aconsequence, even though it is clear that no specific property of the background has beenused in the derivation – and that the final action describes SFT for any background –, thisis not sufficient to establish background independence. Since the theory assumes implicitlya background choice, one cannot guarantee that the physical quantities have no residualdependence in the background, even if the action looks superficially background independ-ent. Background independence in SFT is thus the statement that theories characterized bydifferent CFTs can be related by a field redefinition.

In this chapter, we will sketch the proof of background independence for backgroundsrelated by marginal deformations.3 It is possible to prove it at the level of the action [262,263], or at the level of the equations of motion [252]. The advantage of the second approachis that one can use the 1PI theory, which simplifies vastly the analysis. It also generalizesdirectly to the super-SFT.

Remark 17.1 (Field theory on the CFT space) As mentioned earlier, the string fieldis defined as a functional on the state space of a given CFT and not as a functional on thefield theory space (off-shell states would correspond to general QFTs, only on-shell states areCFTs). In this case, background independence would amount to reparametrization invarianceof the action in the theory space, and would thus almost automatically hold. A completeformulation of SFT following this line is currently out of reach, but some ideas can be foundin [290].

17.2 Problem setupGiven a SFT on a background, there are two ways to describe it on another background:

• deform the worldsheet CFT and express the SFT on the new background;

• expand the original action around the infinitesimal classical solution (to the linearisedequations of motion) corresponding to the deformation.

Background independence amounts to the equivalence of both theories up to a field redefin-ition. The derivation can be performed at the level of the action or of the equations ofmotion. To prove the background independence at the quantum level, one needs to takeinto account the changes in the path integral measure or to work with the 1PI action.

3An alternative approach based on morphism of L∞ algebra is followed in [190].

205

The simplest case is when the two CFTs are related by an infinitesimal marginal deform-ation

δScft = λ

2π

∫d2z ϕ(z, z), (17.1)

with ϕ a (1, 1)-primary operator and λ infinitesimal. The two CFTs are denoted by CFT1and CFT2, and quantities associated to each CFTs is indexed with the appropriate number.

Establishing background independence in this case also implies it for finite marginaldeformation since they can be built from a series of successive deformations. In the lattercase, the field redefinition may be singular, which reflects that the parametrization of oneCFT is not adapted for the other (equivalently, the coordinate systems for the string fieldbreaks down), which is expected if both CFTs are far in the field theory space.

Remember the form of the 1PI action (14.59):

S1[Ψ1] = 1g2s

12 〈Ψ1| c−0 QB |Ψ1〉+

∑′

n≥0

1n! V

1PIn (Ψn

1 )

, (17.2)

where the prime indicates that vertices with n < 3 do not include contributions from thesphere. In all this chapter, we remove the index 1PI to lighten the notations. The equationof motion is:

F1(Ψ1) = QB |Ψ1〉+∑n

1n! `n(Ψn

1 ) = 0. (17.3)

17.3 Deformation of the CFTConsider the case where the theory CFT1 is described by an action Scft,1[ψ1] given in termsof fields ψ1. Then, the deformation of this action by (17.1) gives an action for CFT2:

Scft,2[ψ1] = Scft,1[ψ1] + λ

2π

∫d2z ϕ(z, z). (17.4)

Correlation functions on a Riemann surface Σ in both theories can be related by expandingthe action to first order in λ in the path integral:⟨∏

i

Oi(zi, zi)⟩

2

=⟨

exp(− λ

2π

∫d2z ϕ(z, z)

)∏i

Oi(zi, zi)⟩

1

(17.5a)

≈

⟨∏i

Oi(zi, zi)⟩

1

− λ

2π

∫Σ

d2z

⟨ϕ(z, z)

∏i

Oi(zi, zi)⟩

1

, (17.5b)

where the Oi are operators built from the matter fields ψ1. This expression presents twoobvious problems. First, the correlation function may diverge when ϕ collides with one of theinsertions, i.e. when z = zi in the integration. Second, there is an inherent ambiguity: thecorrelation functions are written in terms of operators in the Hilbert space of CFT1, whichis different from the CFT2 Hilbert space, and there is no canonical isomorphism betweenboth spaces.

Seeing the Hilbert space as a vector bundle over the CFT theory space, the secondproblem can be solved by introducing a connection on this bundle. This allows to relateHilbert spaces of neighbouring CFTs. In fact, the choice of a non-singular connection alsoregularizes the divergences.

The simplest definition of a connection corresponds to cut unit disks around each operatorinsertions [37, 223, 224, 240, 268]. This amounts to define the variation between the two

206

correlation functions as:

δ

⟨∏i

Oi(zi, zi)⟩

1

= − λ

2π

∫Σ−∪iDi

d2z

⟨ϕ(z, z)

∏i

Oi(zi, zi)⟩

1

. (17.6)

The integration is over Σ minus the disks Di = |wi| ≤ 1 where wi is the local coordinatefor the insertion Oi. The divergences are cured because ϕ never approaches another operatorsince the corresponding regions have been removed. The changes in the correlation functionsinduce a change in the string vertices denoted by δVn(V1, . . . ,Vn).

The next step consists in computing the deformations of the operator modes. Since itinvolves only a matter operator, the modes in the ghost sector are left unchanged. TheVirasoro generators change as:

δLn = λ

∮|z|=1

dz2πi z

n+1ϕ(z, z), δLn = λ

∮|z|=1

dz2πi z

n+1ϕ(z, z). (17.7)

As a consequence, the BRST operator changes as

δQB = λ

∮|z|=1

dz2πi c(z)ϕ(z, z) + λ

∮|z|=1

dz2πi c(z)ϕ(z, z). (17.8)

One can prove thatQB , δQB = O(λ2) (17.9)

such that the BRST charge QB + δQB in CFT2 is correctly nilpotent if QB is nilpotent inCFT1.

For the deformation to provide a consistent SFT, the conditions b−0 = 0 and L−0 = 0 mustbe preserved. The first is automatically satisfied since the ghost modes are not modified.Considering an weight-(h, h) operator O, one finds

δL−0 |O〉 = λ

∮|z|=1

dz2πi z

∑p,q

zp−1zq−1 |Op,q〉 − λ∮|z|=1

dz2πi

∑p,q

zp−1zq−1 |Op,q〉 , (17.10)

where Op,q are the fields appearing in the OPE with ϕ:

ϕ(z, z)O(0, 0) =∑p,q

zp−1zq−1Op,q(0, 0). (17.11)

The terms with p 6= q vanish because the contour integrals are performed around circles ofunit radius centred at the origin. Moreover, the terms p = q are identical and cancel witheach other, showing that δL−0 = 0 when acting on states satisfying L−0 = 0.

The SFT action S2[Ψ1] in the new background reads

S2[Ψ1] = S1[Ψ1] + δS1[Ψ1] (17.12)

where the change δS1 in the action is induced by the changes in the string vertices:

δS1[Ψ1] = 1g2s

12 〈Ψ1| c−0 δQB |Ψ1〉+

∑n≥0

1n! δVn(Ψn

1 )

. (17.13)

The equation of motion is:

F2(Ψ1) = F1(Ψ1) + λ δF1(Ψ1) = 0, (17.14)

where F1 is given in (17.3) and

λ δF1(Ψ1) = δQB |Ψ1〉+∑n

1n! δ`n(Ψn

1 ). (17.15)

207

17.4 Expansion of the actionGiven a (1, 1) primary ϕ, a BRST invariant operator is ccϕ. Hence the field

|Ψ1〉 = λ |Ψ0〉 , |Ψ0〉 = c1c1(0) |ϕ〉 (17.16)

is a classical solution to first order in λ since the interactions on the sphere are at leastcubic.

Separating the string field as the contribution from the (fixed) background and a fluctu-ation Ψ′

|Ψ1〉 = λ |Ψ0〉+ |Ψ′〉 , (17.17)

the action expanded to first order in λ reads:

S1[Ψ1] = S1[Ψ0] + S′[Ψ′], (17.18)

where

S′[Ψ′] = 1g2s

(12 〈Ψ

′| c−0 QB |Ψ′〉+∑n

1n!(Vn(Ψ′n) + λVn+1(Ψ0,Ψ′n)

)). (17.19)

The equation of motion is:

F ′(Ψ′) := F1(Ψ′) + λ δF ′(Ψ′) = 0, (17.20)

where F1 is given in (17.3) and

δF ′(Ψ′) =∑n

1n! `n+1(Ψ0,Ψ′n). (17.21)

17.5 Relating the equations of motionIn the previous section, we have derived the equations of motion for two different descriptionsof a SFT obtained after shifting the background: (17.14) arises by deforming the CFT andcomputing the changes in the BRST operator and string products, while (17.20) arises byexpanding the SFT action around the new background. The theory is background independ-ent if both sets of equations (17.14) and (17.20) are related by a (possibly field-dependent)linear transformationM(Ψ′) after a field redefinition of Ψ1 = Ψ1(Ψ′):

F1(Ψ1) + λ δF1(Ψ1) =(1 + λM(Ψ′)

)(F1(Ψ′) + λ δF ′(Ψ′)

), (17.22a)

|Ψ1〉 = |Ψ′〉+ λ |δΨ′〉 . (17.22b)

The zero-order equation is automatically satisfied. To first order, this becomes

ddλF1(Ψ′ + λδΨ′)

∣∣∣∣λ=0

+ δF1(Ψ1)− δF ′(Ψ′) =M(Ψ′)F1(Ψ′). (17.23)

Taking Ψ′ to be a solution of the original action removes the RHS, such that:

λQB |δΨ′〉+ λ∑n

1n! `n+1(δΨ′,Ψ′n) + δQB |Ψ′〉

+∑n

1n! δ`n(Ψ′n)− λ

∑n

1n! `n+1(Ψ0,Ψ′n) = 0.

(17.24)

208

To simplify the computations, it is simpler to consider the inner product of this quantitywith an arbitrary state A (assumed to be even):

∆ := λ 〈A| c−0 QB |δΨ′〉+ λ∑n

1n! Vn+2(A, δΨ′,Ψ′n) +〈A| c−0 δQB |Ψ′〉

+∑n

1n! δVn+1(A,Ψ′n)− λ

∑n

1n! Vn+2(A,Ψ0,Ψ′n).

(17.25)

The goal is to prove the existence of δΨ′ such that ∆ = 0 up to the zero-order equation ofmotion F1(Ψ′) = 0.

17.6 Idea of the proofIn this section, we give an idea of how the proof ends, referring to [252] for the details.

The first step is to introduce new vertices V ′0,3 and V ′n parametrizing the variations ofthe string vertices:

〈A| c−0 δQB |B〉 = λV ′0,3(Ψ0, B,A), δVn(Ψ′n) = λV ′n+1(Ψ0,Ψ′n), (17.26)

where the notation (8.19) has been used. Each subspace V ′g,n is defined such that the LHSis recovered upon integrating the appropriate ωg,n over this section segment. Next, the fieldredefinition δΨ′ is parametrized as:

〈A| c−0 |δΨ′〉 =∑n

1n! Bn+2(Ψ0,Ψ′n, A). (17.27)

The objective is to prove the existence (and if possible the form) of the subspaces Bn+2.Both the vertices V ′n and Bn admit a genus expansion:

V ′n =∑g≥0V ′g,n, Bn =

∑g≥0Bg,n. (17.28)

Plugging the new expressions in (17.25) give:

∆ = −∑n

1n! Bn+2(Ψ0,Ψ′n, QBA) +

∑m,n

1m!n! Bn+2(Ψ0,Ψ′m, `n+1(A,Ψ′n))

+∑n

1n! V

′n+2(A,Ψ0,Ψ′n)−

∑n

1n! Vn+2(A,Ψ0,Ψ′n).

(17.29)

Next, the BRST identity (8.46) and the equation of motion F1(Ψ′) = 0 allow to rewrite thefirst term as:

Bn+2(Ψ0,Ψ′n, QBA) = ∂Bn+2(Ψ0,Ψ′n, A) + nBn+2(Ψ0,Ψ′n−1, QBΨ′, A) (17.30a)

= ∂Bn+2(Ψ0,Ψ′n, A)−∑m

n

m! Bn+2(Ψ0,Ψ′n−1, `m(Ψ′m), A).

(17.30b)In the second term, the sum over n is shifted. Combining everything together gives:

∆ =∑n

1n! ∂Bn+2(Ψ0,Ψ′n, A)−

∑m,n

1m!n! Bn+3(Ψ0,Ψ′n, `m(Ψ′m), A)

+∑m,n

1m!n! Bn+2(Ψ0,Ψ′m, `n+1(A,Ψ′n)) +

∑n

1n! V

′n+2(A,Ψ0,Ψ′n)

−∑n

1n! Vn+2(A,Ψ0,Ψ′n).

(17.31)

209

Solving for ∆ = 0 requires that each term with a different power of Ψ′ vanishes independ-ently:

∂Bn+2(Ψ0,Ψ′n, A) =− V ′n+2(A,Ψ0,Ψ′n) + Vn+2(A,Ψ0,Ψ′n)

+∑m1,m2

m1+m2=n

n!m1!m2! Bm1+3(Ψ0,Ψ′m1 , `m2(Ψ′m2), A)

−∑m1,m2

m1+m2=n

n!m1!m2! Bm1+2

(Ψ0,Ψ′m1 , `m2+1(A,Ψ′m2)

).

(17.32)

In order to proceed, one needs to perform a genus expansion of the various spaces:this allows to solve recursively for all Bg,n starting from B0,3. One can then build |δΨ′〉recursively, which provides the field redefinition. Indeed, the RHS of this equation containsonly Bg′,n′ for g′ < g or n′ < n and the equation for B0,3 contains no Bg,n in the RHS.It should be noted that the field redefinition is not unique, but there is the freedom ofperforming (infinite-dimensional) gauge transformations. Finding an obstruction to solvethese equations mean that the field redefinition does not exist, and thus that the theory isnot background independent

The form of the equation∂B0,3 = V0,3 − V ′0,3 (17.33)

suggests to use homology theory. The interpretation of B0,3 is that it is a space interpolatingbetween V0,3 and V ′0,3. A preliminary step is to check that there is no obstruction: since theLHS is already a boundary one has ∂2B0,3 = 0 and one should check that ∂(RHS) = 0 aswell. It can be shown that it is indeed true. It was proved in [252] that this equation admitsa solution and that the equations for higher g and n can all be solved. Hence, there existsa field redefinition and SFT is background independent.

17.7 References• Proof of the background independence under marginal deformations [252, 262, 263]

(see also [239–241] for earlier results laying foundations for the complete proof).

• L∞ perspective [190, sec. 4] (see also [189, 188, sec. III.B].

• Connection on the space of CFTs [37, 223, 224, 240, 268].

210

Part IV

Superstring theory

211

Chapter 18

Superstring

Superstring theory is generally the starting point for physical model building. It has indeedseveral advantages over the bosonic string, most importantly, the removal of the tachyonand the inclusion of fermions in the spectrum. The goal of this chapter is to introduce themost important concepts needed to generalize the bosonic string to the superstring, bothfor off-shell amplitudes and string field theory. We refer to the review [55] for more details.

18.1 Worldsheet superstring theoryThere are five different superstring theories with spacetime supersymmetry: the types I, IIAand IIB, and the E8 × E8 and SO(32) heterotic models.

In the Ramond–Neveu–Schwarz formalism (RNS), the left- and right-moving sectors ofthe superstring worldsheet are described by a two-dimensional super-conformal field theory(SCFT), possibly with different numbers of supersymmetries. The prototypical example isthe heterotic string with N = (1, 0) and we will focus on this case: only the left-movingsector is supersymmetric, while the right-moving is given by the same bosonic theory as inthe other chapters. Up to minor modifications, the type II theory follows by duplicating theformulas of the left-moving sector to the right-moving one.

18.1.1 Heterotic worldsheetThe ghost super-CFT is characterized by anti-commuting ghosts (b, c) (left-moving) and(b, c) (right-moving) with central charge c = (−26,−26), associated to diffeomorphisms, andby commuting ghosts (β, γ) with central charge c = (11, 0), associated to local supersym-metry. As a consequence the matter SCFT must have a central charge c = (15, 26). Ifspacetime has D non-compact dimensions, then the matter CFT is made of:

• a free theory of D scalars Xµ and D left-moving fermions ψµ (µ = 0, . . . , D− 1) suchthat cfree = 3D/2 and cfree = D;

• an internal theory with cint = 15− 3D/2 and cint = 26−D.

The critical dimension is reached when cint = 0 which corresponds to D = 10.The diffeomorphisms are generated by the energy–momentum tensor T (z); correspond-

ingly, supersymmetry is generated by its super-partner G(z) (sometimes also denoted by

212

TF ). The OPEs of the algebra formed by T (z) and G(z) is:

T (z)T (w) ∼ c/2(z − w)4 + 2T (w)

(z − w)2 + ∂T (w)z − w

, (18.1a)

G(z)G(w) ∼ 2c/3(z − w)3 + 2T (w)

(z − w) , (18.1b)

T (z)G(w) ∼ 32

G(w)(z − w)2 + ∂G(w)

(z − w) . (18.1c)

The superconformal ghosts form a first-order system (see Section 23.2) with ε = −1 andλ = 3/2. Hence, they have conformal weights

h(β) =(

32 , 0), h(γ) =

(−1

2 , 0)

(18.2)

and OPEsβ(z)γ(w) ∼ 1

z − w, γ(z)β(w) ∼ − 1

z − w. (18.3)

The expressions of the ghost energy–momentum tensors are

T gh = −2b ∂c+ c∂b, T βγ = 32 β∂γ + 1

2 γ ∂β. (18.4)

The ghost numbers of the different fields are

Ngh(b) = Ngh(β) = −1, Ngh(c) = Ngh(γ) = 1. (18.5)

The worldsheet scalars satisfy periodic boundary conditions. On the other hand, fermionscan satisfy anti-periodic or periodic conditions: this leads to two different sectors, calledNeveu–Schwarz (NS) and Ramond (R) respectively.

βγ system

The βγ system can be bosonized as

γ = η eφ, β = ∂ξ e−φ, (18.6)

where (ξ, η) are fermions with conformal weights 0 and 1 (this is a first-order system withε = 1 and λ = 1), and φ is a scalar field with a background charge (Coulomb gas). Thisprovides an alternative representation of the delta functions:

δ(γ) = e−φ, δ(β) = eφ. (18.7)

Introducing these operators is necessary to properly define the path integral with bosoniczero-modes. They play the same role as the zero-modes insertions for fermionic fields neededto obtain a finite result (see also Appendix C.1.3):∫

dc0 = 0 =⇒∫

dc0 c0 = 1, (18.8)

because c0 = δ(c0). For a bosonic path integral, one needs a delta function:∫dγ0 =∞ =⇒

∫dγ0 δ(γ0) = 1. (18.9)

213

By definition of the bosonization, one has:

T βγ = T ηξ + Tφ, (18.10)

whereT ηξ = −η ∂ξ, Tφ = −1

2 (∂φ)2 − ∂2φ. (18.11)

The OPE between the new fields are:

ξ(z)η(w) ∼ 1z − w

, eq1φ(z)eq2φ(w) ∼ e(q1+q2)φ(w)

(z − w)q1q2 , ∂φ(z)∂φ(w) ∼ − 1(z − w)2 . (18.12)

The simplest attribution of ghost numbers to the new fields is:

Ngh(η) = 1, Ngh(ξ) = −1, Ngh(φ) = 0. (18.13)

To the scalar field φ is associated another U(1) symmetry whose quantum number iscalled the picture number Npic. The picture number of η and ξ are assigned1 such that βand γ have Npic = 0:

Npic(eqφ) = q, Npic(ξ) = 1, Npic(η) = −1. (18.14)

Because of the background charge, this symmetry is anomalous and correlation functionsare non-vanishing if the total picture number (equivalently the number of φ zero-modes) is:

Npic = 2(g − 1) = −χg. (18.15)For the same reason, the vertex operators eqφ are the only primary operators:

h(eqφ) = −q2(q + 2), (18.16)

and the Grassmann parity of these operators is (−1)q. Special values are

h(eφ) = 32 , h(e−φ) = 1

2 . (18.17)

The superstring theory features a Z2 symmetry called the GSO symmetry. All fields aretaken to be GSO even, except β and γ which are GSO odd and eqφ whose parity is (−1)q.Physical states in the NS sector are restricted to be GSO even: it is required to removethe tachyon of the spectrum and to get a spacetime with supersymmetry. In type II, theRamond sector can be projected in two different ways, leading to the type IIA and type IIBtheories.

The components of the BRST current are:

jB = c(Tm + T βγ) + γG+ bc∂c− 14 γ

2b, (18.18a)

B = cTm + bc∂c. (18.18b)

From there, it is useful to define the picture changing operator (PCO):

X (z) = QB , ξ(z) = c∂ξ + eφG− 14 ∂η e2φ b− 1

4 ∂(η e2φb), (18.19)

which is a weight-(0, 0) primary operator which carries a unit picture number. It is obviouslyBRST exact. This operator will be necessary to saturate the picture number condition: the

1Any linear combination of both U(1) could have been used. The one given here is conventional, butalso the most convenient.

214

naive insertion of eφ ∼ δ(β) breaks the BRST invariance. The PCO zero-mode is obtainedfrom the contour integral:

X0 = 12πi

∮ dzzX (z). (18.20)

It can be interpreted as delocalizing a PCO insertion from a point to a circle, which decreasesthe risk of divergence.

18.1.2 Hilbert spacesThe description in terms of the (η, ξ, φ) fields leads to a subtlety: the bosonization involvesonly the derivative ∂ξ and not the field ξ itself, meaning that the zero-mode ξ0 is absent fromthe original Hilbert space defined from (β, γ). In the bosonized language, the Hilbert spacewithout the ξ zero-mode is called the small Hilbert space and is made of state annihilatedby η0 (the η zero-mode)

Hsmall =|ψ〉 | η0 |ψ〉 = 0

. (18.21)

Removing this condition leads to the large Hilbert space:2

Hsmall = Hlarge ∩ ker η0. (18.22)

A state in Hsmall contains ξ with at least one derivative acting on it.A correlation function defined in terms of the (η, ξ, φ) system is in the large Hilbert space

and will vanish since there is no ξ factor to absorb the zero-mode of the path integral. Asa consequence, correlation functions (and the inner product) are defined with a ξ0 insertion(by convention at the extreme left) or, equivalently, ξ(z). The position does not matter sinceonly the zero-mode contribution survives, and the correlation function is independent of z.Sometimes it is more convenient to work in the large Hilbert space and to restrict later tothe small Hilbert space.

The SL(2,C) invariant vacuum is normalized as

〈k| c−1c−1c0c0c1c1 e−2φ(z) |k′〉 = (2π)Dδ(D)(k + k′). (18.23)

Remark 18.1 (Normalization in type II) In type II theory, the SL(2,C) is normalisedas:

〈k| c−1c−1c0c0c1c1 e−2φ(z)e−φ(w) |k′〉 = −(2π)Dδ(D)(k + k′). (18.24)

The sign difference allows to avoid sign differences between type II and heterotic stringtheories in most formulas [55].

The Hilbert space of GSO even states satisfying the b−0 = 0 and L−0 = 0 conditions isdenoted by HT (ghost and picture numbers are arbitrary). This Hilbert space is the directsum of the NS and R Hilbert spaces:

HT = HNS ⊕HR. (18.25)

The subspace of states with picture number Npic = n is written Hn. The picture numberof NS and R states are respectively integer and half-integer. Two special subspaces of HTplay a distinguished role:

HT = H−1 ⊕H−1/2, HT = H−1 ⊕H−3/2. (18.26)2The relation between the small and large Hilbert spaces is similar to the one between the H and

H0 = b0H Hilbert space from the open string since the (b, c) and (η, ξ) are both fermionic first-ordersystems.

215

To understand this, consider the vacuum |p〉 of the φ field with picture number p:

|p〉 = epφ(0) |0〉 . (18.27)

Then, acting on the vacuum with the βn and γn modes implies

∀n ≥ −p− 12 : βn |p〉 = 0,

∀n ≥ p+ 32 : γn |p〉 = 0.

(18.28)

For p = −1, all positive modes (starting with n = 1/2) annihilate the vacuum in the NSsector. This is a positive asset because positive modes which do not annihilate the vacuumcan create states with arbitrary negative energy (since it is bosonic).3 For p = −1/2 orp = −3/2, the vacuum is annihilated by all positive modes, but not by one of the zero-modeγ0 or β0. Nonetheless, one can show that the propagator in the R sector allows to propagateonly a finite number of states if one chooses H−1/2; the role of H−3/2 will become apparentwhen discussing how to build the superstring field theory.

Basis states are introduced as in the bosonic case:

HT = Span|φr〉, HT = Span|φcr〉 (18.29)

such that〈φcr|φs〉 = δrs. (18.30)

The completeness relations are1 =

∑r

|φr〉〈φcr| (18.31)

HT , and1 =

∑r

(−1)|φr| |φcr〉〈φr| (18.32)

on HT .Finally, the operator G is defined as:

G =

1 NS sector,X0 R sector.

(18.33)

Note the following properties

[G, L±0 ] = [G, b±0 ] = [G, QB ] = 0. (18.34)

It will be appear in the propagator and kinetic term of the superstring field theory.

18.2 Off-shell superstring amplitudesIn this section, we are going to build the scattering amplitudes. The procedure is verysimilar to the bosonic case, except for the PCO insertions and of the Ramond sector. Forthis reason, we will simply state the result and motivate the modifications with respect tothe bosonic case.

3This is not a problem on-shell since the BRST cohomology is independent of the picture number.However, this matters off-shell since such states would propagate in loops and make the theory inconsistent.

216

18.2.1 AmplitudesExternal states can be either NS or R: the Riemann surface corresponding to the g-loopscattering of m external NS states and n external R states is denoted by Σg,m,n. R statesmust come in pairs because they correspond to fermions. As in the bosonic case, the amp-litude is written as the integration of an appropriate p-form Ω(g,m,n)

p over the moduli spaceMg,m,n (or, more precisely, of a section of a fibre bundle with this moduli space as a basis).From the geometric point of view, nothing distinguishes the punctures and thus:

Mg,m,n := dimMg,m,n = 6g − 6 + 2m+ 2n. (18.35)

The form ΩMg,m,nis defined as a SCFT correlation function of the physical vertex operators

together with ghost and PCO insertions.

Remark 18.2 A simple way to avoid making errors with signs is to multiply every Grass-mann odd external state with a Grassmann odd number. These can be removed at the endto read the sign.

The two conditions from the U(1) anomalies on the scattering amplitude are:

Ngh = 6− 6g, Npic = 2g − 2. (18.36)

Given an amplitude with m NS states V NSi ∈ H−1 and n R states V R

j ∈ H−1/2, the abovepicture number can be reached by introducing a certain number of PCO X (yA):

npco := 2g − 2 +m+ n

2 . (18.37)

These PCO are inserted at various positions: while the amplitude does not depend on theselocations on-shell, off-shell it will (because the vertex operators are not BRST invariant).The choices of PCO locations are arbitrary except for several consistency conditions:

1. avoid spurious poles (Section 18.2.3);

2. consistent with factorization (each component of the surface in the degeneration limitsmust saturate the picture number condition).

This parallels the discussion of the choices of local coordinates: as a consequence, thenatural object is a fibre bundle Pg,m,n with the local coordinate choices (up to global phaserotations) and the PCO locations as fibre, and the moduli spaceMg,m,n as base. Forgettingabout the PCO locations leads to a fibre bundle Pg,m,n which is a generalization of theone found in the bosonic case. The coordinate system of the fibre bundle presented in thebosonic case is extended by including the PCO locations yA.

With these information, the amplitude can be written as:

Ag,m,n(V NSi ,V R

j ) =∫Sg,m,n

ΩMg,m,n(V NSi ,V R

j ), (18.38a)

where

ΩMg,m,n= (−2πi)−Mc

g,m,n

⟨Mg,m,n∧λ=1

Bλ dtλnpco∏A=1X (yA)

m∏i=1

V NSi

n∏j=1

V Rj

⟩Σg,n

. (18.38b)

where Sg,m,n is a Mg,m,n-dimensional section of Pg,m,n parametrized by coordinates tλ. The1-form B corresponds to a generalization of the bosonic 1-form. It has ghost number 1

217

and includes a correction to compensate the variation of the PCO locations in terms of themoduli parameters:

Bλ =∑α

∮Cα


∂tλ

(F−1α (σα)

)+∑α

∮Cα


∂tλ

(F−1α (σα)

)−∑A

1X (yA)

∂yA∂tλ

∂ξ(yA).(18.39)

The last factor amounts to consider the combination

X (yA)− ∂ξ(yA) dyA (18.40)

for each PCO insertion:4 the correction is necessary to ensure that the BRST identity (8.46)holds. This can be understood as follows: the derivative acting on the PCO gives a termdX (z) = ∂X (z)dz which must be cancelled. This is achieved by the second term sinceQB , ∂ξ(z) = ∂X (z).

Remark 18.3 While it is sufficient to work with Mg,n for on-shell bosonic amplitudes,on-shell superstring amplitudes are naturally expressed in Pg,m,n (with the local coordinateremoved) since the positions of the PCO must be specified even on-shell.

Remark 18.4 (Amplitudes on the supermoduli space) Following Polyakov’s appro-ach from Chapters 3 and 4 to the superstring would lead to replace the moduli space by thesupermoduli space. The latter includes Grassmann-odd moduli parameters in addition to themoduli parameters from Mg,m+n (in the same way the superspace includes odd coordinatesθ along with spacetime coordinates x). The natural question is whether it is possible to splitthe integration over the even and odd moduli, and to integrate over the latter such that onlyan integral overMg,m+n remains. In view of (18.38a), the answer seems positive. However,this is incorrect: it was proven in [70] that there is no global holomorphic projection of thesupermoduli space to the moduli space. This is related to the problem of spurious polesdescribed below. But, this does not prevent to do it locally: in that case, implementing theprocedure carefully should give the rules of vertical integration [85, 245, 260].

18.2.2 FactorizationThe plumbing fixture of two Riemann surfaces Σg1,m1,n1 and Σg2,m2,n2 can be performed intwo different ways since two NS or two R punctures can be glued.

If two NS punctures are glued, the resulting Riemann surface is Σ(NS)g1+g2,m1+m2−2,n1+n2

.The number of PCO inherited from the two original surfaces is

n(1)pco + n(2)

pco = 2(g1 + g2)− 2 + (m1 +m2 − 2) + n1 + n22 = n(NS)pco , (18.41)

which is the required number for a non-vanishing amplitude. As a consequence, the propag-ator is the same as in the bosonic case:

∆NS = b+0 b−0

1L0 + L0

δ(L−0 ). (18.42)

If two R punctures are glued, the numbers of PCO do not match by one unit:

n(1)pco + n(2)

pco = 2(g1 + g2)− 2 + (m1 +m2) + n1 + n2 − 22 − 1 = n(R)pco − 1. (18.43)

4The sum is formal since it is composed of 0- and 1-forms.

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This means that an additional PCO must be inserted in the plumbing fixture procedure:the natural place for it is in the propagator since this is the only way to keep both verticessymmetric as required for a field theory interpretation. Another way to see the need ofthis modification is to study the propagator (18.42) for Ramond states: since Ramondstates carry a picture number −1/2, the conjugate states have Npic = −3/2 and thus thepropagator has a total picture number −3 instead of −2 (the propagator graph is equivalentto a sphere). Then, to avoid localizing the PCO at a point of the propagator, one insertsthe zero-mode which corresponds to smear the PCO:

∆R = b+0 b−0

X0

L0 + L0δ(L−0 ). (18.44)

Delocalizing the PCO amounts to average the amplitude over an infinite number of points(i.e. to consider a generalized section): this is necessary to preserve the L−0 eigenvalue sinceX0 is rotationally invariant while X (z) is not. Note that the zero-mode can be writtenequivalently as a contour integral around one of the two glued punctures:

X0 = 12πi

∮ dw(1)n

w(1)n

X(w(1)n

)= 1

2πi

∮ dw(2)n

w(2)n

X(w(2)n

). (18.45)

The equality of both expressions holds because X (z) has conformal weight 0.Using the operator G (18.33), the propagator can be written generically as

∆ = b+0 b−0

GL0 + L0

δ(L−0 ). (18.46)

Remark 18.5 (Propagators) NS and R states correspond respectively to bosonic and fer-mionic fields: the operators L+

0 and X0 can be interpreted as the (massive) Laplacian andDirac operators, such that both propagators can be written

∆NS ∼1

k2 +m2 , ∆R ∼i/∂ +m

k2 +m2 . (18.47)

To motivate the identification of X0 with the Dirac operator, remember that X (z) containsa term eφ(z)G(z) (this is the only term which contributes on-shell), where G(z) in turncontains ψµ∂Xµ. But, the zero-modes of ψµ and ∂Xµ correspond respectively to the gammamatrix γµ and momentum kµ when acting on a state.

The PCO zero-mode insertion inside the propagator has another virtue. It was notedpreviously that states with Npic = −3/2 are infinitely degenerate since one can apply β0an arbitrary number of time. These states have large negative ghost numbers. Consideringa loop amplitude, all these states would appear in the sum over the states and lead to adivergence. The problem is present only for loops because the ghost number is not fixed: ina tree propagator, the ghost number is fixed and only a finite number of β0 can be applied.But, the PCO insertion turns these states into Npic = −1/2 states. In this picture number,one cannot create an arbitrarily large negative ghost number since γn0 can only increase theghost number.

18.2.3 Spurious polesA spurious pole corresponds to a singularity of the amplitude which cannot be interpretedas the degeneration limit of Riemann surfaces. As a consequence, they do not correspond toinfrared divergences and don’t have any physical meaning; they must be avoided in order todefine a consistent theory. To achieve this, the section Sg,m,n must be chosen such that it

219

avoids all spurious poles. However, while it is always possible to avoid these poles locally, itis not possible globally (this is related to the results from [70]). Poles can be avoided usingvertical integration: two methods have been proposed, in the small (Sen–Witten) [245, 260]and large (Erler–Konopka) [85] Hilbert spaces respectively. Before describing the essence ofboth approaches, we review the origin of spurious poles.

Origin

Spurious poles arise in three different ways:• two PCOs collide;

• one PCO and one matter vertex collide;

• other singularities of the correlation functions.The last source is the less intuitive one and we focus on it.

A general correlation function of (η, ξ, φ) on the torus5 (satisfying the ghost numbercondition) reads

C(xi, yj , zq) =⟨n+1∏i=1

ξ(xi)n∏j=1

η(yj)m∏k=1

eqkφ(zk)

⟩

=

n∏j′=1

ϑδ

(− yj′ +

∑i

xi −∑j

yj +∑k

qkzk

)n+1∏i′=1

ϑδ

(− xi′ +

∑i

xi −∑j

yj +∑k

qkzk

) ×∏i<i′

E(xi, xi′)∏j<j′

E(yj , yj′)∏i,j

E(xi, yj)∏k,`

E(zk, z`)qkq`.

(18.48)The additional ξ insertion is necessary since it provides the ξ zero-mode, the correlationfunction being defined in the large Hilbert space. On the torus, the picture numbers mustadd to zero and thus the charges qk satisfy∑

k

qk = 0. (18.49)

The function E(x, y) is called the prime form and is a generalization of the function x − yon te torus:

E(x, y) = ϑ1(x− y)ϑ′1(0) ∼x→y x− y. (18.50)

Its presence ensures that C vanishes or diverges appropriately when the operators collide (i.e.that the zeros and poles of C are the expected ones from the OPE). The theta functions areused to make sure that the correlation function satisfies the appropriate boundary conditions(specified by the spin structure δ) for each cycle of the surface.

However, theta functions can also vanish and the ones in the denominator lead to addi-tional singularities (not implied by any OPE) for the correlation function. Since x1 can bechosen arbitrarily, the only theta function which can have poles is

ϑδ

( n+1∑i=2

xi −n∑j=1

yj +m∑k=1

qkzk

)= 0. (18.51)

This defines a complex codimension 1 curve in Pg,m,n, depending on the vertex and PCOlocations, but also on the moduli parameters (appearing in the definition of the theta func-tion). On the other hand, it does not depend on the local coordinate choice. If the sectionSg,m,n intersects this curve, it will be ill-defined (even on-shell).

5The discussion generalizes directly to higher-genus Riemann surfaces.

220

From this formula, several comments can be made. If an operator inserted at z containsnξ fields ∂ξ, nη fields η and a factor epφ, then the dependence in z of the theta functionis of the form (nξ − nη + p)z = Npicz. Then, if the PCO locations are chosen as to avoidspurious poles for a given operator, this will also avoid them for any operator of the samepicture number. This also implies that insertion of β and γ cannot lead to spurious polessince they have Npic = 0; this is important since they appear in the BRST current, thusinsertion of the latter cannot lead to new poles.

Since it is always possible to choose locally a distribution of PCO to avoid spuriouspoles, the idea is to discretize the moduli space in small pieces. But, since the PCO cannotbe distributed continuously along the different components of the moduli space, correctionterms are required. These can be generated in two different ways in both the small andlarge Hilbert spaces. The second is more general while the first may be more adapted sinceit keeps the amplitude in the small Hilbert space.

Vertical integration: large Hilbert space

Consider the n-point amplitude state 〈A(p)| which produces an amplitude with p PCO whencontracted with n external states are specified. The BRST identity implies that the amp-litude state is closed (i.e. gauge invariant)

〈A(p)|Q = 0, (18.52)

whereQ = QB ⊗ 1⊗n−1 + · · ·+ 1⊗n−1 ⊗QB . (18.53)

Moreover, this state is in the small Hilbert space which implies that it is in the kernel of η0:

〈A(p)| η = 0, (18.54)where

η = η0 ⊗ 1⊗n−1 + · · ·+ 1⊗n−1 ⊗ η0. (18.55)The BRST cohomology is trivial in the large Hilbert space: thus, if 〈A(p)| is closed, it

must be exact in this space:〈A(p)| = 〈α(p)|Q, (18.56)

where the state α(p) (called gauge amplitude) must be in the large Hilbert space. This isconsistent with 〈A(p)| η = 0 only if

〈α(p)| ηQ = 0 (18.57)

(Q and η anti-commute); It is then natural to interpret the state on which Q acts as anamplitude with one less PCO

〈A(p−1)| = 〈α(p)| η (18.58)since Npic(η) = −1.

Continuing this procedure leads to an amplitude 〈A(0)| without any PCO insertion, andthus without spurious singularities. Consistency with the picture number anomaly requiresthe external state to have non-canonical picture numbers. But, this should not be a puzzlesince the amplitude states should be viewed as intermediate object to obtain the final amp-litude.

Hence, the amplitude 〈A(p)| can be constructed by starting with 〈A(0)|: inserting ξ(z) inthe amplitude leads to the gauge amplitude 〈α(1)|, whose BRST variation yields 〈A(2)|. Con-tinuing recursively helps to construct the desired amplitude. Moreover, Q and ξ insertionsautomatically take care of the corrections at the interfaces of the components.

Showing that the amplitude is independent of the non-physical data (i.e. gauge invari-ance) is trivial since it is expressed as a BRST exact expression.

221

Vertical integration: small Hilbert space

The section of Pg,m,n is given by a series of discontinuous components linked by verticalsegments. On the vertical segment, the PCO configuration interpolates continuously betweenthe components and the integrand can encounter a spurious pole. Since the integrand is nota total derivative in terms of the fibre coordinates, its integration over a segment dependson the path followed and not only on the end points. This implies that it diverges when itencounters the spurious pole. However, there is a specific prescription which avoids theseproblems. When only one PCO varies, the integrand is a total derivative of the PCO locationand can thus be integrated directly, giving a difference in the two end points. In this case,the result is independent from the specific path and from the presence of the spurious pole.

To be more concrete, given a PCO insertion X (y1), the variation of its location inserts afactor −∂ξ(y1). Integrating this term between two components labelled by i and j – keepingeverything else fixed – leads to a factor ξ(y(j)

1 )− ξ(y(j)1 ).

When several PCOs are involved, it is not sufficient to integrate the vertical segmentalong a path where only one PCO varies at a time. Indeed, because a hole is left in theprocess of the vertical integration. Additional segments must be added and integrated over.This avoids the spurious poles and one can show that it yields a well-defined amplitude.Moreover, it agrees with the large Hilbert space approach.

Finally, it remains to address the question of the Feynman diagrams construction. Inthis case, every graph obtained by plumbing fixture inherits its PCO locations from thelower-dimensional surfaces, and there is no control on the resulting distribution. It can beshown that no spurious singularity is generated in the gluing process if the lower-dimensionalgraphs have no spurious poles. Hence, it is sufficient to ensure that the fundamental graphshave no spurious poles.

18.3 Superstring field theoryThe construction of super-SFT has proceeded along different directions (for reviews, see [55,84, 203]). There are two main strategies for constructing the superstring vertices:

1. brute-force construction: build the vertices recursively from amplitude factorization;

2. dress the bosonic products with superconformal ghosts.

While the second approach is simpler and preferred for explicit construction, the first allowsto derive the general structure as was done for the bosonic string. There are two mainstrategies for dressing the vertices:

1. Munich construction (homotopy algebra bootstrap): use the L∞ and A∞ structuresto derive the superstring vertices from the bosonic vertices (small Hilbert space).

2. Berkovits’ construction (WZW action): generalize Witten’s cubic bosonic open SFT(NS / R in large / small Hilbert space).

As indicated in parenthesis, a super-SFT can be written in the small or large Hilbert space(or a combination). The different approaches have been shown to be equivalent at theclassical level.

The main difficulty in building a super-SFT is to properly describe the Ramond sector.This can be done following two different approaches:

• constraining the Ramond string field;

• using an auxiliary string field.

222

Berkovits’ original SFT cannot describe the Ramond sector in the large Hilbert space, butit is possible to couple Berkovits’ action for the NS field in the large Hilbert space to aRamond field in the small Hilbert space. Another limitation of Berkovits’ approach is thatit works only for the open and heterotic superstrings (but not for type II).

We assume that the problems with PCO are absent (in Berkovits’ and supermoduliconstructions) or that they have been defined using vertical integration.

In the rest of this chapter, we will discuss the kinetic term for each of the first threeapproaches. At the level of the free action, the open and heterotic super-SFT differs only inthe bosonic factors as in Chapter 13.

18.3.1 String field and propagatorAs in the bosonic case, it is natural to consider a string field gathering all possible states

Ψ = Ψ−1 + Ψ−1/2, (18.59)

where Ψ−1 and Ψ−1/2 are respectively the NS and R string fields. If the field is in the smallHilbert space, it satisfies:

η0 |Ψ〉 = 0. (18.60)

The propagator was found in (18.46) to be

∆ = b+0 b−0

GL0 + L0

δ(L−0 ), G =

1 NS,X0 R.

(18.61)

As for the bosonic case, the constraints

b−0 |Ψ〉 = L−0 |Ψ〉 = 0, b+0 |Ψ〉 = 0 (18.62)

must be imposed on the field to ensure that the propagator is invertible.For similar reasons, the PCO insertion implies that the propagator is not invertible since

X0 has zero-modes: this means equivalently that it has a non-empty kernel off-shell or thatit contains derivatives. Two different solutions can be chosen to address this issue: imposingconstraints as for the level-matching condition, or introducing auxiliary fields.

18.3.2 Constraint approachTwo new PCO operators must be introduced:

X = G0 δ(β0) + b0 δ′(β0), Y = −c0 δ′(γ0). (18.63)

The first operator commutes with the BRST operator

[QB , X] = 0. (18.64)

The product of these operators is a projector

XYX = X. (18.65)

Then, the R string field is constrained to satisfy

XY |Ψ−1/2〉 = |Ψ−1/2〉 . (18.66)

A state satisfying this condition is said to be in the restricted Hilbert space. It can be shownthat it reproduces the cohomology of QB on-shell.

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Remark 18.6 Since G0 contains derivatives, the restriction is not purely algebraic as inthe bosonic case. It prevents the degeneration due to γn0 .

Remark 18.7 (Comparison with level-matching) The conditions b−0 = L−0 = 0 canbe rephrased as the statement that the string field Ψ is invariant under the action of theprojector Bc−0

Bc−0 |Ψ〉 = |Ψ〉 , (18.67)

whereB = b−0

∫ 2π

0

dθ2π eiθL−0 = δ(b−0 )δ(L−0 ). (18.68)

The kinetic term (after unfixing the gauge) reads:

S0,2 = −12 〈Ψ−1| c−0 QB |Ψ−1〉 −

12 〈Ψ−1/2| c−0 Y QB |Ψ−1/2〉 . (18.69)

The action is invariant under the gauge transformation

δ |Ψ〉 = QB |Λ〉 (18.70)

whereΛ = Λ−1 + Λ−1/2. (18.71)

Each gauge parameter satisfies the same conditions as the associated field (in particular,Λ−1/2 is in the restricted Hilbert space).

18.3.3 Auxiliary field approachThe disadvantage of the constraint approach is two-fold. First, it treats both components ofthe field on a different footing. Second, the constraint must be imposed by hand and does notfollow from any fundamental principle. Another possibility is to embed the propagator in ahigher-dimensional field space by introducing additional fields: in this way, the propagatorcan be inverted without introducing the inverse of X0.

Let’s introduce the new field

Ψ = Ψ−1 + Ψ−3/2 (18.72)

which satisfies the same conditions as Ψ:

b−0 |Ψ〉 = L−0 |Ψ〉 = 0, b+0 |Ψ〉 = 0. (18.73)

A tentative kinetic term is then:

S0,2 = 12 〈Ψ| c

−0 c

+0 L

+0 G |Ψ〉 −〈Ψ| c−0 c+0 L+

0 |Ψ〉 . (18.74)

The kinetic operator in matrix form for (Ψ,Ψ) reads

K = c−0 c+0 L

+0

(−G 11 0

)(18.75)

and its inverse is∆ = b−0 b

+0

1L+

0

(0 11 G

). (18.76)

This reproduces the expected propagator for (Ψ,Ψ) without needing to invert X0.

224

What is the interpretation of the additional fields? The gauge invariance of the actionis

δ |Ψ〉 = QB |Λ〉 , δ |Ψ〉 = QB |Λ〉 , (18.77)where Λ satisfies the same constraints as Ψ (in particular, it contains more components thanthe Λ of the previous section). Then, the equations of motion are

QB |Ψ〉 = 0, QB |Ψ〉 = 0. (18.78)

This shows that both fields are free and decoupled and that the spectrum is doubled. Topush the interpretation further, one needs to consider the interactions.

Amplitudes involve only the states contained in Ψ and thus the interactions are builtsolely in terms of Ψ. Then, the equations of motion have the form:

QB(|Ψ〉 − G |Ψ〉

)= 0, QB |Ψ〉 = |J(Ψ)〉 , (18.79)

where J(Ψ) is a source term due to the interactions. An equation for Ψ only is obtained bymultiplying the second with G

QB |Ψ〉 = G |J(Ψ)〉 . (18.80)Once Ψ is determined by solving this equation, the auxiliary field Ψ is completely fixed bythe second equation up to free field solutions. This shows that Ψ describes only free fieldseven when Ψ is interacting. Note that this implies that the degrees of freedom containedin Ψ do not even couple to the gravitational field! This can also be shown at the level ofFeynman diagrams.Remark 18.8 The field Ψ is not an auxiliary field strictly speaking since it is propagating(its equation of motion is not algebraic).

18.3.4 Large Hilbert spaceThe last formulation of the kinetic term considers the NS string field to be in the largeHilbert space, i.e. η0 6= 0. The Ramond field must be described with one of the two previousapproach.

Writing the action requires to use a NS field Ψ0 with picture number 0. The kinetic termbecomes

S0,2 = −12 〈〈Ψ0, η0QBΨ0〉〉, (18.81)

where 〈〈·, ·〉〉 is the inner product in the large Hilbert space (contains a ξ0 insertion). Thisaction has an enlarged gauge invariance:

δ |Ψ0〉 = QB |Λ0〉+ η0 |Ω1〉 , (18.82)

and the equation of motion reads

QBη0 |Ψ0〉 = 0. (18.83)

The η0 gauge invariance can be fixed with the condition

ξ0 |Ψ0〉 = 0, (18.84)

and one can introduce a new field Ψ−1 such that

|Ψ0〉 = ξ0 |Ψ−1〉 (18.85)

to satisfy automatically the condition. The equation of motion becomes

QB |Ψ−1〉 = 0, (18.86)

and one recovers the small Hilbert space formulation.

225

18.4 References• General reviews [55, 84, sec. 6].

• Spurious poles and vertical integration:

– small Hilbert space [55, app. C, D, 245, 260]– large Hilbert space [85]

• Constructions of super-SFT:

– “Sen’s” amplitude factorization construction [55, 207, 243, 244, 246, 248]– “Munich” homotopy algebra bootstrap [86–88, 93, 153]– Berkovits’ SFT [20, 82, 93, 156, 166, 180]– supermoduli space [197, 273]– democratic SFT [154, 155]– light-cone SFT [129, 132]

• Relations between different constructions [79, 80, 92, 93, 126].

• Ramond string field:

– constrained field [82, 93, 166, 273]– auxiliary field [55, 93, 244, 248]

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Part V

Spacetime string field theory

227

Chapter 19

Momentum-space SFT

In this chapter, we describe the general properties of SFT actions in the momentum space.This allows to make SFT more intuitive, but also to use standard QFT methods to provevarious properties of string theory. We explain how the Wick rotation is generalized fortheories with vertices diverging at infinite real energies (Lorentzian signature). This allowsto prove important properties of string theory, such as unitarity or crossing symmetry.

19.1 General formSince the explicit expressions of the string vertices are not known, it is not possible to writeexplicitly the SFT action. However, the general properties of the vertices are known: then,one can write a general QFT which contains SFT as a subcase. This is sufficient to alreadyextract a lot of informations. The other advantage is that the QFT language is more familiarand intuitive in many situations. Hence, one can use this general form to built intuitionbefore translating the results in a more stringy language. In a nutshell, SFT is a QFT:

• with an infinite number of fields (of all spins);

• with an infinite number of interactions;

• with non-local interactions ∝ e−#k2 ;

• which reproduces the worldsheet amplitudes (if the latter are well-defined).

The non-locality of the interactions is the most salient property of SFT, beyond theinfinite number of fields. This has a number of consequences:

• the Wick rotation is ill-defined;

• the position representation cannot be used, nor any property relying on it (micro-causality, largest time equation. . . );

• standard assumptions from local QFT (in particular, from the constructive S-matrixprogram, such as micro-causality) break down.

Together, these points imply that the usual arguments from QFTs must be improved. Thishas been an active topic in the recent years and the results will be summarized in Sec-tion 19.2.

We expand the string field in Fourier space using a basis φα(k) as (Chapter 11):

|Ψ〉 =∑j

∫ dDk(2π)D ψα(k) |φα(k)〉 , (19.1)

228

where k is the D-dimensional momentum and α the discrete indices (Lorentz indices, grouprepresentation, KK modes. . . ) of the spacetime fields ψα(k). The action in momentumspace takes the form (in Lorentzian signature):

S = −∫

dDk ψα(k)Kαβ(k)ψβ(−k)

−∑n≥0

∫dDk1 · · · dDkn V (n)

α1···αn(k1, . . . , kn)ψα1(k1) · · ·ψαn(kn).(19.2)

The kinetic matrix Kαβ is usually quadratic in the momentum. In the direct Fourier ex-pansion of the SFT action (14.24), it describes only the classical kinetic term: the quantumcorrections are found in the vertex V (2).

From the action, we can write the Feynman rules (for the path integral weight eiS andS-matrix S = 1 + iT ). The propagator reads:

= Kαβ(k)−1 = −iMαβ

k2 +m2α

Qα(k), (19.3)

where Mαβ is mixing matrix for states of equal mass and Qα a polynomial in k (there isno sum over α). The interactions are obtained by plugging the basis states φα inside thevertices Vn (9.58):

= iV (n)α1···αn(k1, . . . , kn) := iVn

(φα1(k1), . . . , φαn(kn)

)

= i∫

dt e−gαk

ij(t) ki·kj−λ

∑α

m2α

Pα1,...,αn

(k1, . . . , kn; t

),

(19.4)

where t denotes collectively the moduli parameters, Pαi is a polynomial in k, gij is apositive-definite matrix, λ > 0 is a number. There is an implicit sum over the momentumindices.

The terms quadratic in the momenta inside the exponential arise from two sources:

• The correlation functions of the vertex operators 〈∏i eiki·X(zi)〉 is proportional to

e−ki·kjG(zi,zj), where G is the Green function. Additional factors like ∂X contrib-ute to the polynomial Pα1,...,αn .

• It is possible to add stubs to the vertices. The effect is to multiply each leg by a factore−λ(k2

i+m2i ) with λ > 0 (we take λ to be the same for all vertices for simplicity). The

first term of the exponential contributes to the diagonal of the matrix gij . By takingλ sufficiently large, one can enforce that all eigenvalues are positive.

Finally, the exponential term with the masses m2α ensures that the sum over all intermediate

states converge despite an infinite number of states. Indeed, the number of states of massmα grows as ecmα , which is dominated by e−λm2

α for sufficiently large λ. Hence, the additionof stubs make explicit the absence of divergences in SFT.1

1Remember that λ is not a physical parameter and disappears on-shell. This means that the cancellationof the divergences is independent of λ and must always happen on-shell.

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The vertices have no singularity for ki ∈ C finite. As the energy becomes infinite |k0i | →

∞, they behave as:lim

k0→±i∞V (n) = 0, lim

k0→±∞V (n) =∞. (19.5)

The first property is responsible for the soft UV behaviour of string theory in Euclideansignature, while the second prevents from performing the Wick rotation (indeed, the poleat infinity implies that the arcs closing the contour contribute).

The g-loop n-point amputated Green functions are sums of Feynman diagrams, each ofthe form:

Fg,n(p1, . . . , pn) ∼∫

dT∏s

dD`s e−Grs(T ) `r·`s−2Hri(T ) `r·pi−Fij(T ) pi·pj

×∏a

1k2a +m2

a

P(pi, `r;T ),(19.6)

where pi are the external momenta, `r the loop momenta and ki the internal mo-menta, with the latter given by a linear combination of the other. Moreover, T denotes thedependence in the moduli parameters of all the internal vertices, and P is a polynomial in(pi, `r). The matrix Grs is positive definite, which implies that:

• integrations over spatial loop momenta `r converge;

• integrations over loop energies `0r diverge.

As a consequence, the Feynman diagrams in Lorentzian signature are ill-defined: we willexplain in the next section how to fix this problem.

19.2 Generalized Wick rotationWe have seen that loop integrals in Lorentzian signature are divergent because of the largeenergy behaviour of the interactions. But, this is not different from the usual QFT, where theloop integrals are also ill-defined in Lorentzian signature. Indeed, poles of the propagatorssit on the real axis and also give divergent loop integrals (note that the same problem arisealso here). In that case, the strategy is to define the Feynman diagrams in Euclidean spaceand to perform a Wick rotation: the latter matches the expressions in Lorentzian signatureup to the iε-prescription. The goal of the latter is to move slightly the poles away from thereal axis.

Example 19.1 – Scalar fieldConsider a scalar field of mass m with a quartic interaction. The 1-loop 4-point Feyn-man diagram is given in Figure 19.1. The external momenta are pi, i = 1, . . . , 4. Thereare one loop momentum ` and two internal momenta k1 = ` and k2 = p − `, wherep = p1 + p2. The poles in the loop energy `0 are located at:

p± = ±√`2 +m2, q± = p0 ±

√(p− `)2 +m2. (19.7)

The graph is first defined in Euclidean signature, where the external and loop en-ergies are pure imaginary, p0

i , `0 ∈ iR. The poles are shown in Figure 19.2. Then,

the external momenta are analytically continued to real values, p0i ∈ R. At the same

time, the integration contour is also analytically continued thanks to the Wick rotation(Figure 19.3). The contour is closed with arcs, but they don’t contribute since there isno poles in the upper-right and lower-left quadrants, and no poles at infinity. However,one cannot continue the contour such that `0 ∈ R because of the poles on the real axis.

230

The Wick rotation is possible for `0 in the upper-right quadrant, Re `0 ≥ 0, Im `0 > 0,which leads to the iε-prescription `0 ∈ R + iε.

Figure 19.1: 1-loop 4-point function for a scalar field theory.

Figure 19.2: Integration contour for external Euclidean momenta.

Since the Feynman diagram (19.6) is not defined in Lorentzian signature because of thepoles at `0r → ±∞, it is also necessary to start with Euclidean momenta. However, thesame behaviour at infinity prevents from using the Wick rotation since the contributionfrom the arcs does not vanish. It is then necessary to find another prescription for definingthe Feynman diagrams in SFT starting from the Euclidean Green functions. This is givenby the following generalized Wick rotation (Pius–Sen [211]):

1. Define the Green functions for Euclidean internal and external momenta.

2. Perform an analytic continuation of the external energies and of the integration contoursuch that:

• keep poles on the same side;• keep the contour ends fixed at ±i∞.

One can show [211] that the Green functions are analytic in the upper-right quadrant Im p0a >

0,Re p0a ≥ 0, for pa ∈ R, p0

a. Moreover, the result is independent of the contour chosen aslong as it satisfies the conditions described above. In fact, this generalized Wick rotation isvalid even for normal QFT, which raises interesting questions. For example, it seems thatthe internal and external set of states have no intersection, which can be puzzling whentrying to interpret the Cutkosky rules. Nonetheless, everything works as expected.

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Figure 19.3: Integration contour for external Lorentzian momenta after Wick rotation (reg-ular vertices).

Remark 19.1 (Timelike Liouville theory) It has been shown in [14] that this general-ized Wick rotation is also the correct way for defining the timelike Liouville theory.

The fact that the amplitude is analytic only when the imaginary parts of the momentaare not zero, Im p0

a > 0, is equivalent to the usual iε-prescription for QFT. Moreover, it hasbeen shown [253] to be equivalent to the moduli space iε-prescription from [292]. Then, ithas also been used to prove several important properties of string theory shared by localQFTs: Cutkosky rules [211, 212], unitarity [250, 251], analyticity in a subset of the primitivedomain and crossing symmetry [54]. Finally, general soft theorems for string theory (and, infact, any theory of quantum gravity) have been proven in [43, 172, 254, 255]. All together,these properties establish string theory as a very strong candidate for a consistent theoryof everything. The next main question is how to obtain an expression of SFT which isamenable to explicit computations. This will certainly require to understand even betterthe deep structure of SFT, a goal which this review will hopefully help the reader to achieve.

19.3 References• SFT momentum space action [55, 211, 256].

• Consistency properties of string theory [55]:

– generalized Wick rotation, Cutkosky rules and unitarity [211, 212, 249–251, 253].– analyticity and crossing symmetry [54].– soft theorems [43, 172, 254, 255].

232

(a)

(b)

(c)

Figure 19.4: Integration contour after analytic continuation to external Lorentzian momenta.Depending on the values of the external momenta, different cases can happen.

233

Part VI

Classical solutions

234

Chapter 20

Classical solutions

Classical solutions is an important topic in any field theory. Most importantly for SFTit provides the possible backgrounds around which one can expand the action and canthus provide insights to the question of background independence. Indeed given a classicalsolution Φ0 the string field can be separated as a background contribution plus fluctuationsΦ = Φ0 + Φ′, in the same way that the Einstein–Hilbert action can be written by expandingthe metric gµν = gµν+hµν . Classical solutions also provide informations about the dynamicsand the properties of the theory, one of the most important application being the tachyoncondensation.

Several methods have been designed to find solutions. The two major categories are:

• level-truncation (approximate): expand the string field on the subset of all states undera certain level L0 ≤ L, then compute the spacetime action for the corresponding fieldsand derive the equations of motion;

• analytic solution (exact): use the properties of the string field theory (such as algebraicproperties) to provide exact (sometimes formal) solutions.

Solutions are mostly constrained to Witten’s cubic SFT due to its simplicity.If BCFT describes a Dp-brane vacuum then there (p+ 1) non-compact dimensions and

the string field is expanded as

|Φ〉 =∑i

∫ dp+1k

(2π)p+1 φi(k)c1 |i, k〉 . (20.1)

This describes open strings attached to the Dp-brane, with the end points being able tomove only in (p+ 1) dimensions.

Remark 20.1 Note that any operator O corresponding to a state |O〉 defines a string field.For this reason we will use any of these words interchangeably.

20.1 Tachyon condensationBoundary conditions for each spacetime dimension must be imposed on open string endpoints: Neumann (Dirichlet) condition indicates that the end points are free to move (arefixed) along this dimension. In order to conserve momentum, the end points of a string withp Neumann boundary conditions must be attached to a Dp-brane which absorbs the flow ofmomentum. Propagation in 26 dimensions is thus associated with a D25-brane (also calledspace-filling) brane.

235

But the D-branes are unstable objects: since they don’t carry any conserved chargethey can decay to lower-dimensional D-branes, and to open and closed strings. Note thatthe situation is different in superstring theory where at least some branes carry a conservedcharge due to the existence of p-form gauge fields. When a system is in an unstable state, thelinearisation of the dynamical variable around this state displays modes which grow withoutbounds: in field theory the tachyon corresponds precisely to such modes. As a consequenceone expects that the decay of an unstable D-brane will be driven by the tachyon, the latergoing down its potential to a (local) minimum.

Given a SFT formulated in a D-brane background, one expects that there is a classicalsolution corresponding to this background without the brane. Moreover if all branes –including the D25 space filling brane – decay, then the final state must be devoid of openstring excitations since they cannot be attached to anything. Ashoke Sen has formulated aset of conjectures following these intuitions:

1. the effective tachyon potential (obtained by integrating out the other fields) admits alocal minimum equal to the tension of the Dp-brane;

2. there are no open string states in the tachyon vacuum (open strings need to be attachedon D-branes);

3. there are lump solutions describing the decay of the Dp-brane to a Dq-brane withq < p, and the difference of energy corresponds to the difference of the brane tensions.

The first method to attack the problem is using the level truncation, where the stringfield is limited to some states. Studying the properties of the tachyon and its interactionswith the other fields, a consistent truncation to level 2 is

|T 〉 =(T0 + uLm−2 + v c−1b−1 + · · ·

)c1 |0〉 (20.2)

(one considers k = 0). The details are independent of the BCFT chosen because only theproperties of ghosts and of the Virasoro algebra are used. Plugging this expansion into theaction and minimizing the potential one finds that the energy is 96% of the brane tension(against 68% for a level 1 truncation). Numerical computations in the level truncationscheme have been pushed very far and confirmed the conjecture. But a breakthrough wasachieved by Schnabl who found an analytic solution describing exactly the tachyon vacuum.

20.2 Analytic solutionsA direct path to analytic solutions is to use the properties of the ∗-algebra to solve theequation of motion algebraically. While attacking directly this question is doomed to failbecause this algebra is huge, one can restrict the set of string fields forming a subalgebra.

20.2.1 Algebraic toolsThe identity string field 1 is such that

∀Φ : 1 ∗ Φ = Φ ∗ 1 = Φ. (20.3)

Since the goal is to study the tachyon condensation it is useful to consider the c-ghost fieldwhich corresponds to the zero-momentum tachyon vertex operator

|c〉 = c1 |0〉 = c(0) |0〉 . (20.4)

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Next one introduces the non-local operator K through its relation with the SL(2,C) vacuum

|0〉 = e−K |1〉 . (20.5)The operator K can be related to the integral of T (z), and can be seen as L−1 (in anappropriate frame). Finally it is useful to introduce a B field defined similarly from b(z).Then the operator algebra reads

c2 = 0, B2 = 0, c,B = 1,[K,B] = 0, [K, c] = ∂c.

(20.6)

The QB operator acts on these operators as

QBK = 0, QBB = K, QBc = c∂c = cKc. (20.7)

Note that Bc is a projector since

(Bc)2 = Bc (20.8)

using the algebra.Other fields can be constructed from K by considering the Laplace transform

11 +K

=∫ ∞

0dα e−α e−αK , 1√

1 +K= 1√

π

∫ ∞0

dα e−α√α

e−αK . (20.9)

More generally a geometric string field is a field which can be written as the Laplace trans-form of some function f(α)

F (K) =∫ ∞

0dα f(α) e−αK . (20.10)

A contracting homotopy operator A is an operator which satisfies the identity

QB , A = 1. (20.11)

If it exists it implies that the cohomology of QB is empty (and this at all ghost number)since given QBΦ = 0 one finds that Φ is exact by the following computation:

QB(A ∗ Φ) = (QBA) ∗ Φ +A ∗ (QBΦ) = 1 ∗ Φ = Φ. (20.12)

The converse holds: if the cohomology is empty then a state A exists.

20.2.2 SolutionsRecall the equation of motion

QBΦ + Φ2 = 0. (20.13)The energy of a configuration is computed by

E = −S = 16

∫Φ ∗QBΦ = −1

6

∫Φ3. (20.14)

ComputationUpon using the equation of motion:

S =∫ (1

2 Φ ∗QBΦ + 13 Φ3

)=∫ (1

2 Φ ∗QBΦ− 13 Φ ∗QBΦ

). (20.15)

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A simple solution isΦ = α c− cK. (20.16)

Computation

QBΦ = αQBc− (QBc)K + c(QBK) = α cKc− cKcK,Φ2 = (α c− cK)2 = −α cKc+ cKcK.

A more general solution reads

Φ = FcKB

1− F 2 cF = (1− FBcF )QB(

11− FBcF

), (20.17)

where F (K) is an arbitrary function of K. The factor inside the parenthesis can be rewrittenas

11− FBcF = 1 + F

1− F 2 BcF. (20.18)

This relation can be derived by writing

(FBcF )2 = F([Bc, F 2] + F 2Bc

)BcF = F 3BcF. (20.19)

The first term vanishes because B commutes with c inside the commutator, and since it alsocommutes with K it can be extracted from the commutator to it the B outside. Then theexpression is simplified using that (Bc)2 = Bc.

ComputationLet’s show that Φ is a solution. First one finds that

QB(1− FBcF ) = −(1− FBcF )QB(

11− FBcF

)(1− FBcF ) (20.20)

by writing

QB(1− FBcF ) = QB

((1− FBcF ) 1

1− FBcF (1− FBcF ))

= 2QB(1− FBcF ) + (1− FBcF )QB(

11− FBcF

)(1− FBcF ).

Then one can compare with Φ2

Φ2 = (1− FBcF )QB(

11− FBcF

)(1− FBcF )QB

(1

1− FBcF

). (20.21)

An important question is to determine whether this solution is a pure gauge or not. Notethat (20.17) can be rewritten in the same form as a pure gauge solution

Φ = e−ΛQBeΛ, eΛ = 11− FBcF = 1 + F

1− F 2 BcF. (20.22)

For Φ to be a genuine solution the factor eΛ must be singular such that one cannot interpretit as a gauge transformation. For this one needs to have F/(1− F 2) to be ill-defined, whileF/(1− F 2) must be well-defined to ensure that the field itself exists.

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It it also useful to study the cohomology of the BRST operator in the background definedby Φ

QΦ = QB + [Φ, ·], (20.23)

since the fluctuations describe the physical states corresponding to this background. If it isa pure gauge, then one should find the same cohomology as QB . On the other hand, if it isempty it means that there no string excitations, which is expected if this corresponds to atachyon vacuum. One can construct a contracting homotopy operator A for QΨ

A = 1− F 2

KB. (20.24)

This operator exists if K/(1−F 2) is well-defined, which also translates do the non-trivialityof Φ.

Remark 20.2 The explicit form of QΦ acting on an operator is

QΦ = QBA+ Φ ∗A− (−1)|A|A ∗ Φ. (20.25)

The field F (K) can be expanded in powers of K

F (K) = a+ bK + · · · (20.26)

The tachyon vacuum is found for a 6= 1 and b 6= 0 while the original vacuum is a 6= 1.Consider the tachyon vacuum solution

F (K) = 1√1 +K

(20.27)

which is such thatΦ = 1√

1 +KcB(1 +K)c 1√

1 +K. (20.28)

The homotopy operatorA = B

1 +K(20.29)

is regular. The energy of this solution reads

E = − 12π2 = −T, (20.30)

where T is the tension of a D-brane (in the unit where the tachyon coupling constant is setto 1).

Computation

−S = 16

∫Φ ∗QBΦ = 1

6

⟨(c+QB(Bc)

) 11 +K

c∂c1

1 +K

⟩(20.31)

= 16

∫ ∞0

dt1∫ ∞

0dt2 e−(t1+t2)〈c e−t1Kc∂c e−t2K〉 (20.32)

= − 16π2

∫ ∞0

duu3e−u∫ 1

0dv sin2 πv = − 1

2π2 . (20.33)

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Finally one can write the action in terms of fluctuations Ψ′ around the vacuum Φ

S =∫ (1

2 Φ′ ∗QΦΦ′ + 13 Φ′3

)(20.34)

wehreQ′B = QB + [Φ, ·] (20.35)

The cubic interaction is not deformed by changing the background. The form is particularlysimple because no higher-product appears, which is consistent with the idea that Witten’stheory is cubic. This also proves in a simpler manner the background independence for theclassical open string field theory.

20.3 Bibliography• Analytic solutions: reviews [102, 201, 233], papers [75, 94, 200, 232].

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Part VII

Conformal field theory

241

Chapter 21

Conformal field theory in Ddimensions

Starting with this chapter, we discuss general properties of conformal field theories (CFT).The goal is not to be exhaustive, but to provide a short introduction and to gather theconcepts and formulas that are needed for string theory. However, the subject is presentedas a standalone topic such that it can be of interest for a more general public.

The conformal group in any dimension is introduced in this chapter. The specific caseD = 2, which is the most relevant for the current review, is developed in the followingchapters.

21.1 CFT on a general manifoldIn this chapter and in the next one, we discuss CFTs as QFTs living on a spacetime M,independently from string theory (there is no reference to a target spacetime). As such, wewill use spacetime notations together with some simplifications: coordinates are written asxµ with µ = 0, . . . , D − 1 and time is written as x0 = t (x0 = τ) in Lorentzian (Euclidean)signature.

21.1.1 Conformal groupGiven a metric gµν on a D-dimensional manifoldM, the conformal group CISO(M) is theset of coordinate transformations (called conformal symmetries or isometries)

xµ −→ x′µ = x′µ(x) (21.1)

which leaves the metric invariant up to an overall scaling factor:

gµν(x) −→ g′µν(x′) = ∂xρ

∂x′µ∂xσ

∂x′νgρσ(x) = Ω(x)2gµν(x). (21.2)

This means that angles between two vectors u and v are left invariant under the transform-ation:

u · v|u||v|

= u′ · v′

|u′||v′|. (21.3)

It is often convenient to parametrize the scale factor by an exponential

Ω := eω. (21.4)

242

Considering an infinitesimal transformation

δxµ = ξµ, (21.5)

the condition (21.2) becomes the conformal Killing equation

δgµν = Lξgµν = ∇µξν +∇νξµ = 2dgµν∇ρξρ, (21.6)

such that the scale factor isΩ2 = 1 + 2

d∇ρξρ. (21.7)

The vector fields ξ satisfying this equation are called conformal Killing vectors (CKV). Con-formal transformations form a global subgroup of the diffeomorphism group: the generatorsof the transformations do depend on the coordinates, but the parameters do not (for aninternal global symmetry, both the generators and the parameters don’t depend on thecoordinates).

The conformal group contains the isometry group ISO(M) ofM as a subgroup, corres-ponding to the case Ω = 1:

ISO(M) ⊂ CISO(M). (21.8)These transformations also preserve distances between points. The corresponding generatorsof infinitesimal transformations are called Killing vectors and satisfies the Killing equation

δgµν = Lξgµν = ∇µξν +∇νξµ = 0. (21.9)

They form a subalgebra of the CKV algebra.An important point is to be made for the relation between infinitesimal and finite trans-

formations: with spacetime symmetries it often happens that the first cannot be exponenti-ated into the second. The reason is that the (conformal) Killing vectors may be defined onlylocally, i.e. they are well-defined in a given domain but have singularities outside. Whenthis happens, they do not lead to an invertible transformation, which cannot be an elementof the group. These notions are sometimes confused in physics and the term of “group” isused instead of “algebra”. We shall be careful in distinguishing both concepts.

Remark 21.1 (Isometries of M ⊂ Rp,q) In order to find the conformal isometries ofa manifold M which is a subset of Rp,q defined in (21.12), it is sufficient to restrict thetransformations of Rp,q to the subset M [235]. In the process, not all global transformationsgenerically survive. On the other hand, the algebra of local (infinitesimal) transformationsfor M and Rp,q are identical since M is locally like Rp,q.

21.1.2 Conformal field theoryHow to build a conformal field theory (CFT) for some matter fields Ψ, i.e. a QFT on the curvedbackground (M, g) which is invariant under the conformal group CISO(M)?

To answer this question, we explain first how a background theory can be built from a moregeneral theory. A background B is a fixed field configuration which couples to the other fieldsbut which does not have any intrinsic dynamics. Typically, the background is a solution to theequations of motion derived from an action SB [B]

δSBδB

(B) = 0, (21.10)

but this is not necessary.1 One can then consider probe fields Ψ with action Sp[B,Ψ], i.e. fields whichlive on the background without disturbing it (this means that the backreaction is neglected). If

1For example, the Einstein–Hilbert action with a cosmological constant can be expanded around theMinkowski spacetime, even if it is not a solution to the equation of motion.

243

the background theory possesses a gauge symmetry, then any residual symmetry of the backgroundgenerically becomes a global symmetry of the probe action (if the action Sp[B,Ψ] is invariant underthe gauge symmetry). Note that in this case only the fields Ψ transform under the symmetry.

Before considering the conformal transformations, consider first the invariance under the iso-metry of the background metric g (often a solution of Einstein equations). Then, an action S[g,Ψ]which admits ISO(M) as a global symmetry can be constructed by writing a diffeomorphism in-variant action S[g,Ψ] and by freezing the metric g = g. Indeed, isometries does not change thebackground metric and can be used as symmetries of S, where only Ψ transforms. The rest of thediffeomorphisms are not a symmetry of the action.

The extension to CISO(M) requires more work because the conformal transformations modifythe background metric and does not directly give global symmetries of S. The solution is tointroduce an additional gauge symmetry

g′µν(x) = e2ω(x)gµν(x), Ψ′(x) = edΨω(x)Ψ(x), (21.11)called the (local) Weyl symmetry, where dΨ is the dimension of the field Ψ. The group of Weyltransformations is denoted by Weyl(M). If this is a symmetry of the original theory, then the scalingfactor Ω(x) in front of the metric in (21.2) can be compensated with a Weyl transformation. As aconsequence, an action S[g,Ψ] invariant under the conformal group CISO(M) can be obtained froman action S[g,Ψ] invariant under diffeomorphisms and Weyl transformations. Then, the conformalgroup can be understood as the subgroup of the diffeomorphism which transforms the metric like aWeyl transformation. This fact has been encountered in Chapter 3 from a different perspective.Remark 21.2 One may want to reverse the argument by starting with S[g,Ψ] to derive actionsS[g,Ψ] which are invariant under diffeomorphisms and (local) Weyl transformations.

The standard procedure to construct a diffeomorphism invariant theory from S[η, φ] is to usethe minimal coupling of the field Ψ by replacing derivatives with covariant derivatives. But, notall actions S[g,Ψ] can be found in this way: indeed any term in S[g,Ψ] which vanishes upon fixingthe background (for example, a term proportional to the equations of motion if the background is asolution) cannot be recovered from the minimal coupling.

For Weyl transformations, the subject is more complicated: a necessary condition is that S[η,Ψ]be invariant under conformal transformations in flat space, but this condition is sufficient only if theaction is at most quadratic in the first derivatives. On the other hand, global Weyl transformationsrequire only invariance under global scale transformations. These remarks can be important in theconstruction of string worldsheet theories and they are related to Remark 3.17 page 81. Selectedreferences on this topic are [47, 96, 128, 145, 146, 216].Remark 21.3 (Axiomatic formulations) More axiomatic formulations are given in [225, 235].

21.2 CFT on Minkowski spaceIn this section, we consider the case where M = Rp,q (D = p + q) and where g = η is theflat metric with signature (p, q):

η = diag(−1, . . . ,−1︸︷︷︸q

, 1, . . . , 1︸︷︷︸p

). (21.12)

The conformal Killing equation becomes(ηµν∆ + (D − 2)∂µ∂ν

)∂ · ε = 0. (21.13)

The case D = 2 is relegated to the next chapter. For D > 2, one finds the followingtransformations:

translation: ξµ = aµ, (21.14a)rotation: ξµ = ωµνx

ν , (21.14b)dilatation: ξµ = λxµ, (21.14c)

SCT: ξµ = bµx2 − 2b · xxµ, (21.14d)

244

where ωµν is antisymmetric. The rotations include Lorentz transformations and SCT means“special conformal transformation”.

All parameters aµ, ωµν , λ, bµ are constant. The generators are respectively denoted byPµ, Jµν , D,Kµ. The finite translations and rotations form the Poincaré group SO(p, q),while the conformal group can be shown to be SO(p+ 1, q + 1):

ISO(Rp,q) = SO(p, q), CISO(Rp,q) = SO(p+ 1, q + 1). (21.15)

The dimension of this group is

dim SO(p+ 1, q + 1) = 12 (p+ q + 2)(p+ q + 1). (21.16)

Exercise 21.1Prove (21.14), then write the finite transformations.

Exercise 21.2Compute the algebra of the generators and show the equivalence with so(p+ 1, q + 1).

21.3 References• References on higher-dimensional CFTs are [64, 221, 228, 235, 266].

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Chapter 22

Conformal field theory on theplane

Starting with this chapter, we focus on two-dimensional Euclidean CFTs on the complexplane (or equivalently the sphere). We start by describing the geometry of the sphere and therelation to the complex plane and to the cylinder, in order to make contact with the stringworldsheet. Then, we discuss classical CFTs and the Witt algebra obtained by classifyingthe conformal isometries of the complex plane. Then, we describe quantum CFTs andintroduce the operator formalism. This last section is the most important for this reviewas it includes information on the operator product expansion, Hilbert space, Hermitian andBPZ conjugations.

As described at the beginning of Chapter 21, we use spacetime notations for the coordin-ates, but follow otherwise the normalization for the worldsheet. In particular, integrals arenormalized by 2π. However, the spatial coordinate on the cylinder is still written as σ toavoid confusions: xµ = (τ, σ).

22.1 The Riemann sphere22.1.1 Map to the complex planeThe Riemann sphere Σ0, which is diffeomorphic to the unit sphere S2, has genus g = 0 andis thus the simplest Riemann surface. Its most straightforward description is obtained bymapping it to the extended1 complex plane C (also denoted C), which is the complex planez ∈ C to which the point at infinity z =∞ is added:

C = C ∪ ∞. (22.1)

One speaks about “the point at infinity” because all the points at infinity (i.e. the points zsuch that |z| → ∞)

limr→∞

r eiθ :=∞ (22.2)

are identified (the limit is independent of θ).The identification can be understood by mapping (say) the south pole to the origin of

the plane and the north pole to infinity2 (Figure 22.1) through the stereographic projection1This qualification will often be omitted.

246

−−−−−−−−→

Figure 22.1: Map from the Riemann sphere to the complex plane. The south and northpoles are denoted by the letter S and N , and the equatorial circle by E.

z = eiφ cot θ2 , (22.3)

where (θ, φ) are angles on the sphere. Any circle of the sphere is mapped to a circle in thecomplex plane. Conversely, the Riemann sphere can be viewed as a compactification of thecomplex plane.

Exercise 22.1 – Stereographic projection in Cartesian coordinatesFind (22.3) in Cartesian coordinates.

Introducing Cartesian coordinates (x, y) related to the complex coordinates by3

z = x+ iy, z = x− iy, (22.4a)

x = z + z

2 , y = z − z2i , (22.4b)

the metric readsds2 = dx2 + dy2 = dzdz. (22.5)

The relations between the derivatives in the two coordinate systems are easily found:

∂ := ∂z = 12 (∂x − i∂y), ∂ := ∂z = 1

2 (∂x + i∂y). (22.6)

The indexed form will be used when there is a risk of confusion. If the index is omitted thenthe derivative acts directly to the field next to it, for example

∂φ(z1)∂φ(z2) := ∂z1∂z2φ(z1)φ(z2). (22.7)

Generically, the meromorphic and anti-meromorphic parts of a object will be denotedwithout and with a bar, see (22.56) for an example.

2Note that the points are distinguished in order to write the map, but they have nothing special bythemselves (i.e. they are not punctures).

3General formulas can be found in Section 5.1 by replacing (τ, σ) with (x, y). In most cases, φ = 0 inthis chapter.

247

The extended complex plane C can also be described with two coordinates patches z ∈ Cand w ∈ C. In the first, the point at infinity (north pole) is removed, in the second, theorigin (south pole) is removed. On the overlap, the transition function is

w = 1z. (22.8)

This description avoids to work with the infinity: studying the behaviour of f(z) at z =∞is equivalent to study f(1/w) at w = 0.

We have avoided to give any non-flat metric for the different spaces above. The reason isthat any two-dimensional metric is locally conformally equivalent to the flat metric, and itis thus sufficient to work with this metric in each patch. Similarly, the conformal groups forthe Riemann sphere and for the complex plane coincide since the latter covers the sphere.Moreover, CFTs are defined such that the action is insensitive to the conformal factor: theywill take the same form for any metric (some objects will have a dependence in the surface,but it will be easily determined).

22.1.2 Relation to the cylinder – string theoryThe worldsheet of a closed string propagating in spacetime is topologically a cylinder R×S1

of circumference L. In this section, we show that the cylinder can also be mapped to thecomplex plane – and thus to the Riemann sphere – after removing two points. Since thecylinder has a clear physical interpretation in string theory, it is useful to know how totranslate the results from the plane to the cylinder.

It makes also sense to define two-dimensional models on the cylinder independently ofa string theory interpretation since the compactification of the spatial direction from R toS1 regulates the infrared divergences. Moreover, it leads to a natural definition of a “time”and of an Hamiltonian on the Euclidean plane.

Denoting the worldsheet coordinates in Lorentzian signature by (t, σ) with4

t ∈ R, σ ∈ [0, L), σ ∼ σ + L, (22.9)

the metric readsds2 = −dt2 + dσ2 = −dσ+dσ−, (22.10)

where the light-cone coordinatesdσ± = dt± dσ (22.11)

have been introduced. It is natural to perform a Wick rotation from the Lorentzian time tto the Euclidean time

τ = it, (22.12)and the metric becomes

ds2 = dτ2 + dσ2. (22.13)It is logical to introduce the complex coordinates

w = τ + iσ, w = τ − iσ (22.14)

for which the metric isds2 = dwdw. (22.15)

Note that the relation to Lorentzian light-cone coordinates are

w = i(t+ σ) = iσ+, w = i(t− σ) = iσ−. (22.16)4Consistently with the comments at the beginning of Chapter 21, the Lorentzian worldsheet time is

denoted by t instead of τM .

248

Hence, an (anti-)holomorphic function of w (w) depends only on σ+ (σ−) before the Wickrotation: this leads to the identification of the left- and right-moving sectors with the holo-morphic and anti-holomorphic sectors of the theory.

The cylinder can be mapped to the complex plane through

z = e2πw/L, z = e2πw/L, (22.17)

and the corresponding metric is

ds2 =(L

2π

)2 dzdz|z|2

. (22.18)

A conformal transformation brings this metric to the flat metric (22.5). The conventionsfor the various coordinates and maps vary in the different textbooks. We have gathered inTable A.1 the three main conventions and which references use which.

The map from the cylinder to the plane is found by sending the bottom end (corres-ponding to the infinite past t → −∞) to the origin of the plane, and the top end (infinitefuture t→∞) to the infinity. Since the cylinder has two boundaries (its two ends) the mapexcludes the point z = 0 and z = ∞ and one really obtains the space C − 0,∞ = C∗.This space can, in turn, be mapped to the 2-punctured Riemann sphere Σ0,2.

The physical interpretation for the difference between Σ0 and Σ0,2 is simple: since oneconsiders the propagation of a string, it means that the worldsheet corresponds to an amp-litude with two external states, which are the mapped to the sphere as punctures (Fig-ure 22.2, Section 4.1.1). Removing the external states (yielding the tree-level vacuum amp-litude) corresponds to gluing half-sphere (caps) at each end of the cylinder (Figure 22.3).Then, it can be mapped to the Riemann sphere without punctures. As a consequence, theproperties of tree-level string theory are found by studying the matter and ghost CFTs onthe Riemann sphere. Scattering amplitudes are computed through correlation functions ofappropriate operators on the sphere. This picture will generalize to higher-genus Riemannsurfaces, and, since most of the properties are consequence of the algebra, they will becommon to all surfaces.

Mathematically, the difference was to be expected since the cylinder is flat (with χ = 0)but the sphere has a positive curvature (and χ = −2). Punctures contribute negatively tothe curvature (and thus positively to the Euler characteristics).

Remark 22.1 The coordinate z is always used as a coordinate on the complex plane, but thecorresponding metric may be different – compare (22.5) and (22.18). As explained previously,this does not matter since the theory is insensitive to the conformal factor.

22.2 Classical CFTsIn this section, we consider an action S[Ψ] which is conformally invariant. We first identifyand discuss the properties of the conformal algebra and group, before explaining how a CFTis defined.

22.2.1 Witt conformal algebraSince the Riemann sphere is identified with the complex plane, they share the same conformalgroup and algebra. Consider the metric (22.5)

ds2 = dzdz, (22.19)

249

−−−−−→ −−−−−→

Figure 22.2: Map from the cylinder to the sphere with two tubes, to the 2-punctured sphereΣ0,2.

−−−−−−−−→

Figure 22.3: Map from the cylinder with two caps (half-spheres) to the Riemann sphere Σ0.

250

then, any meromorphic change of coordinates

z −→ z′ = f(z), z −→ z′ = f(z) (22.20)

is a conformal transformation since the metric becomes

ds2 = dz′dz′ =∣∣∣∣dfdz

∣∣∣∣2dzdz. (22.21)

However, only holomorphic functions which are globally defined on C are elements ofthe group. At the algebra level, any holomorphic function f(z) regular in a domain D givesa well-defined transformation in this domain D. Hence, the algebra is infinite-dimensional.On the other hand, f(z) is only meromorphic on C generically: it cannot be exponentiatedto a group element. We first characterize the algebra and then obtain the conditions topromote the local transformations to global ones.

Since the transformations are defined only locally, it is sufficient to consider an infinites-imal transformation

δz = v(z), δz = v(z), (22.22)where v(z) is a meromorphic vector field on the Riemann sphere. Indeed, the conformalKilling equation (21.6) in D = 2 is equivalent to the Cauchy–Riemann equations:

∂v = 0, ∂v = 0. (22.23)

The vector field admits a Laurent series

v(z) =∑n∈Z

vnzn+1, v(z) =

∑n∈Z

vnzn+1, (22.24)

and the vn and vn are to be interpreted as the parameters of the transformation. A basis ofvectors (generators) is:

`n = −zn+1∂z, ¯n = −zn+1∂z, n ∈ Z. (22.25)

One can check that each set of generators satisfies the Witt algebra

[`m, `n] = (m− n)`m+n, [¯m, ¯n] = (m− n)¯

m+n, [`m, ¯n] = 0. (22.26)

Since there are two commuting copies of the Witt algebra, it is natural to extend theranges of the coordinates from C to C2 and to consider z and z as independent variables.In particular, this gives a natural action of the product algebra over C2. This procedurewill be further motivated when studying CFTs since the holomorphic and anti-holomorphicparts will generally split, and it makes sense to study them separately. Ultimately, phys-ical quantities can be extracted by imposing the condition z = z∗ at the end (the star isalways reserved for the complex conjugation, the bar will generically denote an independentvariable). In that case, the two algebras are also related by complex conjugation.

Note that the variation of the metric (B.6) under a meromorphic change of coordinates(22.22) becomes

δgzz = ∂v + ∂v, δgzz = δgzz = 0. (22.27)

22.2.2 PSL(2,C) conformal groupThe next step is to determine the globally defined vectors and to study the associated group.

First, the conditions for a vector v(z) to be well-defined at z = 0 are

lim|z|→0

v(z) <∞ =⇒ ∀n < −1 : vn = 0. (22.28)

251

The behaviour at z =∞ can be investigated thanks to the map z = 1/w

v(1/w) = dzdw

∑n

vnw−n−1, (22.29)

where the additional derivative arises because v is a vector. Then, the regularity conditionsat z =∞ are

lim|z|→∞

v(z) = lim|w|→0

dzdw v(1/w) = − lim

|w|→0

v(1/w)w2 <∞ =⇒ ∀n > 1 : vn = 0. (22.30)

As a result, the globally defined generators are

`−1, `0, `1 ∪ ¯−1, ¯0, ¯1 (22.31)

where`−1 = −∂z, `0 = −z∂z, `1 = −z2∂z. (22.32)

It is straightforward to check that they form two copies of the sl(2,C) algebra

[`0, `±1] = ∓`±1, [`1, `−1] = 2`0. (22.33)

The global conformal group is sometimes called Möbius group:

PSL(2,C) := SL(2,C)/Z2 ∼ SO(3, 1), (22.34)

where the additional division by Z2 is clearer when studying an explicit representation. Itcorresponds with kerP1 defined in (3.91):

K0 = PSL(2,C). (22.35)

A matrix representation of SL(2,C) is

g =(a bc d

), a, b, c, d ∈ C, det g = ad− bc = 1, (22.36)

which shows that this group has six real parameters

K0 := dim SL(2,C) = 6. (22.37)

The associated transformation on the complex plane reads

fg(z) = az + b

cz + d. (22.38)

The quotient by Z2 is required since changing the sign of all parameters do not change thetransformation. These transformations have received different names: Möbius, projective,homographic, linear fractional transformations. . .

Holomorphic vector fields are then of the form

v(z) = β + 2αz + γz2, v(z) = β + 2αz + γz2, (22.39)

wherea = 1 + α, b = β, c = −γ, d = 1− α. (22.40)

The finite transformations associated to (21.14) are:

translation: fg(z) = z + a, a ∈ C, (22.41a)rotation: fg(z) = ζ z, |ζ| = 1, (22.41b)

dilatation: fg(z) = λ z, λ ∈ R, (22.41c)

SCT: fg(z) = z

cz + 1 , c ∈ C. (22.41d)

Investigation leads to the following association between the generators and transformations:

252

• translation: `−1 and ¯−1;

• dilatation (or radial translation): (`0 + ¯0);

• rotation (or angular translation): i(`0 − ¯0);

• special conformal transformation: `1 and ¯1.

The inversion is defined by

inversion: I(z) := I+(z) = 1z. (22.42a)

It is not an element of SL(2,C) and, for this reason, only primary fields (defined later)transform as a tensor. However, the inversion with a minus sign

I−(z) := −I(z) = I(−z) = −1z

(22.42b)

is a SL(2,C) transformation.A useful transformation is the circular permutation of (0, 1,∞):

g∞,0,1(z) = 11− z . (22.43)

22.2.3 Definition of a CFTA CFT is characterized by its set of (composite) fields O(z, z) which correspond to anylocal expression constructed from the fields Ψ appearing in the Lagrangian and of theirderivatives. For example, in a scalar field theory, the simplest operators are of the form∂mφn.

Among the operators, two particular categories are distinguished according to their trans-formation laws:

• primary operator:

∀f meromorphic : O(z, z) =(

dfdz

)h(dfdz

)hO(f(z), f(z)

), (22.44)

• quasi-primary (or SL(2,C) primary) operator:

∀f ∈ PSL(2,C) : O(z, z) =(

dfdz

)h(dfdz

)hO(f(z), f(z)

). (22.45)

The parameters (h, h) are the conformal weights of the operator O (both are independentfrom each other), and combinations of them give the conformal dimension ∆ and spin s:

∆ := h+ h, s := h− h. (22.46)

The conformal weights correspond to the charges of the operator under `0 and ¯0. We willuse “(h, h) (quasi-)primary” as a synonym of “(quasi-)primary field with conformal weight(h, h)”. Operators with integer (half-integer) spins are called bosonic (fermionic).

Remark 22.2 While we consider h, h ∈ R, and more specifically h, h ≥ 0 for a unitarytheory (which is the case of string theory except for the reparametrization ghosts), theorieswith h, h ∈ C make perfectly sense (one example is the Liouville theory with c ∈ C [225,227]).

253

Primaries and quasi-primaries are hence operators which have nice transformations re-spectively under the algebra and group. Obviously, a primary is also a quasi-primary. Thesetransformations are similar to those of a tensor with h upper indices and h lower indices(Section 5.1). Another point of view is that the object

O(z, z) dzhdzh (22.47)

is invariant under local / global conformal transformations.The notation f O indicates the complete change of coordinates, including the tensor

transformation law and the possible corrections if the operator is not primary.5 For a primaryfield, we have:

f O(z, z) := f ′(z)hf ′(z)hO(f(z), f(z)

). (22.48)

We stress that it corresponds to the usual function composition only for scalar operators:

f φ(z) = f(φ(z)

). (22.49)

Under an infinitesimal transformations

δz = v(z), δz = v(z), (22.50)

a primary operator changes as

δO(z, z) = (h ∂v + v ∂)O(z, z) + (h ∂v + v ∂)O(z, z). (22.51)

The transformation of a non-primary field contains additional terms, see for example (22.90).

Remark 22.3 (Group versus algebra) The question is whether one requires the theory to beinvariant under the global transformations or rather under local transformations. Theories arisingfrom gauge fixing a local Weyl invariance (which is the case for string theory) leads naturally toa local invariance. More generally, one expects that a local field theory is sensitive only to localproperties. On the other hand, this assumption may be too strong (e.g. in statistical physics orin systems without a Lagrangian formulation). But, making the assumption that only the localproperties matter can be useful for a preliminary study. This is really because the two-dimensionalalgebra is infinite-dimensional that so many models can be solved exactly in two dimensions. Usefuldiscussions can be found in [194, 225, sec. 1.3, 235].

Remark 22.4 (Higher-genus Riemann surfaces) According to Remark 21.1, all Rie-mann surfaces Σg share the same conformal algebra since locally they are all subsets of R2.On the other hand, one finds that no global transformations are defined for g > 1, and onlythe subgroup U(1)×U(1) survives for the torus.

The most important operator in a CFT is the energy–momentum tensor Tab. Accordingto Section 3.1, this tensor is conserved and traceless

∇bTab = 0, gabTab = 0. (22.52)

The traceless equation in components reads

gabTab = 4Tzz = Txx + Tyy = 0 (22.53)

which implies that the off-diagonal component vanishes in complex coordinates

Tzz = 0. (22.54)5In fact, one has f O := f∗O in the notations of Chapter 3.

254

Then, the conservation equation yields

∂zTzz = 0, ∂zTzz = 0, (22.55)

such that the non-vanishing components Tzz and Tzz are respectively holomorphic and anti-holomorphic. This motivates the introduction of the notations:

T (z) := Tzz(z), T (z) := Tzz(z). (22.56)

This is an example of the factorization between the holomorphic and anti-holomorphic sec-tors.

Currents are local objects and thus one expects to be able to write an infinite number ofsuch currents associated to the Witt algebra. Applying the Noether procedure gives

Jv(z) := J zv (z) = −T (z)v(z), Jv(z) := Jzv (z) = −T (z)v(z). (22.57)

22.3 Quantum CFTsThe previous section was purely classical. The quantum theory is first defined through thepath integral

Z =∫

dΨ e−S[Ψ]. (22.58)

We will also develop an operator formalism. The latter is more general than the path integraland allows to work without reference to path integrals and Lagrangians. This is particularlyfruitful as it extends the domain of parameters which can be studied.

22.3.1 Virasoro algebraAs discussed in Section 3.3.3, field measures in path integrals display a conformal anomaly,meaning that they cannot be defined without introducing a scale. As a consequence, the Wittalgebra (22.26) is modified to its central extension, the Virasoro algebra. The generatorsare in both sectors are denoted by Ln and Ln and are called Virasoro operators (ormodes). The algebra is given by:

[Lm, Ln] = (m− n)Lm+n + c

12 m(m− 1)(m+ 1)δm+n, (22.59a)

[Lm, Ln] = (m− n)Lm+n + c

12 m(m− 1)(m+ 1)δm+n, (22.59b)

[Lm, Ln] = 0, [c, Lm] = 0, [c, Lm] = 0, (22.59c)

where c, c ∈ C are the holomorphic and anti-holomorphic central charges. Consistency ofthe theory on a curved space implies c = c, but there is otherwise no constraint on theplane [281].

The sl(2,C) subalgebra is not modified by the central extension. This means that statesare still classified by eigenvalues of (h, h) of (L0, L0).

Remark 22.5 In most models relevant for string theory, one finds that the central chargesare real, c, c ∈ R. Moreover, unitarity requires them to be positive c, c > 0, and onlyreparametrization ghosts do not satisfy this condition. On the other hand, it makes perfectsense to discuss general CFTs for c, c ∈ C (the Liouville theory is such an example [225,227]).

255

22.3.2 Correlation functionsA n-point correlation function is defined by⟨

n∏i=1Oi(zi, zi)

⟩=∫

dΨ e−S[Ψ]n∏i=1Oi(zi, zi). (22.60)

As operators, the fields are time-ordered.Invariance under global transformations leads to strong constraints on the correlation

functions. For quasi-primary fields, they transform under SL(2,C) as⟨n∏i=1Oi(zi, zi)

⟩=

n∏i=1

(dfdz (zi)

)hi (dfdz (zi)

)hi×

⟨n∏i=1Oi(f(zi), f(zi)

)⟩. (22.61)

Considering an infinitesimal variation (22.51) yields a differential equation for the n-pointfunction

δ

⟨n∏i=1Oi(zi, zi)

⟩=

n∑i=1

(hi∂iv(zi) + v(zi)∂i + c.c.

)⟨ n∏i=1Oi(zi, zi)

⟩= 0, (22.62)

where ∂i := ∂zi and v is a vector (22.39) of sl(2,C). These equations are sufficient todetermine completely the forms of the 1-, 2- and 3-point functions of quasi-primaries:

〈Oi(zi, zi)〉 = δhi,0δhi,0, (22.63a)

〈Oi(zi, zi)Oj(zj , zj)〉 = δhi,hjδhi,hjgij

z2hiij z2hi

ij

, (22.63b)

〈Oi(zi, zi)Oj(zj , zj)Ok(zk, zk)〉 = Cijk

zhi+hj−hkij z

hj+hk−hijk z

hi+hk−hjki

× 1zhi+hj−hkij z

hj+hk−hijk z

hi+hk−hjki

,(22.63c)

where we have definedzij = zi − zj . (22.64)

The coefficients Cijk are called structure constants and the matrix gij defines a metric(Zamolodchikov metric) on the space of fields. The metric is often taken to be diagonalgij = δij , which amounts to use an eigenbasis of L0 and L0. The vanishing of the 1-pointfunction of a non-primary quasi-primary holds only on the plane: for example the valueon the cylinder can be non-zero since the map is not globally defined – see in particular(22.169).

Remark 22.6 (Logarithmic CFTs) Logarithmic CFTs display a set of unusual proper-ties [99, 100, 105, 112, 152]. In particular, the correlation functions are not of the formdisplayed above. The most striking feature of those theories is that the L0 operator is non-diagonalisable (but it can be set in the Jordan normal form).

Remark 22.7 (Fake identity) Usually, the only primary operator with h = h = 0 is theidentity 1. While this is always true for unitary theories, there are non-unitary theories(c ≤ 1 Liouville theory, SLE, loop models) where there is another field (called the indicator,marking operator, or also fake identity) with h = h = 0 [14, 56, 114, 127, 206, 225, 227].The main difference between both fields is that the identity is a degenerate field, whereas theother operator with h = h = 0 is not. Such theories will not be considered in this review.Operators with h = ~ = 0 can also be built by comining several CFTs, and they play a veryimportant role in string theory since they describe on-shell states.

256

Finally, the 4-point function is determined up to a function of a single variable⟨ 4∏i=1Oi(zi, zi)

⟩= f(x, x)

∏i<j

1z

(hi+hj)−h/3ij

× c.c. (22.65)

where the cross-ratio x readsx := z12z34

z13z24(22.66)

and

h :=4∑i=1

hi, h :=4∑i=1

hi. (22.67)

The interpretation is that the SL(2,C) invariance allows to fix 3 of the points to an arbitraryvalue, and the final result does not depend on this choice.

Exercise 22.2Prove these formulas.

22.4 Operator formalism and radial quantizationThe description of a CFT on the plane in terms of operators is called “radial quantization”due to the maps given in Section 22.1.2:

z = eτ+iσ = x+ iy. (22.68)

Taking the physical spacetime to be the cylinder (with two caps at its ends), every question isrephrased on the complex plane in order to exploit the powerful tools from complex analysis.The term “radial quantization” comes from the fact that time translation of the cylinder

τ −→ τ + T (22.69)

corresponds to dilatation on the plane

z −→ eT z. (22.70)

Thus, time evolution on the cylinder and radial evolution (from the origin to the complexinfinity) are identified. In particular, the Hamiltonian of the system of the plane is

H = 2πL

(L0 + L0), (22.71)

since the RHS is the dilatation operator. The cylinder length L was defined in (22.9). Thetheory is quantized according to this Hamiltonian. In the string theory language, a statewith H = 0 is said to be on-shell:

on-shell state: h+ h = 0. (22.72)

22.4.1 Radial ordering and commutatorsTime-ordering in τ becomes radial ordering in the plane:

R(A(z)B(w)

)=A(z)B(w) |z| > |w|,(−1)F B(w)A(z) |w| > |z|,

(22.73)

257

where F = 0 (F = 1) for bosonic (fermionic) operators. Radial ordering will often be keptimplicit.

The equal-time (anti-)commutator becomes an equal radius commutator defined bypoint-splitting:

[A(z), B(w)]±,|z|=|w| = limδ→0

(A(z)B(w)||z|=|w|+δ ±B(w)A(z)||z|=|w|−δ

). (22.74)

If A and B are two operators which can be written as the contour integrals of a(z) and b(z)(corresponding to integral over the spatial direction of the cylinder)

A =∮C0

dz2πi a(z), B =

∮C0

dz2πi b(z), (22.75)

then one finds the following commutators:

[A,B] =∮C0

dw2πi

∮Cw

dz2πi a(z)b(w), (22.76a)

[A, b(w)] =∮Cw

dz2πi a(z)b(w). (22.76b)

The contours C0 and Cw are respectively centered around the points 0 and w. For a proof,see Figure 22.4. Since these are contour integrals in the complex plane, the Cauchy–Riemannformula (B.1) can be used to write the result as soon as one knows the poles of the aboveexpression (ultimately, this amounts to pick the sum of residues). In CFTs, the poles ofsuch expressions are given by operator product expansions (OPE).

Figure 22.4: Graphical proof of (22.81).

Given a conserved current jµ

∂µjµ = ∂jz + ∂jz = ∂jz + ∂jz, (22.77)

the associated conserved charge is defined by

Q = 12πi

∮C0

(jzdz − jzdz), (22.78)

where C0 denotes the anti-clockwise contour around z = 0 (equivalently the interior of thecontour is located to the left). The difference of sign in the second term comes from theconjugation of the contour. The additional factor of 1/2π is consistent with the normaliz-ation of spatial integrals in two dimensions. The current components are not necessarilyholomorphic and anti-holomorphic at this level, but in practice this will often be the case(and each component is independently conserved), and one writes

j(z) := jz(z), (z) := jz(z). (22.79)

258

In this case, the charge also splits into a holomorphic and an anti-holomorphic (left- andright-moving6) contributions

Q = QL +QR, QL := 12πi

∮C0

j(z)dz, QR := − 12πi

∮C0

(z)dz. (22.80)

The infinitesimal variation of a field under the symmetry generated by Q reads

δεO(z, z) = −[εQ,O(z, z)] = −ε∮Cz

dw2πi j(w)O(z, z) + ε

∮Cz

dw2πi (w)O(z, z). (22.81)

The contour integrals are easily evaluated once the OPE between the current and the oper-ator is known. This formula gives the infinitesimal variation under a coordinate transform-ation for any field, not only for primaries.

Computation – Equation (22.78)In real coordinates, the charge is defined by integrating the time component of thecurrent jµ over space for fixed time (A.23):

Q = 12π

∫dσ j0.

The first step is to rewrite this formula covariantly. Since the time is fixed on the slice,dτ = 0 and one can write

Q = 12π

∫(dσ j0 − dτ j1) = − 1

2π

∫εµνj

µ dxν .

The last formula is valid for any contour. Moreover, it can be evaluated for complexcoordinates:

Q = − 12π

∮εzz(jz dz − jz dz

)= − i

4π

∮ (jz dz − jz dz

)= − 1

2πi

∮ (jz dz − jz dz

).

One finds a contour integral because τ = cst circles of the cylinder are mapped to|z| = cst contours.

22.4.2 Operator product expansionsThe operator product expansion (OPE) is a tool used repetitively in CFT: it means thatwhen two operators come close to each other, it is possible to replace their product by asum of local operators

Oi(zi, zi)Oj(zj , zj) =∑k

Ckij

zhi+hj−hkij z

hi+hj−hkij

Ok(zj , zj), (22.82)

where the OPE coefficients Ckij are related to the structure constants and the field metricby

Cijk = gk`C`ij . (22.83)

The radius of convergence for the OPE is given by the distance to the nearest operators inthe correlation function. The OPE defines an associative algebra (commutative for bosonicoperators), and the holomorphic sector forms a subalgebra (called the chiral algebra).

6For charges, we use subscript L and R to distinguish both sectors to avoid introducing a new symbol forthe total charge. However, since Q = QL in the holomorphic sector, it is often not necessary to distinguishbetween the two symbols when acting on an operator or a state (however, this is useful for writing modeexpansions). We do not write a bar on QR because the charges don’t depend on the position.

259

Example 22.1 – OPE with the identityThe OPE of a field φ(z) with the identity 1 is found by a direct series expansion

φ(z)1 =∑n∈N

(z − w)nn! ∂nφ(w). (22.84)

Obviously there are no singular terms.

Starting from this point we consider only the holomorphic sector except stated otherwise.The formula for the OPE (22.82) can be rewritten as

A(z)B(w) :=N∑

n=−∞

ABn(z)(z − w)n (22.85)

to simplify the manipulations. The number N ∈ Z is the lowest power of the coordinates.Singular terms are present if N > 0. Generally, only the terms singular as w → z arenecessary in the computations (for example, to use the Cauchy–Riemann formula (B.1)):equality up to non-singular terms is denoted by a tilde

A(z)B(w) ∼N∑n=1

ABn(z)(z − w)n =: A(z)B(w). (22.86)

The RHS of this expression defines the contraction of the operators A and B.While, most of the time, only singular terms are kept

Oi(zi, zi)Oj(zj , zj) ∼∑k

θ(hi+hj−hk)Ckij

(z − w)hi+hj−hk(z − w)hi+hj−hkOk(w, w) (22.87)

(with θ(x) the Heaviside step function), it can happen that one keeps also non-singular terms(the product of two OPE have singular terms coming from non-singular terms multiplyingsingular terms). Explicit contractions of operators through the OPE is also denoted by abracket when there are other operators.

For a primary field φ(z), one finds the OPE with the energy–momentum tensor to be

T (z)φ(w) ∼ hφ(w)(z − w)2 + ∂φ(w)

z − w, (22.88)

where h is the conformal weight of the field. This OPE together with (22.81) for j(z) =−v(z)T (z) correctly reproduces (22.51).


δφ(z) =∮Cz

dw2πi v(w)T (w)φ(z) ∼

∮Cz

dw2πi v(w)

(hφ(z)

(w − z)2 + ∂φ(z)w − z

)= h ∂v(z)φ(z) + v(z)∂φ(z).

For a non-primary operator, the OPE becomes more complicated (as it is reflected bythe transformation property), but the conformal weight can still be identified at the termin z−2. The most important example is the energy–momentum tensor: the central charge isfound as the coefficient of the z−4 term its OPE with itself:

T (z)T (w) ∼ c/2(z − w)4 + 2T (w)

(z − w)2 + ∂T (w)z − w

. (22.89)

260

The OPE indicates that the conformal weight of T is h = 2. Using (22.81) for j(z) =−v(z)T (z), one finds the infinitesimal variation

δT = 2 ∂v T + v ∂T + c

12 ∂3v, (22.90)

The last term vanishes for global transformations: this translates the fact that T is only aquasi-primary. The finite form of this transformation is

T ′(w) =(

dzdw

)−2 (T (z)− c

12 S(w, z))

=(

dzdw

)−2T (z) + c

12 S(z, w) (22.91)

where S(w, z) is the Schwarzian derivative

S(w, z) = w(3)

w′− 3

2

(w′′

w′

)2, (22.92)

where the derivatives of w are with respect to z. This vanishes if the transformation is inSL(2,C), and it transforms as

S(u, z) = S(w, z) +(

dwdz

)2S(u,w) (22.93)

under successive changes of coordinates.


δT (z) =∮Cz

dw2πi v(w)T (w)T (z) ∼

∮Cz

dw2πi v(w)

(c/2

(z − w)4 + 2T (w)(z − w)2 + ∂T (w)

z − w

)= c

2× 3! ∂3v(z) + 2∂v(z)T (z) + v(z)∂T (z).

22.4.3 Hermitian and BPZ conjugationIn this section, we introduce two different notions of conjugations: one is adapted for amp-litudes because it defines a unitary Euclidean time evolution, while the second is morenatural as an inner product of CFT states. Both can be interpreted as providing a mapfrom in-states to out-states on the cylinder.

Given an operator O, we need to define an operation O‡ – called Euclidean adjoint (orsimply adjoint) – which, after Wick rotation from Euclidean to Lorentzian signature, canbe interpreted as the Hermitian adjoint.7 This is necessary in order to define a Hermitianinner-product and to impose reality conditions.

To motivate the definition, consider first the cylinder in Lorentzian signature. SinceHermitian conjugation does not affect the Lorentzian coordinates, the Euclidean time mustreverse its sign:

t† = −iτ † = t =⇒ τ † = −τ. (22.94)Hence, an appropriate definition of the Euclidean adjoint is an Hermitian conjugation to-gether with time reversal.8 Another point of view is that the time evolution operator

7In [218], it is denoted by a bar on top of the operator: we avoid this notation since the bar has alreadytwo meanings in this review (complex conjugation and anti-holomorphic sector). In [297], it is indicated bya subscript hc. Otherwise, in most of the literature, it hasno specific symbol since one directly works withthe modes.

8The Euclidean adjoint can be used to define an inner product: positive-definiteness of the latter iscalled reflection positivity or OS-positive and is a central axiom of constructive QFT.

261

U(τ) := e−τH is not unitary when H is Hermitian H† = H: the solution is to define anew Euclidean adjoint U(τ)‡ := U(−τ)† such that U(τ) is unitary for it.

Time reversal on the cylinder corresponds to inversion and complex conjugation on thecomplex plane:

zτ→−τ−−−−→ e−τ+iσ = 1

z∗= I(z), (22.95)

where I(z) = 1/z is the inversion (22.42).9 On the real surface10 z = z∗, which leads to thedefinition of the Euclidean adjoint in Lorentzian as follows:

O(z, z)‡ :=(I O(z, z)

)† =[

1z2hz2h

O(

1z,

1z

)]†= 1z2hz2h

O†(

1z,

1z

), (22.96)

where I(z) = 1/z. The last equality shows that Euclidean conjugation is equivalent to takethe conjugate of all factors of i but otherwise leaves z and z unaffected. The Euclideanadjoint acts by complex conjugation of any c-number and reverses the order of the operators(acting as a transpose):

(λO1 · · · On)‡ = λ∗O‡n · · · O‡1, λ ∈ C, (22.97)

without any sign.A second operation, called the BPZ conjugation, is useful. It can be defined in two

different ways:

O(z, z)t := I± O(z, z) = (∓1)h+h

z2hz2hO(±1z,±1

z

), (22.98)

where I±(z) = ±1/z is the inversion (22.42). The minus and plus signs are respectivelymore convenient when working with the open and closed strings.11 The BPZ conjugationdoes not complex conjugate c-number nor changes the order of the operators:12

(λO1 · · · On)t = λOt1 · · · Otn, λ ∈ C. (22.99)

The identity is invariant under both conjugation

1‡ = 1t = 1. (22.100)

22.4.4 Mode expansionAny field of weight (h, h) can be expanded in terms of modes Om,n

O(z, z) =∑m,n

Om,nzm+hzn+h

. (22.101)

Note that the modes Om,n themselves are operators. The ranges of the two indices are suchthat

m+ h ∈ Z + ν, n+ h ∈ Z + ν, ν, ν =

0 periodic,1/2 anti-periodic.

(22.102)

9We do not write “z†” because this notation is confusing as one should not complex conjugate the factorof i in the exponential (Section 22.2.1).

10Remember that z is not the complex conjugate of z but an independent variable.11The index t should not be confused with the matrix transpose: it is used in opposition with ‡ and † to

indicate that no complex conjugation is involved.12However, BPZ conjugation reverses the radial ordering: radially ordering the fields leads to additional

signs if there are Grassmann-odd fields [250]. This will not be relevant in most cases since we will work interms of the modes.

262

The values of ν and ν depend on whether the fields satisfies periodic or anti-periodic bound-ary conditions on the plane (for half-integer weights, the periodicity is reversed on thecylinder):

O(e2πiz, z) = e2πiνO(z, z), O(z, e2πiz) = e2πiνO(z, z). (22.103)Depending on whether the weights are integers or half-integers, additional terminology isintroduced:

• If h ∈ Z + 1/2, then one can choose anti-periodic (Neveu–Schwarz or NS) or periodic(Ramond or R) boundary conditions on the cylinder (reversed for the plane):

ν, ν =

0 NS1/2 R

(22.104)

The indices are half-integers (resp. integers) for the NS (R) sector.

• If h ∈ Z, periodic (or untwisted) boundary conditions are more natural, but anti-periodic boundary conditions are also possible and become necessary when consideringorbifolds:

ν, ν =

0 untwisted1/2 twisted

(22.105)

The modes of untwisted (resp. twisted) fields have integer (half-integers) indices.

The mode expansions have no branch cut (fractional power of z or z) for periodic fields(bosonic untwisted or fermionic twisted). We will see explicit examples of such operators inthe rest of this review.

Under Euclidean conjugation (22.96), the modes are related by

(O‡)−m,−n = (Om,n)†. (22.106)

In particular, if the operator is Hermitian (under the Euclidean adjoint), the reality conditionon the modes relates the negative modes with the conjugated positive modes

O‡ = O =⇒ (Om,n)† = O−m,−n. (22.107)

When no confusion is possible (for Hermitian operators), we will write O†m,n instead of(Om,n)†.

For a holomorphic field φ(z), the above expansion becomes

φ(z) =∑

n∈Z+h+ν

φnzn+h . (22.108)

Conversely, the modes are recovered from the field through

φn =∮C0

dz2πi z

n+h−1φ(z), (22.109)

where the integration is counter-clockwise around the origin.If the field is Hermitian, then

φ‡ = φ =⇒ (φn)† = φ−n. (22.110)

The operators φn have a conformal weight of −n (since the weight of z is −1). The BPZconjugate of the modes is

φtn = (I± φ)n = (−1)h(±1)nφ−n. (22.111)

263


φtn = (I± φ)n =∮ dz

2πi zn+h−1I± φ(z)

=∮ dz

2πi zn+h−1

(∓ 1z2

)hφ

(±1z

)= (∓1)h

∮ dz2πi z

n−h−1φ

(±1z

)= (∓1)h

∮ dw2πi

(± 1w

)n−hw−1φ(w)

= (∓1)h(±1)n−h∮ dw

2πi w−n+h−1φ(w),

where we have set w = ±1/z such that

dzz

= ∓ dww2z

= −dww, (22.112)

and the minus sign disappears upon reversing the contour orientation.

The mode expansion of the energy–momentum tensor is

T (z) =∑n∈Z

Lnzn+2 , Ln =

∮ dz2πi T (z)zn+1, (22.113)

where one recognizes the Virasoro operators as the modes. In most situations, the Virasorooperators are Hermitian

L†n = L−n. (22.114)The OPE (22.89) and (22.88) together with (22.76a) help to reconstruct the Virasoro algebra(22.59) and the commutation relations between the Lm and the modes φn of a weight hprimary:

[Lm, φn] =(m(h− 1)− n

)φm+n. (22.115)

This easily gives the commutation relation for the complete field:[Lm, φ(z)] = zm

(z∂ + (n+ 1)h

)φ(z). (22.116)

We will often use (22.59) and (22.115) for m = 0:[L0, L−n] = nL−n, [L0, φ−n] = nφ−n. (22.117)

This means that both φn and Ln act as raising operators for L0 if n < 0, and as loweringoperators if n > 0 (remember that L0 is the Hamiltonian in the holomorphic sector). Whenboth the holomorphic and anti-holomorphic sectors enter, it is convenient to introduce thecombinations

L±n = Ln ± Ln, (22.118)such that L+

0 is the Hamiltonian.Finally, every holomorphic current j(z) has a conformal weight h = 1 and can be expan-

ded asj(z) =

∑n

jnzn+1 . (22.119)

By definition, the zero-mode is equal to the holomorphic chargeQL = j0. (22.120)

264

22.4.5 Hilbert spaceThe Hilbert space of the CFT is denoted by H. The SL(2,C) (or conformal) vacuum13 |0〉is defined by the state which is invariant under the global conformal transformations:

L0 |0〉 = 0, L±1 |0〉 = 0. (22.121)

If the fields are expressed in terms of creation and annihilation operators (which happense.g. for free scalars, free fermions and ghosts), then the Hilbert space has the structure of aFock space.

State–operator correspondence

The state–operator correspondence identifies every state |O〉 of the CFT Hilbert space withan operator O(z, z) through

|O〉 = limz,z→0

O(z, z) |0〉 = O(0, 0) |0〉 . (22.122)

Such a state can be interpreted as an “in” state since it is located at τ → −∞ on thecylinder. Focusing now on a holomorphic field φ(z), the state is defined as

|φ〉 = limz→0

φ(z) |0〉 = φ(0) |0〉 . (22.123)

For this to make sense, the modes which diverge as z → 0 must annihilate the vacuum. Inparticular, for a weight h field φ(z), one finds:

∀n ≥ −h+ 1 : φn |0〉 = 0. (22.124)

Thus, the φn for n ≥ −h+ 1 are annihilation operators for the vacuum |0〉, and converselythe states φn with n < −h + 1 are creation operators. As a consequence, the state |φ〉 isfound by applying the mode n = −h to the vacuum:

|φ〉 = φ−h |0〉 =∮ dz

2πiφ(z)z|0〉 . (22.125)

Since L−1 is the generator of translations on the plane, one finds

φ(z) |0〉 = ezL−1φ(0)e−zL−1 |0〉 = ezL−1 |φ〉 . (22.126)

The vacuum |0〉 is the state associated to the identity 1. Translating the conditions (22.124)to the energy–momentum tensor gives

∀n ≥ −1 : Ln |0〉 = 0. (22.127)

This is consistent with the definition (22.121) since it includes the sl(2,C) subalgebra.If h < 0, some of the modes with n > 0 do not annihilate the vacuum: (22.117) implies

that some states have an energy lower than the one of |0〉. The state |Ω〉 (possibly degenerate)with the lowest energy is called the energy vacuum

∀ |φ〉 ∈ H : 〈Ω|L0 |Ω〉 ≤〈φ|L0 |φ〉 . (22.128)

It is obtained by acting repetitively with the modes φn>0. This vacuum defines a newpartition of the non-zero-modes operators into annihilation and creation operators. If there

13There are different notions of “vacuum”, see (22.128). However, the SL(2,C) vacuum is unique. Indeed,it is mapped to the identity operator under the state–operator correspondence, and the identity is unique(however, there can be other states of weight 0, see Remark 22.7).

265

are zero-modes, i.e. n = 0 modes, then the vacuum is degenerate since they commutewith the Hamiltonian, [L0, φ0] = 0 according to (22.117). The partition of the zero-modesinto creation and annihilation operators depends on the specific state chosen among thedegenerate vacua.

The energy aΩ of |Ω〉, which is also its L0 eigenvalue

L0 |Ω〉 := aΩ |Ω〉 , (22.129)

is called zero-point energy. Bosonic operators with negative h are dangerous because theylead to an infinite negative energy together with an infinite degeneracy (from the zero-mode).

The conjugate vacuum is defined by BPZ or Hermitian conjugation

〈0| = |0〉‡ = |0〉t (22.130)

since both leave the identity invariant. It is also annihilated by the sl(2,C) subalgebra:

〈0|L0 = 0, 〈0|L±1 = 0. (22.131)

Since there are two kinds of conjugation, two different conjugated states can be defined. Theyare also called “out” states since they are located at τ →∞ on the cylinder (Figure 22.2).

Euclidean and BPZ conjugations and inner products

The Euclidean adjoint 〈O‡| of the state |O〉 is defined as

〈O‡| = limw,w→0

〈0| O(w, w)‡ = limw,w→0

1w2hw2h

〈0| O(

1w,

1w

)†(22.132a)

= limz,z→∞

z2hz2h〈0| O†(z, z) (22.132b)

= 〈0| I O†(0, 0), (22.132c)

where the two coordinate systems are related by w = 1/z. From this formula, the definitionof the adjoint of a holomorphic operator φ follows

〈φ‡| = limw→0〈0|φ(w)‡ = lim

w→0

1w2h 〈0|φ

†(

1w

)(22.133a)

= limz→∞

z2h〈0|φ†(z) (22.133b)

= 〈0| I φ†(0). (22.133c)

Then, expanding the field in terms of the modes gives

〈φ‡| = 〈0| (φ†)h. (22.134)

The BPZ conjugated state is

〈φ| := limw→0〈0|φ(w)t (22.135a)

= (±1)h limz→∞

z2h〈0|φ(z) (22.135b)

= 〈0| I± φ(0). (22.135c)

In terms of the modes, one has〈φ| = (±1)h〈0|φh. (22.136)

266

If φ is Hermitian, then the relation between both conjugated states corresponds to a realitycondition:

〈φ‡| = (±1)h〈φ| . (22.137)Taking the BPZ conjugation of the conditions (22.124) tells which modes must annihilate

the conjugate vacuum:∀n ≤ h− 1 : 〈0|φn = 0, (22.138)

and one finds more particularly for the Virasoro operators

∀n ≤ 1 : 〈0|Ln = 0. (22.139)

This can also be derived directly from (22.135) by requiring that applying an operator onthe conjugate vacuum 〈0| is well-defined.

All conditions taken together mean that the expectation value of the energy–momentumtensor in the conformal vacuum vanishes:

〈0|T (z) |0〉 = 0. (22.140)

In particular, this means that the energy vacuum |Ω〉, if different from |0〉, has a negativeenergy.

The Hermitian14 and BPZ inner products are respectively defined by:

〈φ‡i |φj〉 = 〈0| I φj(0)φi(0) |0〉 = limz→∞w→0

z2h〈0|φ†j(z)φi(w) |0〉 , (22.141a)

〈φi|φj〉 = 〈0| I φj(0)φi(0) |0〉 = (±1)h limz→∞w→0

z2h〈0|φj(z)φi(w) |0〉 . (22.141b)

These products can be recast as 2-point correlation functions (22.63b) on the sphere:

〈φi|φj〉 = 〈I φj(0)φi(0)〉, 〈φ‡i |φj〉 = 〈I φ†j(0)φi(0)〉. (22.142)

Developing the state–operator correspondence, the action of one operator on the in-statecan be reinterpreted as the matrix element of this operator using the two external states, oralso as a 3-point function:

〈φi|φj |φk〉 = 〈φiφjφk〉. (22.143)Given a basis of states φi (i can run over both discrete and continuous indices), the

conjugate or dual states φci are defined by:

〈φci |φj〉 = δij (22.144)

(the delta function is discrete and/or continuous according to the indices).

Verma modules

If φ(z) is a weight h primary, then the associated state |φ〉 satisfies:

L0 |φ〉 = h |φ〉 , ∀n ≥ 1 : Ln |φ〉 = 0. (22.145)

Such a state is also called a highest-weight state. The descendant states are defined by allpossible states of the form

|φni〉 :=∏i

L−ni |φ〉 , (22.146)

where the same L−n can appear multiple times. The set of states φni is called a Vermamodule V (h, c). One finds that the L0 eigenvalues of this state is

L0 = h+∑i

ni. (22.147)

14Depending on the normalization, it can also be anti-Hermitian.

267

Normal ordering

The normal ordering of an operator with respect to a vacuum corresponds to placing all cre-ation (resp. annihilation) operators of this vacuum on the left (resp. right). From this defin-ition, the expectation value of a normal ordered operator in the vacuum vanishes identically.The main reason for normal ordering is to remove singularities in expectation values.

Given an operator φ(z), we define two normal orderings:• The conformal normal order (CNO) :O: is defined with respect to the conformal va-

cuum (22.121):〈0| :O: |0〉 = 0. (22.148)

• The energy normal order (ENO) ??O ?

? is defined with respect to the energy vacuum(22.128):

〈Ω| ??O ?? |Ω〉 = 0. (22.149)

We first discuss the conformal normal ordering before explaining how to relate it to theenergy normal ordering.

Given two operators A and B, the simplest normal ordering amounts to subtract theexpectation value:

:A(z)B(w): ?= A(z)B(w)− 〈A(z)B(w)〉. (22.150)This is equivalent to defining the products of two operators at coincident points via point-splitting:

:A(z)B(z): ?= limw→z

(A(z)B(w)− 〈A(z)B(w)〉

). (22.151)

While this works well for free fields, this does not generalize for composite or interactingfields.

The reason is that this procedure removes only the highest singularity in the product: itdoes not work if the OPE has more than one singular term. An appropriate definition is

:A(z)B(w): := A(z)B(w)−A(z)B(w) =∑n∈N

(z − w)nAB−n(z), (22.152)

where the contraction between A and B is defined in (22.86), and the second equality comesfrom (22.85).

Then, the product evaluated at coincident points is found by taking the limit (in thiscase the argument is often indicated only at the end of the product)

:AB(z): := :A(z)B(z): := limw→z

:A(z)B(w): = AB0(z). (22.153)

Indeed, since all powers of (z − w) are positive in the RHS of (22.152), all terms but thefirst one disappear. The form of (22.153) shows that the normal order can also be computedwith the contour integral

:AB(z): =∮Cz

dw2πi

A(z)B(w)z − w

. (22.154)

It is common to remove the colons of normal ordering when there is no ambiguity and, inparticular, to write:

AB(z) := :AB(z):. (22.155)In terms of modes, one has

:AB(z): =∑m

:AB:mzm+hA+hB

, (22.156a)

:AB:m =∑

n≤−hA

AnBm−n +∑

n>−hA

Bm−nAn. (22.156b)

268

This expression makes explicit that normal ordering is non-commutative and non-associative:

:AB(z): 6= :BA(z):, :A(BC)(z): 6= :(AB)C(z):. (22.157)

The product of normal ordered operators can then be computed using Wick theorem. Infact, one is more interested in the contraction of two such operators in order to recover theOPE between these operators: the product is then derived with (22.152).

If Ai (i = 1, 2, 3) are free fields, one has

A1(z) :A2A3(w): = :A1(z)A2A3(w): +A1(z) :A2A3(w):,

A1(z) :A2A3(w): = A1(z)A2(w) :A3(w): +A1(z)A3(w) :A2(w):.(22.158)

If the fields are not free, then the contraction cannot be extracted from the normal ordering.Similarly if there are more fields, then one needs to perform all the possible contractions.

Given two free fields A and B, one has the following identities:

A(z) :B(w)n: = nA(z)B(w) :B(w)n−1:, (22.159a)

A(z) :eB(w): = A(z)B(w) :eB(w):, (22.159b)

:eA(z): :eB(w): = exp(A(z)B(w)

):eA(z)eB(w):. (22.159c)

The last relation generalizes for a set of n fields Ai:

n∏i=1

:eAi : = : exp(

n∑i=1

Ai

): exp

∑i<j

〈AiAj〉, (22.160a)⟨n∏i=1

:eAi :⟩

= exp∑i<j

〈AiAj〉. (22.160b)

Computation – Equation (22.159b)

A(z) :eB(w): = A(z)∑n

1n! :B(w)n: = A(z)B(w)

∑n

1(n− 1)! :B(w)n−1:.

Computation – Equation (22.159c)

:eA(z): :eB(w): =∑m,n

1m!n! :A(z)m: :B(w)n:

=∑m,n,k

k!m!n!

(m

k

)(n

k

)(A(z)B(w)

)k:A(z)m−k: :B(w)n−k:

=∑m,n,k

1k!(m− k)!(n− k)!

(A(z)B(w)

)k:A(z)m−k: :B(w)n−k:.

The factorial k! counts the number of possible ways to contract the two operators.

269

The general properties of normal ordered expressions are identical for both vacua: whatdiffers is the precise computation in terms of the operators (or modes). Hence, the energynormal ordering can be defined in parallel with (22.156), but changing the definitions ofcreation and annihilation operators:

??AB(z) ?? =

∑m

??AB(z) ??nzm+hA+hB

, (22.161a)

??AB

??m =

∑n≤0

AnBm−n +∑n>0

Bm−nAn. (22.161b)

To simplify the definition we assume that A0 is a creation operator and it is thus includedin the first sum (this must be adapted in function of which vacuum state is chosen if thelatter is degenerate).

The relation between the normal ordered modes is

:AB:m = ??AB

??m +

hA−1∑n=0

[Bm+n, A−n]. (22.162)


:AB:m =∑

n≤−hA

AnBm−n +∑

n>−hA

Bm−nAn

=∑n≥hA

A−nBm+n +∑n>0

Bm−nAn +hA−1∑n=0

Bm+nA−n

=∑n≥0

A−nBm+n +∑n>0

Bm−nAn +hA−1∑n=0

[Bm+n, A−n]

= ??AB

??m +

hA−1∑n=0

[Bm+n, A−n].

The choice of the normal ordering for the operators is related to the ordering ambiguitywhen quantizing the system: when the product of two non-commuting modes appears in theclassical composite field, the corresponding quantum operator is ambiguous (generally up toa constant). In practice, one starts with the conformal ordering since it is invariant underconformal transformations and because one can compute with contour integrals. Then, theexpression can be translated in the energy ordering using (22.162). But, knowing how theconformal and energy vacua are related, it is often simpler to find the difference betweenthe two orderings by applying the operator on the vacua.

22.4.6 CFT on the cylinderAccording to (22.44), the relation between the field on the cylinder and on the plane is

φ(z) =(L

2π

)hz−hφcyl(w) (22.163)

(quantities without indices are on the plane by definition). The mode expansion on thecylinder is

φcyl =(

2πL

)h∑n∈Z

φne− 2πL w =

(2πL

)h∑n∈Z

φnzn. (22.164)

270

Using the finite transformation (22.91) for the energy–momentum tensor T , one findsthe relation

Tcyl(w) =(

2πL

)2 (T (z)z2 − c

24

). (22.165)

The vacuum expectation value (Casimir energy) is then proportional to the central charge:

〈Tcyl〉 = − cπ2

6L2 . (22.166)

This energy is provided by the curvature of the cylinder. For the L0 mode, one finds

(L0)cyl = L0 −c

24 , (22.167)

and thus the Hamiltonian is

H = (L0)cyl + (L0)cyl = L0 + L0 −c+ c

24 . (22.168)

More generally, the Schwarzian derivative gives a contribution proportional to the curvature,which is another statement of the conformal anomaly

〈T 〉 = c

12 R. (22.169)

22.5 References• The most complete reference on CFTs is [64] but it lacks some recent developments.

Two excellent complementary books are [32, 235].String theory books generally dedicate a fair amount of pages to CFTs: particularlygood summaries can be found in [31, 149, 218, 219].Finally, a modern and fully algebraic approach can be found in [225, 226]. Other goodreviews are [221, 294].

• There are various other books [118, 135, 147, 192] and reviews [41, 104, 107, 231, 276].

• The maps from the sphere and the cylinder to the complex plane are discussed in [218,sec. 2.6, 6.1].

• Normal ordering is discussed in details in [64, chap. 6] (see also [31, sec. 4.2, 218,sec. 2.2]).

• Hermitian conjugation is discussed in [218, p. 202–3]. For a comparison of Hermitianand BPZ conjugations, see [297, sec. 2.2, 250, p. 11].

• Normal ordering and difference between the different definitions are described in [218,chap. 2, 64, sec. 6.5].

271

Chapter 23

CFT systems

This chapter summarizes the properties of some CFT systems. We focus on the free scalarfield and on the first-order bc system (which generalizes the reparametrization ghosts). Forthe different systems, we first provide an analysis on a general curved background beforefocusing on the complex plane. This is sufficient to describe the local properties on allRiemann surfaces g ≥ 0.

23.1 Free scalar23.1.1 Covariant actionThe Euclidean action of a free scalar X on a curved background gµν is

S = ε

4π`2∫

d2x√g gµν∂µX∂νX, (23.1)

where ` is a scale1 and

ε :=

+1 spacelike−1 timelike

,√ε :=

+1 spacelikei timelikei

(23.2)

denotes the signature of the kinetic term. The field is periodic along σ

X(τ, σ) ∼ X(τ, σ + 2π). (23.3)

The energy–momentum tensor reads

Tµν = − ε

`2

[∂µX∂νX −

12 gµν(∂X)2

], (23.4)

and it is tracelessTµµ = 0. (23.5)

The equation of motion is∆X = 0, (23.6)

where ∆ is the Laplacian (A.28).1To be identified with the string scale, such that α′ = `2.

272

The simplest method for finding the propagator is by using the identity (assuming thatthere is no boundary term)

0 =∫

dX δ

δX(σ)

(e−S[X]X(σ′)

), (23.7)

which yields a differential equation for the propagator:

〈∂2X(σ)X(σ′)〉 = −2πε`2 δ(2)(σ − σ′). (23.8)

This is easily integrated to

〈X(σ)X(σ′)〉 = −ε`2

2 ln |σ − σ′|2. (23.9)

Computation – Equation (23.9)By translation and rotation invariance, one has

〈X(σ)X(σ′)〉 = G(r), r = |σ − σ′|. (23.10)

In polar coordinates, the Laplacian reads

∆G(r) = 1r∂r(rG′(r)). (23.11)

Integrating the differential equation over d2σ = rdrdθ yields

−2πε`2 = 2π∫ r

0dr′ r′ × 1

r′∂r′(r′G′(r′)) = 2πrG′(r). (23.12)

The solution isG′(r) = −ε`2 ln r (23.13)

and the form (23.9) follows by writing

ln r = 12 ln r2 = 1

2 ln |σ − σ′|2. (23.14)

The action (23.1) is obviously invariant under constant translations of X:

X −→ X + a, a ∈ R. (23.15)

The associated U(1) current2 is conserved and reads

Jµ := 2πiε ∂L∂(∂µX) = i

`2gµν∂νX, ∇µJµ = 0. (23.16)

and the charge follows from (A.23):

p = 12π

∫dσ J0 = i

2π`2∫

dσ ∂0X. (23.17)

This charge is called momentum because it corresponds to the spacetime momentum instring theory.

2The group is R but the algebra is u(1) (since locally there is no difference between the real line and thecircle).

273

Moreover, there is a another topological current

Jµ := −i εµνJν = 1`2εµν∂νX, (23.18)

which is automatically conserved:

∇µJµ ∝ εµν [∇µ,∇ν ]X = 0 (23.19)

since [∇µ,∇ν ] = 0 for a scalar field. Note that Jµ is the Hodge dual of Jµ. The conservedcharge is called the winding number :

w = 12π

∫dσ J0 = 1

2π`2∫ 2π

0dσ ∂1X = 1

2π`2(X(τ, 2π)−X(τ, 0)

). (23.20)

Remark 23.1 (Normalization of the current) The definition of the current (23.16) maylook confusing. The factor of i is due to the Euclidean signature, see (A.25a), and the factorof 2π comes from the normalization of the spatial integral. We have inserted ε in order tointerpret the conserved charge p as a component of the momentum contravariant vector instring theory.

To make contact with string theory, consider D scalar fields Xµ(σa) (using string theorynotations). Then, the current becomes

Jaµ = i2π`2 ηµν∂

aXν , (23.21)

where the position of the indices is in agreement with the standard form of Noether’s formula(A.25a) (a current has indices in opposite locations as the parameters and fields). Since wehave η00 = −1 = εX0 , we find that J0a = εX0J

a0 has no epsilon after writing explicitly Ja0 .

The transformation Xµ → Xµ+aµ is a global translation in target spacetime: the chargepµ is identified with the spacetime momentum. The factor of i indicates that pµ is theEuclidean contravariant momentum vector by comparison with (A.7).

The convention of this section is to always work with quantities which will become con-travariant vector to avoid ambiguity.

23.1.2 Action on the complex planeIn complex coordinates, the action on flat space reads

S = ε

2π`2∫

dzdz ∂zX∂zX, (23.22)

giving the equation of motion:∂z∂zX = 0. (23.23)

This indicates that ∂zX and ∂zX are respectively holomorphic and anti-holomorphic suchthat

X(z, z) = XL(z) +XR(z), (23.24)

and we will remove the subscripts when there is no ambiguity (for example, when the positiondependence is written):

X(z) := XL(z), X(z) := XR(z). (23.25)

It looks like XL(z) and XR(z) are unrelated, but this is not the case because of the zero-mode, as we will see below.

274

The U(1) current is written as

J := Jz = i`2∂zX, J := Jz = i

`2∂zX, (23.26)

where we used the relations Jz = J z/2 and Jz = Jz/2. The equation of motion implies thatthe current J is holomorphic, and J is anti-holomorphic:

∂J = 0, ∂J = 0. (23.27)

The momentum splits into left- and right-moving parts:

p = pL + pR, pL = 12πi

∮dz J, pR = − 1

2πi

∮dz J . (23.28)

The components of the topological current (23.18) are related to the ones of the U(1)current:

Jz = i`2∂zX = J, Jz = − i

`2∂zX = −J . (23.29)

As a consequence, the winding number is

w = pL − pR. (23.30)

Note that we have the relations

pL = p+ w

2 , pR = p− w2 , (23.31a)

p2 + w2 = p2L + p2

R, 2pw = p2L − p2

R. (23.31b)

The energy–momentum tensor is

T := Tzz = − ε

`2∂zX∂zX, T := Tzz = − ε

`2∂zX∂zX, Tzz = 0. (23.32)

Since the ∂zX (∂zX) is (anti-)holomorphic, so is T (z) (T (z)). Since the energy–momentumtensor, the current and the field itself (up to zero-modes) split in holomorphic and anti-holomorphic components in a symmetric way, it is sufficient to focus on one of the sector,say the holomorphic one.

The other primary operators of the theory are given by the vertex operators Vk(z):3

Vk(z, z) := :eiεkX(z,z):. (23.33)

Remark 23.2 In fact, it is possible to introduce more general vertex operators

VkL,kR(z, z) := :e2iε(kLX(z)+kRX(z)

):, (23.34)

but we will not consider them in this review.

Remark 23.3 (Plane and cylinder coordinates) The action in w-coordinate (cylinder)takes the same form as a result of the conformal invariance of the scalar field, which in prac-tice results from the cancellation between the determinant and inverse metric (the factor of2 comes from the symmetry of the derivatives). As a consequence, every quantity derivedfrom the classical action (equation of motion, energy–momentum tensor. . . ) will have thesame form in both coordinate systems: we will focus on the z-coordinate, writing the w-coordinate expression when it is insightful to compare. This is not anymore the case at thequantum level: anomalies may translate into differences between quantities: to differentiatebetween the plane and cylinder quantities an index “cyl” will be added when necessary (byconvention, all quantities without qualification are on the plane).

3The ε in the exponential is consistent with interpreting X and k as a contravariant vector.

275

23.1.3 OPEThe OPE between X and itself is directly found from the propagator:

X(z)X(w) ∼ −ε`2

2 ln(z − w). (23.35)

By successive derivations, one finds the OPE between X and ∂X

∂X(z)X(w) ∼ −ε`2

21

z − w, (23.36)

and between ∂X with itself

∂X(z)∂X(w) ∼ −ε`2

21

(z − w)2 . (23.37)

The invariance under the permutation of z and w reflects that X is bosonic.The OPE between ∂X and T allows to verify that the field ∂X is primary with h = 1:

T (z)∂X(w) ∼ ∂X(w)(z − w)2 +

∂(∂X(w)

)z − w

. (23.38)

The OPE of T with itself gives

T (z)T (w) ∼ 12

1(z − w)4 + 2T (w)

(z − w)2 + ∂T (w)z − w

(23.39)

which shows that the central charge is

c = 1. (23.40)

One finds that the operator ∂nX has conformal weight

h = n (23.41)

since the OPE with T is

T (z)∂nX(w) ∼ · · ·+ n∂nX(w)(z − w)2 + ∂(∂nX(w))

z − w(23.42)

where the dots indicate higher negative powers of (z − w). Explicitly, for n = 2, one finds

T (z)∂2X(w) ∼ 2 ∂X(w)(z − w)3 + 2 ∂2X

(z − w)2 + ∂(∂2X(w))z − w

. (23.43)

These states are not primary operators.The OPE of a vertex operator with the current J is

J(z)Vk(w, w) ∼ k

2Vk(w, w)z − w

. (23.44)

This shows that the vertex operators Vk are eigenstates of the U(1) holomorphic currentwith the eigenvalue given by the momentum (with Then, the OPE with T :

T (z)Vk(w, w) ∼ hk Vk(w, w)(z − w)2 + ∂Vk(w, w)

z − w(23.45)

276

together with its anti-holomorphic counterpart show that the Vk are primary operators withweight

(hk, hk) =(ε`2k2

4 ,ε`2k2

4

), ∆k = ε`2k2

2 , sk = 0. (23.46)

Note that classically hk = 0 since ` ∼ ~ [281, p. 81]. The weight is invariant under k → −k.Finally, the OPE between two vertex operators is

Vk(z, z)Vk′(w, , w) ∼ Vk+k′(w, w)(z − w)−εkk′`2/2 . (23.47)

In particular, correlation functions should be computed for εkk′ < 0 in order to avoidexponential growth.


T (z)∂X(w) = − ε

`2:∂X(z)∂X(z): ∂X(w)

∼ −2ε`2

:∂X(z)∂X(z): ∂X(w)

∼ ∂X(z)(z − w)2 .

The result follows by Taylor expanding the numerator.


T (z)∂X(w) = 1`4

:∂X(z)∂X(z): :∂X(w)∂X(w):

∼ 1`4

[:∂X(z)∂X(z): :∂X(w)∂X(w): + :∂X(z)∂X(z): :ioi∂X(w)∂X(w):

+ :∂X(z)∂X(z): :∂X(w)∂X(w): + perms]

∼ 2× 14

1(z − w)4 − 4× 1

2`21

(z − w)2 :∂X(z)∂X(w):

∼ 12

1(z − w)4 −

2`2

1(z − w)2

(:∂X(w)∂X(w): + (z − w) :∂2X(w)∂X(w):

).

277


T (z)∂nX(w) ∼ ∂n−1w

∂X(z)(z − w)2

∼ n! ∂X(z)(z − w)n+1

∼ n!(z − w)n+1

(· · ·+ 1

(n− 1)! (z − w)n−1∂n−1(∂X(w))

+ 1n! (z − w)n∂n(∂X(w))

).

Computation – Equation (23.44)Using (22.159b), one has:

∂X(z)Vk(w, w) ∼ iεk ∂X(z)X(w)Vk(w, w) ∼ iεk(−ε`

2

21

z − w

)Vk(w, w).


T (z)Vk(w, w) ∼ − ε

`2:∂X(z)∂X(z): :eiεkX(w,w):

∼ iεk2

1z − w

∂X(z) :eiεkX(w,w):− 1`2∂X(z) :∂X(z)eiεkX(w,w):

∼ iεk2

1z − w

(:∂X(z) eiεkX(w,w): + ∂X(z) :eiεkX(w,w):

)+ iεk

2:∂X(z)eiεkX(w,w):

z − w

∼ εk2`2

4Vk(w, w)(z − w)2 + ik :∂X(w)eiεkX(w,w):

z − w.

It is also possible to achieve the same result by expanding the exponential.

Computation – Equation (23.47)Using (22.159c) one has:

Vk(z, z)Vk′(w, w) ∼ exp(− kk′X(z, z)X(w, w)

):eiεkX(z,z)eiεk′X(w,w):

∼ (z − w)εkk′`2/2 Vk+k′(w, w).

278

23.1.4 Mode expansionsSince ∂X is holomorphic and of weight h = 1, it can be expanded as:4

∂X = −i√`2

2∑n∈Z

αn z−n−1, ∂X = −i

√`2

2∑n∈Z

αn z−n−1, (23.48)

where an individual mode can be extracted with a contour integral:

αn = i∮ dz

2πi zn−1∂X(z), αn = i

∮ dz2πi z

n−1∂X(z). (23.49)

Integrating this formula gives:

X(z) = xL2 − i

√`2

2 α0 ln z + i√`2

2∑n 6=0

αnnz−n,

X(z) = xR2 − i

√`2

2 α0 ln z + i√`2

2∑n 6=0

αnnz−n.

(23.50)

The zero-modes are respectively α0 and α0 for ∂X and ∂X, and xL and xR for XL andXR. The meaning of the modes will become clearer in Section 23.1.5 where we study thecommutation relations.

First, we relate the zero-modes α0 and α0 to the conserved charges pL and pR (23.28) ofthe U(1) current:

pL = α0√2`2

, pR = α0√2`2

(23.51)

such that

X(z) = xL2 − i`2 pL ln z + i

√`2

2∑n 6=0

αnnz−n, (23.52)

Then, the relations (23.28) and (23.30) allow to rewrite this result in terms of the momentump and winding w:

p = 1√2`2

(α0 + α0

), w = 1√

2`2(α0 − α0

). (23.53)

These relations can be inverted as

α0 =√`2

2 (p+ w), α0 =√`2

2 (p− w). (23.54)

In the same sense that there are two momenta pL and pR conjugated to xL and xR,it makes sense to introduce two coordinates x and q conjugated to p and w. From stringtheory, the operator x is called the center of mass. The expression (23.54) suggests to write:

xL = x+ q, xR = x− q, (23.55)

and conversely:x = 1

2 (xL + xR), q = 12 (xL − xR). (23.56)

4The Fourier expansion is taken to be identical for ε = ±1 fields since ∂X is contravariant. The differencebetween the two cases will appear in the commutators.

279

In terms of these new variables, the expansion of the full X(z, z) reads:

X(z, z) = x− i `2

2

(p ln |z|2 + w ln z

z

)+ i√`2

2∑n 6=0

1n

(αn z

−n + αn z−n). (23.57)

In terms of the coordinates on the cylinder, the part without oscillations becomes:

X(τ, σ) = x− i `2 pτ + `2 wσ + · · · (23.58)

Note how the presence of `2 gives the correct scale to the second term. The mode q does notappear at all, and x is the zero-mode of the complete field X(z, z). As it is well-known, thephysical interpretation of x and p is as the position and momentum of the centre-of-massof the string.5 If there is a compact dimension, then w counts the number of times thestring winds around it, and q can be understood as the position of the centre-of-mass aftera T -duality.6


pL = 12πi

∮dz J = i

`21

2πi

∮dz ∂X = i

`21

2πi

∮dz ∂X

= 1√2`2

12πi

∮dz∑n

αn z−n−1 = 1√

2`2α0.

The computation gives pR after replacing α0 by α0.

If the scalar field is non-compact but periodic on the cylinder, the periodicity condition

X(τ, σ + 2π) ∼ X(τ, σ) (23.59)

translates asX(e2πiz, e−2πiz) ∼ X(z, z). (23.60)

Evaluating the LHS from (23.50) gives a constraint on the zero-modes:

X(e2πiz, e−2πiz) = X(z, z)− i√`2

2 (α0 − α0), (23.61)

which impliesα0 = α0 =⇒ pL = pR = p

2 , w = 0. (23.62)

The other cases will not be discussed in this review, but we still use the general notation tomake the contact with the literature easier. This also implies that XL and XR cannot beperiodic independently. Hence, the zero-mode couples the holomorphic and anti-holomorphicsectors together.

The number operators Nn Nn at level n > 0 are defined by:

Nn = ε

nα−nαn, Nn = ε

nα−nαn. (23.63)

The modes have been normal ordered. They count the number of excitations at the level n:the factor n−1 is necessary because the modes are not canonically normalized. Then, onecan build the level operators

N =∑n>0

nNn. (23.64)

5In Lorentzian signature, this becomes X(τ, σ) = x+ `2 pt+ `2 wσ as expected.6T -duality and compact bosons fall outside the scope of this review and we refer the reader to [300,

chap. 17, 218, chap. 8] for more details.

280

They count the number of excitations at level n weighted by the level itself. This correspondsto the total energy due to the oscillations (the higher the level, the more energy it needs tobe excited).

The Virasoro operators are

Lm = ε

2∑n

:αnαm−n: (23.65)

For m 6= 0, we have

m 6= 0 : Lm = ε

2∑n 6=0,m

:αnαm−n: + ε α0αm, (23.66)

there is no ordering ambiguity and the normal order can be removed. In the case of thezero-mode, one finds

L0 = ε

2∑n

:αnα−n: = N + ε

2 α20 = N + ε`2 p2

L, (23.67)

using (23.64) and (23.51). It is also useful to define L0 which corresponds to L0 strippedfrom the zero-mode contribution:

L0 := N. (23.68)

Similarly, the anti-holomorphic zero-mode is

L0 = N + ε`2 p2R,

L0 := N , (23.69)

such that

L+0 = N + N + ε`2 (p2

L + p2R) = N + N + ε`2

2 (p2 + w2), (23.70a)

L−0 = N − N + ε`2 (p2L − p2

R) = N − N + ε`2 wp, (23.70b)

where L±0 := L0 ± L0 as defined in (22.118). The last equality of each line follows from(23.31b). The expression of L+

0 for N = N = 0 matches the weights (23.46) of the vertexoperators for pL = pR = p/2, which will be interpreted below. It is a good place to stressthat pL, pR, p and w are operators, while k is a number.

23.1.5 CommutatorsThe commutators can be computed from (22.76a) knowing the OPE (23.37). The modes of∂X and ∂X satisfy

[αm, αn] = εm δm+n,0, [αm, αn] = εm δm+n,0, [αm, αn] = 0 (23.71)

for all m,n ∈ Z (including the zero-modes). The appearance of the factor m in the RHSexplains the normalization of the number operator (23.63).

From the commutators of the zero-modes, we directly find the ones for the momentumand winding:

[p, w] = [p, p] = [w,w] = 0, [p, αn] = [p, αn] = [w,αn] = [w, αn] = 0. (23.72)

The OPE (23.36) yields[xL, pL] = iε, [xR, pR] = iε, (23.73)

281

which can be used to determine the commutators of x and q:

[x, p] = [q, w] = iε, [x,w] = [q, p] = 0. (23.74)

This shows that (x, p) and (q, w) are pairs of conjugate variables. Interestingly, the windingnumber w commutes will all other modes except q, but the latter disappears from thedescription. Hence, it can be interpreted as a number which labels different representations:if no other principle (like periodicity) forbids w 6= 0, then one can except to have states withall possible w in the spectrum, each value of w forming a different sector (in the same senseof the NS and R sectors of the superstring). There are other interpretations from the pointof view of T -duality and double field theory [120, 124, 213, 300].

The commutator of the modes with the Virasoro operators is

[Lm, αn] = −nαm+n. (23.75)

as expected from (22.115). For m = 0, this reduces to

[L0, α−n] = nα−n, (23.76)

which shows that negative modes increase the energy. The commutator of the creationmodes α−n with the number operators is

[Nm, α−n] = α−mδm,n. (23.77)

23.1.6 Hilbert spaceThe Hilbert space of the free scalar has the structure of a Fock space.

From (23.76), the momentum p commutes with the Hamiltonian L+0 such that it is a

good quantum number to label the states:7 this translates the fact the action (23.1) doesnot depend on the conjugate variable x. As a consequence, there exists a family of vacua|k〉.

The vacua |k〉 are the states dual to the vertex operators (23.33):

|k〉 := limz,z→0

Vk(z, z) |0〉 = eiεkx |0〉 , (23.78)

where |0〉 is the SL(2,C) vacuum and x is the zero-mode of X(z, z). That this identificationis correct follows by applying the operator p:

p |k〉 = k |k〉 . (23.79)

The notation is consistent with the one of the SL(2,C) vacuum since p |0〉 = 0.The vacuum is annihilated by the action of the positive-frequency modes:

∀n > 0 : αn |k〉 = 0, (23.80)

which is equivalent toNn |k〉 = 0. (23.81)

The different vacua are each ground state of a Fock space (they are all equivalent), but theyare not ground states of the Hamiltonian since they have different energies:

L+0 |k〉 = 2ε`2 k2 |k〉 , L−0 |k〉 = 0, (23.82)

7To simplify the discussion, we do not consider winding but only vertex operators of the form (23.33).

282

using (23.70). The SL(2,C) vacuum is the lowest (highest) energy state if ε = 1 (ε = −1).The Fock space F(k) built from the vacuum at momentum k is found by acting repet-

itively with the negative-frequency modes. A convenient basis, the oscillator basis, is givenby the states:

F(k) = Span|k; Nn〉

, (23.83a)

|k; Nn〉 :=∏n≥1

(α−n)Nn√nNnNn!

|k〉 , Nn ∈ N∗ (23.83b)

(we don’t distinguish the notations between the number operators and their eigenvalues).The full Hilbert space is given by:

H =∫R

dkF(k). (23.84)

Computation – Equation (23.78)We provide a quick argument to justify the second form of (23.78). Take the limit of(23.57) with w = 0:

limz,z→0

eiεkX(z,z) |0〉 = limz,z→0

exp iεk

x− i `2

2 p ln |z|2 + i√`2

2∑n 6=0

1n

(αn z

−n + αn z−n)

= limz,z→0

exp

iεkx+ εk`2

2 p ln |z|2 − εk√`2

2∑n 6=0

1n

(αn z

−n + αn z−n) .

For ε = 1, as z, z → 0, the terms with αn and αn for n < 0 disappear since they areaccompanied with a positive power of zn and zn. The modes with n > 0 diverge but theminus sign makes the exponential to vanish. Finally, the term in p also vanishes becauseln |z|2 → −∞. A more rigorous argument requires to normal order the exponential andthen to use (23.80).


p |k〉 = 1`2

12πi

∮ (dz i∂X(z) + dz i∂X(z)

)Vk(0, 0) |0〉

= 1`2

12πi

∮ (dzz

`2k

2 + dzz

`2k

2

)Vk(0, 0) |0〉

= k Vk(0, 0) |0〉

using (23.44).

Remark 23.4 (Fock space and Verma module isomorphism) Note that, in the ab-sence of null states, there is a one-to-one map between states in the α−n oscillator basisand in the L−n Virasoro basis. This translates an isomorphism between the Fock space andthe Verma module of Vk. One hint for this relation is that applying α−n and L−n changesthe weight (eigenvalue of L0) by the same amount, and there are as many operators in bothbasis.

283

23.1.7 Euclidean and BPZ conjugatesSince X is a real scalar field, it is self-adjoint (22.96) such that

x† = x p† = p, α†n = α−n. (23.85)

As a consequence, the adjoint of the vacuum |k〉 follows from (23.78):

〈k| = |k〉‡ = 〈0| e−iεkx, 〈k| p = 〈k| k. (23.86)

The sign inside the bracket is in agreement with the fact that it is an eigenstate of p witheigenvalue k. From (23.85) it follows that the Virasoro operators are Hermitian:

L†n = L−n. (23.87)

The BPZ conjugate (22.111) of the mode αn is:

αtn = −(±1)nα−n, (23.88)

where the sign depends on the choice of I± in (22.111). Using (23.53), this implies that themomentum operator gets a minus sign:8

pt = −p, 〈−k| = |k〉t . (23.89)

The inner product between two vacua |k〉 and |k′〉 is normalized as:

〈k|k′〉 = 2π δ(k − k′) (23.90)

such that the conjugate state (22.144) of the vacuum reads

〈kc| = 12π 〈k| . (23.91)

The Hermitian and BPZ conjugate states are related as:

|k〉‡ = − |k〉t , (23.92)

which can be interpreted as a reality condition on |k〉.

23.2 First-order bc ghost systemFirst-order systems describe two free fields called ghosts which have a first-order actionand whose conformal weights sum to 1. Commuting (resp. anti-commuting) fields are oftendenoted by β and γ (resp. b and c) and correspondingly first-order systems are also calledβγ or bc systems. We will introduce a sign ε = ±1 to denote the Grassmannality of thefields and always write them as b and c. In string theory, first-order systems describe theFaddeev–Popov ghosts associated to reparametrizations and supersymmetries (Sections 3.4and 18.1).

8Be careful that |k〉 is not a state associated to some operator through the state–operator correspondence:this explains why 〈k| 6= (|k〉)t as in (22.135).

284

23.2.1 Covariant actionA first-order system is defined by two symmetric and traceless fields bµ1···µn and cµ1···µn−1

called ghosts. The dynamics is governed by the first-order action

S = 14π

∫d2x√g gµν bµµ1···µn−1∇νcµ1···µn−1 (23.93)

after taking into account the symmetries of the field indices. Deriving the equations ofmotion and the energy–momentum tensor is done either by writing explicitly the symmet-rization and the traceless conditions, or by using Lagrange multipliers. Obviously, for n = 1,one recovers the reparametrization ghost action (3.145). The action (23.93) is invariant un-der Weyl transformations such that it describes a CFT on flat space.

In fact, one can allow the fields to carry a spinor index, in which case the action containsan additional gamma matrix.

The ghost action (23.93) is invariant under a global U(1) symmetry

bµ1···µn −→ e−iθbµ1···µn , cµ1···µn−1 −→ eiθcµ1···µn−1 . (23.94)

23.2.2 Action on the complex planeThe simplest description of the system is on the complex plane. Due to the conditions im-posed on the fields, they have only two independent components for all n, and the equationsof motion imply that one is holomorphic, and the other anti-holomorphic:

b(z) := bz···z(z), b(z) := bz···z(z), c(z) := cz···z(z), c(z) := cz···z(z). (23.95)

In this language, the action becomes

S = 12π

∫d2z(b∂c+ b∂c). (23.96)

This action gives the appropriate equations of motion

∂b = 0, ∂b = 0, ∂c = 0, ∂c = 0. (23.97)

Since the fields split into holomorphic and anti-holomorphic sectors, it is convenient to studyonly the holomorphic sector as usual. This system is even simpler than the scalar fieldbecause the zero-modes don’t couple both sectors.9 All formulas for the anti-holomorphicsector are directly obtained from the holomorphic one by adding bars on quantities, exceptfor conserved charges which have an index L or R and are both written explicitly.

The action describes a CFT, and the weight of the fields are given by

h(b) = λ, h(c) = 1− λ, h(b) = λ, h(c) = 1− λ, (23.98)

where λ = n if the fields are in a tensor representation, and λ = n + 1/2 if they are in aspinor-tensor representation. The holomorphic energy–momentum reads

T = −λ :b∂c: + (1− λ) :∂b c: (23.99a)= −λ :∂(bc): + :∂b c: (23.99b)= (1− λ) :∂(bc):− :b ∂c:. (23.99c)

Normal ordering is taken with respect to the SL(2,C) vacuum (22.121).9For the scalar field, the coupling of both sectors happened because of the periodicity condition (23.62).

285

Finally, both fields can be classically commuting or anticommuting (see below for thequantum commutators):

b(z)c(w) = −ε c(w)b(z), b(z)b(w) = −ε b(w)b(z), c(z)c(w) = −ε c(w)c(z), (23.100)

where ε denotes the Grassmann parity

ε =

+1 anticommuting,−1 commuting.

(23.101)

Sometimes, if ε = +1, one denotes b and c respectively by β and γ. If b and c are ghostsarising from Faddeev–Popov gauge fixing, then ε = 1 if λ is integer; and ε = −1 if λ ishalf-integer (“wrong” spin–statistics assignment).

The U(1) global symmetry (23.94) reads infinitesimally

δb = −ib, δc = ic, δb = −ib, δc = ic. (23.102)

It is generated by the conserved ghost current with components:

j(z) = −:b(z)c(z):, (z) = −:b(z)c(z): (23.103)

and the associated charge is called the ghost number

Ngh = Ngh,L +Ngh,R, Ngh,L =∮ dz

2πi j(z), Ngh,R = −∮ dz

2πi (z). (23.104)

This charge counts the number of c ghosts minus the number of b ghosts, such that

Ngh(c) = 1, Ngh(b) = −1, Ngh(c) = 1, Ngh(b) = −1. (23.105)

The propagator can be derived from the path integral∫d′bd′c δ

δb(z)

[b(w)e−S[b,c]

]= 0 (23.106)

which gives the differential equation

δ(2)(z − w) + 12π 〈b(w)∂c(z)〉 = 0. (23.107)

Using (B.2), the solution is easily found to be

〈c(z)b(w)〉 = 1z − w

. (23.108)

Remark 23.5 The propagator is constructed with the path integral. For convenience, thezero-modes are removed from the measure: reintroducing them, one finds that the propag-ator is computed not in the conformal vacuum (which has no operator insertion), but in astate with ghost insertions. This explains why the propagator (23.108) is not of the form(22.63b). However, this form is sufficient to extract the OPE as changing the vacuum doesnot introduce singular terms.

286

23.2.3 OPEThe OPEs between the b and c fields are found from the propagator (23.108):

c(z)b(w) ∼ 1z − w

, b(z)c(w) ∼ ε

z − w, (23.109a)

b(z)b(w) ∼ 0, c(z)c(w) ∼ 0. (23.109b)

The OPE of each ghost with T confirms the conformal weights in (23.98):

T (z)b(w) ∼ λ b(w)(z − w)2 + ∂b(w)

z − w, (23.110a)

T (z)c(w) ∼ (1− λ) c(w)(z − w)2 + ∂c(w)

z − w. (23.110b)

The OPE of T with itself is

T (z)T (w) ∼ cλ/2(z − w)4 + 2T (w)

(z − w)2 + ∂T (w)z − w

, (23.111)

where the central charge is:

cλ = 2ε(−1 + 6λ− 6λ2) = −2ε(1 + 6λ(λ− 1)

). (23.112)

Introducing the ghost charge:qλ = ε(1− 2λ), (23.113)

the central charge can also be written as

cλ = ε(1− 3q2λ). (23.114)

This parameter will appear many times in this section and its meaning will become cleareras we proceed.

The OPE between the ghost current (23.103) and the b and c ghosts read

j(z)b(w) ∼ − b(w)z − w

, (23.115a)

j(z)c(w) ∼ c(w)z − w

. (23.115b)

The coefficients of the (z−w)−1 terms correspond to the ghost number of the b and c fields(23.105). More generally, the ghost number Ngh(O) of any operator O(z) is defined by

j(z)O(w) ∼ Ngh(O) O(w)z − w

. (23.116)

The OPE for j with itself is

j(z)j(w) ∼ ε

(z − w)2 . (23.117)

This will be interpreted later in the context of bosonization.Finally, the OPE of the current with T reads:

T (z)j(w) ∼ qλ(z − w)3 + j(w)

(z − w)2 + ∂j(w)z − w

. (23.118)

287

Due to the presence of the z−3 term, the current j(z) is not a primary field if qλ 6= 0, that is,if λ 6= 1/2. In that case, its transformation under changes of coordinates gets an anomalouscontribution:

j(z) = dwdz j

′(w) + qλ2

ddz ln dw

dz = dwdz j

′(w) + qλ2∂2zw

∂zw. (23.119)

This implies in particular that the currents on the plane and on the cylinder (w = ln z) arerelated by:

j(z) = dwdz

(jcyl(w)− qλ

2

), (23.120)

which leads to the following relation between the ghost numbers on the plane and on thecylinder:

Ngh = N cylgh − qλ, Ngh,L = N cyl

gh,L −qλ2 , Ngh,R = N cyl

gh,R −qλ2 . (23.121)

For this reason, it is important to make clear the space with respect to which is given theghost number: if not explicitly stated, ghost numbers in this review are given on the plane.10Due to this anomaly, one finds that the ghost number is not conserved on a curved space:

N c −N b = −ε qλ2 χg = (1− 2λ)(g − 1), (23.122)

where χg is the Euler characteristics (3.9), N b and N c are the numbers of b and c operators.In string theory, where the only ghost insertions are zero-modes, this translates into astatement on the number of zero-modes to be inserted. Hence, this can be interpreted as ageneralization of (3.72). For a proof, see for example [31, p. 397].

Computation – Equation (23.110a)

T (z)b(w) =(− λ :b(z)∂c(z): + (1− λ) :∂b(z) c(z):

)b(w)

∼ −λ :b(z)∂c(z): b(w) + (1− λ) :∂b(z) c(z): b(w)

∼ −λ b(z)∂z1

z − w+ (1− λ) ∂b(z) 1

z − w

∼ λ(b(w) +(((((

((z − w)∂b(w)) 1

(z − w)2 + (1− λ) ∂b(w)z − w

.

10Other references, especially old ones, give it on the cylinder. This can be easily recognized if some ghostnumbers in the holomorphic sector are half-integers: for the reparametrization ghosts, qλ is an integer suchthat the shift in (23.121) is a half-integer.

288


T (z)c(w) =(− λ :b(z)∂c(z): + (1− λ) :∂b(z)c(z):

)c(w)

∼ ελ :∂c(z)b(z): c(w)− ε(1− λ) :c(z)∂b(z): c(w)

∼ λ ∂c(z)z − w

− (1− λ) c(z) ∂z1

z − w

∼ λ ∂c(w)z − w

+ (1− λ)(c(w) + (z − w)∂c(w)

) 1(z − w)2

∼ (1− λ) c(w)(z − w)2 + ∂c(w)

(z − w)2 .

Computation – Equation (23.115a)

j(z)b(w) = −:b(z)c(z): b(w) ∼ −:b(z)c(z): b(w) ∼ − b(z)z − w

∼ − b(w)z − w

.


j(z)c(w) = −:b(z)c(z): c(w) ∼ ε :c(z)b(z): c(w) ∼ c(z)z − w

∼ c(w)z − w

.


j(z)j(w) = :b(z)c(z): :b(w)c(w):

∼ :b(z)c(z): :b(w)c(w): + :b(z)c(z): :b(w)c(w): + :b(z)c(z): :b(w)c(w):

∼ ε

(z − w)2 + ε :c(z)b(w):z − w

+ :b(z)c(w):z − w

∼ ε

(z − w)2 .

23.2.4 Mode expansionsThe b and c ghosts are expanded as

b(z) =∑

n∈Z+λ+ν

bnzn+λ , c(z) =

∑n∈Z+λ+ν

cnzn+1−λ , (23.123)

where ν = 0, 1/2 depends on ε and on the periodicity of the fields, see (22.102). The modesare extracted with the contour formulas

bn =∮ dz

2πi zn+λ−1b(z), cn =

∮ dz2πi z

n−λc(z). (23.124)

289

Ghosts with λ ∈ Z have integer indices and ν = 0 (we don’t consider ghosts with twistedboundary conditions). On the other hand, ghosts with λ ∈ Z + 1/2 have integer indicesand ν = 1/2 in the R sector, and half-integer indices and ν = 0 in the NS sector (seeSection 22.4.4). The choices in the boundary conditions arise from the Z2 symmetry of theaction:

b −→ −b, c −→ −c. (23.125)

The number operators N bn and N c

n are defined to count the numbers of excitations of band c ghosts at level n:

N bn = :b−ncn:, N c

n = ε :c−nbn:. (23.126)

The definitions follow from the commutators (23.134). Then, the level operators N b and N c

are obtained by summing over n:

N b =∑n>0

nN bn, N c =

∑n>0

nN cn. (23.127)

The Virasoro operators are

Lm =∑n

(n− (1− λ)m

):bm−ncn: =

∑n

(λm− n) :bncm−n:. (23.128)

Of particular importance is the zero-mode

L0 = −∑n

n :bnc−n: =∑n

n :b−ncn:. (23.129)

We will give the expression of L0 in terms of the level operators below, see (23.159). Todo this, we will first need to change the normal ordering, which first requires to study theHilbert space.

The modes of the ghost current are

jm = −∑n

:bm−ncn: = −∑n

:bncm−n:. (23.130)

Note that the zero-mode of the current also equals the ghost number

Ngh,L = j0 = −∑n

:b−ncn:. (23.131)

When both the holomorphic and anti-holomorphic sectors enter, it is convenient to in-troduce the combinations

b±n = bn ± bn, c±n = 12 (cn ± cn). (23.132)

The normalization of b±m is chosen to match the one of L±m (22.118), and the one of c±m suchthat (23.135) holds. Note the following useful identities:

b−n b+n = 2bnbn, c−n c

+n = 1

2 cncn. (23.133)

290


T = −λ :b∂c: + (1− λ) :∂bc:

=∑m,n

(λ :bmcn: n+ 1− λ

zm+λzm+2−λ − (1− λ) :bmcn: m+ λ

zm+λ+1zm+1−λ

)=∑m,n

(λ (n+ 1− λ)− (1− λ)(m+ λ)

) :bmcn:zm−n+2

=∑m,n

(λ (n+ 1− λ)− (1− λ)(m− n+ λ)

) :bm−ncn:zm+2

=∑m,n

(n−m+ λm) :bm−ncn:zm+2 =

∑m

Lmzm+2 .

The fourth line follows from shifting m → m − n. The second equality in (23.128)follows by shifting n→ m− n.


j = −:bc: =∑m,n

:bmcn:zm+λzn+1−λ =

∑m,n

:bm−ncn:zm+1 =

∑m

jmzm+1 .

23.2.5 CommutatorsThe (anti)commutators between the modes bn and cn read:

[bm, cn]ε = δm+n,0, [bm, bn]ε = 0, [cm, cn]ε = 0. (23.134)

Therefore, the modes with n < 0 are creation operators and the modes with n > 0 areannihilation operators:

• a b ghost excitation at level n > 0 is created by b−n and annihilated by cn;

• a c ghost excitation at level n > 0 is created by c−n and annihilated by bn.

In terms of b±m and c±m (23.132), we have:

[b+m, c+n ]ε = δm+n, [b−m, c−n ]ε = δm+n. (23.135)

The commutators of the number operators with the modes are:

[N bm, b−n] = b−nδm,n, [N c

m, c−n] = c−nδm,n, (23.136)

while those between the Ln and the ghost modes are:

[Lm, bn] =(m(λ− 1)− n

)bm+n, [Lm, cn] = −(mλ+ n)cm+n, (23.137)

in agreement with (22.115). If n ∈ Z, each ghost field has zero-modes b0 and c0 whichcommutes with L0

[L0, b0] = 0, [L0, c0] = 0. (23.138)

291

The commutator of the current modes reads

[jm, jn] = mδm+n,0. (23.139)

Then, the commutator with the Virasoro operators are

[Lm, jn] = −njm+n + qλ2 m(m+ 1)δm+n,0. (23.140)

Finally, the commutators of the ghost number operator with the ghosts are:

[Ngh, b(w)] = −b(w), [Ngh, c(w)] = c(w). (23.141)


[bm, cn]ε = ε

∮C0

dw2πi w

−1∮Cw

dz2πi z

−1 wn+λzm−λ+1b(z)c(w)

∼ ε∮C0

dw2πi w

−1∮Cw

dz2πi z

−1 wn+λzm−λ+1 ε

z − w

=∮C0

dw2πi w

m+n−1 = δm+n,0.


[Ngh, b(w)] =∮ dz

2πi j(z)b(w) ∼ −∮ dz

2πib(w)z − w

= −b(w).

The computation for c is identical.

23.2.6 Hilbert spaceThe SL(2,C) vacuum |0〉 (22.121) is defined by:

∀n > −λ : bn |0〉 = 0, ∀n > λ− 1 : cn |0〉 = 0. (23.142)

If λ > 1, there are positive modes which do not annihilate the vacuum.To simplify the notations, we consider the case λ ∈ Z, the half-integer case following by

shifting the indices by 1/2. Since the modes c1, . . . , cλ−1 do not annihilate |0〉, one cancreate states

|n1, . . . , nλ−1〉 = cn11 · · · c

nλ−1λ−1 |0〉 (23.143)

which have negative energies:

L0 |n1, . . . , nλ−1〉 = −

λ−1∑j=1

j nj

|n1, . . . , nλ−1〉 , (23.144)

where (23.137) has been used. Moreover, this state is degenerate due to the existence ofzero-modes since they commute with the Hamiltonian – see (23.138). As a consequence, itmust be in a representation of the zero-mode algebra.

If the ghosts are commuting (ε = −1), then it seems hard to make sense of the theorysince one can find a state of arbitrarily negative energy since ni ∈ N. The zero-modes makethe problem even worse. The appropriate interpretation of these states will be discussed inthe context of the superstring theory for λ = 3/2 (superconformal ghosts).

In the rest of this section, we focus on the Grassmann odd case ε = 1.

292

Energy vacuum (Grassmann odd)

Since ni = 0 or ni = 1 for anticommuting ghosts (ε = 1), there is a state of lowest energy.This is the energy vacuum (22.128). Since the zero-modes b0 and c0 commute with L0, it isdoubly degenerate. A convenient basis is

| ↓〉 , | ↑〉, (23.145)

where| ↓〉 := c1 · · · cλ−1 |0〉 , | ↑〉 := c0c1 · · · cλ−1 |0〉 . (23.146)

A general vacuum is a linear combination of the two basis vacua:

|Ω〉 = ω↓ | ↓〉+ ω↑ | ↑〉 , ω↓, ω↑ ∈ C. (23.147)

The algebra of these vacua is the one of a two-state system:

b0 | ↑〉 = | ↓〉 , c0 | ↓〉 = | ↑〉 , b0 | ↓〉 = 0, c0 | ↑〉 = 0. (23.148)

Hence, for the vacuum | ↓〉 (resp. | ↑〉), b0 (resp. c0) acts as an annihilation operator, andconversely c0 (resp. b0) acts as a creation operator. Finally, both states are annihilated byall positive modes:

∀n > 0 : bn | ↓〉 = bn | ↑〉 = 0, cn | ↓〉 = bn | ↓〉 = 0. (23.149)

Note that the SL(2,C) vacuum can be recovered by acting with b−n with n < λ:

|0〉 = b1−λ · · · b−1 | ↓〉 = b1−λ · · · b−1b0 | ↑〉 . (23.150)

The zero-point energy (22.129) of these states is the conformal weight of the vacuum:

L0 | ↓〉 = aλ | ↓〉 , L0 | ↑〉 = aλ | ↑〉 , (23.151)

where aλ can be written in various forms:

aλ = −λ−1∑n=1

n = −λ(λ− 1)2 = cλ

24 + 224 . (23.152)

Taking into account the anti-holomorphic sector leads to a four-fold degeneracy. Thebasis

| ↓↓〉 , | ↑↓〉 , | ↓↑〉 , | ↑↑〉, (23.153)

is built as follows:

| ↓↓〉 := c1c1 · · · cλ−1cλ−1 |0〉 ,| ↑↓〉 := c0 | ↓↓〉 , | ↓↑〉 := c0 | ↓↓〉 , | ↑↑〉 := c0c0 | ↓↓〉 .

(23.154)

The modes b0 and b0 can be used to flip the arrows downward, leading to the followingalgebra:

c0 | ↓↓〉 = | ↑↓〉 , c0 | ↓↓〉 = | ↓↑〉 , c0 | ↓↑〉 = −c0 | ↑↓〉 = | ↑↑〉 ,b0 | ↑↑〉 = | ↓↑〉 , b0 | ↑↑〉 = − | ↑↓〉 , b0 | ↑↓〉 = b0 | ↓↑〉 = | ↓↓〉 ,

(23.155a)

The vacua are annihilated by different combinations of the zero-modes:

b0 | ↓↓〉 = b0 | ↓↓〉 = 0, c0 | ↑↓〉 = b0 | ↑↓〉 = 0,b0 | ↓↑〉 = c0 | ↓↑〉 = 0, c0 | ↑↑〉 = c0 | ↑↑〉 = 0.

(23.155b)

293

In these manipulations, one has to be careful to correctly anti-commute the modes with theones hidden in the definitions of the vacua.

There is a second basis which is more natural when using the zero-modes c±0 and b±0(23.132):

| ↓↓〉 , |+〉 , |−〉 , | ↑↑〉, (23.156)

where the two vacua |±〉 are combinations of the | ↓↑〉 and | ↑↓〉 vacua:

|±〉 = | ↑↓〉 ± | ↓↑〉 . (23.157)

The different vacua are naturally related by acting with c±0 and b±0 which act as raising andlowering operators:

c±0 | ↓↓〉 = 12 |±〉 , c∓0 |±〉 = ± | ↑↑〉 ,

b±0 |±〉 = ±2 | ↓↓〉 , b∓0 | ↑↑〉 = ± |±〉 .(23.158a)

From the previous relations, it follows that the different vacua are annihilated by the zero-modes as follow:

b+0 | ↓↓〉 = b−0 | ↓↓〉 = 0, c−0 |−〉 = b+0 |−〉 = 0,c+0 |+〉 = b−0 |+〉 = 0 c+0 | ↑↑〉 = c−0 | ↑↑〉 = 0,

(23.158b)

This also means that we have

c−0 c+0 | ↓↓〉 = 1

2 | ↑↑〉 , b+0 b−0 | ↑↑〉 = 2 | ↓↓〉 . (23.158c)


2 c+0 |±〉 = (c0 + c0) | ↑↓〉 ± (c0 + c0) | ↓↑〉 = c0 | ↑↓〉 ± c0 | ↓↑〉 = (−1± 1) | ↑↑〉b+0 |±〉 = (b0 + b0) | ↑↓〉 ± (b0 + b0) | ↓↑〉 = b0 | ↑↓〉 ± b0 | ↓↑〉 = (1± 1) | ↓↓〉

2 c±0 | ↓↓〉 = (c0 ± c0) | ↓↓〉 = c0 | ↓↓〉 ± c0 | ↓↓〉 = | ↑↓〉 ± | ↓↑〉 = |±〉b±0 | ↑↑〉 = (b0 ± b0) | ↑↑〉 = b0 | ↑↑〉 ± b0 | ↑↑〉 = | ↓↑〉 ∓ | ↑↓〉 = ∓ |∓〉

Energy normal ordering (Grassmann odd)

We now turn towards the definition of the energy normal ordering (22.149). Ultimately, itwill be found that | ↓〉 is the physical vacuum in string theory. For this reason, the energynormal ordering ?

? · · · ?? is associated to the vacuum | ↓〉 in order to resolve the ambiguityof the zero-modes. In particular, b0 is an annihilation operator in this case, while c0 is acreation operator. In the rest of this section, we translate the normal ordering of expressionsfrom the conformal vacuum to the energy vacuum.

The Virasoro operators Ln for n 6= 0 have no ordering problems since the modes whichcompose them commute. The expression of L0 (23.129) in the energy ordering becomes

L0 =∑n

n??b−ncn

?? + aλ = N b +N c + aλ (23.159)

where aλ is the zero-point energy (23.152) and N b and N c are the ghost mode numbers(23.127). The contribution of the non-zero modes is denoted by:

L0 = N b +N c. (23.160)

294

The expression can be rewritten to encompass all modes:

Lm =∑n

(n− (1− λ)m

)??bm−ncn

?? + aλ δm,0 (23.161)

Similarly, the expression of the ghost number is

Ngh,L = j0 =∑n

??b−ncn

??−

(qλ2 + 1

2

)(23.162a)

=∑n>0

(N cn −N b

n

)+ 1

2(N c

0 −N b0)− qλ

2 , (23.162b)

and thus:jm =

∑n

??bm−ncn

??−

(qλ2 + 1

2

)δm,0. (23.163)

It is useful to define the ghost number without ghost zero-modes:

Ngh,L :=∑n>0

(N cn −N b

n

). (23.164)

One can straightforwardly compute the ghost number of the vacua:

j0 | ↓〉 = (λ− 1) | ↓〉 =(−qλ2 −

12

)| ↓〉 , (23.165a)

j0 | ↑〉 = λ | ↑〉 =(−qλ2 + 1

2

)| ↑〉 . (23.165b)

This confirms that the SL(2,C) vacuum has vanishing ghost number since | ↓〉 containsexactly λ− 1 ghosts:

j0 (23.166)

Using (23.121) allows to write the ghost numbers on the cylinder:

jcyl0 | ↓〉 = −12 | ↓〉 , jcyl0 | ↑〉 = 1

2 | ↑〉 . (23.167)

This result was expected: since the number changes as Ngh → −Ngh when b ↔ c areexchanged, the mean value of the ghost number should be zero.

Remark 23.6 (Ghost number conventions) Since the ghost number is an additive quan-tum number, it is always possible to shift its definition by a constant. This can be used toset the ghost numbers of the vacua to some other values. For example, [31, p. 116] addsqλ/2 to the ghost number in order to get Ngh = ±1/2 on the plane (instead of the cylinder).We do not follow this convention in order to keep the symmetry between the vacuum ghostnumbers on the cylinder.

295

Computation – Equation (23.159)Start with (23.129) and use (22.156):

L0 = −∑n

n :bnc−n: = −∑n≤−λ

n bnc−n + ε∑n>−λ

n c−nbn

=∑n≥λ

n b−ncn + ε∑n>−λ

n c−nbn

=∑n≥λ

n b−ncn + ε∑n>0

n c−nbn + ε

0∑n=−λ+1

n c−nbn

=∑n≥λ


n c−nbn + ε

λ−1∑n=0

n b−ncn + aλ

=∑n>0


n c−nbn + aλ,

=∑n

n??b−ncn

?? + aλ,

using that

0∑n=−λ+1

c−nbn = −λ−1∑n=0

n cnb−n = −λ−1∑n=0

n (−ε b−ncn + 1) = ε

λ−1∑n=0

n b−ncn + aλ.

The result also follows from (22.162).


j0 = −∑n

:b−ncn: = −∑n≥λ

b−ncn + ε∑n>−λ

c−nbn

= −∑n≥λ

b−ncn + ε∑n>0

c−nbn + ε

λ−1∑n=1

cnb−n + ε c0b0

= −∑n≥λ

b−ncn + ε∑n>0

c−nbn −λ−1∑n=1

b−ncn + ε(λ− 1) + ε c0b0

= −∑n>0

b−ncn + ε∑n>0

c−nbn + ε(λ− 1) + ε c0b0.

Finally, one can writeε(λ− 1) = −qλ2 −

ε

2 . (23.168)

The result also follows from (22.162). The second expression is obtained by symmet-rizing the last term such that

ε c0b0 + ε(λ− 1) = ε

2 c0b0 + 12(−b0c0 + ε) + ε(λ− 1)

= 12 (ε c0b0 − b0c0) + ε

(λ− 1

2

).

296

Structure of the Hilbert space (Grassmann odd)

Since the zero-modes commute with the Hamiltonian and with all other negative- andpositive-frequency modes, the Hilbert space is decomposed in several subspaces, each as-sociated to a zero-mode.11

Starting with the holomorphic sector only, the Hilbert space Hgh is:

Hgh = Hgh,0 ⊕ c0Hgh,0, Hgh,0 := Hgh ∩ ker b0, (23.169)

which follows from the 2-state algebra (23.148). Obviously, one has c0Hgh,0 = Hgh ∩ ker c0.The oscillator basis of the Hilbert space Hgh,0 is generated by applying the negative-frequency modes and has the structure of a fermionic Fock space without zero-modes:

Hgh,0 = Span ∣∣↓; N b

n; N cn⟩

, (23.170a)∣∣↓; N bn; N c

n⟩

=∏n≥1

(b−n)Nbn(c−n)N

cn | ↓〉 , N b

n, Ncn ∈ N∗ (23.170b)

(again, number operators and their eigenvalues are not distinguished). This means thatHgh,0 can also be regarded as a Fock space built on the vacuum | ↓〉, for which c0 and b0are respectively creation and annihilation operators. Conversely, c0 and b0 are respectivelyannihilation and creation operators for c0Hgh,0.

In particular, this means that any state can be written as the sum of two states

ψ = ψ↓ + ψ↑, ψ↓ ∈ Hgh,0, ψ↑ ∈ c0Hgh,0, (23.171)

with ψ↓ and ψ↑ built respectively on top of the | ↓〉 and | ↑〉 vacua.

This pattern generalizes when considering both the holomorphic and anti-holomorphicsectors. In that case, the Hilbert space is decomposed in four subspaces:12

Hgh = Hgh,0 ⊕ c0Hgh,0 ⊕ c0Hgh,0 ⊕ c0c0Hgh,0,

Hgh,0 := Hgh ∩ ker b0 ∩ ker b0.(23.172)

Basis states of the Hilbert space Hgh,0 are:∣∣↓↓; N bn; N c

n; N bn; N c

n⟩

=∏n≥1

(b−n)Nbn(b−n)N

bn(c−n)N

cn(c−n)N

cn | ↓↓〉 ,

N bn, N

bn, N

cn, N

cn ∈ N∗.

(23.173)

A general state of Hgh can be decomposed as

ψ = ψ↓↓ + ψ↑↓ + ψ↓↑ + ψ↑↑, (23.174)

where each state is built by acting with negative-frequency modes on the correspondingvacuum.

In terms of the second basis (23.156), the Hilbert space admits a second decomposition:

Hgh = Hgh,0 ⊕ c+0 Hgh,0 ⊕ c−0 Hgh,0 ⊕ c−0 c+0 Hgh,0,

Hgh,0 := Hgh ∩ ker b−0 ∩ ker b+0 .(23.175)

11Due to the specific structure of the inner product defined below, these subspaces are not orthonormalto each other.

12The reader should not get confused by the same symbol Hgh,0 as in the case of the holomorphic sector.

297

In view of applications to string theory, it is useful to introduce two more subspaces:

Hgh,± := Hgh ∩ ker b±0 = Hgh,0 ⊕ c∓0 Hgh,0, (23.176)

and the associated decomposition

Hgh = Hgh,± ⊕ c±0 Hgh,±. (23.177)

In off-shell closed string theory, the principal Hilbert space will be H−gh due to the level-matching condition. In this case, H−gh has the same structure asHgh in the pure holomorphicsector, and c+0 plays the same role as c0. A state in H−gh is built on top of the vacua | ↓↓〉and |+〉.

23.2.7 Euclidean and BPZ conjugatesIn order for the Virasoro operators to be Hermitian, the bn and cn must satisfy the followingconditions:

b†n = εb−n, c†n = c−n. (23.178)Hence, bn is anti-Hermitian if ε = −1. The BPZ conjugates of the modes are:

btn = (−1)λ b−n, ctn = (−1)1−λ c−n, (23.179)

using I+(z) with (22.111).In the rest of this section, we consider only the case ε = 1 and λ ∈ N. The adjoints of

the vacuum read:

| ↓〉‡ = 〈0| c1−λ · · · c−1, | ↑〉‡ = 〈0| c1−λ · · · c−1c0. (23.180)

The BPZ conjugates of the vacua are:

〈↓ | := | ↓〉t = (−1)(1−λ)2〈0| c−1 · · · c1−λ,

〈↑ | := | ↑〉t = (−1)λ(1−λ)〈0| c0c−1 · · · c1−λ.(23.181)

The signs are inconvenient but will disappear when considering both the left and right vacuatogether as in (23.154). We have the following relations:

〈↓ | = (−1)aλ+(1−λ)(2−λ) | ↓〉‡ , 〈↑ | = (−1)aλ | ↑〉‡ , (23.182)

where aλ is the zero-point energy (23.152).

Computation – Equation (23.182)To prove the relation, we can start from the BPZ conjugate 〈↓ | and reorder the modesto bring them in the same order as the adjoint:

〈↓ | = (−1)(1−λ)2+ 12 (2−λ)(1−λ) | ↓〉‡ = (−1)−aλ+(1−λ)(2−λ) | ↓〉‡

The reordering gives a factor (−1) to the power:

λ−2∑i=1

i = 12(2− λ)(1− λ) = −aλ + 1− λ.

Similarly, for the second vacuum:

〈↑ | = (−1)λ(1−λ)− 12λ(1−λ) | ↑〉‡ = (−1) 1

2λ(1−λ) | ↑〉‡ .

298

We can identify the power with (23.152).

Then, we have the following relations:

〈↑ | b0 = 〈↓ | , 〈↓ | c0 = 〈↑ | , 〈↓ | b0 = 0, 〈↑ | c0 = 0. (23.183)

There is a subtlety in defining the inner product because the vacuum is degenerate. Ifwe write the two vacua as vectors

| ↓〉 =(

01

), | ↑〉 =

(10

), (23.184)

then the zero-modes have the following matrix representation:

b0 =(

0 01 0

), c0 =

(0 10 0

). (23.185)

These matrices are not Hermitian as required by (23.178): since Hermiticity follows fromthe choice of an inner product, it means that the vacua cannot form an orthonormal basis.An appropriate choice for the inner products is:13

〈↓ | ↓〉 = 〈↑ | ↑〉 = 0,〈↑ | ↓〉 = 〈↓ | c0 | ↓〉 = 〈0| c1−λ · · · c−1c0c1 · · · cλ−1 |0〉 = 1.

(23.186)

The effect of changing the definition of the inner product or to consider a non-orthonormalbasis is represented by the insertion of c0. The last condition implies that the conjugatestate (22.144) to the SL(2,C) vacuum is:

〈0c| = 〈0| c1−λ · · · c−1c0c1 · · · cλ−1, 〈↓c | = 〈↑ | . (23.187)

23.2.8 SummaryIn this section we summarize the values of the parameters for different theories of interest(Table 23.1).

ε λ qλ cλ aλb, c (diff.) 1 2 −3 −26 −1β, γ (susy.) −1 3/2 2 11 3/8

ψ± 1 1/2 0 1 0η, ξ 1 1 −1 −2 0

Table 23.1: Summary of the first-order systems. Remember that h(b) = λ and h(c) = 1−λ.

23.3 References• Free scalar: general references [281, sec. 4.1.3, 4.3, 4.6.2, 64, sec. 5.3.1, 6.3, 31, sec. 4.2,

218, 149], topological current and winding [123, 300, sec. 17.2–3].

• First-order system: general references [31, chap. 5, sec. 13.1, 149, sec. 4.15, 218,sec. 2.5], ghost vacua [173, sec. 15.3].

13To avoid confusions, let us note that the adjoint in (23.182) are defined only through the adjoint ofthe modes (22.110) but not with respect to the inner product given here, which would lead to exchanging| ↓〉‡ ∼〈↑ | and | ↑〉‡ ∼〈↓ |.

299

Chapter 24

BRST quantization

The BRST quantization can be introduced either by following the standard QFT treatment(outlined in Section 4.2), or by translating it in the CFT language. One can then use all theCFT techniques to extract information on the spectrum, which makes this approach morepowerful. Moreover, this also provides an elegant description of states and string fields.In this chapter, we set the stage of the BRST quantization using the CFT language andwe apply it to string theory. The main results of this chapter are a proof of the no-ghosttheorem and a characterization of the BRST cohomology (physical states).

24.1 BRST for reparametrization invarianceThe BRST symmetry we are interested in results from gauge fixing the reparametrizationinvariance. In this chapter, we focus on the holomorphic sector: since both sectors areindependent, most results follow directly, except those concerning the zero-modes. Weconsider a generic matter CFT coupled to a reparametrization ghosts:

1. matter: central charge cm, energy–momentum tensor Tm and Hilbert space Hm;

2. reparametrization ghosts: bc ghost system (Sections 3.3 and 23.2) with ε = +1 andλ = 2, cgh = −26, energy–momentum tensor T gh and Hilbert space Hgh.

The formulas for the reparametrization ghosts are summarized in Appendix B.3.5. Formodes, the system (m, gh, b or c) is indicated as a superscript to not confuse it with themode index. The total central charge, energy–momentum tensor and Hilbert space aredenoted by:

c = cm + cgh = cm − 26, T (z) = Tm(z) + T gh(z), H = Hm ⊗Hgh. (24.1)

The goal is to find the physical states in the cohomology, that is, which are BRST closed

QB |ψ〉 = 0 (24.2)

but non exact (Section 4.2): the latter statement can be understood as an equivalencebetween closed states under shift by exact states:

|ψ〉 ∼ |ψ〉+QB |Λ〉 . (24.3)

We introduce the BRST current and study its CFT properties. Then, we give a compu-tation of the BRST cohomology when the matter CFT contains at least two scalar fields.

300

24.2 BRST in the CFT formalismThe BRST current can be found from (4.49) to be [218]:

jB(z) = :c(z)(Tm(z) + 1

2 Tgh(z)

): + κ ∂2c(z) (24.4a)

= c(z)Tm(z) + 12 :b(z)c(z)∂c(z): + κ ∂2c(z), (24.4b)

and similarly for the anti-holomorphic sector. This can be derived from (4.52): the generatorof infinitesimal changes of coordinates (given by the Lie derivative) is the energy–momentumtensor. The last piece is a total derivative and does not contribute to the charge: for thisreason, it cannot be derived from (4.52), its coefficient will be determined below. Note thatit is the only term of dimension 1 and of ghost number 1.

The BRST charge is then obtained by the contour integral:

QB = QB,L +QB,R, QB,L =∮ dz

2πi jB(z), QB,R =∮ dz

2πi B(z). (24.5)

As usual, QB ∼ QB,L when considering only the holomorphic sectors such that we generallyomit the index.

24.2.1 OPEThe OPE of the BRST current with T is

T (z)jB(w) ∼(cm

2 − 4− 6κ) c(w)

(z − w)4 + (3− 2κ) ∂c(w)(z − w)3 + jB(w)

(z − w)2 + ∂jB(w)z − w

. (24.6)

Hence, the BRST current is a primary operator only if

cm = 26, κ = 32 . (24.7)

The BRST current must absolutely be primary, otherwise, the BRST symmetry is anomal-ous, which means that the theory is not consistent. This provides another derivation of thecritical dimension. In this case, the OPE becomes

T (z)jB(w) ∼ jB(w)(z − w)2 + ∂jB(w)

z − w. (24.8)

Remark 24.1 (Critical dimension in 2d gravity) The value cm = 26 (critical dimen-sion) was obtained in Section 3.3 by requiring that the Liouville field decouples from thepath integral. In 2d gravity, where this condition is not necessary, (nor even desirable) theLiouville field is effectively part of the matter, such that cL + cm = 26. One can also studythe BRST cohomology in this case.

The OPE of jB(z) with the ghosts are

jB(z)b(w) ∼ 2κ(z − w)3 + j(w)

(z − w)2 + T (w)z − w

, (24.9a)

jB(z)c(w) ∼ :c(w)∂c(w):z − w

. (24.9b)

Similarly, the OPE with any matter weight h primary field φ is

jB(z)φ(w) ∼ h c(w)φ(w)(z − w)2 + :h ∂c(w)φ(w) + c(w)∂φ(w):

z − w. (24.9c)

301

Note that c transforms like a primary field (the first term vanishes since c(w)2 = 0).The OPE with the ghost current is

jB(z)j(w) ∼ 2κ+ 1(z − w)3 −

2∂c(w)(z − w)2 −

jB(w)z − w

, (24.10)

while the OPE with itself is (for κ = 3/2)

jB(z)jB(w) ∼ −cm − 182

:c(w)∂c(w):(z − w)3 − cm − 18

4:c(w)∂2c(w):

(z − w)2 − cm − 2612

:c(w)∂3c(w):z − w

.

(24.11)There is no first order pole if cm = 26: as we will see shortly, this implies that the BRSTcharge is nilpotent.

24.2.2 Mode expansionsThe mode expansion of the BRST charge can be written equivalently

QB =∑m

:cm(Lm−m + 1

2 Lgh−m

): (24.12a)

=∑m

c−mLmm + 1

2∑m,n

(n−m) :c−mc−nbm+n: (24.12b)

In the energy ordering, this expression becomes

QB =∑m

??cm

(Lm−m + 1

2 Lgh−m

)??−

c02 (24.13a)

=∑n

cmLm−m + 1

2∑m,n

(n−m) ??c−mc−nbm+n??− c0, (24.13b)

where the ordering constant is the same as in Lgh0 (as can be checked by comparing both

sides of the anticommutator). The simplest derivation of this term is to use the algebraand to ensure that it is consistent. The only ambiguity is in the second term, when one cdoes not commute with the b: this happens for −n + (m + n) = 0, such that the orderingambiguity is proportional to c0. Then, one finds that it is equal to agh = −1.

The BRST operator can be decomposed on the ghost zero-modes as

QB = c0L0 − b0M + QB (24.14a)

where

QB =∑m 6=0

c−mLmm −

12∑m,n 6=0m+n 6=0

(m− n) ??c−mc−nbm+n?? , (24.14b)

M =∑m 6=0

mc−mcm (24.14c)

The interest of this decomposition is that L0, M and Q do not contain b0 or c0, whichmake it very useful to act on states decomposed according to the zero-modes (23.169). Thenilpotency of the BRST operator implies the relations

[L0,M ] = [QB ,M ] = [QB , L0] = 0, Q2B = L0M. (24.15)

Moreover, one has Ngh(QB) = 1 and Ngh(M) = 2.

302

24.2.3 CommutatorsFrom the various OPE, one can compute the (anti-)commutators of the BRST charge withthe other operators. For the ghosts and a weight h primary field φ, one finds

QB , b(z) = T (z), (24.16a)QB , c(z) = c(z)∂c(z), (24.16b)

[QB , φ(z)] = h ∂c(z)φ(z) + c(z)∂φ(z). (24.16c)

This reproduces correctly (4.52).Two facts will be useful in string theory. First, (24.16c) is a total derivative for h = 1:

[QB , φ(z)] = ∂(c(z)φ(z)

). (24.17)

Second, c(z)φ(z) is closed if h = 1

QB , c(z)φ(z) = (1− h)c(z)∂c(z)φ(z). (24.18)

The commutator with the ghost current is

[QB , j(z)] = −jB(z), (24.19)

which confirms that the BRST charge increases the ghost number by 1

[Ngh, QB ] = QB . (24.20)

One finds that the BRST charge is nilpotent

QB , QB = 0 (24.21)

and commutes with the energy–momentum tensor

[QB , T (z)] = 0 (24.22)

only if the matter central charge corresponds to the critical dimension:

cm = 26. (24.23)

The most important commutator for the modes is

Ln = QB , bn. (24.24)

Nilpotency of QB then implies that QB commutes with Ln:

[QB , Ln] = 0. (24.25)

24.3 BRST cohomology: two flat directionsThe simplest case for studying the BRST cohomology is when the target spacetime has atleast two non-compact flat directions represented by two free scalar fields (X0, X1) (Sec-tion 23.1). The remaining matter fields are arbitrary as long as the critical dimensioncm = 26 is reached. The reason for introducing two flat directions is that the cohomology iseasily worked out by introducing light-cone (or complex) coordinates in target spacetime.

303

The field X0 can be spacelike or timelike ε0 = ±1, while we consider X1 to be alwaysspacelike, ε1 = 1. The oscillators are denoted by α0

m and α1m, and the momenta of the Fock

vacua by k‖ = (k0, k1) such that

k2‖ = ε0(k0)2 + (k1)2. (24.26)

The rest of the matter sector, called the transverse sector ⊥, is an arbitrary CFT withenergy–momentum tensor T⊥, central charge c⊥ = 24 and Hilbert space H⊥. The ghosttogether with the two scalar fields form the longitudinal sector ‖. The motivation for thenames longitudinal and transverse will become clear later: they will be identified with thelight-cone and perpendicular directions in the target spacetime (and, correspondingly, withunphysical and physical states).

The Hilbert space of the theory is decomposed as

H := H‖ ⊗H⊥, H‖ :=∫

dk0 F0(k0)⊗∫

dk1 F1(k1)⊗Hgh, (24.27)

where F0(k0) and F1(k1) are the Fock spaces (23.83a) of the scalar fields X0 and X1, andHgh is the ghost Hilbert space (23.169). As a consequence, a generic state of H reads

|ψ〉 = |ψ‖〉 ⊗ |ψ⊥〉 , (24.28)

where ψ⊥ is a generic state of the transverse matter CFT H⊥ and ψ‖ is built by acting withoscillators on the Fock vacuum of H‖:

|ψ‖〉 = cNc00∏m>0

(α0−m)N

0m(α1

−m)N1m (b−m)N

bm(c−m)N

cm |k0, k1, ↓〉

|k0, k1, ↓〉 := |k0〉 ⊗ |k1〉 ⊗ | ↓〉 , N0m, N

1m ∈ N, N b

m, Ncm = 0, 1.

(24.29)

Since the Virasoro modes commute with the ghost number, eigenstates of the Virasorooperators without zero-modes L0, given by the sum of (23.68) and (23.160), can also betaken to be eigenstates of Ngh. It is also useful to define the Hilbert space of states lying inthe kernel of b0:

H0 = H ∩ ker b0 (24.30)

such thatH = H0 ⊕ c0H0. (24.31)

The full L0 operator reads

L0 = Lm0 + Lgh0 = (Lm0 − 1) +N b +N c, (24.32)

using (23.129) for Lgh0 . A more useful expression is obtained by separating the two sectors

and by extracting the zero-modes using (23.67):

L0 =(L⊥0 −m2

‖,L`2 − 1

)+ L

‖0, (24.33)

where the longitudinal mass and total level operator are:

m2‖,L = −p2

‖,L, L‖0 = N0 +N1 +N b +N c ∈ N. (24.34)

The absolute BRST cohomology Habs(QB) defines the physical states (Section 4.2) andis given by the states ψ ∈ H that are QB-closed but not exact:

Habs(QB) :=|ψ〉 ∈ H

∣∣ QB |ψ〉 = 0,@ |χ〉 ∈ H∣∣ |ψ〉 = QB |χ〉

. (24.35)

304

Since QB commutes with L0, (24.25), the cohomology subspace is preserved under timeevolution.

Before continuing, it is useful to outline the general strategy for studying the cohomologyof a BRST operator Q in the CFT language. The idea is to find an operator ∆ – calledcontracting homotopy operator – which, if it exists, trivializes the cohomology. Conversely,this implies that the cohomology is to be found within states which are annihilated by ∆or for which ∆ is not defined. Then, it is possible to restrict Q on these subspaces: this isadvantageous when the restriction of the BRST charge on these subspaces is a simpler. Infact, we will find that the reduced operator is itself a BRST operator, for which one cansearch for another contracting homotopy operator.1

Given a BRST operator Q, a contracting homotopy operator ∆ for Q is an operator suchthat

Q,∆ = 1. (24.36)Interpreting Q as a derivative operator, ∆ corresponds to the Green function or propagator.The existence of a well-defined ∆ with empty kernel implies that the cohomology is emptybecause all closed states are exact. Indeed, consider a state |ψ〉 ∈ H which is an eigenstateof ∆ and closed QB |ψ〉 = 0. Inserting (24.36) in front of the state gives:

|ψ〉 = QB ,∆ |ψ〉 = QB(∆ |ψ〉

). (24.37)

If ∆ is well-defined on |ψ〉 and |ψ〉 /∈ ker ∆, then ∆ |ψ〉 is another state in H, which impliesthat |ψ〉 is exact. Hence, the BRST cohomology has to be found inside the subspaces ker ∆or on which ∆ is not defined.

24.3.1 Conditions on the statesIn this subsection, we apply explicitly the strategy just discussed to get conditions on thestates. A candidate contracting homotopy operator for QB is

∆ := b0L0

(24.38)

thanks to (24.24):L0 = QB , b0. (24.39)

Indeed, suppose that |ψ〉 is an eigenstate of L0, and that it is closed but not on-shell:

QB |ψ〉 = 0, L0 |ψ〉 6= 0. (24.40)

One can use (24.39) in order to write:

|ψ〉 = QB

(b0L0|ψ〉), (24.41)

Then, the operator inside the parenthesis can be identified with ∆ defined above in (24.38).The formula (24.41) breaks down if ψ is in the kernel of L0 since the inverse is not defined.This implies that a necessary condition for a L0-eigenstate |ψ〉 to be in the BRST cohomologyis to be on-shell:

L0 |ψ〉 = 0. (24.42)Considering explicitly the subset of states annihilated by b0 is not needed at this stage sinceker b0 ⊂ kerL0 for QB-closed states, according to (24.24). Hence, we conclude:

Habs(QB) ⊂ kerL0. (24.43)1A similar strategy shows that there is no open string excitation for the open SFT in the tachyon vacuum.

305

Note that this statement holds only at the level of vector spaces, i.e. when consideringequivalence classes of states |ψ〉 ∼ |ψ〉+Q |Λ〉. This means that there exists a representativestate of each equivalence class inside kerL0, but a generic state is not necessarily in kerL0.For example, consider a state |ψ〉 ∈ kerL0 and closed. Then, |ψ′〉 = |ψ〉 + QB |Λ〉 with|Λ〉 /∈ kerL0 is still in Habs(QB) but |ψ′〉 /∈ kerL0 since [L0, QB ] = 0.

Computation – Equation (24.41)For L0 |ψ〉 6= 0, one has:

|ψ〉 = L0L0|ψ〉 = 1

L0QB , b0 |ψ〉 = 1

L0QB(b0 |ψ〉

)where the fact that |ψ〉 is closed has been used to cancel the second term of the anti-commutator. Note that L0 commutes with both QB and b0 such that it can be movedfreely.

This shows that ∆ = b0/L0 given by (24.38) is not a contracting homotopy operator. Aproper definition involves the projector P0 on the kernel of L0:

|ψ〉 ∈ kerL0 : P0 |ψ〉 = |ψ〉 , |ψ〉 ∈ (kerL0)⊥ : P0 |ψ〉 = 0. (24.44)

Then, the appropriate contracting homotopy operator reads ∆(1−P0) and (24.36) is changedto:

QB ,∆(1− P0) = (1− P0). (24.45)

This parallels completely the definition of the Green function in presence of zero-modes, see(B.3).

We will revisit these aspects later from the SFT perspective. In fact, we will find that QBis the kinetic operator of the gauge invariant theory, while ∆ is the gauge fixed propagatorin the Siegel gauge. This is expected from experience with standard gauge theories: theinverse of the kinetic operator (Green function) is not defined when the gauge invariance isnot fixed.

The on-shell condition (24.42) is already a good starting point. In order to simplify theanalysis further, one can restrict the question of computing the cohomology on the subspace:

H0 := H ∩ ker b0 = Hm ⊗Hgh,0, (24.46)

where Hgh,0 = Hgh ∩ ker b0 was defined in (23.2.6). This subspace contains all states |ψ〉such that:

|ψ〉 ∈ H0 =⇒ b0 |ψ〉 = 0. (24.47)

In this subspace, there is no exact state with L0 6= 0 since such states are identically zeroaccording to (24.41):

b0 |ψ〉 = QB |ψ〉 = 0, L0 |ψ〉 6= 0 =⇒ |ψ〉 = 0. (24.48)

Note that the converse statement is not true: there are on-shell states such that b0 |ψ〉 6= 0.This also makes sense because the ghost Hilbert space can be decomposed with respect tothe ghost zero-modes. The cohomology of QB in the subspace H0 is called the relativecohomology:

Hrel(QB) := H0(QB) =|ψ〉 ∈ H0

∣∣ QB |ψ〉 = 0,@ |χ〉 ∈ H∣∣ |ψ〉 = QB |χ〉

. (24.49)

The advantage of the subspace b0 = 0 is to precisely pick the representative ofHabs whichlies in kerL0. In particular, the operator L0 is simple and has a direct physical interpretation

306

as the worldsheet Hamiltonian. This condition is also meaningful in string theory becausethese states are also mass eigenstates, which have a nice spacetime interpretation, and it willlater be interpreted in SFT as fixing the Siegel gauge. Moreover, it is implied by the choiceof ∆ in (24.38) as the contracting homotopy operator, which is particularly convenient towork with to derive the cohomology. However, there are other possible choices, which areinterpreted as different gauge fixings.

After having built this cohomology, we can look for the full cohomology by relaxing thecondition b0 = 0. In view of the structure of the ghost Hilbert space (23.169), one canexpect that Habs(QB) = Hrel(QB) ⊕ c0Hrel(QB), which is indeed the correct answer. But,we will see (building on Section 4.2.2) that, in fact, it is this cohomology which contains thephysical states in string theory, instead of the absolute cohomology.

As a summary, we are looking for QB-closed non-exact states annihilated by b0 and L0:

QB |ψ〉 = 0, L0 |ψ〉 = 0, b0 |ψ〉 = 0. (24.50)

24.3.2 Relative cohomologyIn (24.14a), the BRST operator was decomposed as:

QB = c0L0 − b0M + QB , Q2B = L0M. (24.51)

This shows that, on the subspace L0 = b0 = 0, QB is nilpotent and equivalent to QB :

|ψ〉 ∈ H0 ∩ kerL0 =⇒ QB |ψ〉 = QB |ψ〉 , Q2B |ψ〉 = 0. (24.52)

Hence, this implies that QB is a proper BRST operator and the relative cohomology of QBis isomorphic to the cohomology of QB :

H0(QB) = H0(QB). (24.53)

Next, we introduce light-cone coordinates in the target spacetime. While it breaks space-time Lorentz covariance, this is helpful mathematically because it allows to introduce agrading of the Hilbert space, for which powerful theorems exist (even if we will need onlybasic facts for our purpose).

Light-cone parametrization

The two scalar fields X0 and X1 are combined in a light-cone (if ε0 = −1) or complex (ifε0 = 1) fashion:

X±L = 1√2

(X0L ±

i√ε0X1L

). (24.54)

The modes of X± are found by following (23.50):2

α±n = 1√2

(α0n ±

i√ε0α1n

), n 6= 0, (24.55a)

x±L = 1√2

(x0L ±

i√ε0x1L

), p±L = 1√

2

(p0L ±

i√ε0p1L

), (24.55b)

The non-zero commutation relations are:

[α+m, α

−n ] = ε0mδm+n,0, [x±L , p

∓L ] = iε0. (24.56)

2For ε0 = 1, this convention matches the ones from [36] for X0 = X and X1 = φ. For ε = −1, thisconvention matches [218].

307

This implies that negative-frequency (creation) modes α±−n are canonically conjugate topositive-frequency (annihilation) modes α∓n . Note the similarity with the first-order system(23.134).

For later purposes, it is useful to note the following relations:

2 p+Lp−L = (p0

L)2 + ε0(p1L)2 = ε0 p

2‖,L, (24.57a)

x+p− + x−p+ = x0p0 + ε0 x1p1, (24.57b)∑

n

α+nα−m−n = 1

2∑n

(α0nα

0m−n + ε0 α

1nα

1m−n

). (24.57c)

In view of the commutators (24.56), the appropriate definitions of the light-cone numberN±n and level operators N± are:

N±n = ε0nα±−nα

∓n , N± =

∑n>0

nN±n . (24.58)

The insertion of ε0 follows (23.63). Then, one finds the following relation:

N+ +N− = N0 +N1. (24.59)

Using these definitions, the variables appearing in L0 (24.33)

L0 =(L⊥0 −m2

‖,L`2 − 1

)+ L

‖0 (24.60)

can be rewritten as:

m2‖,L = −2ε0 p+

Lp−L , L

‖0 = N+ +N− +N b +N c. (24.61)

The expression for the sum of the Virasoro operators (23.65) easily follows from (24.57):

L0m + L1

m = ε0∑n

:α+nα−m−n: = ε0

∑n 6=0,m

:α+nα−m−n: + ε0

(α−0 α

+m + α+

mα−m

). (24.62)

Computation – Equation (24.56)For the modes α±m, we have:

[α+m, α

±n ] = 1

2

[(α0m + i

√ε0α1m

),

(α0n ±

i√ε0α1n

)]= 1

2

([α0m, α

0n]∓ 1

ε0[α1m, α

1n])

= ε02 mδm+n,0(1∓ 1),

where we used (23.72). The other commutators follow similarly from (23.73), for ex-ample:

[x−L , p±L ] = 1

2

[(x0L −

i√ε0x1L

),

(p0L ±

i√ε0p1L

)]= 1

2([x0L, p

0L]± ε0[x1

L, p1L])

= ε02 (1± 1).

308

Computation – Equation (24.57)For the modes α±m, we have:

∑n

α+nα−m−n = 1

2∑n

(α0n + i√ε0α1n

)(α0m−n −

i√ε0α1m−n

)= 1

2∑n

(α0nα

0m−n + ε0 α

1nα

1m−n + i

√ε0

(α0m−nα

1n − α0

nα1m−n)

).

The last two terms in parenthesis cancel as can be seen by shifting the sum n→ m−nin one of the term. Note that, for m 6= 2n, there is no cross-term only after summingover n.

The relations for the zero-modes follow simply by observing that expressions in bothcoordinates can be rewritten in terms of the 2-dimensional (spacetime) flat metric.

Computation – Equation (24.59)Using (24.57), one finds:

N0 +N1 =∑n

n(N0n +N1

n

)=∑n

n(N+n +N−n

)= N+ +N−.

Reduced cohomology

In terms of the light-cone variables, the reduced BRST operator QB reads:

QB =∑m 6=0

c−m

(L⊥m + ε0

∑n

α+nα−m−n

)+ 1

2∑m,n

(n−m) :c−mc−nbm+n:. (24.63)

This operator can be further decomposed. Introducing the degree

deg := N+ −N− + N c − N b (24.64)

such that

∀m 6= 0 : deg(α+m) = deg(cm) = 1, deg(α−m) = deg(bm) = −1, (24.65)

and deg = 0 for the other variables, the operator QB is decomposed as:3

QB = Q0 +Q1 +Q2, deg(Qj) = j, (24.66a)

where

Q1 =∑m6=0

c−mL⊥m +

∑m,n 6=0m+n 6=0

??c−m

(ε0 α

+nα−m−n + 1

2 (m− n) c−mbm+n

)??,

Q0 =∑n6=0

α+0 c−nα

−n , Q2 =

∑n 6=0

α−0 c−nα+n .

(24.66b)

The nilpotency of QB implies the following conditions on the Qj :

Q20 = Q2

2 = 0, Q0, Q1 = Q1, Q2 = 0, Q21 + Q0, Q2 = 0. (24.67)

3The general idea behind this decomposition is the notion of filtration, nicely explained in [8, sec. 3, 46].

309

Hence, Q0 and Q2 are both nilpotent and define a cohomology.One can show that the cohomologies of QB and Q0 are isomorphic4

H0(QB) ' H0(Q0) (24.68)

under general conditions [36], in particular, if the cohomology is ghost-free (i.e. all stateshave Ngh = 1).

The contracting homotopy operator for Q0 is

∆ := B

L0, B := ε0

∑n 6=0

1α+

0α+−nbn. (24.69)

Indeed, it is straightforward to check that

L0 = Q0, B =⇒ Q0, ∆ = 1. (24.70)

As a consequence, a necessary condition for a closed L0-eigenstate |ψ〉 to be in thecohomology of Q0 is to be annihilated by L0:

L0 |ψ〉 = 0, =⇒ N± |ψ〉 = N c |ψ〉 = N b |ψ〉 = 0, (24.71)

since L0 is a sum of positive integers. This means that the state ψ contains no ghost orlight-cone excitations α±−n, b−n and c−n, and lies in the ground state of the Fock space H‖,0.

Then, we need to prove that this condition is sufficient: states with L0 = 0 are closed.First, note that a state |ψ〉 ∈ H0 with L0 has ghost number 1 since there are no ghostexcitations on top of the vacuum | ↓〉, which has Ngh = 1. Second, L0 and Q0 commute,such that:

0 = Q0L0 |ψ〉 = L0Q0 |ψ〉 . (24.72)

Since Q0 increases the ghost number by 1, one can invert L0 = N b + N c + · · · in the lastterm since L0 6= 0 in this subspace. This gives:

Q0 |ψ〉 = 0. (24.73)

Hence, the condition L0 |ψ〉 = 0 is sufficient for |ψ〉 to be in the cohomology. This has to becontrasted with Section 24.3.1 where the condition L0 = 0 is necessary but not sufficient.

In this case, the on-shell condition (24.33) reduces to

L0 = L⊥0 −m2‖,L`

2 − 1 = 0. (24.74)

But, additional states can be found in kerB or in a subspace of H on which B is singular.We have kerB = ker L0 such that nothing new can be found there. However, the operator Bis not defined for states with vanishing momentum α+

0 ∝ p+L = 0. In fact, one must also have

α−0 ∝ p−L = 0 (otherwise, the contracting operator for Q2 is well-defined and can be usedinstead). But, these states do not satisfy the on-shell condition (except for massless stateswith L⊥0 = 1), as it will be clear later (see [278, sec. 2.2] for more details). For this reason,we assume that states have a generic non-zero momentum and that there is no pathology.

4The role of Q0 and Q2 can be reversed by changing the sign in the definition of the degree and the roleof P±n .

310

Full relative cohomology

This section aims to construct states in H0(QB) from states in H(Q0). We follow theconstruction from [36].

Given a state |ψ0〉 ∈ H0(Q0), the state Q1 |ψ0〉 is Q0-closed since Q0 and Q1 anticommute(24.67):

Q0, Q1 |ψ0〉 = 0 =⇒ Q0(Q1 |ψ0〉

)= 0. (24.75)

Since Q1 |ψ0〉 is not in ker L0 (because Q1 increases the ghost number by 1), the state Q1 |ψ0〉is Q0-exact and can be written as Q0 of another state |ψ1〉:

Q1 |ψ0〉 =: −Q0 |ψ1〉 =⇒ |ψ1〉 = − BL0

Q1 |ψ0〉 . (24.76)

Computation – Equation (24.76)Start from the definition and insert (24.70) since L0 is invertible:

Q1 |ψ0〉 =Q0,

B

L0

Q1 |ψ0〉 = Q0

(B

L0Q1 |ψ0〉

).

The state |ψ1〉 is identified with minus the state inside the parenthesis (up to a BRSTexact state).

As for |ψ0〉, apply Q0, Q1 on ψ1:

Q0, Q1 |ψ1〉 = Q0(Q1 |ψ1〉+Q2 |ψ0〉

). (24.77)

This implies that the combination in parenthesis is Q0-closed and, for the same reason asabove, it is exact:

Q1 |ψ1〉+Q2 |ψ0〉 = Q0 |ψ2〉 , |ψ2〉 = − BL0

(Q1 |ψ1〉+Q2 |ψ0〉

). (24.78)


Q0, Q1 |ψ1〉 = Q0Q1 |ψ1〉 −Q21 |ψ0〉 = Q0Q1 |ψ1〉+ Q0, Q2 |ψ0〉 .

The first equality follows from (24.76), the second by using (24.67). The final result isobtained after using that |ψ0〉 is Q0-closed.

Iterating this procedure leads to a series of states:

|ψk+1〉 = − BL0

(Q1 |ψk〉+Q2 |ψk−1〉

). (24.79)

We claim that a state in the relative cohomology |ψ〉 ∈ H0(QB) is built by summing allthese states:

|ψ〉 =∑k∈N|ψk〉 . (24.80)

Indeed, it is easy to check that |ψ〉 is QB-closed:

QB |ψ〉 = 0. (24.81)

311

We leave aside the proof that ψ is not exact (see [36]). Note that ψ and ψ0 have the sameghost numbers

Ngh(ψ) = Ngh(ψ0) = 1 (24.82)since Ngh(BQj) = 0.

In fact, since ψ0 does not contain longitudinal modes, it is annihilated by Q1 and Q2(these operators contain either a ghost creation operator together with a light-cone annihil-ation operator, or the reverse):

Q1 |ψ0〉 = Q2 |ψ0〉 = 0. (24.83)

As a consequence, one has ψk = 0 for k ≥ 1 and ψ = ψ0.


QB |ψ〉 =∑k∈N

QB |ψk〉

= Q0 |ψ0〉+Q1 |ψ0〉+Q0 |ψ1〉︸︷︷︸=0

+Q2 |ψ0〉+Q1 |ψ1〉+Q0 |ψ2〉︸︷︷︸=0

+ · · ·

= 0.

24.3.3 Absolute cohomology, states and no-ghost theoremThe absolute cohomology is constructed from the relative cohomology:

Habs(QB) = Hrel(QB)⊕ c0Hrel(QB). (24.84)

The interested reader is refereed to [36] for the proof. A simple motivation is that the Hilbertspace is decomposed in terms of the ghost zero-modes as in (23.169). Since the zero-modescommute with Q0, linear combination of states in Hrel(QB) and c0Hrel(QB) are expectedto be in the cohomology. Obviously, one has to work out the other terms of QB and provethat there are no other states.

It looks like there is a doubling of the physical states, one built on | ↓〉 and one on | ↑〉.The remedy is to impose the condition b0 = 0 on the states (see also Section 4.2.2 and [278,sec. 2.2] for more details). As already pointed out, states in Habs form equivalence classunder |ψ〉 ∼ |ψ〉+QB |Λ〉, and it is necessary to select a single representative. This is whatthe condition b0 = 0 achieves. Obviously, it is always possible to add BRST exact states towrite another representative (for example, to restore the Lorentz covariance).

The last step is to discuss the no-ghost theorem: the latter states that there is nonegative-norm states in the BRST cohomology of string theory. This follows straightfor-wardly from the condition L0 = 0: it implies that there are no ghost and no light-coneexcitations. The ghosts and the time direction (ifX0 is timelike) are responsible for negative-norm states. Hence, the cohomology has no negative-norm states if the transverse CFT isunitary (which implies that all states in H⊥ have a positive-definite inner-product).

Physical states |ψ〉 ∈ Hrel(QB) are thus of the form:

|ψ〉 = |k0, k1, ↓〉 ⊗ |ψ⊥〉 , |ψ⊥〉 ∈ H⊥, (24.85a)(L⊥0 −m2

‖,L`2 − 1

)|ψ〉 = 0, p2

L,‖ = −m2‖,L`

2. (24.85b)

This form can be made covariant: taking a state of the form |ψ〉⊗| ↓〉 with |ψ〉 ∈ Hm, actingwith QB implies the equivalence with the old covariant quantization:

(Lm0 − 1) |ψ〉 = 0, ∀n > 0 : Lmn |ψ〉 = 0. (24.86)

312

This means that ψ must be a weight 1 primary field of the matter CFT.

Remark 24.2 (Open string) The results of this section provides, in fact, the cohomologyfor the open string after taking pL = p (instead of pL = p/2 for the closed string).

24.3.4 Cohomology for holomorphic and anti-holomorphic sectorsIt remains to generalize the computation of the cohomology when considering both theholomorphic and anti-holomorphic sectors.

In this case, the BRST operator is

QB = c0L0 − b0M + QB + c0L0 − b0M + QB . (24.87)

It is useful to rewrite this expression in terms of L±0 , b±0 and c±0 :

QB = c+0 L+0 − b

+0 M

+ + c−0 L−0 − b

−0 M

− + Q+B , (24.88)

where

L+0 =

(L⊥+

0 −m2‖`

2

2 − 2)

+ L‖+0 , L−0 = L⊥−0 + L

‖−0 (24.89)

andM± := 1

2(M ± M). (24.90)

Because of the relations L±0 = QB , b±0 , we find that states in the cohomology must beon-shell L+

0 = 0 and must satisfy the level-matching condition L−0 = 0:5

L+0 |ψ〉 = L−0 |ψ〉 = 0. (24.91)

Again, it is possible to reduce the cohomology by imposing conditions on the zero-modessuch that the above conditions are automatically satisfied (see also Section 4.2.2). Imposingfirst the condition b−0 = 0 defines the semi-relative cohomology. The relative cohomology isfound by imposing b±0 = 0 and in fact corresponds to the physical space (see [278, sec. 2.3] formore details). The rest of the derivation follows straightforwardly because the two sectorscommute: we find that the cohomology is ghost-free and has no light-cone excitations:

L‖±0 = N0 ± N0 +N1 ± N1 +N b ± N b +N c ± N c = 0. (24.92)

In general, it is simpler to work with a covariant expression and to impose the necessaryconditions. Taking a state |ψ〉 ⊗ | ↓↓〉 with |ψ〉 ∈ Hm, we find that ψ is a weight (1, 1)primary field of the matter CFT:

(Lm0 + Lm0 − 2) |ψ〉 = 0, (Lm0 − Lm0 ) |ψ〉 = 0,∀n > 0 : Lmn |ψ〉 = Lmn |ψ〉 = 0.

(24.93)

An important point is that the usual mass-shell condition k2 = −m2 is provided by the firstcondition only. This also shows that states in the cohomology naturally appears with ccinsertion since

| ↓↓〉 = c(0)c(0) |0〉 = c1c1 |0〉 . (24.94)This hints at rewriting of scattering amplitudes in terms of unintegrated states (4.28) only.

A state is said to be of level (`, ¯) and denoted as ψ`,¯ if it satisfies:

L0 |ψ`,¯〉 = ` |ψ`,¯〉 ,L0 |ψ`,¯〉 = ¯|ψ`,¯〉 . (24.95)

5In the current case, the propagator is less easily identified. We will come back on its definition later.

313

Example 24.1 – Closed string tachyonAs an example, let’s construct the state ψ0,0 with level zero for a spacetime with Dnon-compact dimensions. In this case, the transverse CFT contains D − 2 free scalarswhich combine with X0 and X1 into D scalars Xµ. The Fock space is built on thevacuum |k〉 and we define the mass such that on-shell condition reduces to the standardQFT expression:

k2 = −m2, m2 := 2`2

(N + N − 2), (24.96)

where N and N are the matter level operators. The state in the remaining transverseCFT (without the D − 2 scalars) is the SL(2,C) vacuum with L⊥0 = L⊥0 = 0 (this isthe state with the lowest energy for a unitary CFT). In this case, the on-shell conditionreads

m2`2 = −4 < 0. (24.97)

Since the mass is negative, this state is a tachyon. The vertex operator associated tothe state reads:

V (k, z, z) = c(z)c(z)eik·X(z,z). (24.98)

This also illustrates that the closed string states are product of open string states, up to theexponential term.

24.4 SummaryIn this chapter, we have described the BRST quantization from the CFT point of view. Wehave first considered only the holomorphic sector (equivalently, the open string). We provedthat the cohomology does not contain negative-norm states and we provided an explicit wayto construct the states. Finally, we glued together both sectors and characterized the BRSTcohomology of the closed string.

What is the next step? We could move to computations of on-shell string amplitudes,but this falls outside the scope of this review. We can also start to consider string fieldtheory. Indeed, the BRST equation QB |ψ〉 = 0 and the equivalence |ψ〉 ∼ |ψ〉 + QB |Λ〉completely characterize the states. In QFT, states are solutions of the linearized equationsof motion: hence, the BRST equation can provide a starting point for building the action.This is the topic of Chapter 13.

24.5 References• The general method to construct the absolute cohomology follows [36, 218]. Other

works and reviews include [22, 35, 69, 133, 134, 191, 199].

• String states are discussed in [31, sec. 3.3, 218, sec. 4.1].

314

Part VIII

Riemann surfaces

315

Chapter 25

Riemann surfaces

25.1 LiteratureAn excellent introduction to the idea of Riemann surfaces is [205, chap. 8].

316

Part IX

Mathematics

317

Chapter 26

Homotopy algebras

(Strongly) homotopy algebras are structures which naturally appear in the BV formalismand in string field theories. The three algebra of interest are the A∞ (homotopy associativealgebra), the L∞ (homotopy Lie algebra) and the open-closed homotopy algebra, appearingrespectively in the open, closed and open-closed classical SFT. In this chapter we describethe mathematical properties of these algebras: the relations with the open and closed stringtheory are described in Chapter 16.

26.1 A∞ homotopy associative algebra

26.2 L∞ homotopy Lie algebraA Lm algebra (Lie m-algebra) corresponds to a generalization of Lie algebras for which aset of m multilinear n-products `n (for n ≥ m) satisfy a set of Jacobi-like identities. A L∞is obtained by considering m ∈ N∗.

26.2.1 DefinitionConsider a Z-graded vector space

V =⊕g∈Z

Vg (26.1)

where g denotes the grading, i.e. there is a sequence of vector spaces Vg whose elementsx(g) ∈ Vg are characterized by the same integer g called the degree

deg : Vg −→ Zx 7−→ deg x = g.

(26.2)

An additional Z2-grading called the parity is defined from the Z-grading

| · | : Vg −→ Z2 = Z/2Z = 0, 1x 7−→ |x| = g mod 2.

(26.3)

See Remark 26.1 for a brief translation of the current language to the one of string theory.An arbitrary element of V is denoted by x (with possibly an index if one considers a set

of such elements) and it has a definite degree: it is an element of one of the Vg (and not

318

a sum of elements of these spaces). The vector space V can be promoted to a Z2-gradedcommutative algebra by defining a graded symmetric and associative product

xi ∧ xj = (−1)|xi||xj | xj ∧ xi, (xi ∧ xj) ∧ xk = xi ∧ (xj ∧ xk). (26.4a)

The wedge product is related to the tensor product by

x1 ∧ · · · ∧ xn =∑σ∈S\

(−1)σε(σ, xi)xσ(1) ⊗ · · · ⊗ xσ(n) ∈ V ∧n, (26.5)

where σ runs over all permutations Sn of 1, . . . , n and (−1)σ := sign(σ) = 1 (−1) if thepermutation is even (odd). Moreover the (Koszul) sign ε(σ, xi) is defined to be the signarising by rearranging the xi:

x1 ∧ · · · ∧ xn = ε(σ, xi

)xσ(1) ∧ · · · ∧ xσ(n). (26.6)

If k of the elements are identical this produces k! identical permutations (for even grading) orthis vanishes (for odd grading). The notation V ∧n denotes the symmetrized tensor productsof n copies of V .

The n-products `n are multilinear maps

`n : V ∧n −→ V

x1, . . . , xn 7−→ `n(x1, . . . , xn) := `n(x1 ∧ · · · ∧ xn).(26.7)

Additionally these products are graded symmetric: the interchange of two nearby argumentsgives a sign according to their parities

`n(x1, x2, . . .) = (−1)1+|x1||x2| `n(x2, x1, . . .), (26.8)

and a general permutation σ ∈ Sn follows from (26.6)

`n(xσ(1), . . . , xσ(n)) = (−1)σε(σ, xi

)`n(x1, . . . , xn). (26.9)

The degree of a product applied to n arguments is

deg `n(x1, . . . , xn) = n− 2 +n∑i=1

deg xi, (26.10)

such that the products have an intrinsic degree

deg `n = n− 2. (26.11)

Alternative notations for the first product are

`1(x) = Qx = dx. (26.12)

Remark 26.1 (Translation to SFT language) To help the physics oriented reader, wetranslate some of the concepts into the string field theory language (Chapter 14):

• the vector space V is the CFT Hilbert space H− ∩ kerL−0 ;

• the degree g (Z-grading) corresponds to

g = G− 1 = 1−Ngh (26.13)

where G and Ngh are the spacetime and worldsheet ghost numbers;

319

• the parity (Z2-grading) is opposite to the Grassmann parity;

• the `n products are identified to the brackets [· · · ]0 up to n-dependent signs.

Note that the matching is not completely natural, but this is merely a question of convention:to make a perfect agreement one can introduce the concept of suspension (Section 26.2.3). Inthis case one fins g = G (which motivates the notation) and the signs relating the productsand the string brackets disappear.

To define a L∞ algebra these products must satisfy a set of generalized Jacobi identities

0 = Jn(x1, . . . , xn)

:=∑i,j≥1

i+j=n+1

(−1)i(j−1)∑′

σ∈Sn

(−1)σε(σ, xi) `j(`i(xσ(1), . . . , xσ(i)), xσ(i+1), . . . , xσ(n)

)(26.14)

where the prime on the sum means that σ runs over only over permutations which keep theindices sorted in each product:

σ(1) < · · · < σ(i), σ(i+ 1) < · · · < σ(n), (26.15)

The sign ε(σ, x) was defined in (26.6). The generalized Jacobi identities can be writtenschematically as

Jn :=∑i,j≥0

i+j=n+1

(−1)i(j−1)`i`j = 0, (26.16)

where it is understood that the products are composed and that one sums over all permuta-tions of the arguments.

In order to get some feeling for these equations it is useful to write down explicitly thefirst few:

`1(`1(x)

)= 0, (26.17a)

`1(`2(x1, x2)

)= `2

(`1(x1), x2

)+ (−1)|x1| `2

(x1, `1(x2)

), (26.17b)

`2(`2(x1, x2), x3

)+ (−1)(|x1|+|x2|)|x3| `2

(`2(x3, x1), x2

)+ (−1)(|x2|+|x3|)|x1| `2

(`2(x2, x3), x1

)= `1

(`3(x1, x2, x3)

)+ `3

(`1(x1), x2, x3

)+ (−1)|x1| `3

(x1, `1(x2), x3

)+ (−1)|x1|+|x2| `3

(x1, x2, `1(x3)

).

(26.17c)

The first one means that `1 is nilpotent, and the second implies that it acts like a derivationfor the `2 product; this motivates the notations (26.12). The LHS of the third equation isrecognized to be the (super)-Jacobi identity (sometimes called the Jacobiator), and the RHSindicates that it is violated due the presence of the product `3, such that `2 is not a Liebracket. Conversely the LHS can be interpreted as the fact that `1 fails to be a derivationof the `3 product.

Remark 26.2 (Product `0) It is possible to extend the L∞ algebra by an additional product`0 which does not take any input since it is defined over V ∧0 = C. Indeed for λ ∈ C one has`0(λ) = λ `0(1), such that `0(1) ∈ V defines the product completely and it corresponds to afixed element of V . In this case the sum in (26.14) is simply extended to i, j ≥ 0. Since thiselement is absent in SFT at tree level we will not consider it.

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Example 26.1 – Lie algebraA Lie algebra corresponds to the case

`1 = `3 = 0, `2(x, y) = [x, y] (26.18)

where [·, ·] is the usual Lie bracket. The relations (26.14) imply that all the higher-orderproducts also vanish.

Given two products `m and `n their graded commutator is defined by

[`m, `n] = `m(`n ∧ Im−1)− (−1)|`m||`n|`n(`m ∧ In−1). (26.19)

Finally an inner product on V can be defined as

ω : V ∧2 −→ Cx1, x2 7−→ ω(x1, x2) := 〈x1, x2〉.

(26.20)

The definition follows from Section 26.2.3 and ensures that it will be simply related to theclosed string inner product. In particular it satisfies the properties

ω(x1, x2) = (−1)|x1||x2| ω(x2, x1), (26.21a)ω(x0, `n(x1, . . . , xn)

)= (−1)1+|x0||x1| ω(x1, `n(x0, x2, . . . , xn). (26.21b)

26.2.2 Coalgebra descriptionThe description of the products and of their relations can be unified in terms of the coalgebralanguage. The (graded) symmetric tensor algebra of V

S(V ) =∑n≥0

V ∧n (26.22)

corresponds to the algebra of (graded) symmetric tensors of any rank with the symmetricproduct defining the multiplication. Elements of S(V ) are denoted as X. A group-likeelement of S(V ) is defined as

ex =∑n≥0

1n! x

∧n. (26.23)

and the definition differs from the A∞ case because of the need to symmetrize.Given two maps A and B

A : V ∧k → V ∧`, B : V ∧m → V ∧n, (26.24)

then the tensor product of the two maps is defined by

(A ∧B)(x1 ∧ · · · ∧ xk ∧ xk+1 ∧ · · · ∧ xk+m)

=∑′

σ∈S\

(−1)σε(σ, xi

)(−1)|B|(|xσ(1)|+···+|xσ(k)|)

×A(xσ(1) ∧ · · · ∧ xσ(k)) ∧B(xσ(k+1) ∧ · · · ∧ xσ(k+m)) (26.25)

where the sum is over permutation keeping the entries ordered. The overall sign is found bycommuting B through A and each of its insertions together with all the xσ(i) with i ≤ k.Consider the identity I on V , then the identity on V ∧n is

In := I⊗ · · · ⊗ I = 1n! I ∧ · · · ∧ I, (26.26)

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where there are n products of I. In the last expression it is necessary to divide by n!in order to account for the repetition of identical factors. Note that the generalized Jacobiidentities (26.14) could be written explicitly without vector insertions thanks to the identity,for example

J3 = `1`3 + `3(`1 ∧ I2) + `2(`2 ∧ I) = 0. (26.27)We will soon arrive at an even simpler expression.

Example 26.2 – (26.26) for n = 2Consider applying (26.26) on two different entries x1 and x2:

(I ∧ I)(x1 ∧ x2) = (I⊗ I + I⊗ I)(x1 ⊗ x2 + (−1)|x1||x2|x2 ⊗ x1)= 2(x1 ⊗ x2 + (−1)|x1||x2|x2 ⊗ x1) = 2x1 ∧ x2.

The products `n can be lifted to coderivation `n on S(V )

`n : S(V ) −→ S(V )X 7−→ `n(X)

(26.28)

such that

`n(X) =∑r≥0

(`n ∧ Ir)(X) =

(`n ∧ IN−n)(X) X ∈ V ∧N , N ≥ n,0 X ∈ V ∧N , N < n.

(26.29)

The commutator [`m, `n] is a coderivation derived from [`m, `n] which means that coderiva-tions define a graded Lie algebra. The generalized Jacobi identities (26.14) can be rewrittenin terms of these commutators

Jn =∑i,j≥1

i+j=n+1

[`i, `j ] = 0. (26.30)

For exampleJ3 = [`1, `3] + 1

2 [`2, `2] = 0, (26.31)

where the factor 1/2 comes to take into account identical factors since the product insidethe commutator is symmetrized. One can try to find a recursive solution to the Jacobiequations: indeed if one knows the products `1 to `n+1, then the product `n+2 is the onlyunknown in the equation Jn+2 = 0. Let’s try the ansatz

`n+2 = 1n+ 1

n∑k=0

[`n−k+1,µk+2]. (26.32)

Hence a solution for `n+2 can be found if the dressed-products µk+2 for k ∈ 2, . . . , n canbe determined.

It is useful to gather all coderivations and dressed products in generating functions

`(t) =∑n≥0

tn`n+1 µ(t) =∑n≥0

tnµn+2. (26.33)

The L∞ relations can be summarized by

[`(t), `(t)] = 0, (26.34)

and the recursive ansatz becomes a differential equationddt `(t) = [`(t),µ(t)]. (26.35)

322

26.2.3 SuspensionThe formulation of the previous section, where the product `n has degree n− 2, is the mostnatural one to write the generalized Jacobi identity. But the application to physics is lessnatural and it is convenient to define a new degree such that all products have this degreeto be −1. The procedure which relates both descriptions is called a suspension, and the newproducts in the new one can be directly identified with the closed string products (??).

Let’s introduce new vector space Vg and X = ⊕gVg where g ∈ Z is the degree. Elementsof V are denoted with a tilde, for example x ∈ V . In this representation the products aretaken to have an intrinsic degree of −1

deg ñ = −1, (26.36)

such that the relation (26.10) becomes

deg ñ(x1, . . . , xn) = −1 +

n∑i=1

deg xi. (26.37)

The products are thus Z2-odd. The symmetry property (26.8) has also an additional sign`n(x1, x2, . . .) = (−1)|x1||x2| ˜

n(x2, x1, . . .). (26.38)Is it possible to find a map s between an element of x ∈ V and an element x ∈ V

(and thus between the products) such that the degree (26.10) and (26.37) match? A shortinspection shows that this map should have an intrinsic degree of 1 such that

deg s = 1, deg x = deg x+ 1. (26.39)This map is called a suspension and defines an isomorphism between Vg and Vg+1

s : Vg −→ Vg+1

x 7−→ x = sx.(26.40)

The suspension relates both products asñ(x1, . . . , xn) = (−1)(n−1)|x1|+(n−2)|x2|+···+|xn−1| s`n(x1, . . . , xn). (26.41)

To give explicit examples the first three products are related by˜1(x) = s `1(x), ˜2(x1, x2) = (−1)|x1|s `2(x1, x2),

˜3(x1, x2, x3) = (−1)|x2|s `3(x1, x2, x3).(26.42)

The scalar product on V is taken to agree with the one on V (and is thus denoted by thesame symbol) since it yields a scalar

ω(x1, x2) = ω(x1, x2). (26.43)This implies in particular the relations

ω(x1, x2) = (−1)(|x1|+1)(|x2|+1)ω(x2, x1), (26.44a)ω(x, `n(x1, . . . , xn)

)= (−1)|x||x1|ω(x1, `n(x, x2, . . . , xn). (26.44b)

26.3 Bibliography• L∞ algebra [29, 30, 86, 121, 122] (see [170, 171] for more mathematically oriented

references).

• A∞ algebra [87, 106, 139].

• open-closed homotopy algebra [140–142, 188–190].

323

Part X

Appendices

324

Appendix A

Conventions

Most of the review uses natural units where c = ~ = 1, but the string length `s (or Reggeslope α′) are kept.

A bar is used to denote both complex conjugation and the anti-holomorphic operators.The symbol := (resp. =:) means that the LHS (RHS) is defined by the expression in theRHS (LHS).

A.1 CoordinatesThe number of spacetime (target-space) dimensions is denoted by D = d + 1, where d isthe number of spatial dimensions. The corresponding spacetime and spatial coordinates arewritten with Greek and Latin indices:

xµ = (x0, xi), µ = 0, . . . , D − 1 = d i = 1, . . . , d (A.1)

When time is singled out, one writes x0 = t in Lorentzian signature and x0 = tE in Euclideansignature (or x0 = τ when there is no ambiguity with the worldsheet time).

A p-brane is a (p+1)-dimensional object which worldvolume is parametrized by coordin-ates:

σa = (σ0, σα), a = 0, . . . , p− 1, α = 1, . . . , p. (A.2)The time coordinate can also be singled out as σ0 = τM in Lorentzian signature and asσ0 = τ in Euclidean signature. For the string, the index α is omitted since it takes only onevalue.

The Lorentzian signature is taken to be mostly plus and the flat Minkowski metric reads

ηµν = diag(−1, 1, . . . , 1︸︷︷︸d

). (A.3)

The flat Euclidean metric isδµν = diag(1, . . . , 1︸︷︷︸

D

). (A.4)

Similar notations hold for the worldvolume metrics ηab and δab. The Levi–Civita (completelyantisymmetric) tensor is normalized by

ε01 = ε01 = 1. (A.5)

Wick rotation from Lorentzian time t to Euclidean time τ is defined by

t = −iτ (A.6)

325

(here t and τ are general symbols and should not be confused with the spacetime andworldsheet times). Accordingly, contravariant (covariant) vector transforms with the same(opposite) factor:

V 0M = −iV 0

E , VM,0 = iVE,0. (A.7)

Most computations are performed with both spacetime and worldsheet Euclidean signatures.Expressions are Wick rotated when needed.

Light-cone coordinates are defined by

x± = x0 ± x1. (A.8)

A function depending only on x+ (x−) is said to be left-moving (right-moving) by ana-logy with the displacement of a wave. Under analytic continuation, the left-moving (right-moving) coordinate is mapped to the holomorphic1 (anti-holomorphic) coordinate z (z). Inchiral theories, the left-moving value is written first.

The worldsheet coordinates (τ, σ) on the cylinder are defined by

τ ∈ R, σ ∈ [0, L), σ ∼ σ + L, (A.9)

where typically L = 2π. The integration over the spatial coordinate is normalized such thatthe perimeter of temporal section is normalized to 1 if L = 2π:

L = 12π

∫ L

0dσ = L

2π . (A.10)

This implies that 2d action, conserved charges, etc. are divided by an extra factor of 2π.The coordinates can be written in terms of complex coordinates

w = τ + iσ, w = τ − iσ (A.11)

such that the flat metric isds2 = dτ2 + dσ2 = dwdw. (A.12)

Under Wick rotation, the complex coordinates are mapped to light-cone coordinates asfollows:

w = iσ+, w = iσ−. (A.13)

The cylinder can be mapped to the complex plane through

z = e2πw/L, z = e2πw/L. (A.14)

The definition of the Levi–Civita tensor includes the √g factor, such that

εzz = i2 , εzz = −2i (A.15)

on the complex plane with flat metric.1The terms of holomorphic is simply used to indicate that the object depends only on z, but not on z.

Typically, the objects have singularities and are really meromorphic in z.2In fact, the terms of “left”- and “right”-moving are interchanged in [218, p. 34] to get agreement with

the literature. But, it means that the spatial axis orientation is reversed.Moreover, concerning [64], the first definition agrees with (6.1) but not with (6.53) since the definition of

ξ (our w) is modified in-between. This explains why the definitions of left- and right-moving [64, p. 161] donot agree with the one given in the table.

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refs cylinder plane light-cone left-movinghere, Di Francesco et al.[32, 64, 143, 147, 149, 238, 287, 300] w = τ + iσ z = ew w = iσ+, w = iσ− holomorphic

Blumenhagen et al.[15, 31, 138, 234] w = τ − iσ z = ew w = iσ−, w = iσ+ anti-holomorphic

Polchinski [198, 218, 281] w = σ + iτ z = e−iw w = −σ−, w = σ+ anti-holomorphic2

Table A.1: Conventions for the coordinates.The notations are the following (they can slightlyvary depending on the references): the Euclidean time is obtained by the analytic continu-ation τ = it (denoted also by τ = σ0 = σ2) the spatial direction is σ = σ1, and the light-conecoordinates are σ± = t± σ.

A.2 OperatorsCommutators and anti-commutators are denoted by

[A,B] := [A,B]− = AB −BA, A,B := [A,B]+ = AB +BA. (A.16)

The Grassmann parity of a field A is denoted by |A|

|A| =

+1 Grassmann odd,0 Grassmann even.

(A.17)

Two (anti-)commuting operators satisfy

AB = (−1)|A||B|BA. (A.18)

A.3 QFTEnergy is defined as the first component of the momentum vector

pµ := (E, pi). (A.19)

The following notations are used to denote the number of supersymmetries:

(NL, NR)N , N = NL +NR, (A.20)

where NL and NR are the numbers of left- and right-chirality supersymmetries. The lastform is used when it is not important to know the chirality of the supercharges. The indexmay be omitted for the first form.

The variation of a field φ(x) is defined by

δφ(x) = φ′(x)− φ(x). (A.21)

Given a symmetry with parameter αa independent of the coordinates (internal symmetry),the Noether current in Lorentzian signature is given by:

Jµa = λ∂L

∂(∂µφ)δφ

δαa, ∇µJµa = 0, (A.22)

327

where L is the Lagrangian which does not include the factor √g for curved spaces and λ issome normalization.3 The conserved charges Qa associated to the currents Jµa for a fixedspatial slice t = cst are

Qa = 1λ

∮Σ

dD−1x√hJ0

a , (A.23)

where Σ is a spatial slice and h is the induced metric. One sets λ = 2π in two dimensions,otherwise λ = 1. The variation of a field under a transformation generated by Q is

δαaφ(x) = iαa[Qa, φ(x)] (A.24)

In Euclidean signature, the current and variation are:

Jµa = iλ ∂L∂(∂µφ)

δφ

δαa, (A.25a)

δαaφ(x) = −αa[Qa, φ(x)]. (A.25b)

Note that the charge is still given by (A.23). The factor of i in (A.25a) can be understoodas follows.4 First, the time component J0

a of the current transforms like time such thatJ0a → iJ0

a , which implies that the charge also gets a factor i, Qa → iQa. This explains theminus sign in (A.25b). Then, one needs to make this consistent with the formula (B.9) forthe charge associated to a general surface. Given a spacelike nµ, the integration measureincludes the time which transforms with a factor of i: one can interpret it as coming from thespatial components of the current, J ia → iJ ia, while working with a Euclidean region. Anotherway to understand this factor for the spatial vector is by considering the electromagneticcase, where J contains a time derivative.

The term “zero-mode” has two (related) meanings:

1. given an operator D acting on a space of fields ψ(z), zero-modes ψ0,i(z) of the operatorare all fields with zero eigenvalue Dψ0,i(z) = 0, i = 1, . . . ,dim kerD

2. the zero-mode of a field expansion ψ =∑n ψnz

−n−h is the mode ψ0 for n = 0: on thecylinder, it corresponds to the constant term of the Fourier expansion on the cylinder(hence, a zero-mode of ∂z according to the previous definition)

A prime on an operation on operators (integration measure, determinant, trace. . . ) indicatesthat the zero-modes are excluded. For example, det′D is the product of non-zero eigenvalues.

A.4 Curved space and gravityThe covariant derivative is defined by

∇µ = ∂µ + Γµ (A.26)

where Γµ is the connection. For example, one has for a vector field

∇µAν = ∂µAν + Γ ν

µρ Aρ. (A.27)

The negative-definite Laplacian (or Laplace–Beltrami operator) is defined by

∆ = gµν∇µ∇ν = 1√g∇µ(√ggµν∇ν). (A.28)

3Including the √g would give the current density √gJµa . The simple derivative of the latter vanishes∂µ(√gJµa ) = 0 in view of the identity (B.4).

4We stress that these formulas and arguments do not apply to the energy–momentum tensor.

328

Note that ∇µ does not contain the Christoffel symbol for the index ν because of the identity(B.4) (but it contains a connection for any other index of the field). For a scalar field, bothderivatives become simple derivatives.

The energy–momentum tensor is defined by

Tµν = − 2λ√g

δS

δgµν, (A.29)

where λ = 2π for D = 2 and λ = 1 otherwise.

A.5 Riemann surfacesThe real and complex dimensions of a moduli spaceM are denoted by

M = dimRM, Mc = M2 = dimCM. (A.30)

For example Mg,n = 2Mcg,n = dimRMg,n.

A.6 List of symbolsGeneral:

• D: number of non-compact spacetime dimensions

• g: Riemann surface genus (number of holes / handles); loop order for a scatteringamplitude

• n: number of bulk punctures / marked points; number of external closed string states

• xµ: spacetime non-compact coordinates

• σa = (t, σ): worldsheet coordinates

• gs: closed string coupling

• Zg = Ag,0: genus-g vacuum amplitude

• Ag,n(k1, . . . , kn)α1,...,αn := Ag,n(ki)αi: g-loop n-point scattering amplitude forstates with quantum numbers ki, αi (if connected, amputated Green functions forn ≥ 3)

• Gg,n(k1, . . . , kn)α1,...,αn : g-loop n-point Green function for states with quantum num-bers ki, αi

• T⊥ab: traceless symmetric tensor or traceless component of the tensor Tab

• Ψ: generic (set of) matter field(s)

Hilbert spaces:

• H: generic Hilbert space (in general, Hilbert space of the matter plus ghost CFT)

• H± = H ∩ ker b±0• H0 = H ∩ ker b−0 ∩ ker b+0• H(QB): absolute cohomology of the operator QB inside the space H

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• H−(QB) = H(QB) ∩ H−: semi-relative cohomology of the operator QB inside thespace H

• H0(QB) = H(QB) ∩H0: relative cohomology of the operator QB inside the space H

• A: Grassmann parity of the operator or state A

Riemann surfaces:

• Σg,n: genus-g Riemann surface with n punctures

• Σg = Σg,0: genus-g Riemann surface

• Mg,n: moduli space of genus-g Riemann surfaces with n punctures

• Mg =Mg,0: moduli space of genus-g Riemann surfaces

• Mg,n = dimMg,n

• Mcg,n = dimCMg,n

• Mg = Mg,0 = dimMg = dim kerP †1• Mc

g = Mcg,0 = dimCMg

• Kg,n: conformal Killing vector group of genus-g Riemann surfaces with n punctures

• Kg = Kg,0 = kerP1: conformal Killing vector group of genus-g Riemann surfaces

• Kg,n = dimKg,n

• Kcg,n = dimCKg,n = dimC kerP1

• Kg = Kg,0 = dim kerP1

• KcgKcg,0 = dimC kerP1

• ψi: real basis of kerP1, CKV

• φi: real basis of kerP †1 , real quadratic differentials

• (ψK , ψK): complex basis of kerP1, (anti-)holomorphic CKV

• (φI , φI): complex basis of kerP †1 , (anti-)holomorphic quadratic differentials

• tλ ∈Mg,n: real moduli ofMg,n

• mΛ ∈Mg,n: complex moduli ofMg,n

• ti ∈Mg: real moduli ofMg

• mI ∈Mg: complex moduli ofMg

• z: coordinate on the Riemann surface

• wi: local coordinates around punctures

• za: local coordinates away from punctures

• fi(wi): transition functions from wi to z

• σα: coordinate system on the left of the contour Cα

330

• τα: coordinate system on the right of the contour Cα

CFT:

• Vα(k;σa) := Vk,α(σa): matter vertex operator with5 momentum k and quantum num-bers α inserted at position σa = (z, z)

• Vα(k;σa): unintegrated vertex operator with momentum k and quantum numbers αinserted at position σa

• Vα(k) =∫

d2σ√g Vα(k;σ): integrated vertex operator

• on-shell (closed bosonic string): Vα(k;σa) = ccVα(k;σa) is a (0, 0)-primary, withVα(k;σa) a (1, 1)-primary matter operator

• O: operator O with zero-modes removed

• O†: Hermitian adjoint

• O‡: Euclidean adjoint

• Ot: BPZ conjugation

• 〈O1|O2〉: BPZ inner-product

• 〈O‡1|O2〉: Hermitian inner-product

• |0〉: SL(2,C) (conformal) vacuum

• |Ω〉: energy vacuum (lowest energy state)

• :O: : conformal normal ordering (with respect to SL(2,C) vacuum |0〉)

• ??O ?

? : energy normal ordering (with respect to energy vacuum |Ω〉)

SFT:

• Ψ: closed string field

• Φ: open string field

• φr = φα(k): basis of H (or some subspace)

Indices:

• µ = 0, . . . , D − 1: non-compact spacetime dimensions

• a = 0, . . . , p: worldvolume coordinates (p = 1: worldsheet)

• i = 1, . . . , n: external states, local coordinates

• λ = 1, . . . ,Mg,n: real moduli ofMg,n

• Λ = 1, . . . ,Mcg,n: complex moduli ofMg,n

• i = 1, . . . ,Mg: real moduli ofMg

• I = 1, . . . ,Mcg: complex moduli ofMg

5When the momentum and/or quantum numbers are not relevant, we remove them or simply index theoperators by a number.

331

• i = 1, . . . ,Kg: real CKV of Kg

• K = 1, . . . ,Kcg: complex CKV of Kg

• r = (k, α): index for basis state of H (or some subspaces), α: non-momentum indices

Superstring:• Mg,m,n

• Mg,m,n

• A = 1, . . . , npco: number of PCO

332

Appendix B

Summary of important formulas

This appendix summarizes formulas which appear in the review or which are needed butassumed to be known to the reader (such as formulas from QFT and general relativity).

B.1 Complex analysisThe Cauchy–Riemann formula is∮

Cz

dw2πi

f(w)(w − z)n = f (n−1)(z)

(n− 1)! . (B.1)

One has∂

1z

= 2π δ(2)(z). (B.2)

B.2 QFT, curved spaces and gravityThe Green function G of a differential operator D is defined by

DxG(x, y) = δ(x− y)√g− P (x, y), (B.3)

where P is the projector on the zero-modes of D.The covariant divergence of a vector can be rewritten in terms of a simple derivative:

∇µvµ = 1√g∂µ(√gvµ). (B.4)

Under an infinitesimal change of coordinates

δxµ = ξµ, (B.5)

the metric transforms asδgµν = Lξgµν = ∇µξν +∇νξµ. (B.6)

Stokes’ theorem reads∫V

dDx∇µvµ =∮∂V

dΣµvµ, dΣµ := ε nµ dD−1Σ, (B.7)

333

where V is a spacetime region, S = ∂V its boundary and dD−1Σ the induced integrationmeasure. The vector nµ normal to S points outward and ε := nµn

µ = 1 (−1) if S is timelike(spacelike). If the surface is defined by x0 = cst, then

dD−1Σ = √g dD−1x, nµ = δ0µ. (B.8)

We can write a generalization of (A.23) for a charge associated to a general surface S:

QS = 1λ

∫S

dΣµ Jµa . (B.9)

Stokes’ theorem (B.7) shows that the charge vanishes QS = 0 if S is a closed surface andthat it is conserved QS1 = −QS2 for two spacelike surfaces S1 and S2 extending to infinity(if Jµa vanishes at infinity) (see [214, chap. 3, 300, sec. 8.4] for more details).

B.2.1 Two dimensionsStokes’ theorem (B.7) on flat space reads∫

d2x ∂µvµ =

∮εµν dxνvµ =

∮(v0dσ − v1dτ), (B.10)

since dΣµ = εµνdxν .The integral of the curvature is a topological invariant

χg;b := 14π

∫d2σ√g R+ 1

2π

∮ds k

= 2− 2g − b,(B.11)

called the Euler characteristics and where g is the number of holes and b the number ofboundaries.

B.3 Conformal field theoryIn two dimensions, the energy–momentum tensor is defined by

Tab = − 4π√g

δS

δgab. (B.12)

B.3.1 Complex planeDefining the real coordinates (x, y) from the complex coordinate on the complex plane

z = x+ iy, z = x− iy, (B.13)

334

we have the formulas:

ds2 = dx2 + dy2 = dzdz, gzz = 12 , gzz = gzz = 0, (B.14a)

εzz = i2 , εzz = −2i, (B.14b)

∂ := ∂z = 12 (∂x − i∂y), ∂ := ∂z = 1

2 (∂x + i∂y), (B.14c)

V z = V x + iV y, V z = V x − iV y, (B.14d)

d2x = dxdy = 12 d2z, d2z = dzdz, (B.14e)

δ(z) = 12 δ

(2)(x),∫

d2z δ(2)(z) =∫

d2x δ(2)(x), (B.14f)∫R

d2z (∂zvz + ∂zvz) = −i

∮∂R

(dz vz − dzvz

)= −2i

∮∂R

(vzdz − vzdz). (B.14g)

B.3.2 General propertiesA primary holomorphic field φ(z) of weight h transforms as

f φ(z) =(

dfdz

)hφ(f(z)

)(B.15)

for any local change of coordinates f . A quasi-primary operator transforms like this onlyfor f ∈ SL(2,C). Its mode expansion reads

φ(z) =∑n

φnzn+h , φn =

∮C0

dz2πi z

n+h−1φ(z), (B.16)

where the integration is counter-clockwise around the origin.The SL(2,C) vacuum |0〉 is defined by

∀n ≥ −h+ 1 : φn |0〉 = 0. (B.17)

Its BPZ conjugate 〈0| satisfies:

∀n ≤ h− 1 : 〈0|φn = 0. (B.18)

The state–operator correspondence associates a state |φ〉 to each operator φ(z):

|φ〉 := φ(0) |0〉 = φ−h |0〉 . (B.19)

The operator corresponding to the vacuum is the identity 1.1 The Hermitian and BPZconjugated states are

〈φ‡| := 〈0| I φ†(0) = limz→∞

z2h〈0|φ†(z), 〈φ| := 〈0| I± φ(0) = (±1)h limz→∞

z2h〈0|φ(z).(B.20)

The energy–momentum tensor is a quasi-primary operator of weight h = 2

T (z) =∑n

Lnzn+2 . (B.21)

1Exceptionally, the state |0〉 and the operator 1 does not have the same symbol.

335

The OPE between T and a primary operator h of weight h is

T (z)φ(w) ∼ hφ(w)(z − w)2 + ∂φ(w)

z − w. (B.22)

The OPE of T with itself defines the central charge c

T (z)T (w) ∼ c/2(z − w)4 + 2T (w)

(z − w)2 + ∂T (w)z − w

. (B.23)

B.3.3 Hermitian and BPZ conjugationsBoth conjugations do not change the ghost number of a state.

Hermitian

The Hermitian conjugate of a general state built from n operators Ai and a complex numberλ is

(λA1 · · ·An |0〉)† = λ∗ 〈0|A†n · · ·A†1. (B.24)

The Hermitian conjugation defines an anti-linear inner product

(A,B) = 〈A‡|B〉 , (A,B)† = −(B,A). (B.25)

It has the properties

(A,B + λC) = (A,B) + λ(A,C), (B + λC,A) = (B,A) + λ∗(C,A). (B.26)

The Hermitian adjoint of an operator can then be defined as

(A,OB) = (O†A,B). (B.27)

BPZ

The BPZ conjugate of modes is

φtn = (I± φ)n = (−1)h(±1)nφ−n, (B.28)

where I±(z) = ±1/z. The plus sign is usually used for the closed string, and the minus signfor the open string. Given a general state built from n operators n and a complex number λ,the conjugation does not change the order of the operators and does not conjugate complexnumbers:

(λA1 · · ·An |0〉)t = λ 〈0| (A1)t · · · (An)t. (B.29)However, it reverses radial ordering such that operators must be (anti-)commuted in radialordered expressions.

The BPZ product satisfies

〈A,B〉 = (−1)|A||B|〈B,A〉. (B.30)

Moreover the inner product is non-degenerate, so

∀A : 〈A|B〉 = 0 =⇒ |B〉 = 0. (B.31)

Denoting by |φr〉 a complete basis of states, then the conjugate basis 〈φcr| is definedby the BPZ product as

〈φcr|φs〉 = δrs. (B.32)We have

〈φr|φcs〉 = (−1)|φr|δrs. (B.33)

336

B.3.4 Scalar fieldThe simplest matter CFT is a set of D scalar field Xµ(z, z) such that the i∂Xµ and i∂Xµ

are of weight h = (1, 0) and h = (0, 1)

i∂Xµ =∑n

αµnzn+1 , i∂Xµ =

∑n

αµnzn+1 . (B.34)

The commutation relations between the modes are:

[αµm, ανn] = mδm+n,0ηµν , [αµm, ανn] = mδm+n,0η

µν , [αµm, ανn] = 0. (B.35)

The zero-modes of both operators are equal and correspond to the (centre-of-mass) mo-mentum

αµ0 = αµ0 =√α′

2 pµ. (B.36)

The conjugate of pµ is the centre-of-mass position xµ:

[xµ, pν ] = ηµν . (B.37)

Vertex operators are defined by

Vk(z, z) = :eik·X(z,z):, h = h = α′2k2

4 . (B.38)

The scalar vacuum |k〉 is annihilated by all positive-frequency oscillators and it is char-acterized by its eigenvalue for the zero-mode operator

pµ |k〉 = kµ |k〉 , ∀n > 0 : αµn |k〉 = 0, αµn |k〉 = 0. (B.39)

The vacuum is associated to the vertex operator Vk:

|k〉 = Vk(0, 0) |0〉 = eik·x |0〉 . (B.40)

The conjugate vacuum is

〈k| pµ = 〈k| kµ, 〈k| = |k〉† , 〈−k| = |k〉t . (B.41)

B.3.5 Reparametrization ghostsThe reparametrization ghosts are described by an anti-commuting first-order system withthe parameters (Chapter 23 and table 23.1):

ε = 1, λ = 2, cgh = −26, qgh = −3, agh = −1. (B.42)

We focus on the holomorphic sector.The b and c ghosts have weights:

h(b) = 2, h(c) = −1 (B.43)

such that the mode expansions are:

b(z) =∑n∈Z

bnzn+2 , c(z) =

∑n∈Z

cnzn−1 , (B.44a)

bn =∮ dz

2πi zn+1b(z), cn =

∮ dz2πi z

n−2c(z). (B.44b)

337

The anti-commutators between the modes bn and cn read:

bm, cn = δm+n,0, bm, bn = 0, cm, cn = 0. (B.45)

The energy–momentum tensor and the Virasoro modes are respectively:

T = −2 :b∂c:− :∂b c:, (B.46a)

Lm =∑n

(n+m

):bm−ncn: =

∑n

(2m− n) :bncm−n:. (B.46b)

The expression of the zero-mode is:

L0 = −∑n

n :bnc−n: =∑n

n :b−ncn:. (B.47)

The commutators between the Ln and the ghost modes are:

[Lm, bn] =(m− n

)bm+n, [Lm, cn] = −(2m+ n)cm+n. (B.48)

In particular, L0 commutes with the zero-modes:

[L0, b0] = 0, [L0, c0] = 0. (B.49)

The anomalous global U(1) symmetry for the ghost number Ngh is generated by theghost current:

j = −:bc:, Ngh,L =∮ dz

2πi j(z), (B.50)

such thatNgh(c) = 1, Ngh(b) = −1. (B.51)

Remember that Ngh = Ngh,L in the left sector, such that we omit the index L. The modesof the ghost current are

jm = −∑n

:bm−ncn: = −∑n

:bncm−n:, Ngh,L = j0 = −∑n

:b−ncn:. (B.52)

The commutator of the current modes with itself and with the Virasoro modes are:

[jm, jn] = mδm+n,0, [Lm, jn] = −njm+n −32 m(m+ 1)δm+n,0. (B.53)

Finally, the commutators of the ghost number operator are:

[Ngh, b(w)] = −b(w), [Ngh, c(w)] = c(w). (B.54)

The level operators N b and N c and number operators N bn and N c

n are defined as:

N b =∑n>0

nN bn, N c =

∑n>0

nN cn, (B.55a)

N bn = :b−ncn:, N c

n = :c−nbn:. (B.55b)

The commutator of the number operators with the modes are:

[N bm, b−n] = b−nδm,n, [N c

m, c−n] = c−nδm,n. (B.56)

338

The OPE between the ghosts and different currents are:

c(z)b(w) ∼ 1z − w

, b(z)c(w) ∼ 1z − w

, b(z)b(w) ∼ 0, c(z)c(w) ∼ 0, (B.57a)

T (z)b(w) ∼ 2b(w)(z − w)2 + ∂b(w)

z − w, T (z)c(w) ∼ −c(w)

(z − w)2 + ∂c(w)z − w

. (B.57b)

j(z)b(w) ∼ − b(w)z − w

, j(z)c(w) ∼ c(w)z − w

. j(z)O(w) ∼ Ngh(O) O(w)z − w

, (B.57c)

j(z)j(w) ∼ 1(z − w)2 . (B.57d)

T (z)j(w) ∼ −3(z − w)3 + j(w)

(z − w)2 + ∂j(w)z − w

. (B.57e)

any operator O(z) is defined byThe OPE (B.57e) implies that the ghost number is not conserved on a curved space:

N c −N b = 3− 3g, (B.58)

and leads to a shift between the ghost numbers on the plane and on the cylinder:

Ngh,L = N cylgh,L + 3

2 . (B.59)

The SL(2,C) vacuum |0〉 is defined by

∀n > −2 : bn |0〉 = 0, ∀n > 1 : cn |0〉 = 0. (B.60)

The mode c1 does not annihilate the vacuum and the two degenerate energy vacua are:

| ↓〉 := c1 |0〉 , | ↑〉 := c0c1 |0〉 . (B.61)

The zero-point energy of these states is:

L0 | ↓〉 = agh | ↓〉 , L0 | ↑〉 = agh | ↑〉 , agh = −1. (B.62)

The energy for the normal ordering of the different currents is:

Lm =∑n

(n− (1− λ)m

)??bm−ncn

?? + agh δm,0, (B.63a)

jm =∑n

??bm−ncn

?? + δm,0. (B.63b)

The energy–momentum and ghost current zero-modes are explicitly:

L0 =∑n

n??b−ncn

?? + agh = L0 − 1, (B.64a)

Ngh,L = j0 =∑n

??b−ncn

?? + 1 = Ngh,L + 1

2(N c

0 −N b0)− 3

2 , (B.64b)

L0 = N b +N c, Ngh,L :=∑n>0

(N cn −N b

n

). (B.64c)

Then, one can straightforwardly compute the ghost number of the vacua:

Ngh |0〉 = 0, Ngh | ↓〉 = | ↓〉 , Ngh | ↑〉 = 2 | ↑〉 . (B.65)

339

Using (B.59) allows to write the ghost numbers on the cylinder:

N cylgh | ↓〉 = −1

2 | ↓〉 , N cylgh | ↑〉 = 1

2 | ↑〉 . (B.66)

The bn and cn are Hermitian:

b†n = b−n, c†n = c−n. (B.67)

The BPZ conjugates of the modes are:

btn = (±1)nb−n, ctn = −(±1)nc−n, (B.68)

using I±(z) with (22.111).The conjugates of the vacuum read:

| ↓〉‡ = 〈0| c−1, | ↑〉‡ = 〈0| c−1c0. (B.69)

The BPZ conjugates of the vacua are:

〈↓ | := | ↓〉t = ∓〈0| c−1, 〈↑ | := | ↑〉t = ±〈0| c0c−1. (B.70)

We have the following relations:

〈↓ | = ∓ | ↓〉‡ , 〈↑ | = ∓ | ↑〉‡ . (B.71)

The ghost are normalized with

〈↑ | ↓〉 = 〈↓ | c0 | ↓〉 = 〈0| c−1c0c1 |0〉 = 1, (B.72)

which selects the minus sign in the BPZ conjugation. The conjugate of the ghost vacuum is

〈0c| = 〈0| c−1c0c1. (B.73)

Considering both the holomorphic and anti-holomorphic sectors, we introduce the com-binations:

b±n = bn ± bn, c±n = 12 (cn ± cn). (B.74)

The normalization of b±m is chosen to match the one of L±m (B.78), and the one of c±m suchthat

b+m, c+n = δm+n, b−m, c−n = δm+n. (B.75)

We have the following useful identities:

b−n b+n = 2bnbn, c−n c

+n = 1

2 cncn. (B.76)

B.4 Bosonic stringThe BPZ conjugates of the scalar and ghost modes are

(αn)t = −(±1)n α−n, (bn)t = (±1)n b−n, (cn)t = −(±1)n c−n. (B.77)

Combinations of holomorphic and anti-holomorphic modes:

L±n = Ln ± Ln, b±n = bn ± bn, c±n = 12 (cn ± cn). (B.78)

340

The closed string inner product is defined from the BPZ product by an additional inser-tion of c−0

〈A,B〉 = 〈A| c−0 |B〉 , (B.79)

while the open string inner product is equal to the BPZ product

〈A,B〉 = 〈A|B〉 . (B.80)

The vacuum for the matter and ghosts is

|k, 0〉 := |k〉 ⊗ |0〉 , |k, ↓〉 := |k〉 ⊗ | ↓〉 . (B.81)

The vacuum is normalized as

open: 〈k, ↓ | c0 |k, ↓〉 = 〈k′, 0| c−1c0c1 |k, 0〉 = (2π)Dδ(D)(k + k′), (B.82a)closed: 〈k, ↓↓ | c0c0 |k, ↓↓〉 = 〈k′, 0| c−1c−1c0c0c1c1 |k, 0〉 = (2π)Dδ(D)(k + k′), (B.82b)

B.4.1 Closed stringB.4.2 Open string

341

Appendix C

Quantum field theory

In this appendix, we gather useful information on quantum field theories. The first sectiondescribes how to compute with path integral with non-trivial measures, generalizing tech-niques from finite-dimensional integrals. Then, we summarize the important concepts fromthe BRST and BV formalisms.

C.1 Path integralsIn this section, we explain how analysis, algebra and differential geometry are generalizedto infinite-dimensional vector spaces (fields).

C.1.1 Integration measureIn order to construct a path integral for the field Φ, one needs to define a notion of distanceon the space of fields. The distance between a field Φ and a neighbouring field Φ + δΦ is

|δΦ|2 = G(Φ)(δΦ, δΦ), (C.1)

where G is the (field-dependent) metric on the field tangent space (the field dependence willbe omitted when no confusion is possible). This induces a metric on the field space itself

|Φ|2 = G(Φ)(Φ,Φ), (C.2)

from which the integration measure over the field space can be defined as

dΦ√

detG(Φ). (C.3)

Moreover, the field metric also defines an inner-product between two different elements ofthe tangent space or field space:

(δΦ1, δΦ2) = G(Φ)(δΦ1, δΦ2), (Φ1,Φ2) = G(Φ)(Φ1,Φ2). (C.4)

Remark C.1 (Metric in component form) If one has a set of spacetime fields Φa(x),then a local norm is defined by

|δΦa|2 =∫

dx ρ(x)γab(Φ(x)

)δΦa(x)δΦb(x), (C.5)

which means that the metric in component form is

Gab(x, y)(Φ) = δ(x− y)ρ(x)γab(Φ(x)

). (C.6)

342

Locality means that all fields are evaluated at the same point. On a curved space, it isnatural to write γ only in terms of the metric g and to set ρ(x) =

√det g(x), such that the

inner-product is diffeomorphism invariant.

Since a Gaussian integral is proportional to the squareroot of the operator determinant,the integration measure can be determined by considering the Gaussian integral over thetangent space: ∫

dδΦ e−G(Φ)(δΦ,δΦ) = 1√detG(Φ)

. (C.7)

Note that one needs to work on the tangent space because G(Φ) can depend on the field,which means that the integral ∫

dΦ e−G(Φ)(Φ,Φ). (C.8)

is not Gaussian.Having constructed the Gaussian measure with respect to the metric G(Φ), it is now

possible to consider the path integral of general functional F of the fields:∫dΦ√

detG(Φ)F (Φ). (C.9)

The (effective) action S(Φ) provides a natural metric on the field space by defining√

detG =e−S , or

S = −12 tr lnG(Φ). (C.10)

However, it can be simpler to work with a Gaussian measure by considering only the quad-ratic terms in S, and expanding the rest in a power series. In particular, the partitionfunction is defined from the classical action Scl by

Z =∫

dΦ e−Scl(Φ). (C.11)

Given an operator D, its adjoint D† is defined with respect to the metric as

G(δΦ, DδΦ) = G(D†δΦ, δΦ). (C.12)

The free-field measure is such that the metric on the field space is independent from thefield itself: G(X) = G0. In particular, this implies that the metric is flat and its determinantcan be absorbed in the measure, setting detG0 = 1. In this case, the measure is invariantunder shift of the field:

Φ→ Φ + ε (C.13)such that ∫

dΦ e− 12 |Φ+ε|2 =

∫dΦ e− 1

2 |Φ|2. (C.14)

This property allows to complete squares and shift integration variables (for example togenerate a perturbative expansion and to derive the propagator).

Computation – Equation (C.14)∫dΦ e− 1

2 |Φ+ε|2 =∫

dΦ det δΦδΦ

e− 12 |Φ|

2

=∫

dΦ e− 12 |Φ|

2

(C.15)

The first equality follows by setting Φ = Φ + ε, and the result (C.14) follows by theredefinition Φ = Φ.

343

C.1.2 Field redefinitionsUnder a field redefinition Φ→ Φ′, the norm and the measure are invariant:

dΦ√

detG(Φ) = dΦ√

det G(Φ), G(Φ)(δΦ, δΦ) = G(Φ)(δΦ, δΦ). (C.16)

Conversely, one can find the Jacobian J(Φ, Φ) between two coordinate systems by writing

dΦ = J(Φ, Φ)dΦ, J(Φ, Φ) =∣∣∣∣det ∂Φ

∂Φ

∣∣∣∣ =

√det G(Φ)detG(Φ) . (C.17)

If the measure of the initial field coordinate is normalized such that detG = 1, or equivalently∫dδΦ e−|δΦ|

2= 1, (C.18)

one can determine the Jacobian by performing explicitly the integral

J(Φ)−1 =∫

dδΦ e−G(δΦ,δΦ). (C.19)

Remark C.2 (Identity of the Jacobian for Φ and δΦ) The Jacobian agrees on the spaceof fields and on its tangent space. This is most simply seen by using a finite-dimensionalnotation: considering the coordinates xµ and a vector v = vµ∂µ, the Jacobian for changingthe coordinates to xµ is equivalently

J = det ∂xµ

∂xµ= det ∂v

µ

∂vµ(C.20)

since the vector transforms asvµ = vν

∂xµ

∂xν. (C.21)

In order to be more explicit, one can consider a change of variables

Φ = F (Φ), Φ = F (Φ)−1. (C.22)

where F is invertible and can contain derivatives and be non-polynomial in Φ. The variation ofboth fields are connected by

δΦ = F ′(Φ)δΦ, δΦ = F ′(F (Φ)−1)−1

δΦ, (C.23)

where F ′ is the derivative of F with respect to its argument. It is then possible to compute theJacobian of the transformation

J(Φ) =√

detG′(Φ) = detF ′(F (Φ)−1)−1 (C.24)

where one has taken detG(Φ) = 1 to simplify the analysis. This can be rewritten as an effectiveaction

J(Φ) = e−Seff(Φ), Seff(Φ) = tr lnF ′(F (Φ)−1). (C.25)

Considering a partition function with a generic action S0[Φ], one gets

Z =∫

dΦ e−S0[φ] =∫

dΦ e−S0

(F−1(Φ)

)−Seff(Φ)

. (C.26)

One can imagine using a change of variable to remove derivatives in the action in order to get apurely polynomial action, but the complication would reappear in the form of the determinant.

344

Computation – Equation (C.24)

1√det G(Φ)

=∫

dδΦ e−G(δΦ,δΦ) =∫

dδΦ e−G(δΦ,δΦ)

=∫

dδΦ exp[−G(F ′(F (Φ)−1)−1

δΦ, F ′(F (Φ)−1)−1

δΦ)]

=∫

dδΦ exp[−G(δΦ, F ′

(F (Φ)−1)−1†

F ′(F (Φ)−1)−1

δΦ)]

= 1√detG(Φ)

(detF ′

(F (Φ)−1)−1†

F ′(F (Φ)−1)−1

)−1/2.

Assuming that the operator F ′−1 F−1 and its adjoint have the same spectrum, one gets√det G(Φ) = detF ′

(F (Φ)−1)−1

. (C.27)

These manipulations look complicated for something as simple as a change of variables, but thishelps to ensure that one integrates over the correct variables.

C.1.3 Zero-modesA zero-mode Φ0 of an operator D is a field such that

DΦ0 = 0. (C.28)

In the definition of the path integral over the space of fields Φ, the measure is definedover the complete space. However, this will lead respectively to a divergent or vanishingintegral if the field is bosonic or fermionic, because the integration over the zero-modes canbe factorized from the rest of the integral. Writing the field as

Φ = Φ0 + Φ′, (Φ0,Φ′) = 0, (C.29)

where Φ′ is orthogonal to the zero-mode Φ0, a Gaussian integral of an operator D reads:

Z[D] =∫

dΦ√

detG e− 12 (Φ,DΦ) =

(∫dΦ0

)∫dΦ′ e− 1

2 (Φ′,DΦ′) (C.30)

A first solution could be to simply strip the first factor (for example, by absorbing it in thenormalization), but this is not satisfactory. In particular, the partition function with source

Z[D,J ] =∫

dΦ√

detG e− 12 (Φ,DΦ)−(J,Φ) (C.31)

will depend on the zero-modes through the sources. But, since the zero-modes are stillsingled out, it is interesting to factorize the integration

Z[D,J ] =∫

dΦ0 e−(J,Φ0)∫

dΦ′ e− 12 (Φ′,DΦ′)−(J,Φ′) (C.32)

and to understand what makes it finite. Ensuring that zero-modes are correctly insertedis an important consistency and leads to powerful arguments. Especially, this can help toguess an expression when it cannot be derived easily from first principles.

To exemplify the problem, consider the cases where there is a single constant zero-modedenoted as x (bosonic) or θ (fermionic). The integral over x is infinite:∫

dx =∞. (C.33)

345

Oppositely, the integral of a Grassmann variable θ vanishes:∫dθ = 0. (C.34)

A Grassmann integral satisfies also∫dθ θ =

∫dθ δ(θ) = 1, (C.35)

such that an integral over a zero-mode does not vanish if there one zero-mode in the integrand(due to the Grassmann nature of θ, the integrand can be at most linear). By analogy withthe fermionic case, a possibility for getting a finite bosonic integral is to insert a deltafunction: ∫

dx δ(x) = 1. (C.36)

We will see that this is exactly what happens for the ghosts and super-ghosts in (super)stringtheories.

Since kerD is generally finite-dimensional, it is interesting to decompose the zero-modeon a basis and to integrate over the coefficients in order to obtain a finite-dimensionalintegral. Writing the zero-mode as

θ0(x) = θ0iψi(x), kerD = Spanψi (C.37)

where the coefficients θ0i are constant Grassmann numbers, the change of variables θ →(θ0i, θ

′) implies:

dθ = 1√det(ψi, ψj)

dθ′n∏i=1

dθ0i, (C.38)

where n = dim kerD.Next, according to the discussion above, one can ask if it is possible to rewrite an integ-

ration over dθ′ in terms of an integration over dθ together with zero-mode insertions. Thisis indeed possible and one finds:

dθn∏i=1

θ(xi) = detψi(xj)√det(ψi, ψj)

dθ′. (C.39)

Computation – Equation (C.38)

1 =∫

dθ e−|θ|2

=∫

dθ′dθ0 e−|θ|2−|θ0|2

= J

∫dθ′∏i

dθ0i e−|θ′|2−|θ0iψi|2 = J

√det(ψi, ψj)

Computation – Equation (C.39)The simplest approach is to start with the LHS. This formula is motivated from theprevious discussion: if the integration measure contains n zero-modes, it will vanishunless there are n zero-mode insertions. Moreover, one can replace each of them by the

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complete field since only the zero-mode part can contribute:∫dθ0

n∏j=1

θ(xj) =∫

dθ0

n∏j=1

θ0(xj) = 1√det(ψi, ψj)

∫dnθ0i

n∏j=1

[θ0iψi(xj)

]= detψi(xj)√

det(ψi, ψj)

∫ ∏i

dθ0i θ0i = detψi(xj)√det(ψi, ψj)

.

The third equality follows by developing the product and ordering the θ0i: minus signsresult from anticommuting the θ0i such that one gets the determinant of the basiselements.

C.2 BRST quantizationConsider an action Sm[φi] which depends on some fields φi subject to a gauge symmetry:

δφi = εaδaφi = εaRia(φ), (C.40)

where εa are the (local) bosonic parameters, such that the action is invariant

εaδaSm = 0. (C.41)

The gauge transformations form a Lie algebra with structure coefficients f cab

[δa, δb] = f cabδc. (C.42)

It is important 1) that the algebra closes off-shell (without using the equations of motion),2) that the structure coefficients are field independent and 3) that the gauge symmetry isirreducible (each gauge parameter is independent).

Remark C.3 (Interpretation of the Ria matrices) If the φi transforms in a represent-

ation R of the gauge group, then the transformation is linear in the field

Ria(φ) = (TRa )ijφj , (C.43)

with TRa the generators in the representation R. But, in full generality, this is not the case:

for example the gauge fields Aaµ do not transform in the adjoint representation even if theycarry an adjoint index (only the field strength does), and in this case

Rbaµ = δba∂µ + f cabAbµ. (C.44)

When the fields φi form a non-linear sigma models, the Ria(φ) correspond to Killingvectors of the target manifold.

In order to fix the gauge symmetry in the path integral

Z = Ω−1gauge

∫dφi e−Sm , (C.45)

gauge fixing conditions must be imposed:

FA(φi) = 0. (C.46)

Indeed, without gauge fixing, the integration is performed over multiple identical configura-tions and the result diverges. The index A is different from the gauge index a because they

347

can refer to different representations, but for the gauge fixing to be possible they should runover as many values.

Next, ghost fields ca (fermionic) are introduced for every gauge parameter, anti-ghostsbA (fermionic) and auxiliary (Nakanishi–Laudrup) fields BA (bosonic) for every gauge con-dition. The gauge-fixing and ghost actions are then defined by

Sgh = bAca δaF

A(φi), (C.47a)Sgf = −iBAFA(φi) (C.47b)

such that the original partition function is equivalent to

Z =∫

dφi dbA dca dBA e−Stot (C.48)

whereStot = Sm + Sgf + Sgh. (C.49)

The total action is invariantδεStot = 0. (C.50)

under the (global) BRST transformations

δεφi = iε caδaφi, δεc

a = − i2 ε f

abcc

bcc, δεbA = εBA, δεBA = 0, (C.51)

where ε is an anti-commuting constant parameter. Note that the original action Sm isinvariant by itself since the transformation acts like a gauge transformation with parameterεca. The transformation of ca follows because it transforms in the adjoint representation ofthe gauge group. Direct computations show that this transformation is nilpotent

δεδε′ = 0. (C.52)

These transformations are generated by a (fermionic) charge QB called the BRST charge

δεφi = i [εQB , φi] (C.53)

and similarly for the other fields (stripping the ε outside the commutator turns it to ananticommutator if the field is fermionic). Taking the ghosts to be Hermitian leads to anHermitian charge.

An important consequence is that the two additional terms of the action can be rewrittenas a BRST exact terms

Sgf + Sgh = QB , bAFA. (C.54)

A small change in the gauge-fixing condition δF leads to a variation of the action

δS = QB , bAδFA. (C.55)

The BRST charge should commute with the Hamiltonian in order to be conserved: thisshould hold in particular when changing the gauge fixing condition

[QB , QB , bAδFA] = 0 =⇒ Q2B = 0. (C.56)

Some vocabulary is needed before proceeding further. A state |ψ〉 is said to be BRSTclosed if it is annihilated by the BRST charge

|ψ〉 closed ⇐⇒ |ψ〉 ∈ kerQB ⇐⇒ QB |ψ〉 = 0. (C.57)

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States which are in the image of QB (i.e. they can be written as QB applied on some otherstates) are said to be exact

|ψ〉 exact ⇐⇒ |ψ〉 ∈ ImQB ⇐⇒ ∃ |χ〉 : |ψ〉 = QB |χ〉 . (C.58)

The cohomology H(QB) of QB is the set of closed states which are not exact

|ψ〉 ∈ H(QB) ⇐⇒ |ψ〉 ∈ kerQB , @ |χ〉 : |ψ〉 = QB |χ〉 . (C.59)

Hence the cohomology corresponds to

H(QB) = kerQBImQB

. (C.60)

Two elements of the cohomology differing by an exact state are in the same equivalence class

|ψ〉 ' |ψ〉+QB |χ〉 . (C.61)Considering the S-matrix 〈ψf |ψi〉 between a set of physical initial states ψi and final

states ψf , a small change in the gauge-fixing condition leads to

δF 〈ψf |ψi〉 = 〈ψf | QB , bAδFA |ψi〉 (C.62)

after expanding the exponential to first order. Since the S-matrix should not depend on thegauge this implies that a physical state ψ must be BRST closed (i.e. invariant)

QB |ψ〉 = 0. (C.63)

Conversely, this implies that any state of the form QB |χ〉 cannot be physical because it isorthogonal to every physical state |ψ〉

〈ψ|QB |χ〉 = 0. (C.64)

This implies in particular that the amplitudes involving |ψ〉 and |ψ〉+QB |χ〉 are identical,and any amplitude for which an external state is exact vanishes. As a conclusion, physicalstates are in the BRST cohomology

|ψ〉 physical ⇐⇒ |ψ〉 ∈ H(QB). (C.65)

If there is a gauge where the ghosts decouple from the matter field, then the invarianceof the action and of the S-matrix under changes of the gauge fixing ensures that this state-ment holds in any gauge (but, one still need to check that the gauge preserves the othersymmetries). If such a gauge does not exist, then one needs to employ other methods toshow the desired result.

Note that BA can be integrated out by using its equations of motion

δFA

δφiBA = −δSm

δφi, (C.66)

and this modifies the BRST transformation of the anti-ghost to

δεbA = −ε(δFA

δφi

)−1δSmδφi

. (C.67)

It is also possible to introduce a term

QB , bABBMAB = iBAMABBB (C.68)

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for any constant matrix MAB . Since this is also a BRST exact term, the amplitudes arenot affected. Integrating over BA produces a Gaussian average instead of a delta functionto fix the gauge.

In the previous discussion, the BRST symmetry was assumed to originate from theFaddeev–Popov gauge fixing. But, in fact, it is possible to start directly with an action ofthe form

S[φ, b, c, B] = S0[φ] +QBΨ[φ, b, c, B] (C.69)where Ψ has ghost number −1. It can be proven that this is the most general action invariantunder the BRST transformations (C.51). This can describe gauge fixed action which cannotbe described by the Faddeev–Popov procedure: in particular, the latter yields actions whichare quadratic in the ghost fields (by definition of the Gaussian integral representation of thedeterminant), but this does not exhaust all the possibilities. For example, the backgroundfield method applied to Yang–Mills theory requires using an action quartic in the ghosts.

In this section, several hypothesis have been implicit (off-shell closure, irreducibility andconstant structure coefficients). If one of them breaks, then it is necessary to employ themore general BV formalism.

C.3 BV formalismThe Batalin–Vilkovisky (BV, or also field–antifield) formalism is the most general frameworkto quantize theories with a gauge symmetry. While the BRST formalism (Appendix C.2)is sufficient to describe simple systems, it breaks down when the structure of the gaugesymmetry is more complicated, for example in systems implying gravity. The BV formalismis required in the three following cases (which can occur simultaneously):

1. the gauge algebra is open (on-shell closure);

2. the structure coefficients depend on the fields;

3. the gauge symmetry is reducible (not all transformations are independent).

The BV formalism is also useful for standard gauge symmetries to demonstrate renormaliz-ability and to deal with anomalies.

As explained in the previous section, the ghosts and the BRST symmetry are crucial toensure the consistency of the gauge theory. The idea of the BV formalism is to put on anequal footing the physical fields and all the required auxiliary and ghost fields (before gaugefixing). The introduction of antifields – one for each of the fields – and the description of thefull quantum dynamics in terms of a quantum action (constrained by the quantum masterequation) ensure the consistency of the system. Additional benefits are the presence of a(generalized) BRST symmetry, the existence of a Poisson structure (which allows to bringconcepts from the Hamiltonian formalism), the covariance of the formalism and the simpleinterpretation of counter-terms as corrections to the classical action.

For giving a short intuition, the BV formalism can be interpreted as providing a (anti)ca-nonical structure in the Lagrangian formalism, the role of the Hamiltonian being played bythe action.

C.3.1 Properties of gauge algebraBefore explaining the BV formalism, we review the situations listed above. The classicalaction for the physical fields φi is denoted by S0[φ] and the associated equations of motionby

Fi(φ) = ∂S0∂φi

. (C.70)

350

Then, a gauge algebra is open and has field-dependent structure coefficients F cab(φ) if:

[Ta, Tb] = F cab(φ)Tc + λiabFi(φ). (C.71)

On-shell, Fi = 0 and the second term is absent, such that the algebra closes. The fieldsthemselves are constants from the point of view of the gauge algebra, but their presences inthe structure coefficients complicate the analysis of the theory. Moreover, the path integralis off-shell and for this reason one needs to take into account the last term.

Finally, the gauge algebra can be reducible: in brief, it means that there are gaugeinvariances associated to gauge parameters – and correspondingly ghosts for ghosts –, andthis recursively. Since there is one independent ghost for each generator, there are too manyghosts if the generators are not all independent, and there is a remnant gauge symmetry forthe ghost fields (in the standard Faddeev–Popov formalism, the ghosts are not subject toany gauge invariance). This originates from relations between the generators Ria: denotingby m0 the number of level-0 gauge transformations, the number of independent generatorsis rankRia. Then, the

m1 = m0 − rankRia (C.72)

relations between the generators translate into a level-1 gauge invariance of the ghosts. Thissymmetry can be gauge fixed by performing a second time the Faddeev–Popov procedure,yielding commuting ghosts. This symmetry can also be reducible, and the procedure cancontinue without end. If one finds that the gauge invariance at level n = ` is irreducible, onesays that the gauge invariance is `-reducible. If this does not happen, one defines ` = ∞.The number of generators at level n is denoted by mn.

Example C.1 – p-form gauge theoryA p-form gauge theory is written in terms of a gauge field Ap with a a gauge invariance

δAp = dλp−1. (C.73)

But, due to the nilpotency of the derivative, deformations of the gauge parametersatisfying

δλp−1 = dλp−2 (C.74)

does not translate into a gauge invariance of Ap. Similarly from this should be excludedthe transformation

δλp−2 = dλp−3, (C.75)

and so on until one reaches the case p = 0. Hence, a p-form field has a p-reduciblegauge invariance.

C.3.2 Classical BVDenoting the fields collectively as

ψr = φi, BA, bA, ca, (C.76)

the simplest BV action reads

S[ψr, ψ∗r ] = S0[φ] +QBψr ψ∗r (C.77)

with the antifieldsψ∗r = φ∗i , BA∗, bA∗, c∗a. (C.78)

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The action (C.69) is recovered by writing

ψ∗r = ∂Ψ∂ψr

. (C.79)

This indicates that the general BRST formalism could be rephrased in the BV language.But, in the same way that the BRST formalism generalizes the Faddeev–Popov formalism,it is in turn generalized by the BV formalism. Indeed, the above action is linear in theantifields: this constraint is not required and one can write more general actions. In the restof this section, we explain how this works at the level of the action (classical level) and howthe sets of fields and antifields are defined.

Consider a set of physical fields φi with the gauge invariance

δφi = εa00 Ria0

(φi). (C.80)

Then, associate a ghost field ca0 to each of the gauge parameters εa0 . If the gauge symmetryis reducible, a new gauge invariance is associated to the ghosts

δca00 = εa1

1 Ra0a1

(φi, ca0). (C.81)

This structure is recurring and the ghosts of the level-n gauge invariance are denoted by canand they satisfy

δcann = εan+1n+1 R

anan+1

(φi, ca00 , . . . , cann ). (C.82)

Thus, the set of fields is

ψr = cann n=−1,...,`, c−1 := φ. (C.83)

A ghost number is introduced

Ngh(φi) = 0, Ngh(cann ) = n+ 1, (C.84)

and the Grassmann parity of the ghosts is defined to be opposite (resp. identical) of theparity of the associated gauge parameter for even (resp. odd) n

|cn| = |εann |+ n+ 1. (C.85)

To each of these fields is associated an antifield ψ∗r of opposite parity as ψr and such thattheir ghost numbers sum to −1

Ngh(ψ∗r ) = −1−Ngh(ψr), |ψ∗r | = −|ψr|. (C.86)

The fields and antifields together are taken to define a graded symplectic structure

ω =∑r

dψr ∧ dψ∗r (C.87)

with respect to which they are conjugated to each other

(ψr, ψ∗s ) = δrs, (ψr, ψs) = 0, (ψ∗r , ψ∗s ) = 0. (C.88)

The antibracket (graded Poisson bracket) (·, ·) reads

(A,B) = ∂RA

∂ψr∂LB

∂ψ∗r− ∂RA

∂ψ∗r

∂LB

∂ψr, (C.89)

where the L and R indices indicate left and right derivatives. It is graded symmetric, whichmeans

(A,B) = −(−1)(|A|+1)(|B|+1)(B,A). (C.90)

352

It also satisfies a graded Jacobi identity and the property

Ngh((A,B)) = Ngh(A) +Ngh(B) + 1, |(A,B)| = |A|+ |B|+ 1 mod 2. (C.91)

Moreover, the antibracket acts as a derivative

(A,BC) = (A,B)C + (−1)|B|C(A,C)B. (C.92)

The dynamics of the theory is described by the (classical) master action S[ψr, ψ∗r ] whichsatisfies

Ngh(S) = 0, |S| = 0. (C.93)

In order to reproduce correctly the dynamics of the classical system without ghosts, thisaction is required to satisfy the boundary condition

S[ψr, ψ∗r = 0] = S0[φi], ∂L∂RS

∂c∗n−1,an−1∂cann

∣∣∣∣ψ∗=0

= Ran−1an . (C.94)

Indeed, if the antifields are set to zero, the ghost fields cannot appear because they all havepositive ghost numbers and it is not possible to build terms with vanishing ghost numbersfrom them.

In analogy with the Hamiltonian formalism, the master action can be used as the gen-erator of a global fermionic symmetry, and inspection will show that it corresponds to ageneralization of the BRST symmetry. Writing the generalized and classical BRST operatoras s, the transformations of the fields and antifields read

δθψr = θ sψr = −θ (S, ψr) = θ

∂RS

∂ψ∗r, (C.95a)

δθψ∗r = θ sψ∗r = −θ (S, ψ∗r ) = −θ ∂RS

∂ψr, (C.95b)

where θ is a constant Grassmann parameter. The variation of a generic functional F [ψr, ψ∗r ]is

δθF = θ sF = −θ (S, F ). (C.96)

For the BRST transformation to be a symmetry of the action, the action must satisfy theclassical master equation

(S, S) = 0. (C.97)

This equation can easily be solved by expanding S in the ghosts: the various terms can beinterpreted in terms of properties of the gauge algebra. Then, the Jacobi identity used withtwo S and an arbitrary functional gives

(S, (S, F )) = 0 (C.98)

and this implies that the transformation is nilpotent

s2 = 0. (C.99)

A classical observable O satisfiessO = 0. (C.100)

Due to the BRST symmetry, the action is not uniquely defined and the action

S′ = S + (S, δF ) (C.101)

353

also satisfies the master equation, where δF is arbitrary up to the condition Ngh(δF ) = −1.This can be interpreted as the action S in a new coordinate system (ψ′r, ψ′∗r ) with

ψ′ = ψ − δF

δψ∗, ψ′∗ = ψ∗ + δF

δψ(C.102)

such thatS′[ψ,ψ∗] = S

[ψ − δF

δψ∗, ψ∗ + δF

δψ

]. (C.103)

Indeed, for F = F [ψ,ψ∗], one has

S′[ψ,ψ∗] = S[ψ,ψ∗] + (S, ψ)δFδψ

+ (S, ψ∗) δFδψ∗

= S[ψ,ψ∗]− ∂RS

∂ψ∗δF

δψ+ ∂RS

∂ψ

δF

δψ∗. (C.104)

It can be shown that this transformation preserves the antibracket and the master equation

(ψ′r, ψ′∗s ) = δrs, (S′, S′) = 0. (C.105)

More generally, any transformation preserving the antibracket is called an (anti)canonicaltransformation. One can also consider generating functions depending on both the old andnew coordinates, as is standard in the Hamiltonian formalism. Under a transformation, anyobject depending on the coordinates changes as

G′ = G+ (δF,G). (C.106)

One can consider finite transformation without problems.In order to perform the gauge fixing, one needs to eliminate the antifields. A convenient

condition isSΨ[ψr] = S

[ψr,

∂Ψ∂ψr

], ψ∗r = ∂Ψ

∂ψr, (C.107)

where Ψ[ψr] is called the gauge fixing fermion and satisfies

Ngh(Ψ) = −1, |Ψ| = 1. (C.108)

From the discussion on coordinate transformations this amounts to work in new coordinateswhere ψ′∗r = 0. But such a function Ψ cannot be built from the fields because they all havepositive ghost numbers. One needs to introduce trivial pairs of fields.

A trivial pair (B, c) is defined by the properties

|B| = −|c|, Ngh(B) = Ngh(c) + 1, (C.109a)sc = B, sB = 0 (C.109b)

and the new action readsS = S[ψr, ψ∗r ]−Bc∗ (C.110)

(the position dependence is kept implicit). In this context ψr and ψ∗r are sometimes calledminimal variables. From this, one learns that

(S, S) = (S, S) = 0. (C.111)

At level-0, one introduces the pair

(B0a0 , c0a0) := (B00a0, c00a0

) (C.112)

354

and the associated antifields. The field c0 := b is the Faddeev–Popov anti-ghost associatedto c0 and the trivial pair satisfies

|B0| = |ε0|, |c0| = −|ε0|, Ngh(B0) = 0, Ngh(c0) = −1. (C.113)

For the level 1, two additional pairs are introduced:

(B01a1, c01a1

), (B1a11 , c1a1

1 ) (C.114)

and the corresponding antifields. The motivation for adding an additional pair is that thelevel-0 pair only fixes m0 −m1 of the generators: the additional m1 extra-ghosts c1a1

1 canbe fixed by the residual level-0 symmetry. The first level-1 pair fixes the level-1 symmetry.

Then, the gauge fixed action enjoys a BRST symmetry acting only on the fields

δθψr = θ sψr = θ

∂RS

∂ψ∗r

∣∣∣∣ψ∗r=∂rΨ

. (C.115)

Note that this BRST operator is generically nilpotent only on-shell

s2 ∝ eom. (C.116)

C.3.3 Quantum BVAt the quantum level, one considers the path integral

Z =∫

dψrdψ∗r e−W [ψr,ψ∗r ]/~ (C.117)

where W is called the quantum master action. The reason for distinguishing it from theclassical master action S is that the measure is not necessarily invariant by itself underthe generalized BRST transformation – this translates into a non-gauge invariance of themeasure of the physical fields, i.e. a gauge anomaly.

Quantum BRST transformation are generated by the quantum BRST operator σ

δθF = θ σF = (W,F )− ~∆F, (C.118)

where∆ = ∂R

∂ψ∗r

∂L∂ψr

. (C.119)

Then, the path integral is invariant if W satisfies the quantum master equation

(W,W )− 2~∆W = 0, (C.120)

which can also be written as∆e−W/~ = 0. (C.121)

This can be interpreted as the invariance of Z under changes of coordinates: indeed onefinds that

δW = 12(W,W ), (C.122)

and the integration measure picks a Jacobian

sdet J ∼ 1 + ∆W. (C.123)

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In the limit ~ → 0, one recovers the classical master equation. More generally, the actioncan be expanded in powers of ~

W = S +∑p≥1

~pWp. (C.124)

Observables are given by operators O[ψ,ψ∗] invariant under σ:

σO = 0, (C.125)

which ensures that the expectation value is invariant under changes of Ψ

δ〈O〉 = 0. (C.126)

Note that if O depends just on ψ the condition reduces to sO = 0, but generically there isno such operators (except constants) satisfying this condition for open algebra.

Consider the gauge fixed integral

Z =∫

dψr eWΨ[ψr], WΨ[ψr] = W

[ψr,

∂Ψ∂ψr

]. (C.127)

Varying the gauge fixing fermion by δΨ gives

Z =∫

dψr eWΨ[ψr](∂RS

∂ψ∗r

)ψ∗=∂ψΨ

∂(δΨ)∂ψr

. (C.128)

Integrating by part gives the quantum master equation.

C.4 References• Manipulations of functional integral are given in [115, sec. 15.1, 22.1, 193, chap. 14,

215, 63].

• Zero-modes are discussed in [26].

• A general summary of path integrals for bosonic and fermionic fields can be foundin [218, app. A].

• BRST formalism: most QFT books contain an introduction, more complete referencesare [286, chap. 15, 282, 119];

• BV formalism [286, chap. 15, 103, 109, 282, 119, 59] (several explicit examples aregiven in [109, sec. 3], see [13, 261, 289] for more specific details).

356

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Index

#, 122, 1251PI vertex, 157, 198

region, 1271PR region, 1272d gravity, 79, 301

AdS/CFT, 205algebra

L∞, 320generalized Jacobi, 320

background independence, 204background metric, 59Batalin–Vilkovisky formalism, 196, 350

antibracket, 352BRST transformation, 353classical master equation, 353field redefinition, 353fields and antifields, 351gauge fixing, 354quantum BRST transformation, 355quantum master equation, 355

Beltrami differential, 69Berkovits’ super-SFT, 223bosonic string

fermions, 34bosonic string CFT, 303

L0, 304, 308complex parametrization, 307Hilbert space, 304level operator, 304, 308light-cone parametrization, 307

boundary condition, 262Neveu–Schwarz (NS), 263Ramond (R), 263twisted, 263untwisted, 263

BPZ conjugation, 262mode, 263state, 266

BRST cohomology, 97, 300

absolute, 97, 304, 312relative, 97, 306, 310, 311, 313semi-relative, 97, 313two flat directions, 303–314

BRST current, 301OPE, 301–302

BRST operator, 301commutator, 303fullzero-mode decomposition, 313

mode expansion, 302zero-mode decomposition, 302

BRST quantizationchange of gauge fixing condition, 348,

349charge nilpotency, 348cohomology, 349Nakanishi–Lautrup auxiliary field, 348physical states, 349transformations, 348

Casimir energy, 271central charge, 61, 260CKV, see conformal Killing vectorclassical solution

marginal deformation, 208closed string

dilaton Φ, 33Kalb–Ramond Bµν , 33massless states, 33metric Gµν , 33

closed string amplitudeFeynman diagram decomposition, 150on punctured moduli space, 114tree-level3-point, 1064-point, 108, 135

closed string fundamental vertex, 149g-loop0-point, 156

1-loop

375

0-point, 1561-point, 153

properties, 158recursive construction, 152special vertices, 156tree-level0-point, 1561-point, 1562-point, 156, 1943-point, 1074-point, 110

closed string product, 158g = 0, n = 1, 192, 195ghost number, 158

closed string statestachyon T

physical, 314zero-mode decomposition, 112

closed string vertexfundamental identity, 158

coderivation, 322commutator

[Lm, Ln], 255[Lm, φn], 264CFT, 258

complex coordinates, 100, 247cylinder, 248

conformaldimension, 253spin, 253weight, 253

conformal algebra, 243conformal field theory

classical, 249cylinder, 270definition, 253finite transformation, 254

conformal isometry group, 242conformal Killing

equation, 72, 243group volume, 74vector, 72, 90, 243

conformal structure, 59conjugate state, 267conserved charge, 264, 328, 334

CFT, 258conserved current, 327

CFT, 258mode expansion, 264

contracting homotopy operator, 305, 310contraction, 260conventions

complex coordinates, 327ghost number, 295spacetime momentum current, 274

correlation function, 256quasi-primary operator, 256sphere 1-point (-), 256sphere 2-point (-), 256sphere 3-point (-), 256

covariant SFTbackground independence, 205closed bosonic1PI action, 1981PI gauge symmetry, 199classical action, 194classical equation of motion, 195classical gauge algebra, 195classical gauge transformation, 195fields and antifields, 197gauge fixed action, 192, 193inner product, 185normalization, 193quantum action, 197quantum BV master equation, 197

open bosonicinner product, 173

parameters, 193renormalization, 193

critical dimension, 31, 35, 76, 105, 301, 303superstring, 212

degeneration limit, 126diffeomorphism, 58

group volume, 67, 72, 74infinitesimal (-), 58large (-), 58

dual state, see conjugate state

ε (scalar action sign), 272ηξ ghosts, 213einbein, 53energy vacuum, 265energy–momentum tensor

finite transformation, 261mode expansion, 264

Euclidean adjoint, 261–262mode, 263state, 266

Euler characteristics, 59, 86extended complex plane C, 246

Faddeev–Popovdeterminant, 72

376

field spaceDeWitt metric, 63inner-product, 62, 342norm, 62, 342

field theory space, 205connection, 206Hilbert space bundle, 206

first-order CFT, 284L0, 290, 294U(1) ghost current, 286, 295

mode expansion, 290transformation law, 288

U(1) symmetry, 285, 286action, 285boundary condition, 289BPZ conjugate

modes, 298vacuum, 298

central charge, 287commutator, 291complex components, 285cylinder, 288energy normal ordering, 294energy–momentum tensor, 285mode expansion, 290

equation of motion, 285Euclidean adjointmodes, 298vacuum, 298

Fock space, 297ghost charge, 287ghost number, 286, 295cylinder, 288

Hilbert spacefull, 297holomorphic, 297

inner product, 299level operator, 290mode expansion, 289number operator, 290OPE, 287–289propagator, 286summary, 299vacuum

SL(2,C), 292conjugate, 299energy, 293–294

Virasoro operators, 290, 295weight, 285zero-mode decomposition, 297zero-point energy, 293

free covariant SFT

closed bosonic, 185action, 194classical action, 185equation of motion, 185gauge fixed action, 186, 190gauge fixed equation of motion, 176,

186, 193gauge transformation, 186

open bosonicBV action, 181classical action, 173, 174equation of motion, 172gauge fixed action, 176gauge transformation, 174, 182zero-mode decomposition, 173

free SFT, 306free super-SFT

action, 224, 225gauge transformation, 224, 225

fundamental vertexinterpretation, 193region, 126

ghost number, 80anomaly, 139, 217

gluing compatibility, 125Green function, 87–89, 150

tree-level 2-point, 156group

fundamental domain, 67volume, 66

GSO symmetry, 214

Hermitian adjoint, 261higher-genus Riemann surface

conformal group, 254Hilbert space

CFT, 265holomorphic factorization, 105, 218holomorphic/anti-holomorphic sectors, 249,

326

iε-prescription, 230index

amplitude, 149Riemann surface, 129, 149

inversion map, 253, 262isometry group, 243

Killing vector, 243

L±, 264L∞ algebra, 196, 222

377

left/right-moving sectors, 249, 326level-matching condition, 32, 97, 112, 119,

138, 140, 151, 185, 190, 224, 313light-cone coordinates, 248Liouville

action, 75central charge, 76field, 59, 75free-field measure, 68theory, 232, 253, 255

local coordinates, 106, 114constraints, 119, 125global phase, 119, 137, 139global rescaling, 130reparametrization, 119, 137, 138transition function, 119, 120

logarithmic CFT, 256

marginal deformationaction, 205, 206correlation function, 206

metricgaugeconformal (-), 59conformally flat (-), 60, 100flat (-), 60

gauge decomposition, 67, 69, 71, 78gauge fixing, 59, 65uniformization gauge, 60

mode expansion, 262Hermiticity, 263mode range, 262

modular group, 58, 66moduli space, 65

complex coordinates, 104plumbing fixture decomposition, 125–129with punctures, 113

momentum-space SFTaction, 229consistency, 232Feynman rules, 229finitenessinfinite number of states, 157, 229UV divergence, 157, 229

Green function, 230interaction vertex, 229properties, 228string field expansion, 228

non-locality, 14, 228normal ordering, 268

conformal (-), 268

energy (-), 268mode relation, 270

off-shell closed string amplitude, 116, 134contribution from subspace, 134tree-level3-point, 1074-point, 153

off-shell string amplitude, 113conformal invariance, 107

off-shell superstring amplitude, 217consistency, 217, 219factorization, 218–219PCO insertions, 217supermoduli space, 218

old covariant quantization (OCQ), 312on-shell condition, 32, 87, 98, 257, 305, 310,

312, 313open string states

zero-mode decomposition, 111operator product expansion, 259

GG, 213T -primary, 260TG, 213TT , 260identity-primary, 260

out-of-Siegel gauge constraint, 176

p-brane, 13, 17, 27classification, 17

P1, 69, 70path integral

Faddeev–Popov gauge fixing, 64field redefinition, 344measure, 62, 343free-field, 62, 343ultralocality, 62, 77

Pg,n, 119Pg,n space, 116

p-form, 132, 133BRST identity, 138properties, 136–139

0-form, 1321-form, 133coordinates, 120section, 116, 149vector, 120

Pg,m,n space, 217p-form, 2171-form, 217

picture changing operator, 223picture number, 214

378

anomaly, 214, 217plumbing fixture, 122, 143, 218

p-form, 144ghost 1-form, 144moduli, 122non-separating, 125, 147separating, 122–125, 143, 157vector field, 144

point-particle, 25Polyakov path integral, 57

complex representation, 103Faddeev–Popov gauge fixing, 64, 89

primary operator/state, 253finite transformation, 254weight-0, 256

projectoron-shell, kerL0, 306

propagator, 305closed bosonic, 147, 150–152with stub, 157

closed string, 109NS sector, 218open bosonic, 175R sector, 219Schwinger parametrization, 43, 151superstring, 223

propagator region, 126punctured Riemann surface, 113

Euler characteristics, 113parametrization, 117–118

quadratic differential, 70quasi-primary operator/state, 253

radial quantization, 257reparametrization bc ghost

zero-mode, 97reparametrization bc ghosts, 79

action, 79, 104central charge, 75equation of motion, 80zero-mode, 82

Riemann sphere S2, 246complex plane map, 246cylinder map, 249

Riemann surfacegenus, 57moduli space, 42puncture, 41, 86

scalar field CFTL0, 281

U(1) current, 273, 274U(1) symmetry, 273action, 272, 274boundary conditionperiodic, 280

BPZ conjugatemodes, 284vacuum, 284

central charge, 276commutator, 281–282complex components, 274complex plane, 274cylinder, 275dual position, 279energy–momentum tensor, 272, 275equation of motion, 272, 274Euclidean adjointmodes, 284vacuum, 284

Fock space, 283Hilbert space, 283inner product, 284level operator, 280mode expansion, 32, 279–280momentum, 273, 275, 279, 280normal ordering, 279number operator, 280OPE, 276–278periodic boundary condition, 272position (center of mass), 279propagator, 272topological current, 273, 275vacuum, 282–283conjugate, 284

vertex operator, 275Virasoro operators, 281winding number, 274, 275, 279, 280zero-mode, 279–280

scattering amplitude, 87–89divergence, 23tree-level 2-point (-), 88

Schwarzian derivative, 261section of Pg,n

generalized, 159overlap, 159

Siegel gauge, 175, 183, 186, 307SL(2,C) group, 252SL(2,C) vacuum, 265spacetime description, 17spacetime ghost number

closed string, 189open string, 178

379

spacetime level-truncated actionopen bosonic, 182gauge transformation, 183

spacetime momentum, 274spurious pole, 23, 217, 219–222state (CFT)

bra, 266ket, 265

state–operator correspondence, 265Stokes’ theorem, 103string

composite (-), 16gauge group, 35interactions, 14, 36massive, 30multi-string model, 31orientation, 35parameters, 29, 42properties, 29rotating solution, 30

string amplitudeCKV gauge fixing, 90conformal invariance, 106divergence, 43factorization, 109, 143, 149ghost number, 139g-loop vacuum (-), 62, 78, 80, 82, 83, 104matching QFT, 87–88, 92normalization, 84, 86properties, 139–142pure gauge states decoupling, 141section independence, 141tree-level 2-point (-), 92–95

string amplitudegng-loop n-point (-), 87, 91, 92

string coupling constant, 42string Feynman diagram, 107

1PR diagram, 146, 157change of stub parameter, 157Feynman rules, 148intermediate statesghost number, 147, 148momentum, 147, 148

IR divergence, 193loop diagram, 148

string fieldclosed bosonic, 185classical, 194expansion, 188, 197quantum, 197

expansion, 168Fourier expansion, 46

functional, 166ket representation, 167momentum expansion, 167, 228open bosonic, 172classical, 173expansion, 177Nakanishi–Lautrup auxiliary field, 183parity, 173quantum, 178reality condition, 173

position representation, 167string field path integral

Faddeev–Popov gauge fixing, 179free covariant open bosonic string, 179

string field theory, 45construction, 45

string states, 32, 110–113closed stringtachyon T , 33

dual, 111Fourier expansion, 33gauge invariance, 33off-shell, 110open stringgauge field Aµ, 33tachyon T , 33

resolution of identity, 111string theory

background independence, 204consistency, 92, 232motivations, 13–16

structure constant (CFT), 256stub, 129–130

Feynman diagram, 157parameter, 129

stub parameter, 194, 229super-SFT, 222superconformal βγ ghosts, 213

bosonization, 213conformal weights, 213energy–momentum tensor, 213OPE, 213

superstring, 34, 212N = (2, 0), 35N = (2, 1) heterotic, 35N = (4, 0), 35BRST current, 214heterotic (-), 35, 212Hilbert space, 215large, 215picture number, 215small, 215

380

motivations, 34normalization, 215type I, 35type II (-), 35

superstring fieldauxiliary field, 224constrained Ramond field, 223large Hilbert space, 225Ramond field, 222small Hilbert space, 223

supersymmetry, 34surface state, 135, 144symmetry

Weyl, 244

T-duality, 280tachyon, 33

instability, 34Teichmüller deformation, 67, 69, 70Teichmüller space, 65

vector spacesuspension, 323

Verma module, 267vertex operator

integrated, 86unintegrated, 92Weyl invariance, 89

vertex state, 158vertical integration, 220Virasoro algebra, 255Virasoro operators, 255, 264

Hermiticity, 264

Weylanomaly, 63, 75, 77ghost, 80, 96group volume, 67symmetry, 59, 81

Wick rotation, 230, 325generalized, 231

Wick theorem, 268Witt algebra, 249worldsheet

actionEinstein–Hilbert (-), 84gauge-fixing (-), 96Nambu–Goto (-), 57Polyakov (-), 57sigma model (-), 58

boundary conditions, 31CFT, 31, 34

classification, 31cosmological constant, 77cylinder, 248energy–momentum tensor, 61trace, 61

Nakanishi–Lautrup auxiliary field, 96Riemann surface, 37, 57symmetry, 58background diffeomorphisms, 78background Weyl, 78BRST, 96diffeomorphisms, 58Weyl, 58

worldsheet metric, 57worldvolume description, 15, 17

parametrization, 18

Zamolodchikov metric, 256zero-mode, 328, 345zero-point energy, 266

381

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