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Jørgen E. Andersen Hans U. Boden Atle Hahn Benjamin Himpel Editors

Chern-Simons Gauge Theory: 20 Years After

Chern-Simons Gauge Theory: 20 Years After

Chern-Simons Gauge Theory: 20 Years After

Jørgen E. AndersenHans U. Boden Atle Hahn Benjamin Himpel Editors

Studies inAdvancedMathematics

AMS/IP

Volume 50

American Mathematical Society • International Press

https://doi.org/10.1090/amsip/050

Shing-Tung Yau, General Editor

2000 Mathematics Subject Classification. Primary 11F23, 14E20, 16S10, 19L10, 20C20,20F99, 20L05, 30F60, 32G15, 46E25, 53C50, 53D20, 53D99, 54C40, 55R80, 57M25,57M27, 57M60, 57M50, 57N05, 57N10, 57R56, 58D27, 58D30, 58E09, 58J28, 58J30,

58J52, 58Z05, 70S15, 81T08, 81T13, 81T25, 81T30, 81T45, 83C80.

Library of Congress Cataloging-in-Publication Data

Chern-Simons gauge theory: 20 years after / Jørgen E. Andersen . . . [et al.], editors.p. cm. (AMS/IP studies in advanced mathematics ; v. 50)

Proceedings of a workshop held at the Max Planck Institute for Mathematics in Bonn,Aug. 3–7, 2009.

Includes bibliographical references.ISBN 978-0-8218-5353-5 (alk. paper)1. Number theory—Congresses. 2. Algebraic topology—Congresses. 3. Associative rings—

Congresses. 4. K-theory—Congresses. 5. Group theory—Congresses. I. Andersen, Jørgen E.1965–

QA241.C6355 2010514′.74—dc22

2011012166

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Contents

Preface ix

Remarks on Wilson Loops and Seifert Loops in Chern-Simons Theory 1Chris Beasley

Quantum Field Theory and the Volume Conjecture 19Tudor Dimofte and Sergei Gukov

Computational Aspects in Reidemeister Torsion and Chern–Simons Theories 43Jerome Dubois

Functional Integration and Abelian Link Invariants 65Enore Guadagnini

Chern-Simons Invariants, SO(3) Instantons, and Z/2 Homology Cobordism 83Matthew Hedden and Paul Kirk

Extending the SU(3) Casson Invariant to Rational Homology 3-Spheres 115Christopher M. Herald

Decomposition of Witten–Reshetikhin–Turaev Invariant:Linking Pairing and Modular Forms 131Kazuhiro Hikami

Representations and the Colored Jones Polynomial of a Torus Knot 153Kazuhiro Hikami and Hitoshi Murakami

Eta-Invariants and Anomalies in U(1) Chern-Simons Theory 173Lisa Jeffrey and Brendan McLellan

Delta-Groupoids and Ideal Triangulations 201Rinat M. Kashaev

Invariants of Knots and 3–Manifolds Derived from theEquivariant Linking Pairing 217Christine Lescop

Chern–Simons Theory, the 1/N Expansion, and String Theory 243Marcos Marino

v

vi Contents

Global Lorentzian Geometry from Lightlike Geodesics:What Does an Observer in (2+1)-Gravity See? 261Catherine Meusburger

Spin Foam State Sums and Chern-Simons Theory 277Aleksandar Mikovic and Joao Faria Martins

Representations of the Ptolemy Groupoid, Johnson Homomorphisms,and Finite Type Invariants 285Robert C. Penner

Yang-Mills in Two Dimensions and Chern-Simons in Three 307Ambar N. Sengupta

Intersection Pairings on Spaces of Connections and Chern-SimonsTheory on Seifert Manifolds 317George Thompson

Fermionization and Convergent Perturbation Expansions inChern-Simons Gauge Theory 331Jonathan Weitsman

Analytic Continuation of Chern-Simons Theory 347Edward Witten

Preface

The workshop Chern-Simons gauge theory: 20 years after was held at theMax-Planck-Institute for Mathematics in Bonn from August 3 until August 7,2009. It brought together mathematicians and physicists from many countries, in-cluding Austria, Brazil, Canada, Denmark, France, Germany, Japan, Korea, Lux-embourg, Netherlands, Poland, Portugal, Romania, Spain, Sweden, Switzerland,United Kingdom and the United States of America, to discuss their research pro-grams in geometric topology, stochastic analysis and mathematical physics. Whatwas shared by this diverse group was a strong interest in Chern-Simons theory, andthis is reflected in the talks of the conference as well as in the papers in this volume.

Witten introduced Quantum Chern-Simons theory to knot theory in 1989, whenhe described the Jones polynomial of a link in S3 as a certain (non-rigorous) Feyn-man path integral, more specifically as the quantum field theory whose action is amultiple of the Chern-Simons function. He interpreted these invariants using theaxioms of topological quantum field theory (TQFT) as well as via an asymptoticexpansion.

Since then, there have been several advances in understanding both approachesto Chern-Simons theory, starting with the first rigorous construction of the TQFTversion by Reshetikhin and Turaev. This gave rise to quantum topology, a vibrantfield which is currently moving in a number of directions including:

• categorification,• modular forms, making contact with number theory,• Gromov-Witten theory via Large N duality, and• asymptotic analysis making contact with finite type invariants and hy-perbolic geometry, for example via the volume conjecture and the AJ-conjecture.

Many of these topics were touched upon in the talks delivered at the conferenceand are discussed further in the papers of this volume.

One of the outstanding open problems in Chern-Simons theory is to provide amathematically rigorous treatment for the Feynman path integral, and the difficultyhas to do with the problem of constructing a suitable gauge-invariant measure onthe infinite dimensional space of all connections. A new approach to this difficultproblem is provided by stochastical analysis, and this approach was also discussedat the conference and is the focus for several papers in this volume.

Another important aspect of Chern-Simons gauge theory that was featured atthe conference has to do with Floer homology theory and related invariants for3-manifolds. The starting point is Taubes’ result from 1989 which allows one tointerpret the Casson invariant as the Euler characteristic of the instanton homologydefined by Floer, and some new directions of research in Floer homology and Cassoninvariants are presented in the papers of this volume.

ix

x Preface

In addition to its many scientific activities, the conference featured a num-ber of pleasant social activities, including a boat ride on the Rhine, a hike up theDrachenfels, a visit to the Biergarten in the Rheinau, and a fun wine and pretzelparty at the Hausdorff Institute with musical performances in classical, jazz, androck music by a number of talented individuals, including Dana Fine, Chris Her-ald, Ben Himpel, Diane and Louis Kauffman, Paul Kirk, Markus Land, CristinaMartinez, Justin Roberts, Roland van der Veen and Lucy Zhang. We are especiallygrateful to Matthias Kreck for making the Hausdorff Institute available to us forthis memorable evening.

There were many people who helped to make the conference a success, and wewould like to take this opportunity to recognize their contributions. To begin, theconference received funding through the Hausdorff Center for Mathematics in Bonnand the ITGP network of the European Science Foundation, and we would like tothank all the staff at the Hausdorff Institute and the Max-Planck-Institute in Bonnfor their friendly assistance during the conference. The editors would also like tothank Arthur Greenspoon for his sage advice and Debbie Iscoe for her expertisein preparing the final versions of the manuscripts. Finally, we would like to thankall the authors who contributed their work to this volume, and especially EdwardWitten, who delivered an inspirational lecture at the conference and contributedanother monumental paper that is sure to stimulate further activity in this andrelated fields.

Jørgen E. Andersen, Hans U. Boden, Atle Hahn, and Benjamin HimpelNovember 2010

List of Speakers

Dave Auckly: Gauge-string duality and the structure of largerank Chern-Simons invariants

Dror Bar-Natan: Convolutions on Lie groups and Lie algebras andribbon 2-knots

Dave Auckly: Gauge-string duality and the structure of largerank Chern-Simons invariants

Dror Bar-Natan: Convolutions on Lie groups and Lie algebras andribbon 2-knots

Chris Beasley: Localization for Wilson loops in Chern-Simonstheory

Dana Fine: A geometric alternative to gauge fixing in Chern-Simons theory on S1 × Σ

Stavros Garoufalidis: Chern-Simons theory and arithmeticEnore Guadagnini: Functional integration and abelian link invariantsSergei Gukov: Exact results for perturbative Chern-Simons the-

ory with complex gauge groupChristopher M. Herald: An SU(3) Casson invariant for rational homology

spheresKazuhiro Hikami: WRT invariants and modular formsRinat M. Kashaev: On rings associated with ideal triangulations of

knot complementsLouis H. Kauffman: Khovanov homology and the Potts modelMikhail Khovanov: Categorification of quantum groupsPaul Kirk: Non-abelian representations, homology 3-spheres,

and knot concordanceAlbrecht Klemm: Chern-Simons theory and topological string the-

ory on non-compact Calabi-Yau manifoldsChristine Lescop: On the cube of the equivariant linking pairing for

closed 3-manifolds of rank oneMarcos Marino: Chern-Simons theory, the 1/N expansion, and

string theoryGregor Masbaum: Integral structures in TQFT and the mapping

class groupCatherine Meusburger: Getting physics from 3d gravity: What does an

observer in 3d gravity see?Aleksandar Mikovic: Invariants of spin networks embedded in 3-

manifolds

xi

xii List of Speakers

Hitoshi Murakami: SL(2,C)-representations and asymptotic behav-iors of the colored Jones polynomial of a knot

Robert C. Penner: Fatgraphs and finite type invariantsAlbert Schwarz: Generalizations of Chern-Simons theoryAmbar N. Sengupta: Functional integrals in low-dimensional gauge the-

oriesGeorge Thompson: Chern-Simons theory on Seifert 3-manifoldsJonathan Weitsman: Fermionization and convergent perturbation ex-

pansions in Chern-Simons gauge theoryEdward Witten: Branes and quantization

List of Participants

Sergio Albeverio, University of BonnJørgen E. Andersen, University of AarhusDave Auckly, Kansas State UniversityDror Bar-Natan, University of TorontoStefan Bauer, University of BielefeldChris Beasley, Stony Brook UniversityStefan Behrens, University of BonnAdara Blaga, West University of TimisoaraHans U. Boden, McMaster UniversityFrancesco Costantino, IRMA, University of StrasbourgTien Cuong Dinh, Professor at Paris 6 UniversityJerome Dubois, Universite Paris 7, Denis DiderotMagnus Engenhorst, Institut fur Mathematik, Universitat AugsburgJoao Faria Martins, Centro de Matematica da Universidade do PortoAlexander Felshtyn, University of Szczecin and MPIM, BonnDana Fine, University of MassachusettsStavros Garoufalidis, Georgia Institute of TechnologyMasha Gordina, University of Connecticut/University of BielefeldEnore Guadagnini, University of PisaSergei Gukov, California Institute of TechnologyAtle Hahn, Universidade de LisboaLuiz Hartmann, USP - Universidade de Sao PauloAndriy Haydys, University of BielefeldChris Herald, University of NevadaMichael Heusener, Universite Blaise Pascal Clermont IIKazuhiro Hikami, University of TokyoBenjamin Himpel, University of BonnIulia Elena Hirica, University of BucharestFuji Hiroyuki, Nagoya UniversityLotte Hollands, University of AmsterdamSaeid Jafari, College of Vestsjaelland South (VUC)Franck Jedrzejewski, CEA (French Atomic Commission)Alexander Kahle, University of GottingenUwe Kaiser, Boise State UniversityRinat Kashaev, University of GenevaLouis H. Kauffman, Math Dept, University of Illinois at ChicagoGerald Kelnhofer, University of WienMikhail Khovanov, Columbia UniversityHoil Kim, Kyungpook National UniversityPaul Kirk, Indiana University

xiii

xiv List of Participants

Albrecht Klemm, University of BonnValentin Krasontovitsch, University of Bonn (Math student)Markus Land, University of BonnJonatan Lenells, Cambridge UniversityChristine Lescop, University of Grenoble IPoon Chuan Adrian Lim, University of LuxembourgChristoph Luedeling, University of BonnMarcos Marino, University of GenevaCristina Martinez, Autonoma University of BarcelonaGregor Masbaum, Universite Paris 7Gwenael Massuyeau, CNRS, University of StrasbourgAndrew McIntyre, Bennington CollegeCatherine Meusburger, University of HamburgJouko Mickelsson, University of Helsinki and Royal Inst. of TechnologyAleksandar Mikovic, Universidade Lusofona, LisbonHitoshi Murakami, Tokyo Institute of TechnologyChiara Nappi, Institute for Advanced StudyRobert Penner, University of California, Los AngelesArturo Prat-Waldron, University of California, Berkeley and MPIM BonnJustin Roberts, University of California, San DiegoNobuya Sato, Rikkyo UniversityAlbert Schwarz, UCDavis/MPIMAmbar Sengupta, Louisiana State UniversityJan Swoboda, MPIM, BonnGeorge Thompson, International Centre for Theoretical Physics ICTPGiorgio Trentinaglia, Georg-August-Universitat GottingenAlessandro Valentino, University of GottingenRoland van der Veen, University of AmsterdamJonathan Weitsman, Northeastern UniversityEdward Witten, Institute for Advanced StudyDon Zagier, Max-Planck-Institute for MathematicsJose Miguel Zapata Rolon, University of BonnLucy Liuxuan Zhang, University of Toronto

Titles in This Series

50 Jørgen E. Andersen, Hans U. Boden, Atle Hahn, and Benjamin Himpel,Chern-Simons Gauge Theory: 20 Years After, 2011

49 Matthias Plaue, Alan Rendall, and Mike Scherfner, Advances in LorentzianGeometry, 2011

48 Lizhen Ji, Kefeng Liu, Lo Yang, and Shing-Tung Yau, Fourth InternationalCongress of Chinese Mathematicians, 2010

47 Alexander Grigor’yan, Heat Kernel and Analysis on Manifolds, 2009

46.2 Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Editors,Lagrangian Intersection Floer Theory, 2009

46.1 Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, LagrangianIntersection Floer Theory, 2009

45 Lydia Bieri and Nina Zipser, Extensions of the Stability Theorem of the MinkowskiSpace in General Relativity, 2009

44 Eric Sharpe and Arthur Greenspoon, Editors, Advances in String Theory, 2008

43 Lizhen Ji, Editor, Arithmetic Groups and Their Generalizations, 2008

42.1 Ka-Sing Lau, Zhou-Ping Xin, and Shing-Tung Yau, Editors, Third InternationalCongress of Chinese Mathematicians, 2008

41 Wen-Ching Winnie Li, Editor, Recent Trends in Coding Theory and its Applications,2007

40 Ovidiu Caliin, Der-Chen Chang, and Peter Greiner, Editors, Geometric Analysison the Heisenberg Group and Its Generalizations, 2007

39 Zhijie Chen, Sheng-Li Tan, Jianpan Wang, and Stephen S.-T. Yau, Editors,Proceedings of the International Conference on Complex Geometry and Related Fields, 2007

38 Noriko Yui, Shing-Tung Yau, and James D. Lewis, Editors, Mirror Symmetry V,2006

37 Lizhen Ji, Jian-Shu Li, H. W. Xu, and Shing-Tung Yau, Editors, Lie Groups andAutomorphic Forms, 2006

36 Chuu-Lian Terng, Editor, Integrable Systems, Geometry, and Topology, 2006

35 Felix Finster, The Principle of the Fermionic Projector, 2006

34 Ren-Hong Wong, Editor, Computational Geometry, 2003

33 Eric D’Hoker, Duong Phong, and Shing-Tung Yau, Mirror Symmetry IV, 2002

32 Xi-Ping Zhu, Lectures on Mean Curvature Flows, 2002

31 Kiyoshi Igusa, Higher Franz-Reidemeister Torsion, 2002

30 Weiman Han and Mircea Sofonea, Quasistatic Contact Problems in Viscoelasticityand Viscoplasticity, 2002

29 S. T. Yau and Shuxing Chen, Editors, Geometry and Nonlinear Partial Differential

Equations, 2002

28 Valentin Afraimovich and Sze-Bi Hsu, Lectures on Chaotic dynamical Systems, 2002

27 M. Ram Murty, Introduction to p-adic Analytic Number Theory, 2002

26 Raymond Chan, Yue-Kuen Kwok, David Yao, and Qiang Zhang, Editors,Applied Probability, 2002

25 Donggao Deng, Daren Huang, Rong-Qing Jia, Wei Lin, and Jian Zhong Wong,Editors, Wavelet Analysis and Applications, 2002

24 Jane Gilman, William W. Menasco, and Xiao-Song Lin, Editors, Knots, Braids,and Mapping Class Groups—Papers Dedicated to Joan S. Birman, 2001

23 Cumrun Vafa and S.-T. Yau, Editors, Winter School on Mirror Symmetry, VectorBundles and Lagrangian Submanifolds, 2001

22 Carlos Berenstein, Der-Chen Chang, and Jingzhi Tie, Laguerre Calculus and ItsApplications on the Heisenberg Group, 2001

21 Jurgen Jost, Bosonic Strings: A Mathematical Treatment, 2001

20 Lo Yang and S.-T. Yau, Editors, First International Congress of ChineseMathematicians, 2001

TITLES IN THIS SERIES

19 So-Chin Chen and Mei-Chi Shaw, Partial Differential Equations in Several ComplexVariables, 2001

18 Fangyang Zheng, Complex Differential Geometry, 2000

17 Lei Guo and Stephen S.-T. Yau, Editors, Lectures on Systems, Control, andInformation, 2000

16 Rudi Weikard and Gilbert Weinstein, Editors, Differential Equations and

Mathematical Physics, 2000

15 Ling Hsiao and Zhouping Xin, Editors, Some Current Topics on NonlinearConservation Laws, 2000

14 Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, 2000

13 Vasilios Alexiades and George Siopsis, Editors, Trends in Mathematical Physics,1999

12 Sheng Gong, The Bieberbach Conjecture, 1999

11 Shinichi Mochizuki, Foundations of p-adic Teichmuller Theory, 1999

10 Duong H. Phong, Luc Vinet, and Shing-Tung Yau, Editors, Mirror Symmetry III,1999

9 Shing-Tung Yau, Editor, Mirror Symmetry I, 1998

8 Jurgen Jost, Wilfrid Kendall, Umberto Mosco, Michael Rockner,and Karl-Theodor Sturm, New Directions in Dirichlet Forms, 1998

7 D. A. Buell and J. T. Teitelbaum, Editors, Computational Perspectives on NumberTheory, 1998

6 Harold Levine, Partial Differential Equations, 1997

5 Qi-keng Lu, Stephen S.-T. Yau, and Anatoly Libgober, Editors, Singularities andComplex Geometry, 1997

4 Vyjayanthi Chari and Ivan B. Penkov, Editors, Modular Interfaces: Modular LieAlgebras, Quantum Groups, and Lie Superalgebras, 1997

3 Xia-Xi Ding and Tai-Ping Liu, Editors, Nonlinear Evolutionary Partial DifferentialEquations, 1997

2.2 William H. Kazez, Editor, Geometric Topology, 1997

2.1 William H. Kazez, Editor, Geometric Topology, 1997

1 B. Greene and S.-T. Yau, Editors, Mirror Symmetry II, 1997

American Mathematical

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In 1989, Edward Witten discovered a deep relationship between quantum field theory and knot theory, and this beautiful discovery created a new field of research called Chern-Simons theory. This field has the remarkable feature of intertwining a large number of diverse branches of research in mathematics and physics, among them low-dimen-sional topology, differential geometry, quantum algebra, functional and stochastic analysis, quantum gravity, and string theory.

The 20-year anniversary of Witten’s discovery provided an opportu-nity to bring together researchers working in Chern-Simons theory for a meeting, and the resulting conference, which took place during the summer of 2009 at the Max Planck Institute for Mathematics in Bonn, included many of the leading experts in the field. This volume documents the activities of the conference and presents several original research articles, including another monumental paper by Witten that is sure to stimulate further activity in this and related fields. This collection will provide an excellent overview of the current research directions and recent progress in Chern-Simons gauge theory.