SPECTRAL FLOW IN THE SL(2,R) WZW MODEL

Carmen A. Núñez I.A.F.E. & UBA

WORKSHOP: New Trends in Quantum Gravity   Instituto de Fisica, Sao Paulo       Septembre 2005

  

MOTIVATIONS

I. SL(2) symmetry is rather general

String Theory on AdS3 SL(2,R) WZW model

Black holes in string theory

Liouville theory of 2D quantum gravity

3D gravity

Certain problems in condensed matter

CFT based on affine SL(2)k, not only for k Z and

unitary integrable representations (j Z or Z+½).

RATIONAL vs NON-RATIONAL CFTs

RCFT finite number of representations of modular group

(e.g. c=1 on circle of rational R2 ; extended algebra).

Non-RCFT are qualitatively different

Verma module is reducible; there are null vectors; free field rep.

II. CFTs with SL(2) symmetry simplest models beyond the well studied RCFT

Continuous families of primary fields No highest or lowest weight representations

No singular vectors fusion rules cannot be determined algebraically

OPE of primary fields involves integrals over continuous sets of operators.

STRING THEORY ON AdS3

This string theory is special in many respects:

Simplest string theory in time dependent backgrounds Concept of time in string theory

String theories in more complicated geometries

In the context of AdS/CFT it is special because Worldsheet theory can be studied beyond sugra

It does not require turning on RR backgrounds

BCFT is 2D infinite dimensional algebra

Important lessons from stringy analyses

Observables in spacetime theory Fundamental string excitations

Worldsheet correlation functions Green’s functions of operators (in flat spacetime interpreted as in spacetime CFT S-matrix elements in target space)

Spacetime CFT has Constraints in worldsheet theory non-local features These restrictions are not understood from the

string theory point of view.

Is string theory on AdSIs string theory on AdS33 consistent (unitary)? consistent (unitary)?

Is the OPA closed over unitary states?Is the OPA closed over unitary states?

)(Oi xx, ),,,(2 zzxxzd i

worldsheetett zdxx FF 2arg ),(

STATUS OF STRING THEORY ON AdS3

Unitary spectrum of physical states (spectral flow symmetry) J. Maldacena, H. Ooguri hep-th/0001053

Modular invariant partition function J. Maldacena, H.Ooguri, J. Son; hep-th/0005183 Product of characters of SL(2,R) representations? D. Israel, C. Kounnas, P.Petropoulos; hep-th/0306053

Correlation functions J. Maldacena, H. Ooguri hep-th/0111180

Analytic continuation of J. Teschner, hep-th/0108121

Generalization of bootstrap to

)2(

),2(3 SU

CSLH

SL(2,R) WZW model WZW model

(actions related by analytic continuation of fields)

States in H of SL(2,R) non-normalizable states in H3+

Not all states in the SL(2,R) WZW model can be obtained by analytic continuation from spectral flowed states

AdS/CFT: Consistency of BCFT implies awkward constraints on worldsheet correlators. Factorization of 4-point functions is not unitary unless external states satisfy certain restrictions with no clear interpretation in worldsheet theory.

CORRELATION FUNCTIONS

)2(),2(

SUCSL

)2(),2(

SUCSL

WZW MODEL for SL(2, R)WZW MODEL for SL(2, R)kk

)])()([(Tr12

)])([(Tr8

111ijk3

12

ggggggxdk

ggdk

S

kjiV

WZW

Infinitely many symmetries generated by currents Ja(z), Ja(z),

a=,3

ggk

zJggk

zJ aa )(2

)(,)(2

)( 11

)()2(;),2()( zgzgRSLzg

k : level of the representation

Symmetry Algebra: Virasoro Kac-Moody

0,3

30,

33

2],[

],[;2

],[

mnmnmn

mnmnmnmn

knJJJ

JJJnk

JJ

0,3 )(

12)(],[ nmnmnm mm

cLnmLL

anm

anm nJJL ],[

Sugawara relation:::

)2(1 a

ma

mnn JJk

L

23

k

kc

And similarly for n

an LJ ,

Lie algebra of SL(2,R) can be represented by differential operators

jxx

xDjx

xDx

D 2,, 23

x: isospin coordinate

PRIMARY FIELDSPRIMARY FIELDS ),;,( zzxxj

keep track of SL(2) weights),( xx ),( mm

AdS/CFT interpretation location of operator in dual BCFT

3,,),;,(),;,()(

azzxxwz

DzzxxzJ j

a

ja

),;,(||

),( ,2

2

,;, zzxxxxx

xdzz jj

mjmjmjmj

Form representations of the Lie algebra generated by J0a(z)

0,0| ,; nJ mmjan

jm Unitary representations of SL(2,R)Dj

+: m = j, j+1,…Dj

-: m = –j, – j – 1,…

Cj: , m= , +1,…

21

21 k

j

,21

ij

SPECTRAL FLOWSPECTRAL FLOW

wnnn

nnnn

JJJ

wk

JJJ

~2

~0,

333

23

4

~w

kwJLLL nnnn Sugawara

The transformation

with w Z, preserves the SL(2,R) commutation relations

obey Virasoro algebra with same cnL~

wjD̂ w

jC ,ˆ

The spectral flow automorphism generates new representations

and

Hilbert space of SL(2,R) WZW modelHilbert space of SL(2,R) WZW model

wj

wji

wj

k

wj CCddjDDdj

,

1

0

,2

1

2

1

2/1

wˆˆˆˆH

wjD̂ is an irreducible infinite dimensional representation of the SL(2,R)

algebra generated from highest weight state |j;w> defined by

wjwk

jwjJ

nwjJwjJwjJ nwnwn

;|2

;|

)0(0;|,0;|,0;|

30

31

w Z is the spectral flow parameter or winding number

21

21 k

j

0,ˆ jC

is generated from |j,;w> , 0< <1) and

wjwk

wjJ

nwjJwjJ nwn

;,|2

;,|

)0(0;,|,0;,|

30

3

And the Casimir is

wjjjwjJJJJwk

JwjLk wwww ;|)1(;|21

2;|)2(

2300

0ˆjD and are conventional discrete and continuous represent.

wjC ,ˆ

,21

ij

wjD̂ w

jC ,ˆ and are obtained by spetral flow

CFTs based on affine SL(2)k are well known in the case of

Unitary integrable representations of SU(2) k and integer and half integer spins A.B.Zamolodchikov & V.A.Fateev (1986)

Highest weight representations: k C\{0} and

Admissible representations:

Rational level k+2 = p/q, p,q coprime integers

V.G.Kac & D.A.Kazhdan F.G.Malikov, B.L.Feigin & D.B.Fuchs

H.Awata & Y.Yamada

All these are RCFT Null vector method applies.

0,0)(;0,0)(,,,12 , srIIsrIZsrstrj sr

CORRELATION FUNCTIONSCORRELATION FUNCTIONS

The correlation functions in WZW theory obey linear differentialequations which follow from the Sugawara construction of T(z).Knizhnik-Zamolodchikov equation:

0),()...,()2(

),()...,()(

11011

11

1

1

NNjjaa

NNjj

xzxzJJLk

xzxzzT

N

N

In SU(2) there are null vectors which impose extra constraints andallow to determine the fusion rules. But the space of vectors of theunitary representations of SL(2,R)

21

21 k

j ,21

ijwjD̂ w

jC ,ˆ with and with

contains no null vectors. However the spectral flow plays their rol.

02

;2

|1

km

kjJ

),(2

xzkj

THETHE SPECTRAL FLOW OPERATORSPECTRAL FLOW OPERATOR

),(),(lim),(2

1120

,1, zxzyxyyydzx kj

mjmjmmjwJJ

2,

2k

mJk

mJ

This is an auxiliary field (not physical) which allows to construct operators in sectors w = 1 and w = –1 from operators in w = 0 as follows

),(2

xzk

It satisfies the primary state conditions with

02

;2

|1

km

kjJ

0),(),()(2)(1

12

2

1

N

nnnjk

jnn

nn

N

n n

zxzxxxjx

xxzz n

NULL VECTOR METHODNULL VECTOR METHOD

0)()...()( 1

2

1 1

Njjk zzzJN

),;,(2

),;,()(2

zzxxwz

jxxzzxxzJ j

xj

One can apply the null vector method to correlators containing ),(2

xzk

What information can be obtained from this null vector?

)2

(),(),(),( 212111

221

kjjzxzxzx jjk

This coincides with analytic continuation of Teschner’s result. However it does not determine the fusion rules need 4-point functions

N=2 SL(2,C) conformal invariance of the worldsheet and target space determines the x and z dependence

12

2

12

2

12

2

121221

32

)()()(

)()()(),,2

(),(),(),(

2121

221

22

21213322

2

kkk

zzzzzz

xxxxxxjjk

Czxzxzxjj

kjj

kjj

k

jjk

3-POINT FUNCTIONS3-POINT FUNCTIONS

4-POINT FUNCTIONS4-POINT FUNCTIONS

SL(2,C) conformal invariance of the worldsheet and target space non-trivial dependence on cross ratios

))(())((

;))(())((

2431

4321

2431

4321

xxxxxxxx

xzzzzzzzz

z

Teschner applied generalization of bootstrap for

Maldacena & Ooguri analyzed analytic continuation. Null vector method?

),(

),(),(),(),(

3124214323421

3214142324321

432

31412

4343

31412

4243443322111

xzzzzz

xxxxzxzxzxzx jjjjjjjjjjjjjjjjj

F

),(12

1),( xz

zQ

zP

kxz

zFF

)2(),2(

SUCSL

A closed form for F(z,x) is not known for generic values of ji

KZ reduces to:

Null vector method for 4-point functionsNull vector method for 4-point functions

If one operator is there is one extra equation),(2

xzk

0),(),()(2)(1

3

12

24

1

n

nnjkj

nnn

nn n

zxzxxxjx

xxzz n

and KZ equation simplifies because

02

;2

|1

km

kjJ

The spectral flow operator is not physical.It changes the winding number of another operator by one unit.

This gives a 3-point function violating winding number conservation by one unit.

2|

2| 3

11

kjJ

kjL

N-point functions may violate winding number conservation

up to N-2 units Determined by SL(2,R) algebra

Result agrees with free field approximation (Coulomb gas

formalism). G. Giribet and C.N., JHEP06(2000)010; JHEP06(2001)033

Supersymmetric extension D. Hofman and C.N., JHEP07(2004)019

Need 5-point functions to get information for 4-point function

Comments

Coulomb gas is more practical method than bootstrap of BPZ It works in minimal models and SU(2) CFT due to singular vectors. Extension to SL(2,R) requires analytic continuation in the number of screening operators. It worked for 3-point functions, but this is an experimental fact. There is no theoretical proof.

OPEN PROBLEMSOPEN PROBLEMS

Computation of 4-point functions in w 0 sectors and factorization properties. Closure of OPA on unitary states

Interpretation of unitarity constraints on worldsheet correlators

They do not correspond to well defined objects in BCFT

if nkjn

ii

1

nkjn

ii

1

2

321

kjj

2

343

kjj

Factorization of 4-point functions is not unitary unless

and

j1

j2

j3

j4

J

Non-physical J not well defined objects in BCFT

Each leg imposes additional constraints

Modular properties?

Factorization properties?

Higher genus Riemann surfaces

)()(),(,

RLRL

LRNZ

Verlinde theorem?

THE END