STRING THEORY on ADS3 and ADS3 /CFT2

SOME OPEN PROBLEMS

Carmen A. NúñezI.A.F.E –CONICET-UBA

II Workshop Quantum GravitySao Paulo, September 2009

MOTIVATIONS

String propagation on curved backgrounds has received much attention: own interest and relation to gauge theories

Despite important progress in various directions, several quite elementary questions remain unanswered AdS3

Briefly review these problems and comment on status of the AdS3/CFT2 correspondence

String theory on AdS3

AdS3 geometry + NS-NS two-form =

exact background AdS3 = SL(2,R) group manifold 2d conformally invariant σ-model = WZNW model

AdS3 simplest setting beyond flat space. General non-compact group manifolds define natural

framework to study strings on non-trivial geometries. Restricting to simple groups, only SL(2,R) possesses a

single time direction.

AdS/CFT correspondence Sugra in bulk of AdSn large N SYM on boundary

Low energy limit of more fundamental superstring th. Exact structure of string theory on AdSn? Connection to SYM theory on boundary?

AdS3 plays special role: Asymptotic isometry group is ∞ dimensional Theory on boundary is 2d CFT (≠ 2d sigma model whose

target space is bulk AdS3 on which the string propagates.) So far, only case where duality can be checked beyond

sugra level

Understanding relationship b/ these two CFT has led to set more precisely AdS/CFT correspondence to get feedback on structure of string theory on AdS3

Bosonic string theory on AdS3

Very little is known about WZNW models on

non-compact groups

Most works based on formal extension of the compact case in the framework of current-algebra techniques.

Most works deal with “Euclidean AdS3”= ,

but string spectrum is very different.

3H

Brief review of AdS3 WZNW model

Symmetry generated by SL(2,R)L SL(2,R)R current algebra

Sugawara relation:

Virasoro algebra:

0,3

30,

33

2],[

],[;2

],[

mnmnmn

mnmnmnmn

knJJJ

JJJnk

JJ

::)2(

1 am

amnn JJ

kL

2,,2

3

kk

k

kc

The spectrum

Physical string states must be in unitary representations of SL(2,R)

Hilbert space H : decomposes into unitary rep of current algebra labeled by eigenvalues J. Maldacena and H. Ooguri (2000)

Representations parametrized by j, related to second Casimir as

Principal discrete representations (lowest and highest weight)

m = – j + n m = j – n n=0,1,…

Principal continuous representations

j

j

D

D

ˆ

ˆ

121 23

000002 jjJJJJJc

2

1; jj

Cj: , m= , +1,… (0,1] ,

2

1ij

300 , JL

No-ghost theorem:

Evans, Gaberdiel, Perry (1997)

Representations of the current algebra

Primary states annihilated by Jn3, , n1

Acting with Jn3, , n1 on primaries SL(2,R)

Eigenvalues of L0

bounded below

Weight diagram ofjD̂

2)1(

0

kjj

L

jJ 30

21

2 j

k

Bound on mass of string states

Partition function not modular invariant

Spectral flow symmetry

The transformation

wnnn

nnnn

JJJ

wk

JJJ

~2

~0,

333

with w Z preserves the commutation relations

Sugawara 0,23

4

~nnnnn w

kwJLLL

and obey Virasoro algebra with same cnL

The spectral flow automorphism generates new representations

wj

,ˆ D and w

jC ,ˆ

nL~

Spectral flowed representations

Compact groups (SU(2)): the spectral flow maps positive energy

representation of current algebra into another.

SL(2,R), the spectral flow with w=1: L0 is not bounded belowjD̂

1,ˆ wjD

4~~ 3

000

kJLL

42)1(

0

kj

kjj

L

230

kjJ

wk

JJ2

~30

30

wj

,ˆ D contain ghosts unless:

21

2 j

k

Unitary spectrum

Only case one gets a rep. with L0 bounded below by spectral flow is with w=1

jD̂

1,

2

, ˆˆ

wk

w

jj DD1,ˆ w

jD

21

21 j

k2

)1(0

k

jjL

230

kjJ

Spectral flow symmetry implies are restricted to

AdS/CFT: Operators outside this bound cannot be identified with local operators in BCFT

wj

,ˆ D

Physical spectrum of string theory on AdS3

w Z : winding number

wjD ,ˆ

Long strings

wjC ,ˆ

wj

wji

wj

k

wj CCddjDDdj

,

1

0

,2

1

2

1

2/1

wˆˆˆˆH

Virasoro constraints decouple negative norm states unitary spectrum

Short strings

0142

)1()1( 2

0

hNwk

wmk

jjL

Spectrum verified by partition function J.Maldacena,H. Ooguri, J.Son (2001).

To determine consistency and unitarity of full theory, show that OPE closes on Hilbert space of the theory

correlation functions

Some two- and three-point amplitudes computed J.Maldacena and H. Ooguri (2001)

by analytic continuation from Teschner (1999, 2000)

Spectrum of SL(2,R): non-normalizable states of

No spectral flow representations in

Correlation functions

3H

3H

3H

Some open questions

Despite many efforts and apparent simplicity of the model, several important and elementary issues are still beyond our understanding:

Is the OPE of states closed in the spectrum of the theory?

Scattering amplitudes beyond 3-point functions? construct four-point functions in different sectors and verify that intermediate states in the factorization belong to the

spectrum and it agrees with spectral flow selection rules

Can we apply techniques developed for RCFT?

Fundamental problem: OPE primary states sufficient in RCFT, descendants not strictly

necessary. Spectral flow operation maps primaries into non-primaries

: m= , +1,… (0,1]

Principal continuous rep.

SL(2,R)

ij2

1

3H

ij2

1

No spectral flow or discrete rep.

Eucidean vs. Lorentzian theory

m = – j + n, n = 0,1,…

wjC ,ˆ

21

21 j

k

wjD ,ˆ

To achieve this goal…

Analytic and algebraic structure of SL(2,R) explored further.

Conformal bootstrap approach: requires knowing OPE and structure constants. Then one can construct any n>3-point function in terms of two- and three-point functions.

Coulomb gas approach: Works well for RCFT (minimal models, SU(2)) but requires analytic continuation in models with continuous sets of primary fields.

Studied the OPE in AdS3 by analytic continuation from & adding w W. Baron, C.N. Phys. Rev. D79 (2009) 086004

Reproduced exact three-point functions S. Iguri, C.N. Phys.Rev.D77 (2008) 066015

Constructed w-conserving four-point functions for generic states in AdS3 using Coulomb gas formalism S. Iguri, C.N arXiv:0908.3460

3H

Operator Product Expansion

iP

2

1 2112

2112211212 ,1;2

1)( jjjjjjj

...),(),;(),(),( 22,

,3121222,

,11,

,33

2121

121222

22

11

11

zzmmjAzzdjzzzz wj

mmmmiiiw

w P

wj

mm

wj

mm

Maximal region in which parameters may vary such that no poles hit contour of integration is

132121

132121

032121

2

1},max{

2

1},min{

2

1},min{

w

w

w

Afork

mmmm

Afork

mmmm

Aformmmm

Generalized OPE of including spectral flow

Both spectral flow preserving and non-preserving 3-point functions

Admits analytic continuation to generic j1, j2 defined by deforming the contour. Deformed countour = original + finite # circles

W. Baron, C.N. Phys. Rev. D79 (2009) 0860043H

Fusion rules

P

wwwwj

wjk

wwwwj

w

wj

wj

P

wwwwj

wjk

wwwj

wj

wj

CdjDCC

CdjDCD

2133

3

3

213

3

22

2

1

1

2133

3

3

213

3

22

2

1

1

,3

1

12

1

2

1

,0

1

,,

,3

1

02

1

2

1

,,,

ˆˆˆˆ

ˆˆˆˆ

P

wwwj

jk

jj

wwwj

jjjk

wwwj

wj

wj CdjDDDD 1,

3

2

1

2

1,

2

1

,,, 2133

3

321

213

3

213

213

3

2

2

1

1

ˆˆˆˆˆ

OPE closed in H Verifies several consistency checks:

• k 0 limit classical tensor products of SL(2,R) rep.• w selection rules are reproduced

Full consistency of fusion rules should follow from analysis of factorization and crossing symmetry of four-point functions

Four-point functions

Bootstrap program based on above OPE gives four-point functions with w-preserving and violating channels.

If correlators in AdS3 are obtained from those in by analytic continuation, both channels must give equivalent contributions.

Explicit computation of w-conserving four-point functions using Coulomb gas approach (free fields) confirms this.

S. Iguri, C.N. arXiv:0908.3460

3H

2,

,2,,,,3

,,,,3

)(21

1

,,,,,,

04

|)(||| 43

43

21

21

21

4321

4321

zFAAAzdj

A

jjmm

jjjmmm

wjjjmmm

w

w

jjjjmmmm

w

jjjmw

P

Coulomb gas realization

Successful for minimal models and SU(2)-CFT, j Z/2, but analytic continuation in theories with continous sets of primary fields?

Wakimoto realization in terms of free fields:

2

22

2

2

4

1 keRk

zdS

Interaction term becomes negligible near the boundary . Theory can be treated perturbatively in this region

2121

111

,, )()()( ss

N

bb

N

aa

N

ii

wjmm SSzV ii

ii

Vertex operators Spectral flow operators screening charges

Results

NNwNNk

mm

ZNN

sksj

n

ii

n

ii

n

ii

n

ii

111

211

;)(2

12

)2(

ls

nn

lnls

nnlslnls

nns

l

sl

jjjjmmmm

w

zfjjAjjSBzzzs

A

l

0,

243

,21

2

0

2

,,,,,,

04

|)(|),2,2(),2,12(||||)(

4321

4321

ljjjjjjl 21,21

C

K(x)dxi

lKl 2

1)(

)0(

1

0

Norlund-Rice theorem:

Meromorphic continuationof K(l) with simple poles atx=0,1,2,…

Analytic continuation

jjmm

jjjmmm

wjjjmmm

wjjjjmmmm

w AAAzdjA jjj ,,2

,,,,

03

,,,,

03

)(2,,,,,,

04

43

43

21

21

214321

4321||

P

2112211212 ,1;2

1)( jjjjjjj

Reproduces expression obtained by J. Teschner for

Extends previous work by V. Dotsenko, NPB358 (1991) 547

Some particular four-point functions can also be written as

3H

jjmm

jjjmmm

wjjjmmm

wjjjjmmmm

w AAAzdjA jjjmw ,,2

,,,,

13

,,,,

13

)(2,,,,,,

04

43

43

21

21

214321

4321||

P

Conformal blocks

Relation found between conformal blocks in Liouville theory and Nekrasov’s partition function of N=2 theories revives longstanding idea that all CFT can be described with free fields.

Alday, Gaiotto, Tachikawa, arXiv:0906.3219 Dotsenko-Fateev integrals and Nekrasov’s functions provide a basis

for hypergeometric integrals

)()1()()( 33

22

3221,

1122

2

2

)1(2

zJzFzzzfNk ln

mjnsnsmj

k

jj

k

ljjll

nlnl

),,()()1()(

)1()()(

12

1

1

122

1

2

011 1

132122

321

sn

s

jiji

l

iilsils

ls

i

ji

ji

s

i

ji

z l

iils

ls

ii

ln

yysyyyyz

yzyydydyzJ

jj

Multiple integral realization of conformal blocks.

AdS3/CFT2 correspondence

Type IIB superstring theory on AdS3 x S3 x T4 dual

2d CFT describing the D1-D5 system on T4

J. Maldacena (1997)

Low energy description of D1-D5 system is σ–model with target space (T4)N/SN, SN permutation group of N=N1N5 elements

J. Maldacena, A. Strominger (1998)

Naive evidence from symmetries

Symmetries of the bulk

Isometries of AdS3

SO(2,2) ~ SL(2,R) SL(2,R)

Isometries of S3

SO(4) ~ SU(2) SU(2)

16 supersymmetries

SO(4) of T4

Symmetries of the SCFT

Global Virasoro group

SL(2,R) SL(2,R)

R-symmetry

SU(2) SU(2)

Global charges of N =4 SCFT

SO(4) of T4

The symmetric orbifold

Chiral spectrum of σ–model built on that of single copy of T4 plus operators in twisted sectors

Chiral operators are constructed as

N

nnnn

Shhnhn NnNnO ,

)...1(

2/1,1!)!(

twist field of a single element of SN

n= ± 1, a

Charged under R-symmetry group of SU(2)L SU(2)R

an

Hn

H nnnn

n

,

2;1,

2

Two and three-point functions computed by O. Lunin and S. Mathur (2001)and A. Jevicki, M. Mihailescu, S.Ramgoolam (2000)

The dual string theory

Near horizon geometry of the D1-D5 system AdS3xS3xT4 plus fluxes of RR fields through the S3

Progress in formulation of string propagation on RR backgrounds has been made, but explicit calculations in AdS3

geometry cannot be done yet

Convenient to study string theory on the S-dual background

Near horizon limit of NS1-NS5 system is AdS3 x S3 x T4 with fluxes of NS field B

Worldsheet theory is WZNW model with SL(2,R) SU(2) U(1)4 affine symmetry

Chiral primaries

H = J

Vertex operators of chiral primaries (in -1 picture)

C. Cardona, C.N., JHEP0906, 009 (2009)

Spacetime conformal dimension

SU(2) charge

)(2

'

,',

,1,

2/1',

2,,

'

,',

1'

,1,

'

,','

,1,

1

54 HHi

wMJ

wjMH

wmm

wj

wjMJ

wm

wjMH

wm

j,w

wjMJ

wm

wjMH

wm

j,w

eVSe

Ve

Ve

MJ,M,H,MH,

MJ,MH,MH,

MJ,MH,MH,

R

Y

W

Three-point functions

Factorize into products of SU(2), fermions, SL(2,R) fields.

SU(2) correlators A. Zamolodchikov, V. Fateev (1986)

Unflowed sector M. Gaberdiel, I. Kirsch (2007) A. Dabolkhar, A. Pakman (2007)

Spectral flowed sectors:

Fermionic correlators G. Giribet, A. Pakman, Rastelli (2007) SL(2,R) correlators C. Cardona, C.N. (2009)

AdS3/CFT2 dictionary

k N5 Maldacena, Strominger, 1998

gs2 N5 Vol(T4)/ N1 Giveon, Kutasov, Seiberg, 1999

n 2j+1+kw G. Giribet, A. Pakman, Rastelli, 2007

The matching obtained so far reflects the cancellation of structure constants of AdS3 with those of S3. Fermionic contributions reduce to unity in all cases considered.

Extend the dictionary to four-point functions and descendant states. Consider more general internal spaces.

Conclusions

Despite significant recent progress, string theory on AdS3 (one of the simplest examples beyond flat space) is not well understood. Several consistency checks have been performed to determine consistency but spectral flow sectors have to be studied further.

We determined the fusion rules and computed w-conserving four- point functions using the Coulomb gas approach.

Conclusions

We showed that spectral flow preserving and violating channels give equivalent contributions, thus corroborating that w-conserving amplitudes in AdS3 can be obtained from by analytic continuation.

We verified AdS3/CFT2 correspondence in arbitrary spectral flow sectors up to three-point functions, providing additional verification of AdS3/CFT2 duality conjecture.

3H

Open problems and future directions

It is necessary to understand mechanism determining the decoupling of non-unitary states

Verlinde theorem relating fusion coefficients with modular transformations. Constructing and studying w-violating four-point

functions.

Compute four-point functions in AdS3 x S3 x T4 and compare with symmetric product Pakman, Rastelli, Razamat (2009)

Construct more examples of AdS3/CFT2 duality (other internal geometries)

THE END

Vertex operators & spectral flow selection rules

at least one state in all states in

wjC ,ˆ

wjD ,ˆ

)]()([2

~),(zmzm

ki

mjmmjm ezz

)]()([2

zzk

i

e Spectral flow

operator

Vertex operators for primary states

)]()2

()()2

[(2

~),(zw

kmzw

km

ki

mjmw

mjm ezz

Vertex operators for w ≠ 0 states

11

22

1

1

t

t

N

iid

c

N

iit

wN

NwN

Fusion rules and interactions