1. STRING THEORY on ADS 3 and ADS 3 /CFT 2 SOME OPEN PROBLEMS Carmen A. Núñez I.A.F.E –CONICET-UBA II Workshop Quantum Gravity Sao Paulo, September 2009

2. 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 AdS 3 Briefly review these problems and comment on status of the AdS 3 /CFT 2 correspondence

3. String theory on AdS 3 AdS 3 geometry + NS-NS two-form = exact background  AdS 3 = SL(2,R) group manifold  2d conformally invariant σ-model = WZNW model AdS 3 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.

4. AdS/CFT correspondence Sugra in bulk of AdS n large N SYM on boundary Low energy limit of more fundamental superstring th.  Exact structure of string theory on AdS n ?  Connection to SYM theory on boundary? AdS 3 plays special role:  Asymptotic isometry group is ∞ dimensional  Theory on boundary is 2d CFT (≠ 2d sigma model whose target space is bulk AdS 3 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 AdS 3

5. Bosonic string theory on AdS 3 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 AdS 3 ”=, but string spectrum is very different.

6. Brief review of AdS 3 WZNW model Symmetry generated by SL(2,R) L  SL(2,R) R current algebra Sugawara relation: Virasoro algebra:

7. 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 C j  :, m= ,  +1,…  (0,1]

8.  No-ghost theorem : Evans, Gaberdiel, Perry (1997) Representations of the current algebra Primary states annihilated by J n 3, , n  1 Acting with J n 3, , n  1 on primaries SL(2,R) Eigenvalues of L 0 bounded below Weight diagram of Bound on mass of string states Partition function not modular invariant

9. Spectral flow symmetry The transformation with w  Z preserves the commutation relations Sugawara  and obey Virasoro algebra with same c The spectral flow automorphism generates new representations and

10. 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: L 0 is not bounded below contain ghosts unless:

11. Unitary spectrum Only case one gets a rep. with L 0 bounded below by spectral flow is with w=  1 Spectral flow symmetry implies are restricted to AdS/CFT: Operators outside this bound cannot be identified with local operators in BCFT

12. Physical spectrum of string theory on AdS 3 w  Z : winding number Long strings Virasoro constraints decouple negative norm states unitary spectrum Short strings

13. 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

14. 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

15. : m= ,  +1,…  (0,1] Principal continuous rep. SL(2,R) No spectral flow or discrete rep. Eucidean vs. Lorentzian theory m = – j + n, n = 0,1,…

16. 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 AdS 3 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 AdS 3 using Coulomb gas formalism S. Iguri, C.N arXiv:0908.3460

17. Operator Product Expansion Maximal region in which parameters may vary such that no poles hit contour of integration is Generalized OPE of including spectral flow Both spectral flow preserving and non-preserving 3-point functions Admits analytic continuation to generic j 1, j 2 defined by deforming the contour. Deformed countour = original + finite # circles W. Baron, C.N. Phys. Rev. D79 (2009) 086004

18. Fusion rules  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

19. Four-point functions Bootstrap program based on above OPE gives four-point functions with w -preserving and violating channels. If correlators in AdS 3 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

20. 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: Interaction term becomes negligible near the boundary . Theory can be treated perturbatively in this region Vertex operators Spectral flow operators screening charges

21. Results Norlund-Rice theorem: Meromorphic continuation of K(l) with simple poles at x=0,1,2,…

22. Analytic continuation 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

23. 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 Multiple integral realization of conformal blocks.

24. AdS 3 /CFT 2 correspondence Type IIB superstring theory on AdS 3 x S 3 x T 4 dual 2d CFT describing the D1-D5 system on T 4 J. Maldacena (1997) Low energy description of D1-D5 system is σ–model with target space (T 4 ) N /S N, S N permutation group of N=N 1 N 5 elements J. Maldacena, A. Strominger (1998)

25. Naive evidence from symmetries Symmetries of the bulk Isometries of AdS 3 SO(2,2) ~ SL(2,R)  SL(2,R) Isometries of S 3 SO(4) ~ SU(2)  SU(2) 16 supersymmetries SO(4) of T 4 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 T 4

26. The symmetric orbifold Chiral spectrum of σ–model built on that of single copy of T 4 plus operators in twisted sectors Chiral operators are constructed as twist field of a single element of S N  n = ± 1, a Charged under R-symmetry group of SU(2) L  SU(2) R Two and three-point functions computed by O. Lunin and S. Mathur (2001) and A. Jevicki, M. Mihailescu, S.Ramgoolam (2000)

27. The dual string theory Near horizon geometry of the D1-D5 system AdS 3 xS 3 xT 4 plus fluxes of RR fields through the S 3 Progress in formulation of string propagation on RR backgrounds has been made, but explicit calculations in AdS 3 geometry cannot be done yet Convenient to study string theory on the S-dual background Near horizon limit of NS1-NS5 system is AdS 3 x S 3 x T 4 with fluxes of NS field B  Worldsheet theory is WZNW model with SL(2,R)  SU(2)  U(1) 4 affine symmetry

28. 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

29. 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)

30. AdS 3 /CFT 2 dictionary k N 5 Maldacena, Strominger, 1998 g s 2 N 5 Vol(T 4 )/ N 1 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 AdS 3 with those of S 3. Fermionic contributions reduce to unity in all cases considered. Extend the dictionary to four-point functions and descendant states. Consider more general internal spaces.

31. Conclusions  Despite significant recent progress, string theory on AdS 3 (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.

32. Conclusions  We showed that spectral flow preserving and violating channels give equivalent contributions, thus corroborating that w - conserving amplitudes in AdS 3 can be obtained from by analytic continuation. We verified AdS 3 /CFT 2 correspondence in arbitrary spectral flow sectors up to three-point functions, providing additional verification of AdS 3 /CFT 2 duality conjecture.

33. 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 AdS 3 x S 3 x T 4 and compare with symmetric product Pakman, Rastelli, Razamat (2009) Construct more examples of AdS 3 /CFT 2 duality (other internal geometries)

34. THE END

35. Vertex operators & spectral flow selection rules at least one state in all states in Spectral flow operator Vertex operators for primary states Vertex operators for w ≠ 0 states

36. Fusion rules and interactions