1. 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

2. MOTIVATIONS I. SL(2) symmetry is rather general String Theory on AdS 3  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).

3. RATIONAL vs NON-RATIONAL CFTs  RCFT finite number of representations of modular group (e.g. c=1 on circle of rational R 2 ; 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.

4. STRING THEORY ON AdS 3 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

5. 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 AdS 3 consistent (unitary)? Is the OPA closed over unitary states? Is the OPA closed over unitary states?

6. STATUS OF STRING THEORY ON AdS 3 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

7. 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 H 3 + 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

8. WZW MODEL for SL(2, R) k Infinitely many symmetries generated by currents J a (z), J a (z), a= ,3 k : level of the representation

9. Symmetry Algebra: Virasoro  Kac-Moody Sugawara relation: And similarly for Lie algebra of SL(2,R) can be represented by differential operators x : isospin coordinate

10. PRIMARY FIELDS keep track of SL(2) weights AdS/CFT interpretation location of operator in dual BCFT Form representations of the Lie algebra generated by J 0 a (z)  jm  Unitary representations of SL(2,R) D j + : m = j, j+1,… D j - : m = –j, – j – 1,… C j  :, m= ,  +1,…

11. SPECTRAL FLOW Sugawara  The transformation with w  Z, preserves the SL(2,R) commutation relations obey Virasoro algebra with same c The spectral flow automorphism generates new representations and

12. Hilbert space of SL(2,R) WZW model is an irreducible infinite dimensional representation of the SL(2,R) algebra generated from highest weight state | j;w > defined by w  Z is the spectral flow parameter or winding number

13. is generated from | j,  ; w >, 0<  <1) and And the Casimir is and are conventional discrete and continuous represent. and are obtained by spetral flow

14. 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.

15. CORRELATION FUNCTIONS The correlation functions in WZW theory obey linear differential equations which follow from the Sugawara construction of T(z). Knizhnik-Zamolodchikov equation: In SU(2) there are null vectors which impose extra constraints and allow to determine the fusion rules. But the space of vectors of the unitary representations of SL(2,R) with and with contains no null vectors. However the spectral flow plays their rol.

16. THE SPECTRAL FLOW OPERATOR THE SPECTRAL FLOW OPERATOR 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 It satisfies the primary state conditions with

17. NULL VECTOR METHOD One can apply the null vector method to correlators containing What information can be obtained from this null vector?

18. 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 3-POINT FUNCTIONS

19. 4-POINT FUNCTIONS SL(2,C) conformal invariance of the worldsheet and target space  non-trivial dependence on cross ratios Teschner applied generalization of bootstrap for Maldacena & Ooguri analyzed analytic continuation. Null vector method? A closed form for F( z,x ) is not known for generic values of j i KZ reduces to:

20. Null vector method for 4-point functions If one operator is  there is one extra equation and KZ equation simplifies because  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.

21. 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.

22. OPEN 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

23.  Factorization of 4-point functions is not unitary unless and j1j1 j2j2 j3j3 j4j4 J Non-physical J not well defined objects in BCFT Each leg imposes additional constraints

24.  Modular properties?  Factorization properties? Higher genus Riemann surfaces Verlinde theorem?

25. THE END