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

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 PauloSeptembre 2005

2. MOTIVATIONS CFT based on affine SL(2)k, not only for k  Z and unitary integrable representations (j  Z or Z+½). • 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

3. 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 CFTs with SL(2) symmetry simplest models beyond the well studied RCFT Verma module is reducible; there are null vectors; free field rep. • 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 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

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 spacetimeCFT 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 AdS3 consistent (unitary)? Is the OPA closed over unitary states?

6. 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

7. CORRELATION FUNCTIONS • 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.

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

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 Form representations of the Lie algebra generated by J0a(z) keep track of SL(2) weights AdS/CFT interpretation location of operator in dual BCFT Dj+: m = j, j+1,… Dj-: m = –j, – j – 1,… Cj: , m=  ,  +1,… jm Unitary representations of SL(2,R)

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

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

13. is generatedfrom |j,;w> , 0<  <1) and And the Casimir is andare 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 weightrepresentations: • 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. THESPECTRAL FLOW OPERATOR This isan auxiliary field (not physical) which allows to construct operators in sectors w = 1 andw = –1 from operatorsinw = 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. 3-POINT FUNCTIONS N=2 SL(2,C) conformal invariance of the worldsheet and target space determines the x and z dependence This coincides with analytic continuation of Teschner’s result. However it does not determine the fusion rules  need 4-point functions

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

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. Comments • 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 • 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 onworldsheet correlators • They do not correspond to well defined objects in BCFT if

23. Factorization of4-point functions is not unitary unless • and j3 j1 J j4 j2 Non-physicalJnot well definedobjects in BCFT Each leg imposes additional constraints

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

25. THE END