On the central charge of spacetime current algebras and correlators in string theory on AdS 3

Abstract

Spacetime Virasoro and affine Lie algebras for strings propagating in AdS 3 are known to all orders in α ′ . The central extension of such algebras is a string vertex, whose expectation value can depend on the number of long strings present in the background but should be otherwise state-independent. In hep-th/0106004, on the other hand, a statedependent expectation value was found. Another puzzling feature of the theory is lack of cluster decomposition property in certain connected correlators. This note shows that both problems can be removed by defining the free energy of the spacetime boundary conformal field theory as the Legendre transform of the formula proposed in the literature. This corresponds to pass from a canonical ensemble, where the number of fundamental strings that create the background can fluctuate, to a microcanonical one, where it is fixed.

Supersymmetric Renyi entropy in CFT 2 and AdS 3

“…Indeed, they have dimension (2.4), (2.13) h(σ kŝk ) = 1 2 k n and R-charge (2.11) R k = k n , so (1.10) is satisfied. The resulting dimension and R-charge of S n are 15) strictly larger than (2.9), as expected.…”

“…In particular, the spacetime identity operator, S 1 ≡ I [14,15], is obtained from the identity in N . Consequently, the vertex operators for S n discussed above live purely in the AdS 3 × S 1 factor, and thus do not contain any information about the particular SCFT N in (4.1) that enters the construction, except for the level k, which determines the spacetime central charge of M. Thus, any SCFT with a description in terms of string theory on AdS 3…”

Partition functions of the tensionless string

“…While this term has a clear classical interpretation, it is a bit ad hoc. The same term can be introduced directly on the worldsheet, where it corresponds to the spacetime identity operator [52,53]. It was a puzzle that the spacetime identity operator is proportional to the winding number w i .…”

String dynamics in NS5-F1-P geometries

“…This phenomenon was recently investigated in the closely related context of string excitations around global AdS 3 × S 3 [61]. Worldsheet string theory on SL(2, R) × SU (2) × M provides a perturbative approximation to near-vacuum states and processes in a spacetime CFT with central charge c ST = 6n 5 n 1 , where n 1 is the string winding charge on the y circle.…”

“…This issue was encountered early on in the worldsheet approach to AdS 3 string theory [62] -it was noticed that the central charge in the spacetime Virasoro algebra computed via worldsheet vertex operator methods seemed to depend on the correlation function being evaluated, and thus was not actually a c-number. The resolution of this puzzle came when it was pointed out that calculations involving winding sectors in the SL(2, R) WZW model take place at fixed chemical potential for string winding charge, rather than at fixed charge [61]; and that a Legendre transformation is required to ensure that the charge is kept fixed. When we do this, we are keeping fixed the sum of the flux n 1 in the background and the total winding charge w tot y of perturbative strings.…”