The partition function of the supersymmetric two-dimensional black hole and little string theory

TL;DR

This work computes the torus partition function of the supersymmetric coset $SL(2,\mathbb{R})/U(1)$ and proves its decomposition into (extended) $N=2$ characters, revealing discrete and continuous representations and novel regularization- dependent sectors. By coupling to an $N=2$ minimal model and enforcing integral $N=2$ charges through a $\mathbb{Z}_k$ orbifold, the authors construct the double scaled little string theory (DSLST) background and show it is equivalent to NS5-branes on a circle via a null gauged coset of $SL(2,\mathbb{R})_k \times SU(2)_k$. The spectrum is shown to split into discrete (with $j$ in the improved unitary range) and continuous sectors, with the latter requiring Liouville-type regularization whose finite part matches $N=2$ Liouville reflection amplitudes; the Witten index is computed and found to be 1, consistent with an $A_{k-1}$ ALE base. The paper also connects the DSLST coset to the exact coset CFT describing NS5-branes on a circle, clarifies the role of spacetime supersymmetry in enforcing integral $N=2$ charges, and discusses the interplay between GSO projections and target-space geometry, providing a coherent holographic picture of DSLST and its supergravity limit.

Abstract

We compute the partition function of the supersymmetric two-dimensional Euclidean black hole geometry described by the SL(2,R)/U(1) superconformal field theory. We decompose the result in terms of characters of the N=2 superconformal symmetry. We point out puzzling sectors of states besides finding expected discrete and continuous contributions to the partition function. By adding an N=2 minimal model factor of the correct central charge and projecting on integral N=2 charges we compute the partition function of the background dual to little string theory in a double scaling limit. We show the precise correspondence between this theory and the background for NS5-branes on a circle, due to an exact description of the background as a null gauging of SL(2,R) x SU(2). Finally, we discuss the interplay between GSO projection and target space geometry.