The Partition Function of the Two-Dimensional Black Hole Conformal Field Theory

TL;DR

This work addresses the toroidal partition function of the $SL(2,R)/U(1)$ coset CFT on the Euclidean black hole background. It employs a first-principles path-integral computation using a gauged WZW action, holonomy treatment on the torus, and Ray-Singer analytic torsion to organize the functional integral. The analysis yields a modular-invariant partition function that decomposes into discrete and continuous $SL(2,R)$ representations, establishing the spin bound $\tfrac{1}{2}<j<\tfrac{k-1}{2}$ and a density for the continuous sector. The resulting conformal weights satisfy $h_{cs}=-\frac{j(j-1)}{k-2}+\frac{(n-kw)^2}{4k}$ for discrete and $h_{cs}=\frac{s^2+1/4}{k-2}+\frac{(n-kw)^2}{4k}$ for continuous, with appropriate constraints. These findings confirm the algebraic spectrum and provide a robust foundation for string backgrounds with an $SL(2,R)/U(1)$ factor and for future work on holography, D-branes, and correlation functions.

Abstract

We compute the partition function of the conformal field theory on the two-dimensional euclidean black hole background using path-integral techniques. We show that the resulting spectrum is consistent with the algebraic expectations for the SL(2,R)/U(1) coset conformal field theory construction. In particular, we find confirmation for the bound on the spin of the discrete representations and we determine the density of the continuous representations. We point out the relevance of the partition function to all string theory backgrounds that include an SL(2,R)/U(1) coset factor.