Abstract
We study the semi-classical limit of the reflection coefficient for the SL(2, ℝ)k/U(1) CFT. For large k, the CFT describes a string in a Euclidean black hole of 2-dimensional dilaton-gravity, whose target space is a cigar with an asymptotically linear dilaton. This sigma-model description is weakly coupled in the large k limit, and we investigate the saddle-point expansion of the functional integral that computes the reflection coefficient. As in the semi-classical limit of Liouville CFT studied in [1], we find that one must complexify the functional integral and sum over complex saddles to reproduce the limit of the exact reflection coefficient. Unlike Liouville, the SL(2, ℝ)k/U(1) CFT admits bound states that manifest as poles of the reflection coefficient. To reproduce them in the semi-classical limit, we find that one must sum over configurations that hit the black hole singularity, but nevertheless contribute to the saddle-point expansion with finite action.
Highlights
Virasoro primaries Vα(z, z) in Liouville CFT were studied for general complex values of α
This sigma-model description is weakly coupled in the large k limit, and we investigate the saddle-point expansion of the functional integral that computes the reflection coefficient
As in the semi-classical limit of Liouville CFT studied in [1], we find that one must complexify the functional integral and sum over complex saddles to reproduce the limit of the exact reflection coefficient
Summary
Before coming to the SL(2, R)k/U(1) CFT, in this section we review some aspects of linear-dilaton theories that will be important in what follows, especially the formulation of “asymptotic conditions.” These provide a convenient description of operator insertions in the functional integral via boundary modifications of the action.7
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