2D Black Holes and 2D Gravity

TL;DR

This paper explores the relationship between 2D black holes modeled by the SL(2,R)/U(1) coset and Liouville theory, showing how high boost limits connect black hole models to 2D gravity with Liouville fields.

Contribution

It demonstrates that the high boost limit of 2D black holes reduces to Liouville theory coupled with matter, linking black hole models to 2D gravity in a novel way.

Findings

High boost limit leads to Liouville coupled with matter

Vertex operators of black holes tend to those of 2D gravity

Black hole models simplify to Liouville theory in certain limits

Abstract

\small The SL$(2,R)/U(1)$ coset model, with $U(1)$ an element of the third conjugacy class of $SL(2,R)$ subgroups, is considered. The resulting theory is seen to collapse to a one dimensional field theory of Liouville. Then the 2 dimensional black hole $SL(2,R)/U(1)$, with $U(1)$ a non-compact subgroup boosted by a Lorentz transformation, is considered. In the limit of high boost, the resulting black hole is found to tend to the Liouville field coupled to $a\ C=1$ matter field. The limit of the vertex operators of the 2 dimensional black hole also tend to those of the $C=1$ two dimensional gravity.