A Note on Einstein Gravity on AdS$_3$ and Boundary Conformal Field Theory

TL;DR

This paper explores the connection between the asymptotic metric expansion in AdS$_3$ gravity and the boundary Liouville conformal field theory, revealing explicit relations in specific boundary descriptions.

Contribution

It establishes a simple relation between subleading metric terms in AdS$_3$ and the boundary stress tensor, clarifying the bulk-boundary correspondence in this context.

Findings

Derived a relation between metric expansion terms and boundary stress tensor.

Provided explicit formulas linking bulk metric to boundary free field theory.

Enhanced understanding of AdS$_3$/CFT$_2$ correspondence in Liouville theory.

Abstract

We find a simple relation between the first subleading terms in the asymptotic expansion of the metric field in AdS$_3$, obeying the Brown-Henneaux boundary conditions, and the stress tensor of the underlying Liouville theory on the boundary. We can also provide an more explicit relation between the bulk metric and the boundary conformal field theory when it is described in terms of a free field with a background charge.