Aspects of Diffeomorphism and Conformal invariance in classical Liouville theory

TL;DR

This paper explores the relationship between diffeomorphism and conformal symmetries in two-dimensional classical Liouville theory, revealing how these symmetries are preserved or broken in different regions near black holes.

Contribution

It demonstrates the contrasting behavior of diffeomorphism and conformal invariance in classical Liouville theory near and away from black hole horizons.

Findings

Conformal invariance holds near the black hole horizon.

Diffeomorphism symmetry is broken at the horizon.

Away from the horizon, conformal symmetry is broken while diffeomorphism symmetry remains.

Abstract

The interplay between the diffeomorphism and conformal symmetries (a feature common in quantum field theories) is shown to be exhibited for the case of black holes in two dimensional classical Liouville theory. We show that although the theory is conformally invariant in the near horizon limit, there is a breaking of the diffeomorphism symmetry at the classical level. On the other hand, in the region away from the horizon, the conformal symmetry of the theory gets broken with the diffeomorphism symmetry remaining intact.