The Large-$c$ Virasoro Identity Block is a Semi-Classical Liouville Correlator

TL;DR

This paper demonstrates that the large-c identity block in a mixed heavy-light correlator can be understood as a semi-classical Liouville correlator on a geometry shaped by heavy operators, linking CFT calculations to semi-classical gravity.

Contribution

It provides an analytical derivation showing the identity block corresponds to a Liouville correlator on a geometry generated by heavy operators, connecting CFT and semi-classical AdS$_3$ gravity.

Findings

The background geometry solves the Liouville equation with heavy operator sources.

The light sector of the identity block matches the Liouville correlator in the semi-classical limit.

This approach captures the essence of Einstein gravity as a dynamical geometry in AdS/CFT.

Abstract

It will be shown analytically that the light sector of the identity block of a mixed heavy-light correlator in the large central charge limit is given by a correlation function of light operators on an effective background geometry. This geometry is generated by the presence of the heavy operators. It is shown that this background geometry is a solution to the Liouville equation of motion sourced by corresponding heavy vertex operators and subsequently that the light sector of the identity block matches the Liouville correlation function in the semi-classical limit. This method effectively captures the spirit of Einstein gravity as a theory of dynamical geometry in AdS/CFT. The reason being that Liouville theory is closely related to semi-classical asymptotically AdS$_3$ gravity.