Nearly AdS$_2$ holography in quantum CGHS model

TL;DR

This paper revisits the quantum-corrected CGHS model, revealing how nearly AdS$_2$ geometry emerges and clarifying the role of boundary gravitons and the Virasoro/Schwarzian correspondence in holography.

Contribution

It demonstrates the emergence of nearly AdS$_2$ geometry in the quantum CGHS model and establishes the Virasoro/Schwarzian correspondence at second order perturbation.

Findings

Quantum CGHS model admits nearly AdS$_2$ vacuum with constant dilaton.

Boundary graviton dynamics are described by a Schwarzian-type theory.

Virasoro constraints are equivalent to boundary graviton equations on the vacuum.

Abstract

In light of recent developments in nearly AdS$_2$ holography, we revisit the semi-classical version of two-dimensional dilaton gravity proposed by Callan, Giddings, Harvey, and Strominger (CGHS) in the early 90's. In distinction to the classical model, the quantum corrected CGHS model has an AdS$_2$ vacuum with a constant dilaton. By turning on a non-normalizable mode of the Liouville field, i.e. the conformal mode of the $2d$ gravity, the explicit breaking of the scale invariance renders the AdS$_2$ vacuum nearly AdS$_2$. As a consequence, there emerges an effective one-dimensional Schwarzian-type theory of pseudo Nambu-Goldstone mode - the boundary graviton - on the boundary of the nearly AdS$_2$ space. We go beyond the linear order perturbation in non-normalizable fluctuations of the Liouville field and work up to the second order. As a main result of our analysis, we clarify the role of the boundary graviton in the holographic framework and show the Virasoro/Schwarzian correspondence, namely that the $2d$ bulk Virasoro constraints are equivalent to the graviton equation of motion of the $1d$ boundary theory, at least, on the $SL(2,R)$ invariant vacuum.