A theory of reparameterizations for AdS$_3$ gravity

TL;DR

This paper develops a boundary quantum field theory for AdS$_3$ gravity, based on reparametrizations and coadjoint orbits of the Virasoro group, providing new tools for computing Virasoro blocks and understanding boundary gravitons.

Contribution

It introduces a novel boundary QFT for AdS$_3$ gravity as a path integral over Virasoro coadjoint orbits, connecting to Schwarzian theory and enabling large $c$ computations.

Findings

The boundary theory is ultraviolet-complete and non-modular invariant.

The torus partition function equals the vacuum Virasoro character, exact at one-loop.

The theory reduces to Schwarzian gravity upon compactification.

Abstract

We rewrite the Chern-Simons description of pure gravity on global AdS$_3$ and on Euclidean BTZ black holes as a quantum field theory on the AdS boundary. The resulting theory is (two copies of) the path integral quantization of a certain coadjoint orbit of the Virasoro group, and it should be regarded as the quantum field theory of the boundary gravitons. This theory respects all of the conformal field theory axioms except one: it is not modular invariant. The coupling constant is $1/c$ with $c$ the central charge, and perturbation theory in $1/c$ encodes loop contributions in the gravity dual. The QFT is a theory of reparametrizations analogous to the Schwarzian description of nearly AdS$_2$ gravity, and has several features including: (i) it is ultraviolet-complete; (ii) the torus partition function is the vacuum Virasoro character, which is one-loop exact by a localization argument; (iii) it reduces to the Schwarzian theory upon compactification; (iv) it provides a powerful new tool for computing Virasoro blocks at large $c$ via a diagrammatic expansion. We use the theory to compute several observables to one-loop order in the bulk, including the "heavy-light" limit of the identity block. We also work out some generalizations of this theory, including the boundary theory which describes fluctuations around two-sided eternal black holes.