A theory of reparameterizations for AdS$_3$ gravity

TL;DR

The paper shows that pure AdS$_3$ gravity can be reformulated as two copies of a boundary quantum field theory obtained from the geometric quantization of coadjoint orbits of the Virasoro group. This boundary theory is ultraviolet complete and equivalent to a Weyl/anomaly-driven reparameterization dynamics, not modular invariant, with $1/c$ acting as the coupling. It yields exact or one-loop exact results for torus and sphere partition functions, reproduces Virasoro blocks in the large-$c$ limit (including heavy-light and light-light regimes), and provides a consistent framework for analyzing BTZ black holes and conical defects. The approach aligns with prior geometric quantization and localization results while offering a streamlined, diagrammatic route to CFT data and boundary-graviton dynamics, and it opens avenues for supersymmetric extensions and higher-genus observables.

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.

Figures (4)

• Figure 1: A graphical representation of the Kruskal extended BTZ black hole \ref{['E:kruskal']}. Here $u$ and $v$ are the standard light-cone coordinates in the plane and $\theta$ is suppressed. The solid black lines are the asymptotic AdS boundaries, the dashed grey lines are the horizons, and the dotted red lines are the past and future singularities. The blue lines indicate curves of constant $z$.
• Figure 2: The exchange diagram which computes the $O(1/C)$ correction to the identity block. The blobs refer to the bilocal operators, and the vertex factors $\mathcal{J}_{12}^{(1)}$ and $\mathcal{J}_{34}^{(1)}$ are the linear couplings of the reparameterized two-point function to the $\epsilon$ field and are given in \ref{['E:J1']}. The $\epsilon$ propagator is given in \ref{['E:eps2']}.
• Figure 3: The diagram with $n$ exchanges of the $\epsilon$ field between two bilocals.
• Figure 4: The one-loop diagram which gives the $O(1/C)$ correction to the identity block in the heavy-light limit. The coupling to the reparameterization field is given by $\frac{h_L^2}{2C}(\mathcal{J}_{12}'^{1}\cdot\epsilon)^2 + h_L \mathcal{J}_{12}'^{(2)}\cdot\epsilon$.