Holography for bulk states in 3D quantum gravity

TL;DR

This paper explores the holographic description of bulk states in 3D quantum gravity, linking gravitational solutions to classical Virasoro blocks and Liouville theory, with implications for understanding quantum gravity in lower dimensions.

Contribution

It introduces a formulation where bulk gravitational equations reduce to Liouville equations with sources, connecting wavefunctions to Virasoro blocks and extending Polyakov's conjecture.

Findings

Bulk states described by classical Virasoro vacuum blocks.

Reduction of gravity equations to decoupled Liouville equations.

Modified results for closed universes with compact slices.

Abstract

In this work we discuss the holographic description of states in the Hilbert space of (2+1)-dimensional quantum gravity, living on a time slice in the bulk. We focus on pure gravity coupled to pointlike sources for heavy spinning particles. We develop a formulation where the equations for the backreacted metric reduce to two decoupled Liouville equations with delta-function sources under pseudosphere boundary conditions. We show that both the semiclassical wavefunction and the gravity solution are determined by a universal object, namely a classical Virasoro vacuum block on the sphere. In doing so we derive a version of Polyakov's conjecture, as well as an existence criterion, for classical Liouville theory on the pseudosphere. We also discuss how some of these results are modified when considering closed universes with compact spatial slices.