A universal sum over topologies in 3d gravity

TL;DR

This paper investigates the sum over topologies in 3D AdS quantum gravity, linking bulk manifold surgeries with boundary conformal bootstrap principles to understand the ensemble of geometries contributing to the gravitational path integral.

Contribution

It introduces a systematic approach using surgery moves on 3-manifolds that reflect boundary ensemble properties, revealing a broad class of manifolds consistent with crossing symmetry and typicality.

Findings

Identifies surgery moves generating on-shell hyperbolic manifolds

Shows the large variety of manifolds compatible with boundary ensemble constraints

Demonstrates the connection between 3-manifold topology and boundary statistical properties

Abstract

We explore the sum over topologies in AdS$_3$ quantum gravity and its relationship with the statistical interpretation of the boundary theory. We formulate a statistical version of the conformal bootstrap that systematizes the universal statistical properties of high-energy CFT$_2$ data. We identify a series of surgery moves on bulk manifolds that precisely reflect the requirements of typicality and crossing symmetry of the boundary ensemble. These surgery moves generate a large number of bulk manifolds that have to be included in any reasonable definition of the gravitational path integral. We show that this procedure generates only on-shell (hyperbolic) manifolds, although it does not produce all of them. These proofs rely on structure theorems of 3-manifolds, which non-trivially interact with the requirements of the statistical boundary ensemble. We illustrate the application of this procedure with many examples, such as Euclidean wormholes, twisted $I$-bundles and handlebody-knots. Our findings reveal a large space of possible choices of which manifolds can be included in the gravitational path integral, reflecting a wide range of possible statistical ensembles consistent with crossing symmetry and typicality.