A universal sum over topologies in 3d gravity

TL;DR

The paper develops a structured program to understand the sum over bulk topologies in AdS$_3$ gravity as a statistical ensemble of CFT$_2$ data. It shows that handling crossing symmetry and typicality naturally leads to a large class of on-shell hyperbolic bulk manifolds generated by a gravitational machine built from cylinder and Dehn surgeries, including Euclidean wormholes, twisted I-bundles, and handlebody-knots. While the handlebody sum alone enforces crossing symmetry, typicality requires many non-handlebody saddles, rendering pure 3d gravity non-minimal and leaving room for alternative ensembles. The work further demonstrates that the gravitational machine preserves hyperbolicity but does not generate all hyperbolic 3-manifolds (e.g., acylindrical ones), and discusses implications for unitarity, spectral statistics, and connections to JT gravity and tensor-model approaches. Overall, it proposes a concrete, topological mechanism for enforcing boundary statistical constraints in the bulk while highlighting open questions about completeness and uniqueness of the bulk dual.

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.

Figures (8)

• Figure 1: Example of a crossing transformation $\gamma$ on a genus-$2$ Riemann surface. The two sets of contractible cycles (the thick closed curves) lead to two distinct genus-$2$ handlebodies upon filling in the bulk.
• Figure 2: In section \ref{['sec:minimal completion']}, we show that index contractions, in a statistical ensemble of heavy OPE coefficients, correspond to 3-manifold surgery where the gluing surface is a cylinder. Combining this surgery with crossing transformations generates many non-handlebodies.
• Figure 3: Examples of non-handlebodies with genus-2 boundary. These non-handlebodies are all necessary in the sum over topologies, in order for the statistical description of the CFT to be consistent. $a)$ Complement of a handlebody-knot in $\mathrm{S}^3$. $b)$ Twisted $I$-bundle. $c)$ Dehn surgery on the complement of a handlebody embedded in $\mathrm{S}^3$.
• Figure 4: The three basic Moore-Seiberg moves, acting on embedded bordered surfaces in a given pair-of-pants decomposition of a Riemann surface. From left to right, these are the braiding transformation, the fusion move and the modular S-transform.
• Figure 5: For a closed Riemann surface of genus $g\geqslant 2$, the mapping class group $\text{MCG}(\Sigma_g)$ is generated by Dehn twists along the $2g+1$ cycles drawn in red.