[论文解读] A universal sum over topologies in 3d gravity

Alexandre Belin, Scott Collier+3 more|arXiv (Cornell University)|Jan 12, 2026

Noncommutative and Quantum Gravity Theories被引用 0

一句话总结

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.

研究动机与目标

• Define a statistical ensemble for 2D CFT data dual to pure AdS3 gravity at large central charge.
• Explain how the bulk sum over topologies is organized to satisfy crossing symmetry, modular invariance, and typicality.
• Determine which bulk manifolds are required for a consistent gravitational path integral and how non-handlebodies arise.
• Show that the gravitational machine produces hyperbolic, non-handlebody manifolds and analyze their role in the ensemble.
• Illustrate the framework with explicit examples like Euclidean wormholes, twisted I-bundles, and handlebody-knots.

提出的方法

• Formulate a statistical distribution µ over CFT data consisting of primary dimensions and OPE coefficients.
• Decompose genus-g partition functions into Virasoro conformal blocks and express averages via conformal-block graphs.
• Impose crossing symmetry and modular invariance on the average partition functions across all genera.
• Use crossing kernels of the Virasoro TQFT to translate bootstrap constraints into integral equations for moments of µ.
• Introduce typicality inspired by Eigenstate Thermalization Hypothesis to fix non-Gaussian moments and index contractions.
• Define and apply the gravitational machine, a set of surgery moves that generate non-handlebody topologies from seed handlebodies.

实验结果

研究问题

• RQ1 What is the role of the bulk sum over topologies in encoding a universal statistical ensemble for boundary CFT data?
• RQ2 How do crossing symmetry, modular invariance, and typicality constrain the moments of the CFT data distribution µ?
• RQ3 Can non-handlebody bulk manifolds be systematically generated and required for a consistent gravitational path integral?
• RQ4 What is the nature of the bulk manifolds produced by the gravitational machine, and are they all hyperbolic or cylindrical?
• RQ5 How do Euclidean wormholes, twisted I-bundles, and handlebody-knots fit into the proposed framework?

主要发现

• The handlebody sum can render the average genus-g partition function crossing symmetric, and non-Gaussian OPE statistics are required for full crossing symmetry across all genera.
• The gravitational machine generates a large class of non-handlebody hyperbolic 3-manifolds from seed handlebodies via three basic surgery moves, ensuring compatibility with typicality and crossing symmetry.
• All manifolds produced by the machine are hyperbolic and cylindrical, i.e., contain embedded essential cylinders, but not all hyperbolic non-handlebodies are generated.
• The approach provides a rigorous link between boundary OPE statistics and on-shell bulk saddles, implemented in Virasoro TQFT to finite-c partition functions.
• Explicit examples including Euclidean wormholes, twisted I-bundles, and handlebody-knots illustrate the generated topology space and its implications for the gravitational path integral.