Crossing symmetry of OPE statistics

TL;DR

This paper investigates the crossing symmetry of large-$c$ 2D CFTs via 3D gravity, revealing strong correlations among OPE coefficient moments and introducing pseudo-hyperbolic manifolds to demonstrate consistency across OPE channels.

Contribution

It introduces the concept of pseudo-hyperbolic manifolds and explains their role in ensuring crossing symmetry in the ensemble of large-$c$ 2D CFTs.

Findings

OPE coefficient moments are strongly correlated.

Pseudo-hyperbolic manifolds relate to hyperbolic partition functions.

Crossing symmetry is guaranteed by these correlations and manifold structures.

Abstract

We study the crossing symmetry of the ensemble of large-$c$ 2D CFTs defined through 3D gravity. A central observation is that statistical moments of OPE coefficients are not independent; rather, lower and higher moments are strongly correlated. Using Virasoro TQFT, we clarify how these correlations arise and how they guarantee consistency across OPE channels. Our analysis introduces the new notion of pseudo-hyperbolic manifolds, which are a certain class of non-hyperbolic manifolds whose partition functions are nevertheless related to those of hyperbolic ones. These manifolds serve as bridges that help manifest the crossing symmetry of the CFT ensemble.