Crossing symmetry of OPE statistics

TL;DR

The work addresses how crossing symmetry emerges in an ensemble of 2D CFTs at large $c$ defined by 3D gravity, focusing on the correlated moments of OPE coefficients. It introduces pseudo-hyperbolic manifolds and leverages Virasoro TQFT, fusion kernels, and modular kernels to construct a gravity-based demonstration of both spherical and modular crossing across OPE channels. By linking channel changes to geometric surgeries and using joining/splitting procedures, the paper shows how disconnected contributions can be reorganized into pseudo-hyperbolic geometries that reproduce crossing across the ensemble. The results suggest a self-consistent, gravity-backed picture of crossing symmetry in ensembles, with potential extensions to BCFTs and finite-$c$ regimes.

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.