TQFT gravity and ensemble holography

TL;DR

This paper proposes a general holographic duality between 3d topological quantum field theory (TQFT) gravity, summed over topologies, and an ensemble of 2d conformal field theories (CFTs), using genus reduction techniques.

Contribution

It provides a generic derivation of TQFT gravity and ensemble holography without relying on explicit partition functions, ensuring unitarity and including all topologies in the bulk sum.

Findings

Duality trivializes at very high genus Riemann surfaces.

Bulk sum reduces to a finite sum over topologies with handlebodies.

Boundary ensemble weights are shown to be equal in Abelian Chern-Simons theory.

Abstract

We outline a general derivation of holographic duality between "TQFT gravity" - the path integral of a 3d TQFT summed over different topologies - and an ensemble of boundary 2d CFTs. The key idea is to place the boundary ensemble on a Riemann surface of very high genus, where the duality trivializes. The duality relation at finite genus is then obtained by genus reduction. Our derivation is generic and does not rely on an explicit form of the bulk or boundary partition functions. It guarantees unitarity and suggests that the bulk sum should include all possible topologies. In the case of Abelian Chern-Simons theory with compact gauge group we show that the weights of the boundary ensemble are equal, while the bulk sum reduces to a finite sum over equivalence classes of topologies, represented by handlebodies with possible line defects.