Holographic Equidistribution

TL;DR

This paper demonstrates that Hecke operators' equidistribution in large N limits simplifies the partition function to light states, offering a holographic interpretation as a sum over geometries.

Contribution

It applies an equidistribution theorem for Hecke operators to connect modular functions, conformal field theories, and holography, revealing a new perspective on large N limits.

Findings

Heavy sector contributions are integrated out in large N limits.

Partition functions reduce to Poincaré series of light states.

Holographic interpretation as a sum over semiclassical geometries.

Abstract

Hecke operators acting on modular functions arise naturally in the context of 2d conformal field theory, but in seemingly disparate areas, including permutation orbifold theories, ensembles of code CFTs, and more recently in the context of the AdS$_3$/RMT$_2$ program. We use an equidistribution theorem for Hecke operators to show that in each of these large $N$ limits, an entire heavy sector of the partition function gets integrated out, leaving only contributions from Poincar\'e series of light states. This gives an immediate holographic interpretation as a sum over semiclassical handlebody geometries. We speculate on further physical interpretations for equidistribution, including a potential ergodicity statement.