Harmonic analysis of 2d CFT partition functions

TL;DR

This paper employs harmonic analysis on the moduli space of 2D CFTs to decompose partition functions, revealing insights into Narain theories, ensemble averages, and connections to AdS$_3$ gravity, with implications for the operator spectrum.

Contribution

It introduces a harmonic analysis framework for 2D CFT partition functions, extending spectral theory methods to general theories and linking to gravitational duals.

Findings

Decomposition of Narain lattice partition functions into Laplacian eigenfunctions.

Identification of properties of Narain theories through spectral analysis.

Operator spectrum determined by specific degeneracies.

Abstract

We apply the theory of harmonic analysis on the fundamental domain of $SL(2,\mathbb{Z})$ to partition functions of two-dimensional conformal field theories. We decompose the partition function of $c$ free bosons on a Narain lattice into eigenfunctions of the Laplacian of worldsheet moduli space $\mathbb H/SL(2,\mathbb Z)$, and of target space moduli space $O(c,c;\mathbb Z)\backslash O(c,c;\mathbb R)/O(c)\times O(c)$. This decomposition manifests certain properties of Narain theories and ensemble averages thereof. We extend the application of spectral theory to partition functions of general two-dimensional conformal field theories, and explore its meaning in connection to AdS$_3$ gravity. An implication of harmonic analysis is that the local operator spectrum is fully determined by a certain subset of degeneracies.