Harmonic analysis of 2d CFT partition functions

TL;DR

This work develops a harmonic-analysis framework for 2d CFT partition functions by performing a Roelcke-Selberg decomposition on the modular fundamental domain. It first applies the method to Narain lattice CFTs, revealing how the primary partition functions separate into moduli-averaged (Eisenstein) pieces and moduli-dependent fluctuations, with explicit results at low central charge and surprising links to Maass cusp forms. For generic CFTs, the authors show how to render partition functions square-integrable via modular completion, interpret the resulting spectral data in gravity-language terms (including half-wormhole analogies), and establish a spectral-determinacy result (fixing the full spectrum from light, scalar, and spin-1 data under a non-degeneracy assumption). The analysis uncovers deep connections between modular geometry, number theory (cusp forms, L-functions), and AdS$_3$ physics, and suggests concrete avenues to bound scalar gaps and understand chaos in CFT spectra through spectral data. Overall, the paper provides a rigorous, modular-invariant toolkit to extract and interpret the operator content of 2d CFTs and their holographic implications.

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.

Figures (1)

• Figure 1: In a 2d CFT, the full Virasoro primary spectrum is determined by the primary spectrum in blue. Circles represent the unitarity bound, $\Delta\geq j$, where $\Delta=h+\overline{h}$ and $j=|h-\overline{h}|$. This statement assumes that the cuspidal eigenspectrum is non-degenerate, an unproven but widely held property of $SL(2,\mathbb{Z})$. The $j=1$ data may, subject to a further mild assumption about Maass cusp forms, be replaced by the data of any fixed integer spin $j>0$ without affecting this conclusion. This result is explained in Section \ref{['secspecdet']}.