Rademacher Expansions and the Spectrum of 2d CFT

TL;DR

This paper applies Rademacher expansions from number theory to analyze the spectrum of 2D conformal field theories, deriving a spectral density expression constrained by modular invariance, and addressing issues like negative density of states.

Contribution

It introduces a Rademacher expansion-based formula for the spectral density of 2D CFTs with no extended chiral algebra and c>1, linking light spectrum data to the full spectrum.

Findings

Derived a convergent Rademacher expansion for spectral density

Matched the expansion with existing Poincare constructions for finite light operators

Proposed a scenario to resolve negative density of states in pure gravity duals

Abstract

A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and $c>1$. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin $j \neq 0$. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.