# Modular Invariance, Tauberian Theorems, and Microcanonical Entropy

## TL;DR

This paper establishes rigorous bounds on operator counts and entropy in 2D conformal field theories using modular invariance and tauberian theorems, confirming the Cardy formula and revealing universal contributions.

## Contribution

It introduces new universal bounds and error estimates for microcanonical entropy and operator spacing, extending the understanding of modular invariance in 2D CFTs.

## Key findings

- Derived bounds on operator counts at high energies
- Confirmed the Cardy formula with optimal error estimates
- Identified a universal contribution to entropy controlled by central charge

## Abstract

We analyze modular invariance drawing inspiration from tauberian theorems. Given a modular invariant partition function with a positive spectral density, we derive lower and upper bounds on the number of operators within a given energy interval. They are most revealing at high energies. In this limit we rigorously derive the Cardy formula for the microcanonical entropy together with optimal error estimates for various widths of the averaging energy shell. We identify a new universal contribution to the microcanonical entropy controlled by the central charge and the width of the shell. We derive an upper bound on the spacings between Virasoro primaries. Analogous results are obtained in holographic 2d CFTs. We also study partition functions with a UV cutoff. Control over error estimates allows us to probe operators beyond the unity in the modularity condition. We check our results in the 2d Ising model and the Monster CFT and find perfect agreement.