## High-Energy Conformal Bootstrap and Tauberian Theory

Baurzhan Mukhametzhanov
Member, School of Natural Sciences, IAS
November 1, 2019

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. Finally, we identify a new universal