Modular Invariance, Tauberian Theorems, and Microcanonical Entropy

Understand

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.