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Journal of High Energy Physics

, 2019:261 | Cite as

Modular invariance, tauberian theorems and microcanonical entropy

  • Baur MukhametzhanovEmail author
  • Alexander Zhiboedav
Open Access
Regular Article - Theoretical Physics
  • 7 Downloads

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.

Keywords

Conformal Field Theory AdS-CFT Correspondence 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsHarvard UniversityCambridgeU.S.A.
  2. 2.CERN, Theoretical Physics DepartmentGeneva 23Switzerland

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