Modular Invariance, Tauberian Theorems, and Microcanonical Entropy

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.

arXiv comment39 pages, 9 figures

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• Bounds on the density of states and the spectral gap in CFT${}_2$
• On Triality Defects in 2d CFT
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• Bound on asymptotics of magnitude of three point coefficients in 2D CFT
• Discreteness and Integrality in Conformal Field Theory
• AdS/BCFT from Conformal Bootstrap: Construction of Gravity with Branes and Particles
• Scalar Modular Bootstrap and Zeros of the Riemann Zeta Function
• Hartle-Hawking state and its factorization in 3d gravity
• 3d Gravity as a random ensemble
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• Conformal line defects at finite temperature
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• Running of the Number of Degrees of Freedom in Quantum Conformal Gravity