Bounds on the density of states and the spectral gap in CFT${}_2$

Abstract

We improve the recently discovered upper and lower bounds on the $O(1)$ correction to the Cardy formula for the density of states integrated over an energy window (of width $2\delta$), centered at high energy in 2 dimensional conformal field theory. We prove optimality of the lower bound for $\delta \to 1^{-}$. We prove a conjectured upper bound on the asymptotic gap between two consecutive Virasoro primaries for a central charge greater than $1,$ demonstrating it to be $1.$ Furthermore, a systematic method is provided to establish a limit on how tight the bound on the $O(1)$ correction to the Cardy formula can be made using bandlimited functions. The techniques and the functions used here are of generic importance whenever the Tauberian theorems are used to estimate some physical quantities.

arXiv commentv2: 26 pages, 10 figures, optimality of the lower bound included, new section on connection to sphere packing v3: approximates the published version, reference added

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• High Energy Modular Bootstrap, Global Symmetries and Defects
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• A Basis of Analytic Functionals for CFTs in General Dimension
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• EFT Asymptotics: the Growth of Operator Degeneracy
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• Discreteness and Integrality in Conformal Field Theory
• Hartle-Hawking state and its factorization in 3d gravity
• Lightcone Modular Bootstrap and Tauberian Theory: A Cardy-like Formula for Near-extremal Black Holes
• Universal Dynamics in Non-Orientable CFT${}_2$
• Universal fine grained asymptotics of free and weakly coupled Quantum Field Theory