Rademacher Expansions and the Spectrum of 2d CFT

Abstract

A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and $c>1$. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin $j\ne 0$. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.

arXiv comment30 pages, 5 figures

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• Bootstrapping Non-Invertible Symmetries
• Harmonic analysis of 2d CFT partition functions
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• Beurling-Selberg Extremization and Modular Bootstrap at High Energies
• Wormholes and Spectral Statistics in the Narain Ensemble
• Narain to Narnia
• Comments on the holographic description of Narain theories
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• Evaluating one-loop string amplitudes
• Discreteness and Integrality in Conformal Field Theory
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• Scalar Modular Bootstrap and Zeros of the Riemann Zeta Function
• Holographic description of Narain CFTs and their code-based ensembles
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• Spinning Particle Geometries in AdS${}_3$/CFT${}_2$
• Rademacher expansion of modular integrals
• Three-dimensional quantum gravity according to ST modular bootstrap
• Aspects of three-dimensional higher-curvature gravities