Automorphic Spectra and the Conformal Bootstrap

Petr Kravchuk; Dalimil Mazáč; Sridip Pal

doi:10.1090/cams/26

Automorphic Spectra and the Conformal Bootstrap

Petr Kravchuk, Dalimil Mazáč , Sridip Pal

Institute for Advanced Study

Research output: Contribution to journal › Article › peer-review

Abstract

We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of representation theory of PSL2(R) and semi-definite programming, the method yields rigorous upper bounds on the Laplacian spectral gap. In several examples, the bound is nearly sharp. For instance, our bound on all genus-2 surfaces is λ₁≤3.8388976481, while the Bolza surface has λ₁≈3.838887258. The bounds also allow us to determine the set of spectral gaps attained by all hyperbolic 2-orbifolds. Our methods can be generalized to higher-dimensional hyperbolic manifolds and to yield stronger bounds in the two-dimensional case. The ideas were closely inspired by modern conformal bootstrap.

Original language	English
Pages (from-to)	1-63
Number of pages	63
Journal	Communications of the American Mathematical Society
Volume	4
Early online date	17 Jan 2024
DOIs	https://doi.org/10.1090/cams/26
Publication status	E-pub ahead of print - 17 Jan 2024

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https://arxiv.org/abs/2111.12716Licence: CC BY

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AB - We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of representation theory of PSL2(R) and semi-definite programming, the method yields rigorous upper bounds on the Laplacian spectral gap. In several examples, the bound is nearly sharp. For instance, our bound on all genus-2 surfaces is λ1≤3.8388976481, while the Bolza surface has λ1≈3.838887258. The bounds also allow us to determine the set of spectral gaps attained by all hyperbolic 2-orbifolds. Our methods can be generalized to higher-dimensional hyperbolic manifolds and to yield stronger bounds in the two-dimensional case. The ideas were closely inspired by modern conformal bootstrap.

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JO - Communications of the American Mathematical Society

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Automorphic Spectra and the Conformal Bootstrap

Abstract

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