Automorphic Spectra and the Conformal Bootstrap

TL;DR

This paper introduces a novel method combining automorphic forms and semi-definite programming to rigorously bound Laplacian spectra of hyperbolic surfaces, nearly matching known examples and applicable to higher dimensions.

Contribution

It develops a new approach inspired by conformal bootstrap to constrain spectral gaps of hyperbolic surfaces and orbifolds using representation theory and optimization techniques.

Findings

Bound on genus-2 surfaces: λ₁ ≤ 3.8389 (close to Bolza surface)

Determined spectral gaps for all hyperbolic 2-orbifolds

Method can be extended to higher-dimensional hyperbolic manifolds

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 $\mathrm{PSL}_2(\mathbb{R})$ 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 $\lambda_1\leq 3.8388976481$, while the Bolza surface has $\lambda_1\approx 3.838887258$. 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.