Automorphic Spectra and the Conformal Bootstrap

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 ${\lambda }_1\le 3.8388976481$, 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.

v2: various improvements, especially in Section 3.9; v3: published version

References in corpus (4)

• Bootstrapping Mixed Correlators in the 3D Ising Model
• Bounds in 4D Conformal Field Theories with Global Symmetry
• Bootstrapping Closed Hyperbolic Surfaces
• Properties of Eigenvalues on Riemann Surfaces with Large Symmetry Groups

Cited by in corpus (4)

• Bootstrapping the 3d Ising Stress Tensor
• Bootstrapping Closed Hyperbolic Surfaces
• Conformal Fields from Neural Networks
• RG flows in de Sitter: c-functions and sum rules