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.

arXiv commentv2: various improvements, especially in Section 3.9; v3: published version

References in corpus (27)

• Structure Constants and Conformal Bootstrap in Liouville Field Theory
• Bounding scalar operator dimensions in 4D CFT
• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Solving the 3D Ising Model with the Conformal Bootstrap
• Conformal Partial Waves and the Operator Product Expansion
• Precision Islands in the Ising and $O(N)$ Models
• Bootstrapping the O(N) Vector Models
• Analytic Continuation of Liouville Theory
• New upper bounds on sphere packings I
• Bootstrapping Mixed Correlators in the 3D Ising Model
• The sphere packing problem in dimension 24
• The sphere packing problem in dimension 8
• Carving Out the Space of 4D CFTs
• The subconvexity problem for $\GL_{2}$
• Radial Coordinates for Conformal Blocks
• Bounds in 4D Conformal Field Theories with Global Symmetry
• Analyticity and Unitarity for Cosmological Correlators
• Analytic Bounds and Emergence of ${\text{AdS}}_2$ Physics from the Conformal Bootstrap
• The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices
• Sphere Packing and Quantum Gravity
• Searching for gauge theories with the conformal bootstrap
• An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces
• Bootstrap Bounds on Closed Einstein Manifolds
• Bootstrapping Closed Hyperbolic Surfaces
• Bootstrap Bounds on Closed Hyperbolic Manifolds
• Properties of Eigenvalues on Riemann Surfaces with Large Symmetry Groups
• Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian

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