Automorphic Spectra and the Conformal Bootstrap

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

P S L 2 ( R ) \mathrm {PSL}_2(\mathbb {R})

and semidefinite 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

λ 1 ≤ 3.8388976481 \lambda _1\leq 3.8388976481

, while the Bolza surface has

λ 1 ≈ 3.838887258 \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.