IISER MOHALI

Abstract: A classical problem in spectral geometry is to understand how the geometry and, for arithmetic quotients, the arithmetic structure of a hyperbolic manifold constrains the spectrum of its Laplacian. A central example is Selberg’s eigenvalue conjecture, which predicts that for congruence arithmetic hyperbolic surfaces there are no nonzero Laplace eigenvalues below $1/4$. In this talk, I will describe how ideas originating in conformal field theory in theoretical physics provide a new approach to such questions.
Let $\Gamma$ be a cocompact lattice in $G = \mathrm{PSL}_2(\mathbb{R})$. The decomposition of $L^2(\Gamma\backslash G)$ into irreducible representations encodes the Laplace spectrum of the hyperbolic surface $\Gamma\backslash \mathbb{H}^2$, while $G$-invariant trilinear functionals encode the trilinear periods of Laplace eigenfunctions. I will explain how consistency of different spectral decompositions of products of automorphic forms (i.e. vectors inside $L^2(\Gamma\backslash G)$) leads to a system of identities closely analogous to what is known as the crossing equations of the conformal bootstrap program in conformal field theory.

Combined with positivity and semidefinite programming, these identities give rigorous, nearly sharp bounds on Laplace eigenvalues. I will discuss the bass-note spectrum (a question spiritually similar to Selberg’s eigenvalue conjecture) and then describe a recent application of the same bootstrap framework to obtain Weyl bounds for trilinear periods and, in the arithmetic setting, triple-product $L$-functions. I will conclude by briefly mentioning extensions to spectral bounds for hyperbolic $3$-manifolds.

No background in conformal field theory will be assumed.