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 $λ_1\leq 3.8388976481$, while the Bolza surface has $λ_1\approx 3.838887258$. 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.

Figures (5)

• Figure 1: The operators that we construct for a hyperbolic orbifold are labeled by points on the Riemann sphere. Depending on the type of the operator, it is labeled by a point in the upper or the lower hemisphere, or by a point on the equator.
• Figure 2: The conjectured structure of the set of the eigenvalues $\lambda_1$ attained in hyperbolic orbifolds. There are several discrete points at large $\lambda_1$ and a continuum below.
• Figure 3: A schematic representation of the space $\Gamma\backslash G$ as a principal $K$-bundle over base $X$. Here $K$ is the circle group and $X=\Gamma\backslash G/K$ is a hyperbolic orbifold.
• Figure 4: Artist's impression of a $[0;k_1,k_2,k_3]$ orbifold (not to scale).
• Figure 5: Plot of the extremal functional $P_\alpha^n(\lambda)$ for $n=6$ and $\Lambda=41$, multiplied by $e^{-\lambda/50}$ for clarity. Besides the simple zero at $\lambda\approx 44.88835$, the functional has a sequence of local minima (almost double zeros). The location of the local minima are in a very good agreement with higher eigenvalues of the $[0;2,3,7]$ orbifold.

Theorems & Definitions (36)

• Theorem 1.1
• Definition 2.1: Coherent state
• Definition 2.2: Correlator
• Theorem 2.3
• Theorem 3.1: See, e.g. Knapp
• Proposition 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• Proposition 3.5