2023/11/22 by Elliott Gesteau, Sridip Pal, Gesteau, Elliott +5 · 1 citation
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Number Theory (math.NT) #Quantum chaos and dynamical systems #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2311.13330
openalex publication_date 2023/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We describe a method for constraining Laplacian and Dirac spectra of two dimensional compact orientable hyperbolic spin manifolds and orbifolds. The key ingredient is an infinite family of identities satisfied by the spectra. These spectral identities follow from the consistency between 1) the spectral decomposition of functions on the spin bundle into irreducible representations of SL(2,ℝ) and 2) associativity of pointwise multiplication of functions. Applying semidefinite programming methods to our identities produces rigorous upper bounds on the Laplacian spectral gap as well as on the Dirac spectral gap conditioned on the former. In several examples, our bounds are nearly sharp; a numerical algorithm based on the Selberg trace formula shows that the [0;3,3,5] orbifold, a particular surface with signature [1;3], and the Bolza surface nearly saturate the bounds at genus 0, 1 and 2 respectively. Under additional assumptions on the number of harmonic spinors carried by the spin-surface, we obtain more restrictive bounds on the Laplacian spectral gap. In particular, these bounds apply to hyperelliptic surfaces. We also determine the set of Laplacian spectral gaps attained by all compact orientable two-dimensional hyperbolic spin orbifolds. We show that this set is upper bounded by 12.13798; this bound is nearly saturated by the [0;3,3,5] orbifold, whose first non-zero Laplacian eigenvalue is λ(0)1≈ 12.13623.