Bounds on spectral gaps of Hyperbolic spin surfaces

Journal Article · Thu Jan 23 00:00:00 EST 2025 · Journal of the Association for Mathematical Research

DOI:https://doi.org/10.56994/JAMR.003.001.003· OSTI ID:3005952

Gesteau, Elliott ^[1]; Pal, Sridip ^[1]; Simmons–Duffin, David ^[1]; Xu, Yixin ^[1]

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1. California Institute of Technology, Pasadena, CA (United States)

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.

• Research Organization: California Institute of Technology, Pasadena, CA (United States)
• Sponsoring Organization: USDOE Office of Science (SC), High Energy Physics (HEP)
• Grant/Contract Number: SC0011632
• OSTI ID: 3005952
• Journal Information: Journal of the Association for Mathematical Research, Journal Name: Journal of the Association for Mathematical Research Journal Issue: 1 Vol. 3; ISSN 2998-4114
• Publisher: Association for Mathematical Research
• Country of Publication: United States
• Language: English

Related Subjects

Spectral theory
automorphic representations
eigenvalue problems on manifolds
harmonic analysis on homogeneous spaces
representation theoretic methods