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Automorphic spectra and the conformal bootstrap

Journal Article · · Communications of the American Mathematical Society
DOI:https://doi.org/10.1090/cams/26· OSTI ID:2587152
 [1];  [2];  [3]
  1. Institute for Advanced Study, Princeton, NJ (United States); King’s College London (United Kingdom)
  2. Institute for Advanced Study, Princeton, NJ (United States); Université Paris-Saclay, Gif-sur-Yvette (France)
  3. Institute for Advanced Study, Princeton, NJ (United States); California Institute of Technology, Pasadena, CA (United States)
+ Show Author Affiliations
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 PSL2(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 λ1 ≤ 3.8388976481, while the Bolza surface has λ1 ≈ 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.
Research Organization:
Institute for Advanced Study, Princeton, NJ (United States)
Sponsoring Organization:
USDOE
Grant/Contract Number:
SC0009988
OSTI ID:
2587152
Journal Information:
Communications of the American Mathematical Society, Journal Name: Communications of the American Mathematical Society Journal Issue: 1 Vol. 4; ISSN 2692-3688
Publisher:
American Mathematical Society (AMS)Copyright Statement
Country of Publication:
United States
Language:
English

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