Automorphic Spectra and the Conformal Bootstrap
Communications of the American Mathematical Society(2021)
摘要
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 PSL_2(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≤ 3.8388976481,
while the Bolza surface has λ_1≈ 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.
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