Automorphic Spectra and the Conformal Bootstrap
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| Format: | Preprint |
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2021
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| author | Kravchuk, Petr Mazac, Dalimil Pal, Sridip |
| author_facet | Kravchuk, Petr Mazac, Dalimil Pal, Sridip |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_12716 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Automorphic Spectra and the Conformal Bootstrap Kravchuk, Petr Mazac, Dalimil Pal, Sridip High Energy Physics - Theory Mathematical Physics Representation Theory Spectral Theory 58J50, 11F70, 43A85, 81T05, 58C40 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. |
| title | Automorphic Spectra and the Conformal Bootstrap |
| topic | High Energy Physics - Theory Mathematical Physics Representation Theory Spectral Theory 58J50, 11F70, 43A85, 81T05, 58C40 |
| url | https://arxiv.org/abs/2111.12716 |