Automorphic Spectra and the Conformal Bootstrap
Abstract
We point out that the spectral geometry of hyperbolic manifolds provides a remarkably precise model of the modern conformal bootstrap. As an application, we use conformal bootstrap techniques to derive rigorous computerassisted upper bounds on the lowest positive eigenvalue $\lambda_1(X)$ of the LaplaceBeltrami operator on closed hyperbolic surfaces and 2orbifolds $X$. In a number of notable cases, our bounds are nearly saturated by known surfaces and orbifolds. For instance, our bound on all genus2 surfaces $X$ is $\lambda_1(X)\leq 3.8388976481$, while the Bolza surface has $\lambda_1(X)\approx 3.838887258$. Our methods can be generalized to higherdimensional hyperbolic manifolds and to yield stronger bounds in the twodimensional case.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.12716
 Bibcode:
 2021arXiv211112716K
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Representation Theory;
 Mathematics  Spectral Theory
 EPrint:
 53+9 pages, 5 figures