Bootstrap bounds on closed hyperbolic manifolds
Abstract
The eigenvalues of the LaplaceBeltrami operator and the integrals of products of eigenfunctions must satisfy certain consistency conditions on compact Riemannian manifolds. These consistency conditions are derived by using spectral decompositions to write quadruple overlap integrals in terms of products of triple overlap integrals in multiple ways. In this paper, we show how these consistency conditions imply bounds on the Laplacian eigenvalues and triple overlap integrals of closed hyperbolic manifolds, in analogy to the conformal bootstrap bounds on conformal field theories. We find an upper bound on the gap between two consecutive nonzero eigenvalues of the LaplaceBeltrami operator in terms of the smaller eigenvalue, an upper bound on the smallest eigenvalue of the rough Laplacian on symmetric, transversetraceless, rank2 tensors, and bounds on integrals of products of eigenfunctions and eigentensors. Our strongest bounds involve numerically solving semidefinite programs and are presented as exclusion plots. We also prove the analytic bound λ_{i+1} ≤ 1/2 + 3λ_{i} + √{λ_{i}^{2}+2 λ_{i}+1 /4 } for consecutive nonzero eigenvalues of the LaplaceBeltrami operator on closed orientable hyperbolic surfaces. We give examples of genus2 surfaces that nearly saturate some of these bounds. To derive the consistency conditions, we make use of a transversetraceless decomposition for symmetric tensors of arbitrary rank.
 Publication:

Journal of High Energy Physics
 Pub Date:
 February 2022
 DOI:
 10.1007/JHEP02(2022)025
 arXiv:
 arXiv:2107.09674
 Bibcode:
 2022JHEP...02..025B
 Keywords:

 Differential and Algebraic Geometry;
 Conformal Field Theory;
 Field Theories in Higher Dimensions;
 High Energy Physics  Theory;
 Mathematics  Differential Geometry;
 Mathematics  Spectral Theory
 EPrint:
 32 pages, 10 figures