Bootstrap bounds on closed hyperbolic manifolds
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Peer-reviewed
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Abstract
The eigenvalues of the Laplace-Beltrami 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 Laplace-Beltrami
operator in terms of the smaller eigenvalue, an upper bound on the smallest
eigenvalue of the rough Laplacian on symmetric, transverse-traceless, rank-2
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
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1029-8479