Bootstrap Bounds on Closed Hyperbolic Manifolds
arXiv:2107.09674 · doi:10.1007/JHEP02(2022)025
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 for consecutive nonzero eigenvalues of the Laplace-Beltrami operator on closed orientable hyperbolic surfaces. We give examples of genus-2 surfaces that nearly saturate some of these bounds. To derive the consistency conditions, we make use of a transverse-traceless decomposition for symmetric tensors of arbitrary rank.
arXiv comment32 pages, 10 figures
References in corpus (27)
- Bounding scalar operator dimensions in 4D CFT
- The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
- Solving the 3D Ising Model with the Conformal Bootstrap
- Bootstrapping Mixed Correlators in the 3D Ising Model
- Carving Out the Space of 4D CFTs
- Compact Hyperbolic Extra Dimensions: Branes, Kaluza-Klein Modes and Cosmology
- Universal Constraints on Conformal Operator Dimensions
- Bootstrapping 3D Fermions
- Rigorous Limits on the Interaction Strength in Quantum Field Theory
- A Semidefinite Program Solver for the Conformal Bootstrap
- Decoding a Three-Dimensional Conformal Manifold
- Modular Constraints on Calabi-Yau Compactifications
- Scalar CFTs and Their Large N Limits
- Hyperbolic compactification of M-theory and de Sitter quantum gravity
- Power spectrum of the geodesic flow on hyperbolic manifolds
- Unitarization from Geometry
- Computation of eigenmodes on a compact hyperbolic 3-space
- Kaluza-Klein Towers on General Manifolds
- Scaling the semidefinite program solver SDPB
- N=4 Superconformal Bootstrap of the K3 CFT
- Random hyperbolic surfaces of large genus have first eigenvalues greater than
- Pour toute surface hyperbolique de genre , $λ_{2g-2}>\frac 14}$
- An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces
- Bootstrap Bounds on Closed Einstein Manifolds
- Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds
- On the eigenmodes of compact hyperbolic 3-manifolds
- Properties of Eigenvalues on Riemann Surfaces with Large Symmetry Groups