Bootstrap Bounds on Closed Einstein Manifolds

Abstract

A compact Riemannian manifold is associated with geometric data given by the eigenvalues of various Laplacian operators on the manifold and the triple overlap integrals of the corresponding eigenmodes. This geometric data must satisfy certain consistency conditions that follow from associativity and the completeness of eigenmodes. We show that it is possible to obtain nontrivial bounds on the geometric data of closed Einstein manifolds by using semidefinite programming to study these consistency conditions, in analogy to the conformal bootstrap bounds on conformal field theories. These bootstrap bounds translate to constraints on the tree-level masses and cubic couplings of Kaluza-Klein modes in theories with compact extra dimensions. We show that in some cases the bounds are saturated by known manifolds.

42 pages, 17 figures; v2: minor changes; v3: corrected the value of the smallest Lichnerowicz eigenvalue of transverse-traceless rank-2 tensors on ${\mathbb{CP}}^n$ with $n>2$

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• Bootstrapping Mixed Correlators in the 3D Ising Model
• Numerical Calabi-Yau metrics
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• Massive Spin-2 Scattering Amplitudes in Extra-Dimensional Theories
• Sum Rules for Massive Spin-2 Kaluza-Klein Elastic Scattering Amplitudes

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• Bootstrapping Closed Hyperbolic Surfaces
• Bootstrap Bounds on Closed Hyperbolic Manifolds
• From Noncommutative Geometry to Random Matrix Theory
• Automorphic Spectra and the Conformal Bootstrap
• Symmetry and Unification from Soft Theorems and Unitarity
• Cheeger bounds on spin-two fields