Bootstrap Bounds on Closed Einstein Manifolds

TL;DR

This paper applies bootstrap methods to derive bounds on the geometric data of closed Einstein manifolds, linking spectral properties to physical parameters in theories with extra dimensions.

Contribution

It introduces a novel bootstrap approach to constrain geometric data of Einstein manifolds using semidefinite programming, inspired by conformal bootstrap techniques.

Findings

Bounds on eigenvalues and overlap integrals derived

Known manifolds saturate some of the bounds

Constraints on Kaluza-Klein masses and couplings obtained

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.