Cheeger bounds on spin-two fields

TL;DR

This paper establishes bounds on the mass of spin-two fields arising from gravity compactifications with small bridges, using geometric and synthetic Ricci curvature techniques, with implications for holography and the swampland conjecture.

Contribution

It introduces a novel approach linking Cheeger bounds, Bakry-Émery geometry, and synthetic Ricci bounds to analyze spin-two fields in complex compactifications.

Findings

Massive spin-two fields with small mass are shown to arise in certain compactifications.

New bounds on higher eigenvalues are derived, consistent with the spin-two swampland conjecture.

Examples are provided where bounds are in tension with the conjecture in non-scale-separated regimes.

Abstract

We consider gravity compactifications whose internal space consists of small bridges connecting larger manifolds, possibly noncompact. We prove that, under rather general assumptions, this leads to a massive spin-two field with very small mass. The argument involves a recently-noticed relation to Bakry--\'Emery geometry, a version of the so-called Cheeger constant, and the theory of synthetic Ricci lower bounds. The latter technique allows generalizations to non-smooth spaces such as those with D-brane singularities. For AdS$_d$ vacua with a bridge admitting an AdS$_{d+1}$ interpretation, the holographic dual is a CFT$_d$ with two CFT$_{d-1}$ boundaries. The ratio of their degrees of freedom gives the graviton mass, generalizing results obtained by Bachas and Lavdas for $d=4$. We also prove new bounds on the higher eigenvalues. These are in agreement with the spin-two swampland conjecture in the regime where the background is scale-separated; in the opposite regime we provide examples where they are in naive tension with it.