Can you hear the Planck mass?

TL;DR

This paper derives the Weyl law for Laplacian eigenvalues on Riemannian manifolds using physical principles related to gravitational behavior in compactifications, and explores implications for gravity localization and spectral properties in warped spaces.

Contribution

It introduces the concept of weighted quantum ergodicity to reconcile the Weyl law with warped compactifications and rigorously proves the law for certain singular spaces using RCD theory.

Findings

Weyl law relates eigenvalues to volume and warping functions in compactifications.

Weighted quantum ergodicity explains eigenfunction oscillations in warped spaces.

Spectral discreteness depends on Dp-brane singularities for p=6,7,8.

Abstract

For the Laplacian of an $n$-Riemannian manifold $X$, the Weyl law states that the $k$-th eigenvalue is asymptotically proportional to $(k/V)^{2/n}$, where $V$ is the volume of $X$. We show that this result can be derived via physical considerations by demanding that the gravitational potential for a compactification on $X$ behaves in the expected $(4+n)$-dimensional way at short distances. In simple product compactifications, when particle motion on $X$ is ergodic, for large $k$ the eigenfunctions oscillate around a constant, and the argument is relatively straightforward. The Weyl law thus allows to reconstruct the four-dimensional Planck mass from the asymptotics of the masses of the spin 2 Kaluza--Klein modes. For warped compactifications, a puzzle appears: the Weyl law still depends on the ordinary volume $V$, while the Planck mass famously depends on a weighted volume obtained as an integral of the warping function. We resolve this tension by arguing that in the ergodic case the eigenfunctions oscillate now around a power of the warping function rather than around a constant, a property that we call weighted quantum ergodicity. This has implications for the problem of gravity localization, which we discuss. We show that for spaces with D$p$-brane singularities the spectrum is discrete only for $p =6,7,8$, and for these cases we rigorously prove the Weyl law by applying modern techniques from RCD theory.