Can you hear the Planck mass?

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 Dp-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.