Path integral of free fields and the determinant of Laplacian in warped space-time

TL;DR

This paper investigates the spectral properties of the Klein-Gordon operator on Euclidean $AdS_3/Z$, revealing the non-normalizability of eigenfunctions and introducing a related weighted Laplacian operator with similar determinant properties.

Contribution

It clarifies the spectral structure of the Klein-Gordon operator in warped space-times and introduces a weighted Laplacian that shares determinant dependence, offering new insights for warped compactifications.

Findings

Eigenfunctions of $ abla^2$ are non-normalizable on $H_3/Z$.

A weighted Laplacian $ ilde abla^2$ has the same determinant dependence as $ abla^2$.

Green's function of $ abla^2$ can be decomposed into eigenfunctions of $ ilde abla^2$.

Abstract

We revisit the problem of computing the determinant of Klein-Gordon operator $\Delta = -\nabla^2 + M^2$ on Euclideanized $AdS_3$ with the Euclideanized time coordinate compactified with period $\beta$, $H_3/Z$, by explicitly computing its eigenvalues and computing their product. Upon assuming that eigenfunctions are normalizable on $H_3/Z$, we found that there are no such eigenfunctions. Upon closer examination, we discover that the intuition that $H_3/Z$ is like a box with normalizable eigenfunctions was false, and that there is, instead, a set of eigenfunctions which forms a continuum. Somewhat to our surprise, we find that there is a different operator $\tilde \Delta = r^2 \Delta$, which has the property that (1) the determinant of $\Delta$ and the determinant of $r^2 \Delta$ have the same dependence on $\beta$, and that (2) the Green's function of $\Delta$ can be spectrally decomposed into eigenfunctions of $\tilde \Delta$. We identify the $\tilde \Delta$ operator as the ``weighted Laplacian'' in the context of warped compactifications, and comment on possible applications.