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

Abstract

We revisit the problem of computing the determinant of Klein-Gordon operator Delta=-del(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 (Delta) over tilde =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 (Delta) over tilde. We identify the (Delta) over tilde operator as the weighted Laplacian'' in the context of warped compactifications, and comment on possible applications.