Functional determinants of radial operators in AdS2

Abstract

We study the zeta-function regularization of functional determinants of Laplace and Dirac-type operators in two-dimensional Euclidean AdS 2 space. More specifically, we consider the ratio of determinants between an operator in the presence of background fields with circular symmetry and the free operator in which the background fields are absent. By Fourier-transforming the angular dependence, one obtains an infinite number of one-dimensional radial operators, the determinants of which are easy to compute. The summation over modes is then treated with care so as to guarantee that the result coincides with the two-dimensional zeta-function formalism. The method relies on some well-known techniques to compute functional determinants using contour integrals and the construction of the Jost function from scattering theory. Our work generalizes some known results in flat space. The extension to conformal AdS 2 geometries is also considered. We provide two examples, one bosonic and one fermionic, borrowed from the spectrum of fluctuations of the holographic 1 4-BPS latitude Wilson loop.

Toward precision holography in type IIA with Wilson loops

“…The ζ-function regularization method is non-perturbative but does not seem to lead to the expected field theory answer as it stands. We have previously developed the ζ-function approach in [34] and applied it to the N = 4 context in [35] motivated by the goal of constructing a regularization that is explicitly diffeomorphic invariant. The key new ingredient in this work that introduces extra ambiguities with respect to our earlier efforts is the fact that some of the modes correspond to massless fermions.…”

“…In this section we follow our previous work [34,35] where we developed a regularization in the case of radial determinants that coincides with ζ-function regularization in various cases. There are various reasons to tackle the problem using these methods.…”

“…Although these latter methods have proven to be quite effective it is conceptually satisfying to deal with the problem directly on the disk. The main outcome of [34] is a prescription for computing ζ-function regularized determinants of radial operators in asymptotically AdS 2 spacetimes. The result for bosons is ln detO detO free = ln…”

Zeta-function regularization of holographic Wilson loops

“…In this section we recall a number of results for determinants of Laplace-and Dirac-like operators in AdS 2 that appear in a companion paper [12] to which we refer the reader for the full derivation. The method applies to operators defined on the AdS 2 geometry (4) and in the presence of external fields.…”

“…where we have set A ρ = 0 and A τ = A(ρ), as well as V = V (ρ), W = W (ρ) and Ω = Ω(ρ). Appropriate regularity conditions at the origin and fall-off conditions at infinity are required for the background fields (see [12] for further details).…”