Functional Determinants of Radial Operators in $AdS_2$

TL;DR

This paper develops a method for calculating functional determinants of Laplace and Dirac operators in $AdS_2$, using zeta-function regularization, Fourier analysis, and scattering theory, with applications to holographic Wilson loops.

Contribution

It introduces a systematic approach to compute determinants in $AdS_2$ backgrounds, extending known flat space results and including conformal geometries.

Findings

Derived explicit formulas for determinants in $AdS_2$

Validated the method with examples from Wilson loop spectra

Extended techniques to conformal $AdS_2$ geometries

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 $\frac{1}{4}$-BPS latitude Wilson loop.