One-Loop Partition Function, Gauge Accessibility and Spectra in AdS$_3$ Gravity

TL;DR

This paper analyzes the one-loop partition function of AdS$_3$ gravity, emphasizing the role of square-integrability of fields, and explores the spectrum of Laplacians, supporting analytic continuation methods.

Contribution

It rederives the partition function via Laplacian determinants, clarifies gauge accessibility for ghost fields, and characterizes the spectrum as purely essential, aligning with recent mathematical results.

Findings

Partition function expressed as ratio of Laplacian determinants.

Gauge conditions are accessible for square-integrable ghost fields.

Spectrum of Laplacians in thermal AdS$_3$ has only essential spectrum.

Abstract

We continue the study of the one-loop partition function of AdS$_3$ gravity with focus on the square-integrability condition on the fluctuating fields. In a previous work we found that the Brown-Henneaux boundary conditions follow directly from the $L^2$ condition. Here we rederive the partition function as a ratio of Laplacian determinants by performing a suitable decomposition of the metric fluctuations. We pay special attention to the asymptotics of the fields appearing in the partition function. We also show that in the usual computation using ghost fields for the de Donder gauge, such gauge condition is accessible precisely for square-integrable ghost fields. Finally, we compute the spectrum of the relevant Laplacians in thermal AdS$_3$, in particular noticing that there are no isolated eigenvalues, only essential spectrum. This last result supports the analytic continuation approach of David, Gaberdiel and Gopakumar. The purely essential spectra found are consistent with the independent results of Lee and Delay of the essential spectrum of the TT rank-2 tensor Lichnerowickz Laplacian on asymptotically hyperbolic spaces.