Holographic formula for the determinant of the scattering operator in thermal AdS

TL;DR

This paper tests a holographic formula relating the determinant of the scattering operator in thermal AdS space to a scalar Laplacian determinant, using explicit bulk computations and zeta-functions in hyperbolic quotients.

Contribution

It provides a concrete verification of the holographic formula in thermal AdS and BTZ geometries through explicit calculations.

Findings

The holographic formula holds for thermal AdS and BTZ geometries.

Bulk computations match the predictions from the zeta-function approach.

The method can be extended to other hyperbolic quotients.

Abstract

A 'holographic formula' expressing the functional determinant of the scattering operator in an asymptotically locally anti-de Sitter(ALAdS) space has been proposed in terms of a relative functional determinant of the scalar Laplacian in the bulk. It stems from considerations in AdS/CFT correspondence of a quantum correction to the partition function in the bulk and the corresponding subleading correction at large N on the boundary. In this paper we probe this prediction for a class of quotients of hyperbolic space by a discrete subgroup of isometries. We restrict to the simplest situation of an abelian group where the quotient geometry describes thermal AdS and also the non-spinning BTZ instanton. The bulk computation is explicitly done using the method of images and the answer can be encoded in a (Patterson-)Selberg zeta-function.