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

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 sub-leading 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 a 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.