Functional determinants, generalized BTZ geometries and Selberg zeta function

Abstract

We continue the study of a special entry in the AdS/CFT dictionary, namely a holographic formula relating the functional determinant of the scattering operator in an asymptotically locally anti-de Sitter space to a relative functional determinant of the scalar Laplacian in the bulk. A heuristic derivation of the formula involves a one-loop quantum effect in the bulk and the corresponding sub-leading correction at large N on the boundary. We presently explore the formula in the background of a higher dimensional version of the Euclidean BTZ black hole, obtained as a quotient of hyperbolic space by a discrete subgroup of isometries generated by a loxodromic (or hyperbolic) element consisting of dilation (temperature) and torsion angles (twist). The bulk computation is done using heat-kernel techniques and fractional calculus. At the boundary, we acquire a recursive scheme that allows us to successively include rotation blocks in spacelike planes in the embedding space. The determinants are compactly expressed in terms of an associated (Patterson–)Selberg zeta function and a connection to quasi-normal frequencies is discussed.