# Functional determinants, generalized BTZ geometries and Selberg zeta function

## TL;DR

This paper explores a higher-dimensional spinning BTZ black hole in AdS space, deriving a holographic formula relating functional determinants of the scattering operator and scalar Laplacian, using heat-kernel techniques and Selberg zeta functions.

## Contribution

It extends the holographic determinant formula to higher-dimensional spinning BTZ geometries using advanced mathematical tools.

## Key findings

- Derived a recursive scheme for boundary determinants from non-spinning to spinning geometries.
- Expressed determinants in terms of Selberg zeta functions.
- Connected the determinants to quasi-normal frequencies.

## 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 (ALAdS) 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 a higher-dimensional version of the spinning 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 (spinning). The bulk computation is done using heat-kernel techniques and fractional calculus. At the boundary, we get a recursive scheme that allows us to range from the non-spinning to the full-fledged spinning geometries. The determinants are compactly expressed in terms of an associated (Patterson-)Selberg zeta function and a connection to quasi-normal frequencies is discussed.