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 (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.

13 pages

References in corpus (3)

• Double-Trace Deformations, Mixed Boundary Conditions and Functional Determinants in AdS/CFT
• Holographic formula for the determinant of the scattering operator in thermal AdS
• A Gauge-Gravity Relation in the One-loop Effective Action

Cited by in corpus (6)

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• Normal modes in thermal AdS via the Selberg zeta function
• Berry curvature, horocycles and scattering states in $Ad{S}_3/CF{T}_2$
• Quasinormal Corrections to Near-Extremal Black Hole Thermodynamics
• A Selberg zeta function for warped AdS${}_3$ black holes
• Higher spin partition functions via the quasinormal mode method in de Sitter quantum gravity