Functional determinants, generalized BTZ geometries and Selberg zeta function

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2010-05, Vol.43 (20), p.205402-205402
• Main Authors: Aros, R, Díaz, D E
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Black holes (astronomy); Boundaries; Determinants; Exact sciences and technology; Mathematical analysis; Mathematical models; Physics; Planes; Scattering
• ISSN: 1751-8121; 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/43/20/205402

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.