Determinants and conformal anomalies of GJMS operators on spheres

Abstract

The conformal anomalies and functional determinants of the Branson--GJMS operators, P_{2k}, on the d-dimensional sphere are evaluated in explicit terms for any d and k such that k < d/2+1 (if d is even). The determinants are given in terms of multiple gamma functions and a rational multiplicative anomaly, which vanishes for odd d. Taking the mode system on the sphere as the union of Neumann and Dirichlet ones on the hemisphere is a basic part of the method and leads to a heuristic explanation of the non--existence of `super--critical' operators, 2k>d for even d. Significant use is made of the Barnes zeta function. The results are given in terms of ratios of determinants of operators on a (d+1)-dimensional bulk dual sphere. For odd dimensions, the log determinant is written in terms of multiple sine functions and agreement is found with holographic computations, yielding an integral over a Plancherel measure. The N-D determinant ratio is also found explicitly for even dimensions. Ehrhart polynomials are encountered.

arXiv comment20 pages. Last major revision. Section on holographic aspects added

References in corpus (7)

• Partition functions and double-trace deformations in AdS/CFT
• Vacuum energy on orbifold factors of spheres
• Effective action in spherical domains
• Polyakov formulas for GJMS operators from AdS/CFT
• Large-order Perturbation Theory and de Sitter/Anti de Sitter Effective Actions
• The Barnes zeta-function, sphere determinants and Glaisher- Kinkelin-Bendersky constants
• Massive-Scalar Effective Actions on Anti-de Sitter Spacetime

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