Determinants and conformal anomalies of GJMS operators on spheres

TL;DR

This paper computes the conformal anomalies and determinants of GJMS operators on spheres, revealing their structure via gamma and zeta functions, and confirms results with holographic duality in odd dimensions.

Contribution

It provides explicit formulas for GJMS operator determinants on spheres using advanced special functions, and explains the absence of super-critical operators.

Findings

Determinants expressed with multiple gamma functions and anomalies vanish for odd dimensions.

Agreement with holographic calculations for odd-dimensional spheres.

Explicit ratios of determinants on (d+1)-dimensional bulk dual 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.