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Determinants and conformal anomalies of GJMS operators on spheres

The conformal anomalies and functional determinants of the Branson-GJMS operators, P2k, on the d-dimensional sphere are evaluated in explicit terms for any d and k such that k ⩽ d/2 (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.