Determinants and conformal anomalies of GJMS operators on spheres

Understand

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.