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Abstract: The conformal anomalies and functional determinants of the Branson-GJMSoperators, P {2k}, on the d-dimensional sphere are evaluated in explicit termsfor any d and k such that k < d-2+1 if d is even. The determinants are givenin 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 ofNeumann and Dirichlet ones on the hemisphere is a basic part of the method andleads to a heuristic explanation of the non-existence of `super-critical-operators, 2k>d for even d. Significant use is made of the Barnes zetafunction. The results are given in terms of ratios of determinants of operatorson a d+1-dimensional bulk dual sphere. For odd dimensions, the logdeterminant is written in terms of multiple sine functions and agreement isfound with holographic computations, yielding an integral over a Plancherelmeasure. The N-D determinant ratio is also found explicitly for evendimensions. Ehrhart polynomials are encountered.



Autor: J.S.Dowker

Fuente: https://arxiv.org/







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