Numerical evaluation of spherical GJMS determinants for even dimensions

TL;DR

This paper numerically evaluates the determinants of GJMS scalar operators on even-dimensional spheres, revealing extrema in the effective action as the parameter k varies, with implications for understanding conformal geometry.

Contribution

It provides a numerical method for computing GJMS determinants in even dimensions, including critical cases, and analyzes their behavior across real values of k.

Findings

Determinants computed using Barnes gamma functions and digamma functions.

Extrema in the effective action occur at specific k values, especially integers in odd dimensions.

Multiplicative anomalies are expressed as odd polynomials in k.

Abstract

The functional determinants of the GJMS scalar operators, P_{2k}, on even-dimensional spheres are computed via Barnes multiple gamma functions relying on the numerical availability of the digamma function. For the critical k=d/2 case, it is necessary to calculate the Stirling moduli. The multiplicative anomalies are given as odd polynomials in $k$ and it is emphasised that that the Dirichlet--to--Robin factorisation, P_{2l+1}, gives the same results as P_{2k} if k=l+1/2.The results are presented as graphs and show a series of extrema in the effective action as k is varied in the reals. For odd dimensions these extrema occur at integer k.