Spherical Dirac GJMS operator determinants

TL;DR

This paper extends the calculation of conformal anomalies and determinants of GJMS operators to spin-half fields in the context of AdS/CFT, using Barnes zeta functions and exploring factorization properties.

Contribution

It introduces a novel extension to spin-half of scalar GJMS determinant calculations, including new factorization and eigenvalue analysis methods.

Findings

Determinants for GJMS operators are computed and shown to be equal for different factorizations.

The scalar and spinor determinants are expressed in terms of each other in odd dimensions.

Numerical results support the theoretical findings.

Abstract

Motivated by AdS/CFT, the extension is made to spin-half of a scalar calculation of the conformal anomalies and functional determinants of GJMS operators. The formal aspects are heuristic but sufficient. A Barnes zeta function representation again proves effective. The determinants are calculated for the two factorisations of the gamma-function (intertwiner) form of the GJMS operator, and shown to be equal, even including the multiplicative anomaly. A comment is made on the general eigenvalue problem and a few numerical results are presented. An alternative approach is detailed for odd dimensions and it is shown that the scalar determinants are expressed in terms of the spinor ones, and vice versa. An explicit, general form is given.