On heat coefficients, multiplicative anomaly and 4D Casimir energy for GJMS operators

This note aims to verify a prediction on the total derivative term of the 4D trace anomaly, and the corresponding heat coefficient, for GJMS operators. It stems from the explicit computation of an {\it improved} Casimir (or vacuum) energy on the sphere that takes into account the multiplicative anomaly among the (shifted) Laplacian factors and connects, via the Cappelli-Coste relation, with both the type A central charge and the total derivative term of the 4D trace anomaly. The present heat coefficient computation is based on Juhl’s explicit formula for GJMS operators, Gilkey’s formula for the integrated heat coefficient of higher-order Laplacians, and the $\mathit{\text{conformal}}\mathit{\text{principle}}$ by Branson and Ørsted.