Comment on `Index-free Heat Kernel Coefficients'

TL;DR

This paper analyzes high-order heat-kernel coefficients in curved backgrounds with gauge connections and critiques a foundational result by van de Ven. It identifies two precise issues in the $a_5$ coefficient—a mis-specified index structure in a penultimate factor and an improper treatment of $ extsf{Z}$-operator actions in flat space—that require a mixed-symmetry correction and a careful $n o 0$ limit, respectively. The authors argue that these corrections can be validated by direct computation of $a_5$ using recent resummations and cross-checks with other literature, and they illustrate the impact by presenting explicit diagonal expressions for the first five modified heat-kernel coefficients $c_0$ through $c_5$ in the abelian, Fock–Schwinger gauge setup. Overall, the work strengthens the reliability of higher-order heat-kernel expansions used for effective actions and vacuum expectations in nontrivial backgrounds.

Abstract

The article by Anton E. M. van de Ven, Class. Quantum Grav. \textbf{15} (1998), is one of the fundamental references for higher-order heat kernel coefficients in curved backgrounds and with non-abelian gauge connections. In this manuscript, we point out two errors and ambiguities in the $\mathsf{a}_5$ coefficient, which may also affect the higher-order ones.