Complete Computation of DeWitt-Seeley-Gilkey Coefficient E_4 for Nonminimal Operator on Curved Manifolds

TL;DR

This work delivers the first complete computation of the DeWitt-Seeley-Gilkey coefficient $E_4$ for a nonminimal elliptic operator on curved manifolds with gauge fields, valid for general dimension $n$ and explicitly for $n=4$. The authors employ Widom's covariant pseudodifferential calculus, representing the heat operator via the resolvent and solving recursion relations for the amplitude coefficients $\sigma_m$, then integrating to obtain $E_m$, with a dedicated C implementation (COLIM and DWSGCOEF). They present the general $\text{tr}_L E_4$ in terms of $C_1,\dots,C_{13}$ and provide the $n=4$ specialization featuring $\ln(1-a)$-dependent terms, along with the full $E_4$ expression in the Appendix (comprising $73$ tensor monomials). The results refine previous work by clarifying gauge-parameter and dimension dependencies and illustrate the computational complexity of nonminimal operators, highlighting the need for canonical tensor bases and automated reduction techniques. This advancement impacts quantum field theory, quantum gravity, and spectral geometry by enabling exact heat-kernel coefficients for nonminimal operators, which are essential for effective actions, anomalies, and index theorems.

Abstract

Asymptotic heat kernel expansion for nonminimal differential operators on curved manifolds in the presence of gauge fields is considered. The complete expressions for the fourth coefficient E_4 in the heat kernel expansion for such operators are presented for the first time. The expressions were computed for general case of manifolds of arbitrary dimension n and also for the most important case n=4. The calculations have been carried out on PC with the help of a program written in C.