Index-free Heat Kernel Coefficients

TL;DR

This paper computes explicit index-free formulas for the first six heat kernel coefficients of Laplace-type operators on compact Riemannian manifolds, revealing structural insights and potential implications for quantum field theory anomalies.

Contribution

It provides the first explicit formulas for the fifth heat kernel coefficient and extends the calculation to the sixth in flat space with gauge connection, using a noncovariant approach.

Findings

Explicit formulas for the first five heat kernel coefficients.

The sixth coefficient is derived for flat space with gauge connection.

Structural analysis of heat kernel coefficients and their implications for anomalies.

Abstract

Using index-free notation, we present the diagonal values of the first five heat kernel coefficients associated with a general Laplace-type operator on a compact Riemannian space without boundary. The fifth coefficient appears here for the first time. For a flat space with a gauge connection, the sixth coefficient is given too. Also provided are the leading terms for any coefficient, both in ascending and descending powers of the Yang-Mills and Riemann curvatures, to the same order as required for the fourth coefficient. These results are obtained by directly solving the relevant recursion relations, working in Fock-Schwinger gauge and Riemann normal coordinates. Our procedure is thus noncovariant, but we show that for any coefficient the `gauged' respectively `curved' version is found from the corresponding `non-gauged' respectively `flat' coefficient by making some simple covariant substitutions. These substitutions being understood, the coefficients retain their `flat' form and size. In this sense the fifth and sixth coefficient have only 26 and 75 terms respectively, allowing us to write them down. Using index-free notation also clarifies the general structure of the heat kernel coefficients. In particular, in flat space we find that from the fifth coefficient onward, certain scalars are absent. This may be relevant for the anomalies of quantum field theories in ten or more dimensions.