Index-free heat kernel coefficients

Abstract

Using index-free notation, we present the diagonal values a j (x, x) of the first five heat kernel coefficients a j (x, x ′ ) associated with a general Laplace-type operator on a compact Riemannian space without boundary. The fifth coefficient a 5 (x, x) appears here for the first time. For the special case of a flat space, but 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.

Heat kernel coefficients on the sphere in any dimension

“…Several consistency checks have been performed. We compared the first five heat kernel coefficients and their full dimensional dependence to the known results for heat kernel coefficients on general manifolds [11][12][13][14][15][16][17][18][19], and found complete agreement. We have also derived the heat kernel coefficients on spheres with the help of spectral sums by exploiting the known eigenspectra of Laplacians on the sphere.…”

“…For the endomorphism (28) we can now read off the heat kernel coefficients b d+2n by comparison with (19). To distinguish them, we note that the coefficients c d+2n ) are linear in (independent of) the parameters κ i (d).…”

“…This asymptotic expansion at early proper time works well for small space-time separations, which makes it a convenient tool to study short distance divergences in quantum field theory. On general manifolds, the first six heat kernel coefficients are known [11][12][13][14][15][16][17][18][19]. Calculations of heat kernel coefficients are greatly simplified on specific manifolds such as maximally symmetric backgrounds where they can be obtained from Green's function [9].…”

Untitled

“…The P B;V j ;l are related in a very simple way to the coefficients of the heat expansion; it is possible to compute the P B;V j ;l from the knowledge of the a l for j C 1 Ä l Ä 3j . This is enough to recompute the coefficient of " 2 and also, in principle, the coefficients of " 4 in the expansion, because the a l are known up to l D 6 in the case of a flat metric (see [Ven 1998]).…”

“…The a l are given explicitly in [Gilkey 2004, p. 201] for l Ä 3, and are known for l Ä 5 [Avramidi 1990;Ven 1998]. See also the related works [Hitrik 2002;Hitrik and Polterovich 2003a;2003b;Polterovich 2000].…”