Heat trace for Laplacian type operators with non-scalar symbols

TL;DR

This paper develops a computational method to determine heat-trace coefficients for Laplacian-type operators with matrix-valued symbols on fiber bundles, exemplified in four dimensions, and explores conditions for explicit formula derivation.

Contribution

It introduces a new computational approach for heat-trace coefficients of non-scalar Laplacian operators and analyzes when explicit formulas can be obtained.

Findings

Computed heat-trace coefficient a_1 in 4D using invariants.

Established a method to derive heat-trace coefficients via computational machinery.

Identified conditions under which explicit formulas for coefficients are possible.

Abstract

For an elliptic selfadjoint operator $P =-[u^{\mu\nu}\partial_\mu \partial_\nu +v^\nu \partial_\nu +w]$ acting on a fiber bundle over a Riemannian manifold, where $u,v^\mu,w$ are $N\times N$-matrices, we develop a method to compute the heat-trace coefficients $a_r$ which allows to get them by a pure computational machinery. It is exemplified in dimension 4 by the value of $a_1$ written both in terms of $u,v^\mu,w$ or diffeomorphic and gauge invariants. We also answer to the question: when is it possible to get explicit formulae for $a_r$?