Heat trace for Laplace type operators with non-scalar symbols

Abstract

For an elliptic selfadjoint operator P = − [ u μ ν ∂ μ ∂ ν + v ν ∂ ν + w ] acting on a fiber bundle over a compact Riemannian manifold, where u μ ν , v μ , w are N × 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 any even dimension by the value of a 1 written both in terms of u μ ν = g μ ν u , v μ , w or diffeomorphic and gauge invariants. We also address the question: when is it possible to get explicit formulae for a r ?