Heat coefficient [formula omitted] for non minimal Laplace type operators

Abstract

Given a smooth hermitean vector bundle V of fiber ℂN over a compact Riemannian manifold and ∇ a covariant derivative on V, let P=−(|g|−1∕2∇μ|g|1∕2gμνu∇ν+pμ∇μ+q) be a non minimal Laplace type operator acting on smooth sections of V where u,pν,q are MN(ℂ)-valued functions with u positive and invertible. For any a∈Γ(End(V)), we consider the asymptotics Trae−tP∼t↓0∑r=0∞ar(a,P)t(r−d)∕2 where the coefficients ar(a,P) can be written as an integral of the functions ar(a,P)(x)=tr[a(x)ℛr(x)].This paper revisits the previous computation of ℛ2 by the authors and is mainly devoted to a computation of ℛ4. The results are presented with u-dependent operators which are universal (i.eP-independent) and which act on tensor products of u, pμ, q and their derivatives via (also universal) spectral functions which are fully described.