Heat asymptotics for nonminimal Laplace type operators and application to noncommutative tori

Let P be a Laplace type operator acting on a smooth hermitean vector bundle V of fiber CN over a compact Riemannian manifold given locally by P=−[gμνu(x)∂μ∂ν+vν(x)∂ν+w(x)] where u,vν,w are MN(C)-valued functions with u(x) positive and invertible. For any a∈Γ(End(V)), we consider the asymptotics Tr(ae−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)]. The computation of R2 is performed opening the opportunity to calculate the modular scalar curvature for noncommutative tori.