Heat coefficient $a_4$ for nonminimal Laplace type operators

Abstract

Given a smooth hermitean vector bundle $V$ of fiber ${\mathbb{C}}^N$ over a compact Riemannian manifold and $\nabla$ a covariant derivative on $V$, let $P=-(|g{|}^{-1/2}{\nabla }_{\mu }|g{|}^{1/2}{g}^{\mu \nu }u{\nabla }_{\nu }+p^{\mu }{\nabla }_{\mu }+q)$ be a nonminimal Laplace type operator acting on smooth sections of $V$ where $u,p^{\nu },q$ are $M_N(\mathbb{C})$-valued functions with $u$ positive and invertible. For any $a\in \Gamma (\text{End}(V))$, we consider the asymptotics $\text{Tr}a{e}^{-tP}{\sim }_{t\downarrow 0}{\sum }_{r=0}^{\infty }{a}_r(a,P)t^{(r-d)/2}$ where the coefficients $a_r(a,P)$ can be written as an integral of the functions $a_r(a,P)(x)=\text{tr}[a(x){\mathcal{R}}_r(x)]$. This paper revisits the previous computation of ${\mathcal{R}}_2$ by the authors and is mainly devoted to a computation of ${\mathcal{R}}_4$. The results are presented with $u$-dependent operators which are universal (\textsl{i.e.} $P$-independent) and which act on tensor products of $u$, $p^{\mu }$, $q$ and their derivatives via (also universal) spectral functions which are fully described.

arXiv comment30 pages. Published version

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Cited by in corpus (1)

• Local invariants of non-commutative tori