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

TL;DR

This paper develops heat asymptotics for nonminimal Laplace type operators on vector bundles over manifolds, enabling the calculation of modular scalar curvature for noncommutative tori, extending geometric analysis tools.

Contribution

It derives explicit heat kernel coefficient formulas for nonminimal Laplace type operators, facilitating the computation of scalar curvature in noncommutative geometry.

Findings

Computed the second heat kernel coefficient for a class of nonminimal operators.

Enabled the calculation of modular scalar curvature for noncommutative tori.

Extended classical heat kernel asymptotics to nonminimal, matrix-valued operators.

Abstract

Let $P$ be a Laplace type operator acting on a smooth hermitean vector bundle $V$ of fiber $\mathbb{C}^N$ over a compact Riemannian manifold given locally by $P= - [g^{\mu\nu} u(x)\partial_\mu\partial_\nu + v^\nu(x)\partial_\nu + w(x)]$ where $u,\,v^\nu,\,w$ are $M_N(\mathbb{C})$-valued functions with $u(x)$ positive and invertible. For any $a \in \Gamma(\text{End}(V))$, we consider the asymptotics $\text{Tr} (a e^{-tP}) \underset{t \downarrow 0^+}{\sim} \,\sum_{r=0}^\infty a_r(a, P)\,t^{(r-d)/2}$ where the coefficients $a_r(a, P)$ can be written locally as $a_r(a, P)(x) = \text{tr}[a(x) \mathcal{R}_r(x)]$. The computation of $\mathcal{R}_2$ is performed opening the opportunity to calculate the modular scalar curvature for noncommutative tori.