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

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.
Type de document :
Pré-publication, Document de travail
31 pages. Ancillary Mathematica notebook file which proves, by direct computations, some results .. 2017
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https://hal.archives-ouvertes.fr/hal-01573594
Contributeur : Thierry Masson <>
Soumis le : jeudi 10 août 2017 - 09:26:32
Dernière modification le : vendredi 11 août 2017 - 01:07:42

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  • HAL Id : hal-01573594, version 1
  • ARXIV : 1707.09657

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Bruno Iochum, Thierry Masson. Heat asymptotics for nonminimal Laplace type operators and application to noncommutative tori. 31 pages. Ancillary Mathematica notebook file which proves, by direct computations, some results .. 2017. 〈hal-01573594〉

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