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

2 CPT - E2 Géométrie, Physique et Symétries
CPT - Centre de Physique Théorique - UMR 7332
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
Domaine :
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-01573594
Contributeur : Thierry Masson <>
Soumis le : jeudi 10 août 2017 - 09:26:32
Dernière modification le : jeudi 18 janvier 2018 - 01:30:08

### Identifiants

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

### Citation

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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