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.

32 pages. v2: small modifications in the text, added the missing ancillary Mathematica notebook file which proves, by direct computations, some results established in the paper

References in corpus (7)

• The Gauss-Bonnet Theorem for the noncommutative two torus
• Modular curvature and Morita equivalence
• Heat asymptotics for nonminimal Laplace type operators and application to noncommutative tori
• The term a_4 in the heat kernel expansion of noncommutative tori
• Heat trace for Laplacian type operators with non-scalar symbols
• Modular curvature for toric noncommutative manifolds
• On the Scalar Curvature for the Noncommutative Four Torus

Cited by in corpus (12)

• Heat asymptotics for nonminimal Laplace type operators and application to noncommutative tori
• Spectral Metric and Einstein Functionals
• Spectral Action in Noncommutative Geometry
• Noncommutative Residue and Canonical Trace on Noncommutative Tori. Uniqueness Results
• Spectral geometry of functional metrics on noncommutative tori
• Hypergeometric function and Modular Curvature II. Connes-Moscovici functional relation after Lesch's work
• Heat coefficient $a_4$ for nonminimal Laplace type operators
• Local invariants of non-commutative tori
• Effective actions in supersymmetric gauge theories: heat kernels for non-minimal operators
• Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori
• Zeta-regularization and the heat-trace on some compact quantum semigroups
• Heat kernel approach to the one-loop effective action for nonlinear electrodynamics