Local invariants of non-commutative tori
arXiv:1910.00758
Abstract
We consider a generic curved non-commutative torus extending the notion of conformally deformed non-commutative torus from \cite{Connes-Tretkoff}. In general, a curved non-commutative torus is no longer represented by a spectral triple, not even by a twisted spectral triple. Therefore, the geometry of this manifold is governed by a positive second order differential operator (Laplace-Betrami operator) rather than a first order differential operator (Dirac operator). For this manifold, we prove an asymptotic expansion of the heat semi-group generated by Laplace-Beltrami operator and provide an algorithm to compute the local invariants which appear as coefficients in the expansion. This allows to extend the results of \cite{Connes-Tretkoff}, \cite{Connes-Moscovici}, \cite{FaKh} (beyond conformal case and/or for multi-dimensional tori).
arXiv comment31 pages
References in corpus (11)
- The Gauss-Bonnet Theorem for the noncommutative two torus
- Levi-Civita's Theorem for Noncommutative Tori
- Modular curvature and Morita equivalence
- Curved noncommutative torus and Gauss--Bonnet
- Heat asymptotics for nonminimal Laplace type operators and application to noncommutative tori
- The term a_4 in the heat kernel expansion of noncommutative tori
- Scalar curvature in conformal geometry of Connes-Landi noncommutative manifolds
- Laplace-Beltrami Operators on Noncommutative Tori
- Spectral Action in Noncommutative Geometry
- Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals
- Heat coefficient for nonminimal Laplace type operators