Local invariants of non-commutative tori

2019/10/02 by Fedor Sukochev, Dmitriy Zanin, Sukochev, Fedor +1
Mathematics · #46L87 #58B34 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1910.00758

openalex publication_date 2019/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a generic curved non-commutative torus extending the notion of conformally deformed non-commutative torus from \citeConnes-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 \citeConnes-Tretkoff, \citeConnes-Moscovici, \citeFaKh (beyond conformal case and/or for multi-dimensional tori).

Citations

• The Gauss-Bonnet Theorem for the noncommutative two torus
• Mathematica notebook for the explicit functions and functional relations for ``The term a₄ in the heat kernel expansion of noncommutative tori"

Cited by

• Hypergeometric function and Modular Curvature II. Connes-Moscovici functional relation after Lesch's work

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