[Paper Review] Local invariants of non-commutative tori

Fedor Sukochev, Dmitriy Zanin|arXiv (Cornell University)|Oct 2, 2019

Advanced Operator Algebra Research25 references4 citations

TL;DR

This paper establishes an asymptotic heat trace expansion for generic curved non-commutative tori, extending the Minakshisundaram-Plejel theorem beyond the conformal case. It provides an algorithm to compute local invariants as coefficients in the expansion, using a Laplace-Beltrami operator instead of a Dirac operator, and generalizes prior results on scalar and higher curvature invariants in non-commutative geometry.

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).

Motivation & Objective

• To generalize the Minakshisundaram-Plejel theorem to non-commutative tori with generic, non-flat Riemannian metrics.
• To establish the existence of an asymptotic expansion of the heat trace for the Laplace-Beltrami operator on curved non-commutative tori.
• To provide an algorithmic method for computing local invariants (coefficients in the heat expansion) beyond the conformal case.
• To extend previous results on scalar curvature and higher-order invariants in non-commutative geometry to multi-dimensional, non-conformally deformed tori.
• To define local invariants via heat trace expansion in the absence of a spectral triple or twisted spectral triple structure.

Proposed method

• Construct a positive second-order differential operator (Laplace-Beltrami operator) as the geometric realization of the Riemannian metric on the non-commutative torus.
• Use the heat semigroup generated by this operator to derive an asymptotic expansion of the trace of the heat kernel.
• Define local invariants $ a_k(x) $ as coefficients in the expansion $ ext{Tr}( ho(x)e^{-t riangle_g}) hicksim t^{-d/2} ext{sum} t^{k/2} a_k(x) $, with $ k $ even.
• Introduce a regularization procedure via a metric perturbation $ g' $, embedding the original torus into a higher-dimensional non-commutative torus to exploit analytic continuation.
• Apply complex analysis techniques, including contour integration and holomorphicity of correlation functions $ ext{corr}_k(u,z) $, to relate traces on different metric structures.
• Use the Poisson summation formula and trace identities to relate the heat trace on the extended metric to the original one, enabling extraction of coefficients.

Experimental results

Research questions

• RQ1 Can the Minakshisundaram-Plejel asymptotic expansion be extended to non-commutative tori with generic, non-conformally deformed Riemannian metrics?
• RQ2 How can local invariants (coefficients in the heat trace expansion) be systematically computed in the absence of a spectral triple structure?
• RQ3 What is the structure of the heat trace expansion for the Laplace-Beltrami operator on a curved non-commutative torus?
• RQ4 How do the coefficients $ a_k(x) $ relate to geometric invariants such as scalar and Riemann curvature in the non-commutative setting?
• RQ5 Can the algorithmic computation of $ a_k(x) $ be generalized beyond the 2D conformal case to higher-dimensional non-commutative tori?

Key findings

• An asymptotic expansion of the heat trace $ ext{Tr}( ho(x)e^{-t riangle_g}) $ exists for generic curved non-commutative tori, with coefficients $ a_k(x) $ corresponding to local invariants.
• The coefficients $ a_k(x) $ are given by $ a_k(x) = au(x u^{-1/2} I_k u^{1/2}) $, where $ I_k $ are explicitly computable functionals derived from the metric tensor.
• The method generalizes previous results on scalar curvature ($ a_2 $) and higher-order invariants ($ a_4 $) to non-conformal, multi-dimensional non-commutative tori.
• The heat trace expansion is shown to hold via a perturbation argument embedding the original torus into a higher-dimensional non-commutative torus with extended metric $ g' $, enabling trace comparison.
• The trace on the extended metric satisfies $ ext{Tr}(e^{-t riangle}) = (rac{ au}{t})^{(d'-d)/2}(1 + O(t^ u)) $, which allows recovery of the original expansion up to higher-order corrections.
• The final result confirms that $ ext{Tr}( ho(x)e^{-t riangle_g}) hicksim t^{-d/2} ext{sum}_{k ext{ even}} t^{k/2} au(x u^{-1/2} I_k u^{1/2}) $, providing a complete algorithm for computing local invariants.