The term a_4 in the heat kernel expansion of noncommutative tori

Abstract

We consider the Laplacian associated with a general metric in the canonical conformal structure of the noncommutative two torus, and calculate a local expression for the term a_4 that appears in its corresponding small-time heat kernel expansion. The final formula involves one variable functions and lengthy two, three and four variable functions of the modular automorphism of the state that encodes the conformal perturbation of the flat metric. We confirm the validity of the calculated expressions by showing that they satisfy a family of conceptually predicted functional relations. By studying these functional relations abstractly, we derive a partial differential system which involves a natural action of cyclic groups of order two, three and four and a flow in parameter space. We discover symmetries of the calculated expressions with respect to the action of the cyclic groups. In passing, we show that the main ingredients of our calculations, which come from a rearrangement lemma and relations between the derivatives up to order four of the conformal factor and those of its logarithm, can be derived by finite differences from the generating function of the Bernoulli numbers and its multiplicative inverse. We then shed light on the significance of exponential polynomials and their smooth fractions in understanding the general structure of the noncommutative geometric invariants appearing in the heat kernel expansion. As an application of our results we obtain the a_4 term for noncommutative four tori which split as products of two tori. These four tori are not conformally flat and the a_4 term gives a first hint of the Riemann curvature and the higher dimensional modular structure.

Preliminary version, 134 pages, 21 figures

References in corpus (4)

• Type III and spectral triples
• The Gauss-Bonnet Theorem for the noncommutative two torus
• The Ricci flow on noncommutative two-tori
• On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^{\ast }$-Dynamical Systems

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• Laplace-Beltrami Operators on Noncommutative Tori
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• Gauss-Bonnet for matrix conformally rescaled Dirac
• Spectral geometry of functional metrics on noncommutative tori
• The Ricci Curvature for Noncommutative Three Tori
• Pseudodifferential calculus on noncommutative tori, II. Main properties
• Local invariants of non-commutative tori
• Heat coefficient $a_4$ for nonminimal Laplace type operators
• Zeta-regularization and the heat-trace on some compact quantum semigroups
• A twisted local index formula for curved noncommutative two tori