The term a_4 in the heat kernel expansion of noncommutative tori

TL;DR

This paper computes the a_4 term in the heat kernel expansion for noncommutative tori, revealing symmetries, functional relations, and geometric insights relevant to noncommutative geometry and curvature analysis.

Contribution

It provides an explicit local formula for a_4 in noncommutative tori, explores its symmetries, and extends results to four-dimensional cases with non-flat metrics.

Findings

Derived explicit a_4 formula involving variable functions.

Identified symmetries under cyclic group actions.

Extended results to noncommutative four tori with curvature insights.

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.