General Rearrangement Lemma for Heat Trace Asymptotic on Noncommutative Tori

TL;DR

This paper derives an explicit formula for the second heat coefficient in the small time heat trace asymptotics of second order elliptic operators on noncommutative tori, extending previous conformal case results using hypergeometric integrals.

Contribution

It generalizes the rearrangement operators to noncommutative tori with a new explicit formula, including additional functional relations and combinatorial verifications.

Findings

Explicit formula for the second heat coefficient.

Recovery of previous conformal case results.

Validation of new functional relations through combinatorial analysis.

Abstract

We study a technical problem arising from the spectral geometry of noncommutative tori: the small time heat trace asymptotic associated to a general second order elliptic operator. We extend the rearrangement operators in the conformal case to the general setting using hypergeometric integrals over Grassmannians. The main result is the explicit formula of the second heat coefficient in terms of the coefficients. When specializing to examples in conformal case, we not only recover results in previous works but also obtain some extra functional relations whose validation provides experimental support to the main results. At last, we verify the relations based on combinatorial properties derived from the hypergeometric features.