Wodzicki residue and minimal operators on a noncommutative 4-dimensional torus

TL;DR

This paper calculates the Wodzicki residue of the inverse of a conformally rescaled Laplace operator on a 4D noncommutative torus, revealing that the naive generalization of the Laplace-Beltrami operator is not minimal.

Contribution

It provides the first explicit computation of the Wodzicki residue in this noncommutative setting and clarifies the nature of minimal operators.

Findings

Wodzicki residue of the inverse is computed explicitly.

The standard Laplace-Beltrami operator is not minimal in this context.

Insights into noncommutative geometric operators are gained.

Abstract

We compute the Wodzicki residue of the inverse of a conformally rescaled Laplace operator over a 4-dimensional noncommutative torus. We show that the straightforward generalization of the Laplace-Beltrami operator to the noncommutative case is not the minimal operator.