Equivalence of Two Approaches to Yang-Mills on Non-commutative Torus

TL;DR

This paper demonstrates the equivalence of two different formulations of Yang-Mills theory on noncommutative tori and provides a structural theorem for modules over certain subalgebras, advancing understanding in noncommutative geometry.

Contribution

It proves the equivalence of two Yang-Mills notions on noncommutative tori and establishes a structure theorem for modules over spectrally invariant subalgebras.

Findings

Both notions of Yang-Mills action agree on noncommutative n-tori.

A structure theorem for Hermitian modules over spectrally invariant subalgebras.

Enhanced understanding of module structures in noncommutative geometry.

Abstract

There are two notions of Yang-Mills action functional in noncommutative geometry. We show that for noncommutative n-torus both these notions agree. We also prove a structure theorem on the Hermitian structure of a finitely generated projective modules over spectrally invariant subalgebras of $C^*$-algebras.