Classification of Connections on Higher-Dimensional Non-Commutative Tori

TL;DR

This paper classifies flat connections on modules over higher-dimensional non-commutative tori, linking the Yang-Mills functional's minima to these connections, advancing understanding of non-commutative geometry.

Contribution

It provides a classification of flat connections on modules with integrable connections over higher-dimensional non-commutative tori, connecting geometric analysis with algebraic structures.

Findings

Yang-Mills functional minimized by flat connections

Classification of flat connections on modules with integrable connections

Extension of non-commutative geometric analysis

Abstract

If X is a full, finitely generated, projective module over a non-commutative torus, the Yang-Mills functional attains its minimum exactly on the flat connections on X. We classify the flat connections on modules admitting integrable connections.