Subadditivity and additivity of the Yang-Mills action functional in Noncommutative Geometry

TL;DR

This paper explores the properties of subadditivity and additivity of the Yang-Mills action functional within noncommutative geometry, establishing conditions under which these properties hold and analyzing their implications.

Contribution

It introduces a hypothesis that guarantees subadditivity of the Yang-Mills functional and provides a necessary and sufficient condition for its additivity, with applications to noncommutative tori.

Findings

Yang-Mills functional is always subadditive under the hypothesis

Additivity implies subadditivity and is characterized by specific conditions

Examples include noncommutative tori, spin manifolds, and quantum Heisenberg manifolds

Abstract

We formulate notions of subadditivity and additivity of the Yang-Mills action functional in noncommutative geometry. We identify a suitable hypothesis on spectral triples which proves that the Yang-Mills functional is always subadditive, as per expectation. The additivity property is much stronger in the sense that it implies the subadditivity property. Under this hypothesis we obtain a necessary and sufficient condition for the additivity of the Yang-Mills functional. An instance of additivity is shown for the case of noncommutative $n$-tori. We also investigate the behaviour of critical points of the Yang-Mills functional under additivity. At the end we discuss few examples involving compact spin manifolds, matrix algebras, noncommutative $n$-torus and the quantum Heisenberg manifolds which validate our hypothesis.