Subadditivity and additivity of the noncommutative Yang-Mills action functional

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. The additivity property is stronger in the sense that it implies the subadditivity property. Under our hypothesis, we also 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 behavior of critical points of the Yang-Mills functional under additivity. In the end, we discuss examples involving compact spin manifolds, matrix algebras, noncommutative n-torus, and the quantum Heisenberg manifolds to validate our hypothesis.