The action functional for Moyal planes

TL;DR

This paper demonstrates that Moyal planes can be formulated as nonunital spectral triples in noncommutative geometry, and computes the associated noncommutative Yang-Mills action, integrating Moyal gauge theory into the geometric framework.

Contribution

It establishes Moyal planes as nonunital spectral triples and explicitly computes their action functional, connecting Moyal gauge theory with noncommutative geometric formalism.

Findings

Moyal planes are nonunital spectral triples.

The noncommutative Yang-Mills action for Moyal planes is derived.

Moyal gauge theory fits into noncommutative geometry framework.

Abstract

Modulo some natural generalizations to noncompact spaces, we show in this letter that Moyal planes are nonunital spectral triples in the sense of Connes. The action functional of these triples is computed, and we obtain the expected result, ie the noncommutative Yang-Mills action associated with the Moyal product. In particular, we show that Moyal gauge theory naturally fit into the rigorous framework of noncommutative geometry.