Noncommutative Induced Gauge Theories on Moyal Spaces

TL;DR

This paper explores the construction and analysis of renormalisable gauge theories on Moyal noncommutative spaces, including one-loop effective actions and potential models ensuring renormalisability.

Contribution

It provides a detailed examination of the challenges and methods for constructing renormalisable gauge theories on Moyal spaces, including explicit one-loop calculations and candidate actions.

Findings

One-loop effective gauge action includes noncommutative Yang-Mills and harmonic-like terms.

Additional gauge-invariant terms may improve renormalisability.

Candidate actions for renormalisable gauge theories on Moyal space are proposed and discussed.

Abstract

Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, which remains so far a challenging problem, is then closely examined. The computation in 4-D of the one-loop effective gauge theory generated from the integration over a scalar field appearing in a renormalisable theory minimally coupled to an external gauge potential is presented. The gauge invariant effective action is found to involve, beyond the expected noncommutative version of the pure Yang-Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic term, which for the noncommutative $\phi^4$-theory on Moyal space ensures renormalisability. A class of possible candidates for renormalisable gauge theory actions defined on Moyal space is presented and discussed.