Poisson gauge theory

TL;DR

This paper formulates a Poisson gauge theory as a semi-classical limit of non-commutative gauge algebra, exploring its structure, field equations, and examples with coordinate-dependent non-commutativity.

Contribution

It introduces a dynamical Poisson gauge theory framework that captures semi-classical features of non-commutative gauge theories with variable non-commutativity.

Findings

Derived covariant derivatives and field strength for Poisson gauge theory.

Established Bianchi identities and gauge-invariant field equations.

Provided explicit examples with linear and non-linear Poisson structures.

Abstract

The Poisson gauge algebra is a semi-classical limit of complete non-commutative gauge algebra. In the present work we formulate the Poisson gauge theory which is a dynamical field theoretical model having the Poisson gauge algebra as a corresponding algebra of gauge symmetries. The proposed model is designed to investigate the semi-classical features of the full non-commutative gauge theory with coordinate dependent non-commutativity $\Theta^{ab}(x)$, especially whose with a non-constant rank. We derive the expression for the covariant derivative of matter field. The commutator relation for the covariant derivatives defines the Poisson field strength which is covariant under the Poisson gauge transformations and reproduces the standard $U(1)$ field strength in the commutative limit. We derive the corresponding Bianchi identities. The field equations for the gauge and the matter fields are obtained from the gauge invariant action. We consider different examples of linear in coordinates Poisson structures $\Theta^{ab}(x)$, as well as non-linear ones, and obtain explicit expressions for all proposed constructions. Our model is unique up to invertible field redefinitions and coordinate transformations.