Poisson gauge models and Seiberg-Witten map

TL;DR

This paper revises Poisson gauge theory, exploring its geometric structures, deriving explicit Lagrangians for linear non-commutativity, and utilizing Seiberg-Witten maps to address construction ambiguities.

Contribution

It provides a geometric reformulation of Poisson gauge theory, explicit gauge Lagrangians for linear non-commutativity, and applies Seiberg-Witten maps to resolve construction ambiguities.

Findings

Explicit gauge Lagrangian for linear non-commutativity.

Recovery of known non-commutative examples from the general formula.

Use of Seiberg-Witten maps to address model arbitrariness.

Abstract

The semiclassical limit of full non-commutative gauge theory is known as Poisson gauge theory. In this work we revise the construction of Poisson gauge theory paying attention to the geometric meaning of the structures involved and advance in the direction of a further development of the proposed formalism, including the derivation of Noether identities and conservation of currents. For any linear non-commutativity, $\Theta^{ab}(x)=f^{ab}_c\,x^c$, with $f^{ab}_c$ being structure constants of a Lie algebra, an explicit form of the gauge Lagrangian is proposed. In particular a universal solution for the matrix $\rho$ defining the field strength and the covariant derivative is found. The previously known examples of $\kappa$-Minkowski, $\lambda$-Minkowski and rotationally invariant non-commutativity are recovered from the general formula. The arbitrariness in the construction of Poisson gauge models is addressed in terms of Seiberg-Witten maps, i.e., invertible field redefinitions mapping gauge orbits onto gauge orbits.