Noncommutative Lorentz Symmetry and the Origin of the Seiberg-Witten Map

TL;DR

This paper explores how noncommutative Yang-Mills theory exhibits invariance under conformal transformations and derives the Seiberg-Witten map as a covariant splitting of these transformations, linking noncommutative geometry with symmetry principles.

Contribution

It demonstrates the invariance of noncommutative Yang-Mills action under conformal transformations and derives the Seiberg-Witten map from this symmetry structure.

Findings

Noncommutative Yang-Mills forms an irreducible representation of the Lie algebra of rigid transformations.

The action is invariant under combined conformal transformations of the field and noncommutativity parameter.

The Seiberg-Witten differential equation is derived from covariant splitting of conformal transformations.

Abstract

We show that the noncommutative Yang-Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The noncommutative Yang-Mills action is invariant under combined conformal transformations of the Yang-Mills field and of the noncommutativity parameter \theta. The Seiberg-Witten differential equation results from a covariant splitting of the combined conformal transformations and can be computed as the missing piece to complete a covariant conformal transformation to an invariance of the action.