Singularities of the Seiberg-Witten map

TL;DR

This paper constructs explicit solutions to the Seiberg-Witten map for linear gauge fields and reveals that these solutions diverge at certain points in the non-commutativity parameter space, indicating limitations in deforming Yang-Mills theories.

Contribution

It provides explicit solutions to the Seiberg-Witten map and analyzes their divergence structure in the non-commutative parameter space.

Findings

Solutions diverge at specific theta values

Deformation from Yang-Mills to non-commutative theory is limited to one connected component

Identifies the presence of 'event horizons' in theta-space

Abstract

We construct an explicit solution of the Seiberg-Witten map for a linear gauge field on the non-commutative plane. We observe that this solution as well as the solution for a constant curvature diverge when the non-commutativity parameter theta reaches certain event horizon in the theta-space. This implies that an ordinary Yang-Mills theory can be continuously deformed by the Seiberg-Witten map into a non-commutative theory only within one connected component of the theta-space.