Towards an explicit expression of the Seiberg-Witten map at all orders

TL;DR

This paper investigates the explicit form of the Seiberg-Witten map at all orders, revealing recursive structures and summation formulas in special cases, thereby clarifying its gauge transformation ambiguities.

Contribution

It provides a recursive formulation of the Seiberg-Witten map in the abelian case and sums up the map for pure gauge configurations, enhancing understanding of its structure.

Findings

Recursive pattern in the abelian Seiberg-Witten map

Explicit summation for pure gauge cases

Clarification of gauge ambiguity in the map

Abstract

The Seiberg-Witten map links noncommutative gauge theories to ordinary gauge theories, and allows to express the noncommutative variables in terms of the commutative ones. Its explicit form can be found order by order in the noncommutative parameter theta and the gauge potential A by the requirement that gauge orbits are mapped on gauge orbits. This of course leaves ambiguities, corresponding to gauge transformations, and there is an infinity of solutions. Is there one better, clearer than the others ? In the abelian case, we were able to find a solution, linked by a gauge transformation to already known formulas, which has the property of admitting a recursive formulation, uncovering some pattern in the map. In the special case of a pure gauge, both abelian and non-abelian, these expressions can be summed up, and the transformation is expressed using the parametrisation in terms of the gauge group.