Chern-Simons in the Seiberg-Witten map for non-commutative Abelian gauge theories in 4D

TL;DR

This paper provides a cohomological framework for understanding the Seiberg-Witten map in non-commutative gauge theories, revealing its topological and geometrical origins through BRST cohomology and Chern-Simons forms.

Contribution

It introduces a cohomological BRST characterization of the Seiberg-Witten map and links its coefficients to topological invariants, extending the understanding to both Abelian and non-Abelian theories.

Findings

Coefficients of the SW map are elements of BRST cohomology.

First coefficients of the SW map relate to Chern-Simons three form.

The existence of the SW map is established for Abelian and non-Abelian cases.

Abstract

A cohomological BRST characterization of the Seiberg-Witten (SW) map is given. We prove that the coefficients of the SW map can be identified with elements of the cohomology of the BRST operator modulo a total derivative. As an example, it will be illustrated how the first coefficients of the SW map can be written in terms of the Chern-Simons three form. This suggests a deep topological and geometrical origin of the SW map. The existence of the map for both Abelian and non-Abelian case is discussed. By using a recursive argument and the associativity of the $\star$-product, we shall be able to prove that the Wess-Zumino consistency condition for non-commutative BRST transformations is fulfilled. The recipe of obtaining an explicit solution by use of the homotopy operator is briefly reviewed in the Abelian case.