Enveloping algebra-valued gauge transformations for non-abelian gauge groups on non-commutative spaces

Abstract

An enveloping algebra valued gauge field is constructed, its components are functions of the Lie algebra valued gauge field and can be constructed with the Seiberg-Witten map. This allows the formulation of a dynamics for a finite number of gauge field components on non-commutative spaces.

Seiberg-Witten maps and noncommutative Yang-Mills theories for arbitrary gauge groups

“…In fact (4.11) and (4.13) take the same form as the corresponding equations in the U(N) case. It is also evident that (4.12) reproduces the first order expressions of the Seiberg-Witten maps for the fermions derived in [1,2]. Using a solution to these equations in (4.10), the latter provides the commutative effective action S eff [A, ψ; ϑ] in (4.5) [notice that (4.10) is still expressed in terms of hatted fields] according to…”

“…Finally we set all commutative gauge fields and gauge parameters to zero except for a subset corresponding to a Lie subalgebra g of U. In this way one can easily construct noncommutative gauge theories of the same type as in [1,2] to all orders in the deformation parameter and for all choices of g. The inclusion of matter fields is also straightforward, as we shall demonstrate.…”

Seiberg-Witten Map for Superfields onN=(1/2,0) andN=(1/2,1/2) Deformed Superspace

“…This question is automatically answered by solving the Seiberg-Witten equation in terms of superfields. For this we will apply the method developed by Wess and collaborators in [7][8][9][10] to determine the Seiberg-Witten maps for the superfield case.…”

“…This map has become known as the Seiberg-Witten map. In [7][8][9][10] gauge theory on noncommutative space was formulated using the Seiberg-Witten map. In contrast to earlier approaches [11][12][13][14], this method works for arbitrary gauge groups.…”

Comments on the U(2) noncommutative instanton

“…To begin with, it is well known that consistency requires that the gauge group for noncommutative gauge theories has to be U(N) (or certain subgroups of U(N) [24]- [25]). We then consider the U(2) case and write…”