Construction of non-Abelian gauge theories on noncommutative spaces

Abstract

We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with a constant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge theories.

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.…”

“…The main difference of our approach from [1,2] is that we use unconstrained commutative gauge fields and gauge parameters for all elements of a basis of U (rather than only of g) and drop the unwanted fields and parameters only at the very end. An advantage of this approach is that the existence of the models to all orders is manifest.…”

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.…”