Noncommutative Yang-Mills from equivalence of star products

Abstract

It is shown that the transformation between ordinary and noncommutative Yang-Mills theory as formulated by Seiberg and Witten is due to the equivalence of certain star products on the D-brane world-volume.

Gauge theory on noncommutative spaces

“…This expression has appeared in string theory contexts related to noncommutative Yang-Mills theory mainly as a coordinate transformation [12,13,14].…”

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

“…We shall show, however, that all the component fields can be obtained from a Lie algebra-valued connection by a SeibergWitten map [3,4,6]. This was also observed in [16], where a result in this direction has been obtained for SO(n) and Sp(n).…”

“…The construction of the dependent coefficients is based on the Seiberg-Witten map [3]. The existence of this map can be proven in general [4,5,6], here we demonstrate this map by explicitely calculating the expansion to first order in a parameter that characterizes the deviation from commuting coordinates.…”

On the general structure of the non-abelian Born–Infeld action

“…In order to analyze the independence of the action from the non-commutativity parameter 0, it is convenient, as will become clear later, to rewrite S in terms of the ^-independent abelian potential A. To this end, we follow the analysis of Jurco and Schupp [8], whose work describes the Seiberg-Witten map Asw in cm invariant way, which is best suited for our purposes. Most of this section is then nothing but a review of the ideas of [8], rewritten in the notation of this paper.…”

“…To this end, we follow the analysis of Jurco and Schupp [8], whose work describes the Seiberg-Witten map Asw in cm invariant way, which is best suited for our purposes. Most of this section is then nothing but a review of the ideas of [8], rewritten in the notation of this paper.…”