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

Abstract

Seiberg-Witten maps and a recently proposed construction of noncommutative Yang-Mills theories (with matter fields) for arbitrary gauge groups are reformulated so that their existence to all orders is manifest. The ambiguities of the construction which originate from the freedom in the Seiberg-Witten map are discussed with regard to the question whether they can lead to inequivalent models, i.e., models not related by field redefinitions.

Seiberg-Witten maps to all orders

“…It is worth mentioning that the SW differential equation (30) can also be obtained in an elegant way by studying the BRST cohomology of the NC YM theory [16,17]. In this setting, it is shown that the most general solution including the homogeneous solutions of Eq.…”

Hybrid Seiberg-Witten map, itsθ-exact expansion, and the antifield formalism

“…The method in question produces a solution to the "evolution" Seiberg-Witten map equation, an equation which was obtained in Refs. [38][39][40][41] by using the antifield formalism techniques-see Refs. [42][43][44], for alternative cohomogical approaches and also Ref.…”

The Seiberg-Witten Map for Noncommutative Gauge Theories

“…This formulation is sufficient for Yang-Mills theories, but for gauge theories with reducible gauge transformations, such as theories with gauge potentials which are differential forms of degree higher than one, it is appropriate to use the antifield formalism of Batalin and Vilkoviskii (BV). The deformation of the gauge structure should then be studied by defining generalized Seiberg-Witten maps in the context of the BV formalism [12,13,14]. The use of the master equation couples intimately the gauge transformations and the dynamics, i.e.…”

“…One sees already from (14) and (15) that Λ and A i cannot be Lie algebra valued in general, and we follow [5,6] by allowing them to be in the enveloping algebra of the Lie algebra of λ and a i . A representation of this Lie algebra lifts naturally to a representation of its enveloping algebra.…”