A cohomological approach to the non-abelian Seiberg-Witten map

Abstract

We present a cohomological method for obtaining the non-Abelian Seiberg-Witten map for any gauge group and to any order in θ. By introducing a ghost field, we are able to express the equations defining the Seiberg-Witten map through a coboundary operator, so that they can be solved by constructing a corresponding homotopy operator. * email address: dmbrace@lbl.gov † email address: BLCerchiai@lbl.gov ‡

Noncommutative GUTs, Standard Model and C,P,T

“…SW map was initially introduced for U(N) gauge fields in [9], in the context of open string theory (and the zero slope limit α ′ → 0 [9]). It has then been studied in the case of arbitrary gauge groups [10,11,[13][14][15][16] and space-time dependent parameters θ µν [17]. The SW map and the ⋆ product allow us to expand the noncommutative action order by order in θ and to express it in terms of ordinary commutative fields, so that one can then study the property of this θ-expanded commutative action.…”

On the All Order Solutions of Seiberg-Witten Map for Noncommutative Gauge Theories

“…and therefore one can explicitly study the underlying cohomological structure in the NC case 14,16 . Moreover, by requiring SW-map respects the gauge equivalence (5)ŝ µ (A; θ) = s µ (A; θ).…”

“…Recently, it has been shown that these solutions can also be rewritten in a geometric setting and they are compatible with hermiticity and charge conjugation conditions 13 . Our aim in this short note, is first to review the results given in Ref.12 in a slightly different setting 14 and then to comment on the general structure of the homogeneous solutions of the SW-map to all orders. Moreover, we also show that the contribution of the first order homogeneous solution to the second order can be written again in terms of the first order homogeneous solutions, similar to the inhomogeneous case.…”

The background field method and the linearization problem for Poisson manifolds

“…It was introduced in [43] in order to formulate the WZ consistency condition for the Seiberg-Witten map. By construction ∆ is nilpotent.…”

“…The question of the existence of a homotopy for ∆ has been addressed in [43]. On general grounds one can prove [13,14] that such an operator exists provided that the structure constants f ij k are those of a semisimple Lie algebra.…”