Comments on gauge equivalence in noncommutative geometry

Abstract

We investigate the transformation from ordinary gauge field to noncommutative one which was introduced by N. . It is shown that the general transformation which is determined only by gauge equivalence has a path dependence in 'θ-space'. This ambiguity is negligible when we compare the ordinary Dirac-Born-Infeld action with the noncommutative one in the U(1) case, because of the U(1) nature and slowly varying field approximation. However, in general, in the higher derivative approximation or in the U(N) case, the ambiguity cannot be neglected due to its noncommutative structure. This ambiguity corresponds to the degrees of freedom of field re-Gauge theories on noncommutative spaces have been investigated for many years from mathematical and physical viewpoint ([Connes] and the references in [SW]). Especially in string theory, the worldvolume theory of D-branes in a background B-field is described by noncommutative Yang-Mills or Dirac-Born-Infeld theory.Recently, Seiberg and Witten [SW] argued the equivalence between ordinary gauge field theory and the noncommutative one as the low energy effective theories of open strings: they arise from the same two-dimensional field theory regularlized in different ways, so that there must be a transformation among them. In [SW], this transformation is uniquely given by the gauge equivalence relation, and this implies the equivalence between ordinary Dirac-Born-Infeld action and the noncommutative one.In this short note, we re-examine the validity of above arguments and point out that the transformation of [SW] has in general ambiguities. In section 2, we begin with the gauge equivalence relation between two nearby points in the 'θ-space' and show that there is ambiguity with arbitrary constant parameters. Then, we discuss the path dependence of the transformation in the 'θ-space', which is found by considering the commutator of two transformations. This implies the existence of another ambiguity. In section 3 we investigate these ambiguities from different viewpoint. Next in section 4, we consider the U(1) case in slowly varying field approximation. This is the situation of [SW] in comparing the ordinary Dirac-Born-Infeld action with the noncommutative one. In this case the ambiguities do not affect the result of [SW], because of the U(1) nature and of neglecting the higher derivative terms. In section 5, we summarize the paper and give some discussions. In "Note Added", we argue that the path dependence can be reduced to the field redefinition.

Derivative corrections to the D-brane Born-Infeld action: non-geodesic embeddings and the Seiberg-Witten map

“…It is easy to see that a tensor will generically not remain a tensor after a field redefinition. The ambiguity of the SW map and the relation to field redefinitions was first pointed out in [21] (see also [10]). The possibility of derivative corrections to the SW map will be discussed later on.…”

Nonequivalent Seiberg?Witten maps for noncommutative massive U(N) gauge theory

“…We are considering a theory involving the pure gauge field b µ besides the usual gauge field a µ . So, the space of solutions for ǫ (1) , G (1) , A (1) µ , B (1) µ representing the non commutative field extensions is actually greater than the one studied in detail in [10]. One can check that now instead of (4.1) we get…”

“…Interesting aspects of the general form of this map can be found in [10]. The mapped Lagrangian is usually written as a nonlocal infinite series of ordinary fields and their space-time derivatives but the noncommutative Noether identities are however kept by the Seiberg-Witten map.…”

Singularities of the Seiberg-Witten map

“…That is, the result of action of two infinitesimal shifts δ 1 θ and δ 2 θ on A i or F ij depends on their order [4]. Analogous statement holds for a gauge group element g(x) even in the zero curvature case [5].…”

“…In the U (1) case a constant curvature F ij is gauge-invariant whereas A i is not. Furthermore, a solution of the SW map for the gauge field A i depends on the choice of a deformation path in the θ-space even in the zero curvature case [4,5]. These technical problems are minimized if we consider a linear gauge field on R d θ :…”