Noncommutative Wess–Zumino–Witten actions and their Seiberg–Witten invariance

Abstract

We analyze the noncommutative two-dimensional Wess–Zumino–Witten model and its properties under Seiberg–Witten transformations in the operator formulation. We prove that the model is invariant under such transformations even for the noncritical (non-chiral) case, in which the coefficients of the kinetic and Wess–Zumino terms are not related. The pure Wess–Zumino term represents a singular case in which this transformation fails to reach a commutative limit. We also discuss potential implications of this result for bosonization.

Introduction

The Wess–Zumino–Witten (WZW) model has been of fundamental importance in various physical contexts. Generalizations of this model applying to different systems, such as theories defined on a noncommutative spacetime [1], [2], [3], [4], [5], [6], higher-dimensional quantum Hall theories [7], [8] or higher-dimensional bosonization [9], [10], have also appeared.

One of the most interesting applications is in standard bosonization theory [11]. In its basic form, the WZW model provides a bosonic representation of a free fermionic system (with non-Abelian symmetries) on a two-dimensional spacetime. Further, a suitable noncommutative version of the model has been proposed as an exact bosonization of fermion systems in dimensions higher than two [9]. As a result, the study of the WZW model on noncommutative spaces as a tool for bosonization has acquired new interest.

The WZW model on two noncommutative spacetime dimensions (NCWZW) should provide a bosonic description of free fermions on the same noncommutative spacetime. Such a free fermionic theory, however, does not manifest any observable differences from the standard (commutative) free fermionic theory, and hence this NCWZW model should be, somehow, equivalent to the commutative WZW model. In [5], Moreno and Schaposnik showed that there is a transformation that maps the (critical) NCWZW model to the commutative one. Such mappings, first introduced for gauge theories, are called Seiberg–Witten (SW) transformations [12]. Subsequently, the related Chern–Simons action on three noncommutative dimensions was also shown to be invariant under an SW transformation [13].

Noncommutative field theories can be written either in the star-product formulation or in the operator formulation. Although the two formulations are equivalent, their formalisms often differ substantially in their detail and each gives a different perspective. The work of [5] was in the star-product formulation. A direct demonstration of the SW invariance of the NCWZW action proves hard to obtain in this formulation and, instead, the invariance of the variation of the action was examined. It is interesting to examine the same problem in the operator formulation and see if a direct proof of the invariance of the action itself is attainable.

Another question that naturally arises is whether the criticality of the NCWZW model is crucial for its SW invariance. In other words, it is interesting to know whether there is a similar SW mapping for the non-critical NCWZW model, in which the kinetic and Wess–Zumino terms have different coefficients.

In this work we address the above two issues. We demonstrate that the operator formulation manifests an explicit simplification for the critical theory and affords a direct proof of the SW invariance of the action (as well as a straightforward proof of the quantization of the coefficient of the Wess–Zumino term). We further show, in both the star-product and the operator formulation, that there is an SW transformation that maps any noncritical NCWZW model to the commutative one, thus establishing a correspondence for the entire family of models. The only singular point in this family is the “bare” Wess–Zumino term (without kinetic term), for which the corresponding transformation becomes singular for vanishing noncommutativity parameter and thus fails to reach a commutative limit.

The plan of this paper is as follows: in Section 2 we introduce the star-product formulation, review the work of [5] and generalize it to show the SW invariance of the noncritical NCWZW. In Section 3 we review the operator formulation of the model and of SW transformations, apply it to the model at hand and prove the invariance of the action of the NCWZW model for the noncritical case. Finally, in Section 4 we present some conclusions and directions for further research.