Comments on Open Wilson Lines and Generalized Star Products

We consider an open Wilson line as a momentum representation of a boundary state which describes a D-brane in a constant B-field background. Using this picture, we study the Seiberg-Witten map which relates the commutative and noncommutative gauge fields, and determine the products of fields appearing in the general terms in the expansion of this map. In the recent study of noncommutative gauge theories, the open Wilson lines play an important role. They are used to construct gauge invariant operators [1,2,3,4], and also appear in the couplings between open and closed string modes [5,6,7,8,9]. The correlation functions of open Wilson lines are studied in [10,11,12]. In the expansion of an open Wilson line as a power series of the noncommutative gauge field, there appears the so-called generalized star products [13]. These products also appear in the scattering amplitudes of string theories and noncommutative field theories open Wilson lines were used to construct the Seiberg-Witten map which relates the commutative and noncommutative gauge fields [20]. In this map, there also appears the structure of the generalized star products. In this short note, we consider an open Wilson line as a momentum representation of a boundary state describing a D-brane in a constant B-field background. Using this picture, we study the Seiberg-Witten map and determine the products of fields which appear in the general terms in the expansion of this map. This paper is organized as follows: In section 2, we review the open Wilson lines from the viewpoint of boundary state. In section 3, we show that the generalized star products can be written as correlation functions on S 1. In section 4, we construct the Seiberg-Witten map up to O(A 3) from the gauge equivalence condition and compare this result with that obtained from the open Wilson line. Section 5 is devoted to the conclusion and discussion. 2. Open Wilson Lines from Boundary State In this section, we review how open contours appear in gauge invariant objects in noncommutative gauge theories [1,4] from the viewpoint of boundary state. Let us consider a gauge theory on noncommutative R