Moyal Representation of the String Field Star Product in the Presence of a B-background

TL;DR

This paper demonstrates that Witten's star algebra for open strings in a B-background can be represented as a product of Moyal pairs, revealing how noncommutativity arises from both string overlap and the B-field, with implications for algebra contraction.

Contribution

It introduces a Moyal representation of the string field star product in the presence of a B-background, detailing the structure and B-dependence of Moyal pairs and their contraction behavior.

Findings

Star algebra retains Moyal pair structure with B-background.

B-dependent Moyal parameters are only for discrete pairs.

Large-B limit simplifies the star algebra to a center-of-mass noncommutative algebra.

Abstract

In this paper we show that in the presence of an anti-symmetric tensor $B$-background, Witten's star algebra for open string fields persists to possess the structure of a direct product of commuting Moyal pairs. The interplay between the noncommutativity due to three-string overlap and that due to the $B$-background is our main concern. In each pair of noncommutative directions parallel to the $B$-background, the Moyal pairs mix string modes in the two directions and are labeled, in addition to a continuous parameter, by {\it two} discrete values as well. However, the Moyal parameters are $B$-dependent only for discrete pairs. We have also demonstrated the large-$B$ contraction of the star algebra, with one of the discrete Moyal pairs dropping out while the other giving rise to the center-of-mass noncommutative function algebra.