Non-commutativity from the double sigma model

TL;DR

This paper explores how non-commutativity naturally emerges from the double sigma model, showing that $O(D,D)$ symmetry unifies commutative and non-commutative theories and relates to the Seiberg-Witten map.

Contribution

It demonstrates the intrinsic non-commutative nature of the double sigma model and links $O(D,D)$ symmetry to the unification of commutative and non-commutative theories.

Findings

Non-commutativity arises from the double sigma model's structure.

$O(D,D)$ symmetry unifies commutative and non-commutative theories.

The open-closed relation is an $O(D,D)$ rotation leading to the Seiberg-Witten map.

Abstract

We show how non-commutativity arises from commutativity in the double sigma model. We demonstrate that this model is intrinsically non-commutative by calculating the propagators. In the simplest phase configuration, there are two dual copies of commutative theories. In general rotated frames, one gets a non-commutative theory and a commutative partner. Thus a non-vanishing $B$ also leads to a commutative theory. Our results imply that $O\left(D,D\right)$ symmetry unifies not only the big and small torus physics, but also the commutative and non-commutative theories. The physical interpretations of the metric and other parameters in the double sigma model are completely dictated by the boundary conditions. The open-closed relation is also an $O(D,D)$ rotation and naturally leads to the Seiberg-Witten map. Moreover, after applying a second dual rotation, we identify the description parameter in the Seiberg-Witten map as an $O(D,D)$ group parameter and all theories are non-commutative under this composite rotation. As a bonus, the propagators of general frames in double sigma model for open string are also presented.