Noncommutativity and nonassociativity of type II superstring with coordinate dependent RR field -- the general case

TL;DR

This paper investigates the noncommutative and nonassociative structures arising from T-duality in type II superstring theory with coordinate-dependent RR fields, employing a generalized Buscher procedure to analyze non-geometric backgrounds.

Contribution

It extends the Buscher T-duality method to coordinate-dependent backgrounds, revealing non-local, non-geometric structures and deriving the Poisson brackets of the T-dual theory.

Findings

T-duality leads to non-commutative and non-associative structures.

The T-dual theory is non-local and defined in a non-geometric double space.

Poisson brackets of the T-dual theory are explicitly derived.

Abstract

In this paper we consider non-commutativity that arises from T-duality of bosonic coordinates of type II superstring in presence of coordinate dependent Ramond-Ramond field. Action with such choice of the background fields is not translational invariant. Consequently, we will employ generalization of Buscher procedure that can be applied to cases that have coordinate dependent fields and that do not possess translational isometry. Bosonic part of newly obtained T-dual theory is non-local and defined in non-geometric double space spanned by Lagrange multipliers $y_\mu$ and double coordinate ${\Delta}V_\mu$. We will apply Buscher procedure once more on T-dual theory to check if original theory can be salvaged. Finally, we will use T-dual transformation laws along with Poisson brackets of original theory to derive Poisson bracket structure of T-dual theory.