Noncommutativity and nonassociativity of type II superstring with coordinate dependent RR field

TL;DR

This paper explores how coordinate-dependent Ramond-Ramond fields induce noncommutativity and nonassociativity in type II superstring theory through generalized T-duality, revealing complex algebraic structures in non-geometric backgrounds.

Contribution

It introduces a generalized Buscher procedure for coordinate-dependent backgrounds and demonstrates how T-duality leads to noncommutative and nonassociative structures in superstring theory.

Findings

Noncommutativity depends on supercoordinates $x^$, $ heta^$, $ar heta^$

Nonassociativity parameter is a constant tensor with infinitesimal RR field derivative

Original theory can be reconstructed from T-dual theory using transformation laws

Abstract

In this paper we will consider noncommutativity that arises from bosonic T-dualization of type II superstring in presence of Ramond-Ramond (RR) field, which linearly depends on the bosonic coordinates $x^\mu$. The derivative of the RR field $C^{\alpha\beta}_\mu$ is infinitesimal. We will employ generalized Buscher procedure that can be applied to cases that have coordinate dependent background fields. Bosonic part of newly obtained T-dual theory is non-local. It is defined in non-geometric space spanned by Lagrange multipliers $y_\mu$. We will apply generalized Buscher procedure once more on T-dual theory and prove that 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 and nonassociativity relation. Noncommutativity parameter depends on the supercoordinates $x^\mu$, $\theta^\alpha$ and $\bar\theta^\alpha$, while nonassociativity parameter is a constant tensor containing infinitesimal $C^{\alpha\beta}_\mu$.