# Directly from $H$-flux to the family of three nonlocal $R$-flux theories

## TL;DR

This paper explores the T-dualization of a 3D bosonic string in a weakly curved background, revealing a sequence of three nonlocal theories with increasing noncommutative and nonassociative properties, independent of T-dualization order.

## Contribution

It systematically derives a family of three nonlocal string theories with distinct noncommutative and nonassociative features through sequential T-dualizations.

## Key findings

- First T-dual theory is commutative and associative.
- Second T-dual theory exhibits noncommutativity but remains associative.
- Final T-dual theory is both noncommutative and nonassociative.

## Abstract

In this article we consider T-dualization of the 3D closed bosonic string in the weakly curved background - constant metric and Kalb-Ramond field with one non-zero component, $B_{xy}=Hz$, where field strength $H$ is infinitesimal. We use standard and generalized Buscher T-dualization procedure and make T-dualization starting from coordinate $z$, via $y$ and finally along $x$ coordinate. All three theories are {\it nonlocal}, because variable $\Delta V$, defined as line integral, appears as an argument of background fields. After the first T-dualization we obtain commutative and associative theory, while after we T-dualize along $y$, we get, $\kappa$-Minkowski-like, noncommutative and associative theory. At the end of this T-dualization chain we come to the theory which is both noncommutative and nonassociative. The form of the final T-dual action does not depend on the order of T-dualization while noncommutativity and nonassociativity relations could be obtained from those in the $x\to y\to z$ case by replacing $H\to - H$.