In this article we consider closed bosonic string in the presence of constant metric and Kalb‐Ramond field with one non‐zero component, Bxy=Hz, where field strength H is infinitesimal. Using Buscher T‐duality procedure we dualize along x and y directions and using generalized T‐duality procedure along z direction imposing trivial winding conditions. After first two T‐dualizations we obtain Q flux theory which is just locally well defined, while after all three T‐dualizations we obtain nonlocal R flux theory. Origin of non‐locality is variable ΔV defined as line integral, which appears as an argument of the background fields. Rewriting T‐dual transformation laws in the canonical form and using standard Poisson algebra, we obtained that Q flux theory is commutative one and the R flux theory is noncommutative and nonassociative one. Consequently, there is a correlation between non‐locality and closed string noncommutativity and nonassociativity.

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 xμ$x^\mu$. The derivative of the RR field Cμαβ$C^{\alpha \beta }_\mu$ 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 yμ$y_\mu$. 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 xμ$x^\mu$, θα$\theta ^\alpha$ and trueθ¯α$\bar{\theta }^\alpha$, while nonassociativity parameter is a constant tensor containing infinitesimal Cμαβ$C^{\alpha \beta }_\mu$.

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 nonzero 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 nonlocal, because variable ∆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 Tdualize along y, we get, κ-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 Tdualization while noncommutativity and nonassociativity relations could be obtained from those in the x → y → z case by replacing H → −H.