Noncommutativity and nonassociativity of closed bosonic string on T-dual toroidal backgrounds

In this article we consider closed bosonic string in the presence of constant metric and Kalb-Ramond field with one non-zero component, $B_{xy}=Hz$, 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 $\Delta V$ 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.