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In this article we consider closed bosonic string in the presence of constant metric and Kalb‐Ramond field with one non‐zero component,B   x  y    =  H  z   , 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    Δ  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.

Noncommutativity and Nonassociativity of Closed Bosonic String on T‐dual Toroidal Backgrounds