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, 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 article a closed bosonic string is considered in the presence of constant metric and Kalb‐Ramond field with one non‐zero component, Bx y = Hz, where field strength H is infinitesimal. Using Buscher T‐duality procedure one dualizes along x and y directions and applies generalized T‐duality procedure along z direction imposing trivial winding conditions. After first two T‐dualizations the authors obtain a Q flux theory which is just locally well defined, while after all three T‐dualizations one obtains a nonlocal R flux theory. Rewriting T‐dual transformation laws in the canonical form and using standard Poisson algebra, the result is that the Q flux theory is a 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.