Abstracts

Bojan Nikolic

From 3D torus with H-flux to torus with R-flux and back

We consider 3D closed bosonic string propagating in the constant metric and Kalb-Ramond field with one non-zero component, B_{xy}=Hz, where field strength H is infinitesimal. In the first part of the article, applying Buscher T-dualization procedure and generalized one, we T-dualize along line x -> y -> z, which means that we T-dualize first along x coordinate, then along y and, finally, along z coordinate. 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.

In the second part of the article, we reverse the T-dualization line and T-dualize along z -> y -> x. All three theories are nonlocal, because variable \Delta V appears as an argument of background fields. After the first T-dualization we obtain commutative and associative theory, while after we T-dualize once more, along y, we get noncommutative and associative theory. At the end, dualizing along x, 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 -> y -> z case by replacing H -> -H.