Canonical approach to the closed string non-commutativity

Abstract

We consider the propagation of the closed bosonic string in the weakly curved background. We show that the closed string non-commutativity is essentially connected to the T-duality and nontrivial background. From the T-duality transformation laws, connecting the canonical variables of the original and T-dual theory, we find the structure of the Poisson brackets in the T-dual space corresponding to the fundamental Poisson brackets in the original theory. We find that the commutative original theory is equivalent to the non-commutative T-dual theory, in which Poisson brackets close on winding and momenta numbers and the coefficients are proportional to the background fluxes.