Canonical approach to the closed string noncommutativity

TL;DR

This paper explores how T-duality transforms a commutative closed string theory into a non-commutative one in weakly curved backgrounds, revealing non-geometric structures like R-flux in string theory.

Contribution

It derives the Poisson bracket structure of the T-dual theory, showing the emergence of non-commutativity proportional to background fluxes and winding/momentum numbers.

Findings

Original theory is commutative; T-dual is non-commutative.

Poisson brackets in T-dual space depend on fluxes and winding/momentum.

Non-commutative T-dual theory is more nongeometrical than T-folds.

Abstract

We consider the closed string moving in the weakly curved background and its totally T-dualized background. Using T-duality transformation laws, we find the structure of the Poisson brackets in the T-dual space corresponding to the fundamental Poisson brackets in the original theory. From this structure we obtain that the commutative original theory is equivalent to the non-commutative T-dual theory, whose Poisson brackets are proportional to the background fluxes times winding and momenta numbers. The non-commutative theory of the present article is more nongeometrical then T-folds and in the case of three space-time dimensions corresponds to the nongeometric space-time with $R$-flux.