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.
Highlights
It is well known that the open string endpoints, attached to a Dp-brane, are non-commutative [1,2,3,4,5,6,7,8,9,10,11,12]
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 see 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 momentum numbers
Summary
It is well known that the open string endpoints, attached to a Dp-brane, are non-commutative [1,2,3,4,5,6,7,8,9,10,11,12]. We performed the T-dualization procedure along all the coordinates, and we obtained the T-duality transformation yμ = yμ(xμ) of the locally nongeometric background (the end of the chain (1.2) with yμ and fD-flux) and the geometric background (torus with H -flux in the beginning of the chain (1.2)). In both approaches it was assumed that the geometric backgrounds (described by Y a in [16] and by X a in our paper) have the standard commutation relations. The third appendix contains the mathematical details regarding the transition from PB { X, Y } to PB {X, Y }
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