Abstract
We consider the closed string propagating in the weakly curved background which consists of constant metric and Kalb-Ramond field with infinitesimally small coordinate dependent part. We propose the procedure for constructing the T-dual theory, performing T-duality transformations along coordinates on which the Kalb-Ramond field depends. The obtained theory is defined in the non-geometric double space, described by the Lagrange multiplier $y_\mu$ and its $T$-dual $\tilde{y}_\mu$. We apply the proposed T-duality procedure to the T-dual theory and obtain the initial one. We discuss the standard relations between T-dual theories that the equations of motion and momenta modes of one theory are the Bianchi identities and the winding modes of the other.
Highlights
Duality symmetry was for the first time described in the context of toroidal compactification in [1,2,3]
We show that the T -dual of the T -dual is the original theory
We will investigate the closed bosonic string moving in the weakly curved background, with the goal to find the generalization of the Buscher construction of the T -dual theory
Summary
Duality symmetry was for the first time described in the context of toroidal compactification in [1,2,3] (thoroughly explained in [4,5]). To preserve the physical meaning of the original theory, one requires that the new fields vαμ do not carry the additional degrees of freedom. To obtain a theory physically equivalent to the original one, all degrees of freedom carried by the gauge fields vαμ should be eliminated. The first one is that the target space of the T dual theory in the weakly curved background turns out to be a non-geometrical one [15,16,17,18,19,20,21,22,23,24,25,26,27,28] This is a doubled space with two coordinates, one of them being the Lagrange multiplier as in the case of the flat background. We show that the T -dual of the T -dual is the original theory
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