Abstract
We introduce the 2D dimensional double space with the coordinates ZM = (xμ, yμ), whose components are the coordinates of initial space xμ and its T-dual yμ. We shall show that in this extended space the T-duality transformations can be realized simply by exchanging the places of some coordinates xa, along which we want to perform T-duality, and the corresponding dual coordinates ya. In such an approach it is evident that T-duality leads to the physically equivalent theory and that a complete set of T-duality transformations forms a subgroup of the 2D permutation group. So, in double space we are able to represent the backgrounds of all T-dual theories in a unified manner.
Highlights
T-duality of the closed string has been investigated for a long time [1, 2, 3, 4]
Permutation of the initial coordinates xa with its T-dual ya we can realize by multiplying double space coordinate (2.14), written as by the constant symmetric matrix (T a)T = T a 1 − Pa Pa Pa 1 − Pa
Introducing the 2D dimensional space, which beside initial D dimensional space-time coordinates xμ contains the corresponding T-dual coordinates yμ, we offered simple formulation for T-duality transformations
Summary
T-duality of the closed string has been investigated for a long time [1, 2, 3, 4]. It transforms the theory of a string moving in a toroidal background into the theory of a string moving in different toroidal background. From the canonical point of view considered in Ref.[8], there is similarity between open and closed string non-commutativity In both cases, the initial coordinates are given as a functions of some effective coordinates but as a linear combination of the effective coordinates and the effective momenta. The basic tools in our approach are T-duality transformations connected beginning and end of the chain Rewriting these transformations in the double space we obtain the fundamental expression, where the generalized metric relate derivatives of the extended coordinates. We will show that this expression is enough to find background fields from every nodes of the chain and T-duality transformations between arbitrary nodes In such a way we unify the beginning and all corresponding T-dual theories of the chain (1.1)
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