T-duality as coordinates permutation in double space

TL;DR

This paper presents a framework where T-duality in string theory is represented as coordinate permutations within a doubled space, simplifying the understanding of dualities and unifying backgrounds of T-dual theories.

Contribution

The authors introduce a double space with coordinates for original and dual variables, showing T-duality as coordinate permutation, which unifies all T-dual backgrounds.

Findings

T-duality transformations are realized as coordinate exchanges in double space.

Complete T-duality transformations form a subgroup of the permutation group.

Unified representation of all T-dual backgrounds in double space.

Abstract

We introduce the $2D$ dimensional double space with the coordinates $Z^M= (x^\mu, y_\mu)$ which components are the coordinates of initial space $x^\mu$ and its T-dual $y_\mu$. We shall show that in this extended space the T-duality transformations can be realized simply by exchanging places of some coordinates $x^a$, along which we want to perform T-duality and the corresponding dual coordinates $y_a$. In such approach it is evident that T-duality leads to the physically equivalent theory and that complete set of T-duality transformations form subgroup of the $2D$ permutation group. So, in the double space we are able to represent the backgrounds of all T-dual theories in unified manner.