T-Duality on Torus

1. Aug 21, 2015

"pi"mp

I think I have a relatively decent grasp on T-duality where we've compactified $S^{1}$. However, when compactifying a 2-torus, is the T-duality transformation where you invert both radii of the two circles simultaneously, or is the claim that you can invert one of the two, leaving the other fixed, and yield the same physics?

I suspect that it's the second choice. If indeed this is the case, is there anything at all special about the case where you invert both radii simultaneously? I believe this is equivalent to merely interchanging the A-cycle and B-cycle on the torus.

On a $d$ torus, the T-duality is actually enlarged a group $O(d,d;\mathbb{Z})$ of transformations. Some of these are analogous to $R\rightarrow 1/R$, but others involve a change of basis of the lattice of the torus (which includes the coordinate redefinition that swaps the A and B-cycles.) So we can perform these transformations singly or compose them to form more complicated transformations.