# The equivalence of two worlds related by T-duality

T-duality in string theory relates a world containing open and closed strings with a D$p$-brane with a compact dimension with radius $R$ with a dual world with a D$(p-1)$-brane with a radius $R'=\alpha'/R$, where the D$(p-1)$-brane is located at an certain angle $\theta$ on the dual circle.

If you apply the duality a second time on the dual world containing a D$(p-1)$ (and imagine this brane has still a compact dimension), you get a dual dual world with a D$(p-2)$.

• By duality I understand an isomorphism relating two different worlds. In the case that all $p$ dimensions of the brane are compact. Is this world containing than isomorphic to one with a D$0$-brane?
• I also suppose that my understanding of a duality as an isomorphic is somehow wrong: if you apply the 'duality-map' you do not get the original D$p$-brane but a D$(p-2)$-brane.
• Yes, if all the dimensions are compact, well, we really mean that all spatial dimensions are compactified on a torus $T^9$, then (multiple) T-duality may map any simple D$p$-brane aligned with some dimensions to a D0-brane.
• Under T-duality, D$p$-brane is mapped either to a D$(p+1)$-brane or a D$(p-1)$-brane, so its dimension either increases or decreases. Which scenario occurs depends on whether the T-dualized direction of space is one along the D-brane world volume or not. If it is along the world volume, then the dual D-brane will become localized in the dual dimension, and the dual dimension to the T-dualized one will therefore be transverse, and not parallel, to the new D-brane, and therefore its dimensionality drops by one. If we T-dualize a dimension transverse to the D-brane, the T-dual D-brane becomes extended, so its dimension is higher than the original one by one. If we T-dualize the "same" dimension (it's really two dimensions T-dual to each other) twice, then clearly the brane is wrapped on the dimension in one step and localized in the dimension in the other step, so the D-brane dimension goes once up and once down, returning where you were before.