 The equivalence of two worlds related by T-duality

+ 3 like - 0 dislike
3815 views

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$(p-2)$.

I'm wondering about two issues:

• 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.

This post imported from StackExchange Physics at 2014-08-10 13:48 (UCT), posted by SE-user Anne O'Nyme
asked Aug 9, 2014

• 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.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysic$\varnothing$OverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.