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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 11th 2010

Added to T-duality a section with the discussion of the usual path-integral heuristics for why the two sigma-models on T-dual backgrounds yield equivalent quantum field theories.

1. Dear Urs,

the path integral heuristics behind path-integral sigma-models T-duality should actually be simpler (at least at the level of the naive idea). Namely, everything boils down to saying that the 2-torus obtained by opposite sides identification of $[0,1]\times [0,T]$ and the one obtained by opposite sides identification of $[0,1]\times [0,1/T]$ (both with the standard flat metric obtained by restriction from $\mathbb{R}^2$) are conformally equivalent.

A way of seeing this is to recall that up to conformal equivalence a 2-torus can be seen as a parallelogram in $\mathbb{C}=\mathbb{R}^2$ with a vertex in $0$, a vertex in $1$ and the other vertex in $\tau\in\mathbb{H}$, where $\mathbb{H}$ denotes the upper half-plane. The torus of parameter $\tau$ and the one of parameter $\tau'$ are conformally equivalent iff

$\tau'=\frac{a\tau+b}{c\tau+d}$

with

$\left( \array{ a&b\\ c&d }\right)$

in $SL(2;\mathbb{Z})$. The two parameters $i T$ and $i/T$ are related by the matrix

$S=\left( \array{ 0&-1\\ 1&0 }\right)$
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 11th 2010

True, but I do not quite see how you’d derive T-duality of the target space this way.

Also, one point of the computation I posted is that it also applies to the open string and shows the action of T-duality on D-branes.

• CommentRowNumber4.
• CommentAuthordomenico_fiorenza
• CommentTimeJun 11th 2010
• (edited Jun 11th 2010)

Right, I misunderstood what you were referring to. now I see that what I wrote could have something to do with what you were saying, but at the moment I could not say exactly what.. :)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 11th 2010
• (edited Jun 11th 2010)

I think you are certainly right that the basic mechanism underlying T-duality is at least morally that of the $S$-transformation on a torus. But I don’t quite see how one can upgrade this observation to a proof that two T-dual backgrounds give equivalent sigma-model QFTs. Possibly there is a way, though.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 14th 2012

Correcting some typos at T-duality, I don’t know how to fix

A quick way to get an indication for this is to notice that the center-of-mass energy of the string in such a circle-bundle background is In terms of the worldsheet theory.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeFeb 14th 2012

Thanks for catching this. Not sure what happened there. I have changed the sentence to

A quick way to get an indication for this is to consider the center-of-mass energy of the string in such a circle-bundle background.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJan 3rd 2019