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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]×[0,T][0,1]\times [0,T] and the one obtained by opposite sides identification of [0,1]×[0,1/T][0,1]\times [0,1/T] (both with the standard flat metric obtained by restriction from 2\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 = 2\mathbb{C}=\mathbb{R}^2 with a vertex in 00, a vertex in 11 and the other vertex in τ\tau\in\mathbb{H}, where \mathbb{H} denotes the upper half-plane. The torus of parameter τ\tau and the one of parameter τ\tau' are conformally equivalent iff

    τ=aτ+bcτ+d \tau'=\frac{a\tau+b}{c\tau+d}


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

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

    S=(0 1 1 0) 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 SS-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

    added pointer to today’s

    diff, v45, current

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 9th 2019

    Added a new reference

    • Mark Bugden, A Tour of T-duality: Geometric and Topological Aspects of T-dualities, (arXiv:1904.03583)

    diff, v46, current

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