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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2018

    started a bare minimum at Poisson-Lie T-duality, for the moment just so as to have a place to record the two original references

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 12th 2018
    • (edited Apr 12th 2018)

    added more references, in particular Ossa-Quevedo 92, which is maybe the first one.

    Made “nonabelian T-duality” a redirect

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 12th 2018

    Are you looking at this with regard to Higher T-duality? Any interest in higher non-abelian T-duality for, say, String(G)String(G)?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 12th 2018
    • (edited Apr 12th 2018)

    Yes and no: The question in the background was whether the “geometric” complement of “topological” 3-spherical T-duality could be Poisson-Lie T-duality for isometry group SU(2)SU(2). I still don’t know on the math side. But at least the physics contexts do not match: The Poisson-Lie T-duality is based on string sigma-models, while the realization of spherical T-duality that we found replaces strings by 5-branes.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2018
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 16th 2018

    added pointer to

    • Branislav Jurco, Jan Vysoky, Poisson-Lie T-duality of String Effective Actions: A New Approach to the Dilaton Puzzle, Journal of Geometry and Physics Volume 130, August 2018, Pages 1-26 (arXiv:1708.04079)

    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2018

    added pointer to yesterday’s

    diff, v6, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 9th 2019

    added the following warning, based on discussion we had at the meeting String and M-theory, Singapre 2018. I am not aware of any written reference that would make this statement, but if anyone has one, it should be added.

    It has been proven that these generalized T-dualities are are/induce equivalences of the corresponding string sigma-models at the level of classical field theory. However it seems to be open, and in fact questionable, that away from standard T-duality this yields an equivalence at the level of worldsheet quantum field theories, hence it is open whether Poisson-Lie/non-abelian T-duality is really a duality operation on perturbative string theory vacua.

    diff, v7, current

  1. Added the reference to Double Field Theory on group manifolds

    Anonymous

    diff, v8, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2019

    Thanks! I moved your paragraph from the References-section to the main text, and have it cite your reference there.

    diff, v9, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2019

    added pointer to today’s discussion of nonabelian T-folds in

    diff, v10, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeFeb 14th 2019
    • (edited Feb 14th 2019)

    My contact person gave me the following more detailed assessment of the non-duality status of non-abelian T-duality by email. For the moment I have no citations for this, apart from this private correspondence, but since it seems to be very worthwhile information, I have put it into the entry, like so:


    While ordinary abelian T-duality is supposedly a full duality in string theory, in particular in that it is an equivalence on the string perturbation series to all orders of the string length/Regge slope α\alpha' and the string coupling constant g sg_s, it has apparntly been shown by Martin Roček (citation?) that there are topological obstructions at higher genus for non-abelian T-duality, letting it break down in higher orders of g sg_s; and already a genus-0 (tree level) it apparently breaks down for the open string (i.e. on punctured disks) at some order of α\alpha'.


    If any expert sees this, please feel invited to correct/expand.

    diff, v11, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2019

    Added pointer to today’s

    diff, v13, current

    • CommentRowNumber14.
    • CommentAuthorLuigi
    • CommentTimeSep 19th 2019

    Does this new duality reduce to higher T-duality in the abelian case? At first sight it looks like the dual algebra being shifted h=h *[n2]h' = h^\ast[n-2] does not allow it

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2019

    It would be interesting to find a relation, but in the present form I’d think neither can reduce to the other, simply because they formulate different aspects of T-duality:

    Here the Poisson-Lie perspective focuses on the local geometric content, while the higher T-duality we described is the higher generalization of topological T-duality, albeit rationalized.

    To see how the two higher generalizations relate to each other, it would hence be useful to first have a concrete relation between ordinary Poisson-Lie T-duality and ordinary topological T-duality. Did anyone look into that?

    Another vague thought one might have here is that, due to S 3SU(2)S^3 \simeq SU(2), the higher 3-spherical topological T-duality is somehow related to non-higher but 𝔰𝔲(2)\mathfrak{su}(2)-nonabelian T-duality. But beyond the coincidental S 3SU(2)S^3 \simeq SU(2) there may be no real hint for this, and in fact the physics pictures behind the two sides are disparate (M5-brane physics on one hand, perturbative string sigma models on the other.)

    So in summary, I don’t know. And I am not thinking about it. But it seems like something worth exploring.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2019

    added pointer to:

    • Ladislav Hlavatý, Ivo Petr, Poisson-Lie plurals of Bianchi cosmologies and Generalized Supergravity Equations (arxiv:1910.08436)

    diff, v14, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 19th 2019

    added pointer to today’s

    • Emanuel Malek, Daniel C. Thompson, Energy Physics - Theory Poisson-Lie U-duality in Exceptional Field Theory (arxiv:1911.07833)

    diff, v15, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2019
    • (edited Dec 17th 2019)

    added pointer to today’s

    discussion of non-abelian T-duality from a comprehensive picture of higher differential geometry, relating Kaluza-Klein compactification on principal ∞-bundles to double field theory, T-folds, type II geometry, exceptional geometry, etc.:

    diff, v17, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeApr 21st 2020

    added pointer to today’s

    diff, v19, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2020

    added pointer to today’s

    • Francesco Bascone, Franco Pezzella, Patrizia Vitale, Poisson-Lie T-Duality of WZW Model via Current Algebra Deformation (arXiv:2004.12858)

    diff, v22, current

    • CommentRowNumber21.
    • CommentAuthorLuigi
    • CommentTimeJun 23rd 2020

    Added a pointer to today

    • Chris D. A. Blair, Daniel C. Thompson, Sofia Zhidkova, Exploring Exceptional Drinfeld Geometries (arxiv:2006.12452)

    diff, v23, current

    • CommentRowNumber22.
    • CommentAuthorLuigi
    • CommentTimeJun 23rd 2020
    • (edited Jun 23rd 2020)

    I was also thinking something about this.

    They limited themselves to “Exceptional Drinfel’d doubles” representable by TDTG 2T *GT D \simeq T G \oplus \wedge^2T^\ast G for some subgroup GG, in analogy with what happens for classical Drinfel’d doubles (i.e. TDTGT *GT D\simeq T G \oplus T^\ast G).

    Now, as far as I understand the Drinfel’d double D(G)D(G) of a group GG is related to a cocycle ωH 3(G,)\omega\in H^3(G,\mathbb{Z}) (in particular, a flat gerbe(?)).

    It looks to me that they are trying to define something analogous by using ωH 4(G,)\omega\in H^4(G,\mathbb{Z}).

    Maybe then a full working definition of an “Exceptional Drinfel’d double” would require a to be in correspondence an element of the cohomotopy π 4(G)\pi^4(G), in the spirit of Exceptional geometry as given by differential cohomotopy theory.

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