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nLab > Latest Changes: symmetry protected trivial order

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 10th 2013
    • (edited Oct 10th 2013)

    Xiao-Gang Wen has started an entry symmetry protected trivial order.

    He mentions ’group super-cohomology theory’ as describing fermionic SPT orders. Given our super-slick account of group cohomology, do we just change the ambient ∞\infty-topos to Super ∞\infty Grpd?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2013

    Thanks for highlighting, I would have missed it otherwise.

    Yes, if this means (as it seems it does) group cohomology of super-Lie groups, then, yes, this is just mapping spaces in Super∞GrpdSuper\infty Grpd.

    Right now we are talking about just such super Lie group cohomology at our Super Gerbes meeting. And that’s why right now I have to run and quit reading here. But I’ll try to get back to this later this evening. Thanks again for the heads up.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 10th 2013
    • (edited Oct 10th 2013)

    Started a page for Xiao-Gang Wen, who is now at the Perimeter Institute. The description there mentions ’condensed matter’ theory. We could do with an entry on that. Someone at Princeton gives it a go here. So solid-state physics is now seen as a branch of condensed matter physics.

    EDIT: Oh, that’s just taken from wikipedia Condensed matter physics.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2013

    We have had a stub solid state physics for some time. I have added more redirects.

    • CommentRowNumber5.
    • CommentAuthorxgwen
    • CommentTimeOct 11th 2013
    Group super-cohomology theory is a term that Gu and I invented. It is not the group cohomology for supergroup. It is a "group cohomology" with anticommuting coefficient. The usual group cohomology with U(1) coefficient is denoted as H^d[G,U(1)].
    In group super-cohomology we replace U(1) by something that contain anticommuting Grassman numbers. We really want to know is there such kind of group cohomology theory in math (maybe under a different name). I would like to thank Urs for editting the SPT entry.
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 11th 2013

    Thanks for the information. I would expect that also your definition of cohomology with super-geometric coefficients is still given by maps in the higher supergeometric topos.

    Could you point me to the precise page of an article where the group super-cohomology in your sense is defined? Thanks!

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 11th 2013

    Presumably it’s Symmetry-protected topological orders for interacting fermions – fermionic topological non-linear sigma-models and a group super-cohomology theory

    In this paper, we introduce a (special) group super-cohomology theory

    See Appendix C on p. 35.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 7th 2015
    • (edited Oct 7th 2015)

    I have finally begun to cross-linke symmetry protected topological order with higher dimensional WZW model, due to the article

    • Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013) arXiv:1106.4772; A short version in Science 338, 1604-1606 (2012) pdf

    which argues that the bosonic SPT phases are described just by such higher WZW models.

    This needs to be expanded on.

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