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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 5th 2018
    • (edited Jan 5th 2018)

    In the discussion of Saemann and Schmidt’s article ’An M5-brane model’, David Ben-Zvi remarks:

    Anyway the point is from what I’ve understood the object needed for the fields of the (2,0) theory is something different - it’s roughly a bundle for the group BG. This group doesn’t exist, but say in the abelian setting when G is a torus we need something like a BT bundle (or T-gerbe), not something like a bundle for a BU(1) extension of T, which seems to me the natural abelian counterpart of a string connection. And indeed the Higgs mechanism for the nonabelian tensor field (or moving out on the Coulomb branch) produces exactly the (now perfectly understood) theory of T-gerbes.

    So that’s my confusion - it seems to me the point of whether we centrally extend G or not (in a higher sense) is not the problem, the problem is we need to somehow deloop G, and we can only do that for G abelian.

    Doesn’t groupoid-principal infinity-bundle provide the resources for this?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 5th 2018
    • (edited Jan 5th 2018)

    Sure, if we have a (non-abelian) 2-group, then there is the corresponding concept of principal 2-bundle. But the above is about the question what that 2-group should be. Christian has argued that it is the String-2-group, with a copy of a non-abelian GG in degree 0 and an abelian group in degree 1. David BZ in the quote is saying that he thinks somehow the nonabelian GG itself must move to degree 1.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 5th 2018
    • (edited Jan 5th 2018)

    I fixed the typo, thanks.

    I had the impression that David BZ doesn’t think the construction possible for non-abelian GG “…we can only do that for GG abelian”.

    Perhaps we’ll find out whether this is so if he replies to my comment pointing to the construction. Of course, as you say, there’s also then the matter of choosing a suitable 2-group or 2-groupoid.

    Before the quoted portion he writes,

    I’m happy calling the objects you’re considering built out of the string 2-group nonabelian gerbes. The issue is not if they’re abelian or not, but if they’re the RIGHT nonabelian object, and my feeling is that the sought-for nonabelian object doesn’t exist in the current language.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 5th 2018
    • (edited Jan 5th 2018)

    I had the impression that David BZ doesn’t think the construction possible for non-abelian GG “…we can only do that for GG abelian”.

    The crossed module G1G \to 1 (with GG in degree 1) exists and hence is a 2-group precisely only of GG is abelian. So one cannot move a non-abelian group to degree 1 in this naive way.

    But crossed modules GHG \to H with GG non-abelian do exist if HH and the structure maps are chosen suitably.

    For instance the string 2-group of GG under debate is equivalent to the crossed module which has not GG but its (still non-abelian) centrally extended loop group Ω^G\hat \Omega G in degree 1 and its (equally non-abelian) path group in degree 0: Ω^GPG\hat \Omega G \to P G.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 5th 2018

    But as another way of going ’non-abelian’, is there anything to prevent the passage from non-abelian group GG to delooped groupoid BGB G, and then to form a BGB G-principal infinity-bundle.

    Why restrict to 2-groups?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 5th 2018
    • (edited Jan 5th 2018)

    Any ideas on which 2-group (or nn-groupp) to use for coincident M5s should be closely guided by non-trivial plausibility checks, otherwise it’s hard to see if we get anywhere. For instance the comments that you quote in #1 by themselves are equally consistent with Christian’s proposal, since of course we also have string-like extension by BTB T for TT a torus group larger than U(1)U(1). (For instance the T-duality 2-group is of this form.) The result is something which looks like “TT shifted up in degree and made non-abelian”.

    I thought this was good about Christian’s proposal, that they claimed to haved checked e.g. that the instantons in 6d come out right as resolved self-dual string solitons, that proposals for the (1,0) theory are reproduced, etc. But I haven’t followed closely. When I am free again in a month or two I hope to look into it.