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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?
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 $G$ in degree 0 and an abelian group in degree 1. David BZ in the quote is saying that he thinks somehow the nonabelian $G$ itself must move to degree 1.
I fixed the typo, thanks.
I had the impression that David BZ doesn’t think the construction possible for non-abelian $G$ “…we can only do that for $G$ 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.
I had the impression that David BZ doesn’t think the construction possible for non-abelian $G$ “…we can only do that for $G$ abelian”.
The crossed module $G \to 1$ (with $G$ in degree 1) exists and hence is a 2-group precisely only of $G$ is abelian. So one cannot move a non-abelian group to degree 1 in this naive way.
But crossed modules $G \to H$ with $G$ non-abelian do exist if $H$ and the structure maps are chosen suitably.
For instance the string 2-group of $G$ under debate is equivalent to the crossed module which has not $G$ but its (still non-abelian) centrally extended loop group $\hat \Omega G$ in degree 1 and its (equally non-abelian) path group in degree 0: $\hat \Omega G \to P G$.
But as another way of going ’non-abelian’, is there anything to prevent the passage from non-abelian group $G$ to delooped groupoid $B G$, and then to form a $B G$-principal infinity-bundle.
Why restrict to 2-groups?
Any ideas on which 2-group (or $n$-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 $B T$ for $T$ a torus group larger than $U(1)$. (For instance the T-duality 2-group is of this form.) The result is something which looks like “$T$ 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.
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