Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
  
  
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 5th 2018
   • (edited Jan 5th 2018)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIn the discussion of Saemann and Schmidt's article 'An M5-brane model', David Ben-Zvi [remarks](https://golem.ph.utexas.edu/category/2017/12/an_m5brane_model.html#c053256): >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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 5th 2018
   • (edited Jan 5th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSure, 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.

   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 in degree 0 and an abelian group in degree 1. David BZ in the quote is saying that he thinks somehow the nonabelian itself must move to degree 1.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 5th 2018
   • (edited Jan 5th 2018)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI 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 fixed the typo, thanks.

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 5th 2018
   • (edited Jan 5th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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: $\hat \Omega G \to P G$.

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

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

   But crossed modules with non-abelian do exist if and the structure maps are chosen suitably.

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

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 5th 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexBut 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?

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

   Why restrict to 2-groups?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJan 5th 2018
   • (edited Jan 5th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAny 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.

   Any ideas on which 2-group (or -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 for a torus group larger than . (For instance the T-duality 2-group is of this form.) The result is something which looks like “ 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.