Puzzle Pieces Falling Into Place
Posted by Urs Schreiber
There should be a 3group $G_3$ governing ChernSimons theory for gauge group $G$. Which one is it?
I would like to present evidence that it should be the strict 3group #
which is a sub3group of the nonstrict automorphism 3group #
of the $\mathrm{String}_G$ # 2group #
Moreover, the canonical lax 2representation #
for $C_2 = \mathrm{Hilb}_\mathbb{C}$ should extend canonically to a lax 3representation
on endomorphisms of $C_2$ #.
Unless I am mixed up  which is your task to find out  this suggests to relate the correspondence
to higher Schreier theory #.
Here is my evidence.

First of all, $G_3 = (U(1) \to \hat \Omega G \to P G)$ is indeed a sub3group of $\mathrm{AUT}(\hat \Omega G \to P G)$.
In fact, I think that for any strict 2group $(H \stackrel{t}{\to} G)$ we get a strict 3group $\mathrm{ker}(t) \to H \stackrel{t}{\to} G$ which is a sub3group of $\mathrm{AUT}(H \to G) \,,$ by restricting all vertical morphisms $f(\bullet)$ in this calculation to identities.
 It is thought to be known that the obstruction for a $G$bundle on $X$ to lift to a $\mathrm{String}_G$2bundle # on $X$ is a ChernSimons 2gerbe # classified by (half of) the first Pontryagin class. I think the Deligne 4cocycle of that ChernSimons 2gerbe # is precisely a nonabelian transition cocycle for the 3group $(U(1) \to \hat \Omega G \to P G)$.
 The 3representation $\tilde \rho : \mathrm{Aut}(\Sigma(G_2)) \to \mathrm{End}(\Sigma(C_2))$ is obtained from the lax $\rho : \Sigma(G_2) \to \Sigma(C_2)$ by noticing that the relevant constructions in $\mathrm{Aut}(\Sigma(G_2))$ # and $\mathrm{End}(\Sigma(C_2))$ # involve the same diagrams. The central $U(1)$ is in both cases realized in terms of the modifications of pseudonatural transformations of auto/endomorphisms of a 2category.
 3transport with values in $G_3 = (U(1) \to \hat \Omega G \to P G)$ (as well as the 3vector transport associated under $\tilde \rho$) associates to 1, 2 and 3paths essentially the sort of data that people like Freed # and Stolz & Teichner (see the table on p. 78 of their text ) have identified. Here I say “essentially” because there is an issue with different equivalent incarnations of $\mathrm{String}_G$. This is a point that requires more detailed discussion.
 This seems to indicate a connection between $\tilde \rho$associated # 3transport and the 3transport which seems to underlie # the FRS description # of ChernSimons/CFT.
 Finally, if we allow ourselves to think of the 2D/3D QFTs here as strings/membranes, then the identification $G_3 = \mathrm{AUT}(String)_G$ matches exactly the proposed identification # of the gauge 3group of the corresponding target space theory.
Re: Puzzle Pieces falling into Place
Just a comment on the strict 3group $ker(t) \to H \to G$; I think it might equivalent (biequivalent) to $H \to G$, but I suppose that’s sort of ok  we are looking for inner automorphisms. I’ll have to think about this one