## September 28, 2006

### Puzzle Pieces Falling Into Place

#### Posted by Urs Schreiber There should be a 3-group $G_3$ governing Chern-Simons theory for gauge group $G$. Which one is it?

I would like to present evidence that it should be the strict 3-group #

(1)$G_3 = (U(1) \to \hat \Omega G \to P G)$

which is a sub-3-group of the non-strict automorphism 3-group #

(2)$\mathrm{AUT}(\mathrm{String}_G)$

of the $\mathrm{String}_G$ # 2-group #

(3)$\mathrm{String}_G = (\hat \Omega G \to P G) \,.$

Moreover, the canonical lax 2-representation #

(4)$\rho : \Sigma(\mathrm{String}_G) \to \Sigma(C_2)$

for $C_2 = \mathrm{Hilb}_\mathbb{C}$ should extend canonically to a lax 3-representation

(5)$\tilde \rho : \Sigma(G_3) \to \mathrm{End}(\Sigma(C_2))$

on endomorphisms of $C_2$ #.

Unless I am mixed up - which is your task to find out - this suggests to relate the correspondence

(6)$\text{2D CFT} \leftrightarrow \text{3D TFT}$

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 sub-3-group of $\mathrm{AUT}(\hat \Omega G \to P G)$.

In fact, I think that for any strict 2-group $(H \stackrel{t}{\to} G)$ we get a strict 3-group $\mathrm{ker}(t) \to H \stackrel{t}{\to} G$ which is a sub-3-group 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 $\mathrm{String}_G$-2-bundle # on $X$ is a Chern-Simons 2-gerbe # classified by (half of) the first Pontryagin class. I think the Deligne 4-cocycle of that Chern-Simons 2-gerbe # is precisely a nonabelian transition cocycle for the 3-group $(U(1) \to \hat \Omega G \to P G)$.
• The 3-representation $\tilde \rho : \mathrm{Aut}(\Sigma(G_2)) \to \mathrm{End}(\Sigma(C_2))$ 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 2-category.
• 3-transport with values in $G_3 = (U(1) \to \hat \Omega G \to P G)$ (as well as the 3-vector transport associated under $\tilde \rho$) associates to 1-, 2- and 3-paths 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 # 3-transport and the 3-transport which seems to underlie # the FRS description # of Chern-Simons/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 3-group of the corresponding target space theory.
Posted at September 28, 2006 1:53 PM UTC

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### Re: Puzzle Pieces falling into Place

Just a comment on the strict 3-group $ker(t) \to H \to G$; 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

Posted by: David Roberts on September 29, 2006 2:33 AM | Permalink | Reply to this

### Re: Puzzle Pieces falling into Place

I think it might equivalent (biequivalent) to $H \to G$

Yes, right. I should have commented on that. We had run into this issue before, for instance here (and also by private email), going back to the considerations on $n$-curvature # and the description of $n$-connections in terms of higher Schreier theory #.

Let’s look at the analogous situation one level down.

Say we have a central extension

(1)$1 \to U(1) \to \hat H \to H \to 1$

of an ordinary group $H$.

Say we want to understand lifts of principal $H$-bundles to $\hat H$-bundles. These lifts are obstructed by the corresponding $U(1)\to 1$ lifting gerbes. If this lifting gerbe is nontrivial, the lift does not exist.

Notice this:

These $(U(1)\to 1)$-lifting gerbes together with the $H$-bundles they correspond to, provide exactly the data of a trivializable nonabelian gerbe with 2-group

(2)$U(1) \to \hat H \,.$

If you write it down, you see immediately that the 3-cocycle of a $(U(1) \to \hat H)$-gerbe is the same as that of the (possibly twisted) bundle obtained by lifting a 2-cocycle for an $H$-bundle to $\hat H$.

So this is the same sort of situation as with the 3-group above:

the 2-group

(3)$(U(1) \to \hat H)$

should be equivalent to

(4)$(1 \to H) \,.$

Correspondingly, all $(U(1)\to \hat H)$-gerbes are trivializable.

In fact, they are trivializable because they are all manifestly trivialized by a twisted $\hat H$-bundle.

But, if you take the $(U(1)\to \hat H)$ 3-cocycle and forget the $\hat H$-part, just remembering the $U(1)$-part, you do get a $(U(1)\to 1)$ 3-cocycle, which is (in general) non-trivializable as a $(U(1)\to 1)$-cocycle.

I think this is what is going on in the Chern-Simons theory, too.

As supportive evidence, consider this: Stolz and Teichner emphasize that a string bundle, i.e. a $(\hat \Omega G \to P G)$-gerbe, provides a trivialization for Chern-Simons theory (pp. 79-80).

For these reasons I think it does make sense to consider these “blown up” $n$-groups such as $U(1)\to \hat H$ and $\mathrm{ker}(t) \to H \stackrel{t}{\to} G$.

While the $n$-bundles with these structure groups will be trivializable, there is interesting information in how they are trivialized, i.e. which morphism (twisted $(n-1)$-bundle) trivializes them.

Posted by: urs on September 29, 2006 10:19 AM | Permalink | Reply to this

### Re: Puzzle Pieces falling into Place

It’s been a while, but what I said in 2006 was wrong. Writing $t\colon \widehat{K} \to L$ for the crossed module, $A = ker(t)$ and $K = \widehat{K}/A \trianglelefteq L$, I believe we have weak equivalences $(A \hookrightarrow \widehat{K} \to L)\quad \xrightarrow{\sim}\quad (1 \to K \hookrightarrow L)\quad \xrightarrow{\sim}\quad (1 \to 1 \to L/K).$ Hence the 3-group Urs mentions in his post is up to equivalence of smooth group stacks, just an ordinary Lie group $G = L/K$. What this does, however, is give us a representative $G \xleftarrow{\sim} (A \hookrightarrow \widehat{K} \to L) \to (A \to 1 \to 1)$ for the map of 3-group stacks $G \to \mathbf{B}^2 A$ and hence, after delooping, the classifying map $\mathbf{B}G \to \mathbf{B}^3 A$ for the multiplicative $A$-gerbe on $G$.

Posted by: David Roberts on October 3, 2017 2:15 AM | Permalink | Reply to this

### Re: Puzzle Pieces falling into Place

As the pieces of the puzzle fall into place, have you gained a better view of what the picture is? Could you describe it in terms of the “X is a Y-structure in the context Z” story you told us about here?

Posted by: David Corfield on September 29, 2006 11:04 AM | Permalink | Reply to this

### Re: Puzzle Pieces falling into Place

As the pieces of the puzzle fall into place, have you gained a better view of what the picure is?

What made puzzle pieces fall - and apparently even into place - was, for me, the observation (stated here and supported - or in fact explained - there) that the right notion of $n$-transport with values in $T$ is not, in general, an $n$-functor with values in $T$, but an $(n+1)$-functor with values in $\mathrm{Aut}(T)$ (or, more generally, $\mathrm{End}(T)$).

For the relevant applications, we may find $T$ sitting inside $\mathrm{End}(T)$ as part of the “inner” endomorphisms. Let me call these $\mathrm{Inn}(T)$.

But even if we restrict to these inner morphisms, there are more degrees of freedom in $\mathrm{Inn}(T)$ than in $T$. There are injections $T \stackrel{\subset}{\to} \mathrm{Inn}(T) \stackrel{\subset}{\to} \mathrm{End}(T) \,.$

More precisely - for $T = \Sigma(C)$ the suspension of a monoidal category - $\mathrm{Inn}(\Sigma(T))$ looks essentially like $\Sigma(T)$, only that 2-morphisms of $\Sigma(T)$, which are 2-globes $\array{ \bullet &\stackrel{R}{\to}& \bullet \\ &\;\Downarrow f& \\ \bullet &\stackrel{R'}{\to}& \bullet }$ are replaced by squares $\array{ \bullet &\stackrel{R}{\to}& \bullet \\ v_f \downarrow\;\; &\;\Downarrow f& \;\;\downarrow u_f \\ \bullet &\stackrel{R'}{\to}& \bullet } \,,$ and that there are now 3-morphisms going between the vertical sides of these squares.

Now, unless I am mixed up, the new freedom provided by these vertical arrows $v_f$ and $u_f$ accounts in particular for the following structures:

• For 2-functors into $T = \Sigma(G_2)$, for $G_2$ a strict 2-group, we have $v_f = \mathrm{Id}$ and $u_f$ provides # the fake curvature #.
• In particular, for 2-functors into $T = \Sigma(H \stackrel{t}{\to} G)$, we have $(\mathrm{ker}(t) \to H \stackrel{t}{\to} G) \subset \mathrm{Aut}(H \to G)$, with $u_f = \mathrm{Id}$ and the $\mathrm{ker}(t)$-degree of freedom in the morphisms $u_f \to u'_f$ which run “perpendicular to $(H \to G)$”.
• For lax 2-functors into $T = \Sigma(C_2)$, for $C_2$ a monoidal category with duals on objects, $v_f$ and $u_f$ seem to provide # the “Wilson lines” running perpendicular to the boundary in Chern-Simons 3D TFT.

The third point can be understood as the image under a 3-representation of the second point.

Could you describe it in terms of the “X is a Y-structure in the context Z” story you told us about here?

Very roughly, my impression is this:

The “Y-structure” we are dealing with is:

- an $n$-transport pseudofunctor $\mathrm{tra} : P_n \to \mathrm{End}(T)$ with associated curvature $\mathrm{curv}_\mathrm{tra} : P_{n+1} \to \mathrm{End}(T) \,.$

Then

• A non-fake-flat principal $(H \to G)$-gerbe with connection is a Y-structure of the above sort for $T$ a 2-groupoid with vertex 2-group equivalent to $\Sigma(H \to G)$
• Chern-Simons theory with gauge group $G$ is the quantum theory of # a Y-structure as above, which is associated to a principal $\mathrm{String}_G = (\hat \Omega G \to P G)$-gerbe example above by a canonical 3-rep.

In particular, the relation $\text{2D CFT} \leftrightarrow \text{3D TFT}$ is induced by $\mathrm{tra} \leftrightarrow \mathrm{curv}_\mathrm{tra}$

The first of these two items I think I understand sufficiently. For the second item I have so far no full description. But I do have the evidence provided in the entry above. There are probably refinements necessary in my statement of this second item.

But that’s the picture that I thought I see emerging.

Posted by: urs on September 29, 2006 12:17 PM | Permalink | Reply to this
Read the post WZW as Transition 1-Gerbe of Chern-Simons 2-Gerbe
Weblog: The n-Category Café
Excerpt: How the WZW 1-gerbe arises as the transition 1-gerbe of the Chern-Simons 2-gerbe.
Tracked: October 29, 2006 5:11 PM
Read the post A 3-Category of twisted Bimodules
Weblog: The n-Category Café
Excerpt: A 3-category of twisted bimodules.
Tracked: November 3, 2006 2:20 PM

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