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September 28, 2006

Puzzle Pieces Falling Into Place

Posted by Urs Schreiber

There should be a 3-group G 3G_3 governing Chern-Simons theory for gauge group GG. Which one is it?

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

(1)G 3=(U(1)Ω^GPG) 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)AUT(String G) \mathrm{AUT}(\mathrm{String}_G)

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

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

Moreover, the canonical lax 2-representation #

(4)ρ:Σ(String G)Σ(C 2) \rho : \Sigma(\mathrm{String}_G) \to \Sigma(C_2)

for C 2=Hilb C_2 = \mathrm{Hilb}_\mathbb{C} should extend canonically to a lax 3-representation

(5)ρ˜:Σ(G 3)End(Σ(C 2)) \tilde \rho : \Sigma(G_3) \to \mathrm{End}(\Sigma(C_2))

on endomorphisms of C 2C_2 #.

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

(6)2D CFT3D TFT \text{2D CFT} \leftrightarrow \text{3D TFT}

to higher Schreier theory #.

Here is my evidence.

  • First of all, G 3=(U(1)Ω^GPG)G_3 = (U(1) \to \hat \Omega G \to P G) is indeed a sub-3-group of AUT(Ω^GPG)\mathrm{AUT}(\hat \Omega G \to P G).

    In fact, I think that for any strict 2-group (HtG) (H \stackrel{t}{\to} G) we get a strict 3-group ker(t)HtG \mathrm{ker}(t) \to H \stackrel{t}{\to} G which is a sub-3-group of AUT(HG), \mathrm{AUT}(H \to G) \,, by restricting all vertical morphisms f()f(\bullet) in this calculation to identities.

  • It is thought to be known that the obstruction for a GG-bundle on XX to lift to a String G\mathrm{String}_G-2-bundle # on XX 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)Ω^GPG)(U(1) \to \hat \Omega G \to P G).
  • The 3-representation ρ˜:Aut(Σ(G 2))End(Σ(C 2))\tilde \rho : \mathrm{Aut}(\Sigma(G_2)) \to \mathrm{End}(\Sigma(C_2)) is obtained from the lax ρ:Σ(G 2)Σ(C 2)\rho : \Sigma(G_2) \to \Sigma(C_2) by noticing that the relevant constructions in Aut(Σ(G 2))\mathrm{Aut}(\Sigma(G_2)) # and End(Σ(C 2))\mathrm{End}(\Sigma(C_2)) # involve the same diagrams. The central U(1)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)Ω^GPG)G_3 = (U(1) \to \hat \Omega G \to P G) (as well as the 3-vector transport associated under ρ˜\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 String G\mathrm{String}_G. This is a point that requires more detailed discussion.
  • This seems to indicate a connection between ρ˜\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=AUT(String) GG_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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5 Comments & 2 Trackbacks

Re: Puzzle Pieces falling into Place

Just a comment on the strict 3-group ker(t)HGker(t) \to H \to G; I think it might equivalent (biequivalent) to HGH \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 HGH \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 nn-curvature # and the description of nn-connections in terms of higher Schreier theory #.

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

Say we have a central extension

(1)1U(1)H^H1 1 \to U(1) \to \hat H \to H \to 1

of an ordinary group HH.

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

Notice this:

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

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

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

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

the 2-group

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

should be equivalent to

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

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

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

But, if you take the (U(1)H^)(U(1)\to \hat H) 3-cocycle and forget the H^\hat H-part, just remembering the U(1)U(1)-part, you do get a (U(1)1)(U(1)\to 1) 3-cocycle, which is (in general) non-trivializable as a (U(1)1)(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 (Ω^GPG)(\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” nn-groups such as U(1)H^U(1)\to \hat H and ker(t)HtG\mathrm{ker}(t) \to H \stackrel{t}{\to} G.

While the nn-bundles with these structure groups will be trivializable, there is interesting information in how they are trivialized, i.e. which morphism (twisted (n1)(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:K^Lt\colon \widehat{K} \to L for the crossed module, A=ker(t)A = ker(t) and K=K^/ALK = \widehat{K}/A \trianglelefteq L, I believe we have weak equivalences (AK^L)(1KL)(11L/K). (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/KG = L/K. What this does, however, is give us a representative G(AK^L)(A11) G \xleftarrow{\sim} (A \hookrightarrow \widehat{K} \to L) \to (A \to 1 \to 1) for the map of 3-group stacks GB 2AG \to \mathbf{B}^2 A and hence, after delooping, the classifying map BGB 3A\mathbf{B}G \to \mathbf{B}^3 A for the multiplicative AA-gerbe on GG.

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 nn-transport with values in TT is not, in general, an nn-functor with values in TT, but an (n+1)(n+1)-functor with values in Aut(T)\mathrm{Aut}(T) (or, more generally, End(T)\mathrm{End}(T)).

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

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

More precisely - for T=Σ(C)T = \Sigma(C) the suspension of a monoidal category - Inn(Σ(T))\mathrm{Inn}(\Sigma(T)) looks essentially like Σ(T)\Sigma(T), only that 2-morphisms of Σ(T)\Sigma(T), which are 2-globes R f R \array{ \bullet &\stackrel{R}{\to}& \bullet \\ &\;\Downarrow f& \\ \bullet &\stackrel{R'}{\to}& \bullet } are replaced by squares R v f f u f R , \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 fv_f and u fu_f accounts in particular for the following structures:

  • For 2-functors into T=Σ(G 2)T = \Sigma(G_2), for G 2G_2 a strict 2-group, we have v f=Idv_f = \mathrm{Id} and u fu_f provides # the fake curvature #.
  • In particular, for 2-functors into T=Σ(HtG)T = \Sigma(H \stackrel{t}{\to} G), we have (ker(t)HtG)Aut(HG)(\mathrm{ker}(t) \to H \stackrel{t}{\to} G) \subset \mathrm{Aut}(H \to G), with u f=Idu_f = \mathrm{Id} and the ker(t)\mathrm{ker}(t)-degree of freedom in the morphisms u fu fu_f \to u'_f which run “perpendicular to (HG)(H \to G)”.
  • For lax 2-functors into T=Σ(C 2)T = \Sigma(C_2), for C 2C_2 a monoidal category with duals on objects, v fv_f and u fu_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 nn-transport pseudofunctor tra:P nEnd(T) \mathrm{tra} : P_n \to \mathrm{End}(T) with associated curvature curv tra:P n+1End(T). \mathrm{curv}_\mathrm{tra} : P_{n+1} \to \mathrm{End}(T) \,.


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

    In particular, the relation 2D CFT3D TFT \text{2D CFT} \leftrightarrow \text{3D TFT} is induced by tracurv tra \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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