## April 3, 2007

### Oberwolfach CFT, Tuesday Morning

#### Posted by Urs Schreiber

Q-systems in $C^*$-categories, the Drinfeld double and its modular tensor representation category and more on John Roberts’ ideas on higher nonabelian cohomology in quantum field theory, all on one Tuesday morning at this CFT workshop.

I have to be brief, more details might follow later.

Q-systems

Early in the morning Pinhas Grossmann reviewed the central concepts that revolve around the notion of a Q-system in a $C^*$-category. Luckily, with categorical theoretic hindsight, the gist of this can be summarized in a few words:

A $C^*$-category is simply the obvious many-object version of a $C^*$-algebra.

For instance, let $A$ be a von Neumann algebra, and consider the monoidal subcategory $\mathrm{AQFTBim}(A) \subset \mathrm{Bim}(A)$ that contains only those bimodules which are induced from endomorphisms of $A$ and only those morphisms of bimodules that come from intertwiners of such endomorphisms.

Then $\mathrm{AQFTBim}(A)$ is a $C^*$-category, with the star operation simply coming from the Hilbert space adjoint of bounded operators.

By making use of that $*$-structure it is possible to give something like half the definition of a Frobenius algebra internal to $\mathrm{AQFTBim}(A)$, with the remaining half of morphisms being provided by taking the star of those appearing in the definition. This is called a Q-system. It turns out that the internal Frobenius algebras obtained this way are actually “special” and symmetric.

I tried to find out what is known about how Q-systems sit inside the 2-category of all special symmetric Frobenius algebras internal to $\mathrm{AQFTBim}(A)$. the bottom line seemed to be that special symmetric Frob algebras internal to $\mathrm{AQFTBim}(A)$ turn out in examples considered to always be Q-systems, and that for well studied choices of $A$ the classification of $Q$-systems coincides with that of the corresponding special symmetric Frobenius algebras.

But a general theorem clarifying the inclusion is apparently unknown.

Modular tensor categories from the Drinfeld double of a finite group

WZW models correspond to conformal field theories coming from strings propagating on Lie groups. There is a finite group analog of most everything of the technology involved in this context.

Jürgen Müller reviewed for us some basics of modular tensor categories, of the algebra that is called the “Drinfeld double” of a finite group, and how its representation category provides a well-studied example for a modular tensor categoriy.

Again I am lucky, in that slides for the entire talk are available from the author’s website:

Modular tensor categories and quantum doubles

After the talk there was a question about how this compares to the Lie group case. I advertized the beautiful insight by Simon Willerton that the entire Drinfeld double issue becomes much more transparent, and in fact enjoyable, after we realize that underlying it is really the action groupoid of the finite group on itsef, which, one observes, is nothing but the groupoid of functors from the circle to the group, which, in turn, can be shown to play exactly the role of the loop group of $G$.

Later Zoran Skoda told me that Bressler has been working for quite some time on exactly this picture, too, with the results still not available publicly. Zoran said that Bressler’s work contains Willerton’s insight as a special case. But I haven’t seen it yet.

John Roberts on QFT and higher nonabelian cohomology

The above was the official part. Behind the scenes I am having a very interesting and very helpful conversation with Roberto Conti on possible higher categorical structures in algebraic quantum field theory.

He pointed me to the interesting paper

J. Roberts & G. Ruzzi
A cohomological description of connections and curvature over posets
math/0604173

which extends John Roberts’ old ideas on nonabelian cohomology and its relation to quantum field theory.

This is about higher cocycles with values in $n$-groups that appear as iterated inner automorphisms $n$-groups of some ordinary 1-group $G$ $G_2 = \mathrm{INN}(G)$ $G_3 = \mathrm{INN}(\mathrm{INN}(G))$ $G_4 = \mathrm{INN}(\mathrm{INN}(\mathrm{INN}(G))) \,.$ (I’ll comment on these iterated inner automorphism group further below.)

Actually, the description is pretty closely related to Toby Bartels’ notion of 2-bundles, one difference being that Roberts and Ruzzi take care of formulating everything on arbitrary $n$-simplices, not necessarily those coming from some covering space.

In fact, the motivation for them is application of this formalism to algebraic quantum field theory (“structural characterization of gauge theory”), where they would take the underlying simplicial domain category to be induced by open subset in some Minkowski spacetime, partially ordered by inclusion.

They also discuss connections for these cocycles. At a first glance (and I have not yet had time for more than that) it is a little hard to decide exactly how this is related this is to our Cocycles of parallel Transport 2-Functors with values in a 2-group, but I think that their cocycle description would essentially yield the 2-anafunctor , which is an ordinary functor on 2-paths in the transition groupoid.

Another interesting parallel to what we have been talking about is the use of iterated inner automorphism groups. I emphasized how this is what we want to look at in $n$-Transport and Higher Schreier Theory and $n$-Curvature.

Here at the $n$-Café we also once talked about how iterated inner automorphism groups should correspond to the sequence of topological field theories that starts WZW, CS, BF, …

Posted at April 3, 2007 1:23 PM UTC

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### Re: Oberwolfach CFT, Tuesday Morning

Here at the n-Café we also once talked about how iterated inner automorphism groups should correspond to the sequence of topological field theories that starts WZW, CS, BF, …

That doesn’t seem to be the place where I asked if WZW is really the beginning of the series, or if there’s one or two that come before it. So I’ll ask now (with the understanding that this won’t distract you from Oberwolfach, of course).

Posted by: Allen Knutson on April 3, 2007 4:08 PM | Permalink | Reply to this

### Re: Oberwolfach CFT, Tuesday Morning

I asked if WZW is really the beginning of the series, or if there’s one or two that come before it

I do remember that you asked that before, and that I said I would have to think a little more about the right answer.

Back then John mentioned that there is one standard way do go down the dimensional ladder of TQFTs, namely simply by integrating everything over the circle.

Might be that this is the answer, but if so I do not quite see it yet.

In fact, my above statement is itself already imprecise. What I really have an argument for (namely this one) is that CS theory at level $k$ is a theory of connections with values in $\mathrm{INN}(\mathrm{String}_k(G))$.

While it is true that WZW theory at level $k$ is governed just by $\mathrm{String}_k(G)$ itself, the role this 2-group play here is different from the role the 3-group $\mathrm{INN}(\mathrm{String}_k(G))$ should play in CS theory.

And it is not clear to me at the moment, what the analogous next step down the dimensional ladder would be. I am not sure why and if we’d have a reason to expect it to be present.

On the other hand, there are rough plausibility considerations which indicate that $\mathrm{INN}(\mathrm{INN}(\mathrm{String}_k(G)))$ would be the right structure 4-group of 4D BF theory, I think.

Posted by: urs on April 4, 2007 12:01 AM | Permalink | Reply to this

### Re: Oberwolfach CFT, Tuesday Morning

Long ago, Allen wrote:

I asked if WZW is really the beginning of the series, or if there’s one or two that come before it.

As Urs mentioned, there’s a standard way to get an $(n-1)$-dimensional TQFT from an $n$-dimensional one. Namely, given an $n$-dimensional TQFT, say $Z$, we define an $(n-1)$-dimensional one, say $Z'$, by

$Z'(X) = Z(X \times S^1)$

If we apply this ‘looping’ construction to the 3d TQFT called Chern–Simons theory we get a 2d TQFT which could be called the topological Wess–Zumino–Witten model (but is usually called the $G/G$ gauged WZW model).

From this, in turn, we can get a 1d TQFT, which is just a Hilbert space. And from that we can get a 0d TQFT, which is just a complex number (the dimension of our Hilbert space). This not particularly thrilling.

In the other direction, one can hope that the 3d TQFT called Chern–Simons theory comes from a 4d TQFT with the 2nd Chern class as action. That would be the logical end of the line, since the 2nd Chern class is where this whole series of field theories comes from, and it’s a 4d cohomology class. There is a candidate 4d TQFT, namely $B F$ theory. However, I have never been able to obtain Chern–Simons theory from $B F$ theory by ‘looping’.

So, there’s a puzzle.

Another perhaps more exciting puzzle is: can we generalize this whole story to higher Chern classes? Or, more generally: can we get a tower of TQFTs going up to dimension $2 n$ or maybe $2n -1$ starting from a compact simple Lie group and a class in $H^{2 n}(B G, \mathbb{Z})$?

Posted by: John Baez on January 13, 2009 2:03 AM | Permalink | Reply to this

### Re: Oberwolfach CFT, Tuesday Morning

Unless manifolds are essential, why not the suspension $\Sigma X$ instead of the product of X and a circle?

Posted by: jim stasheff on January 13, 2009 3:15 PM | Permalink | Reply to this

### Re: Oberwolfach CFT, Tuesday Morning

Hi Urs, these descriptions of what’s going on over at Oberwolfach are really useful, keep up the good work.

Posted by: Bruce Bartlett on April 3, 2007 5:44 PM | Permalink | Reply to this

### Re: Oberwolfach CFT, Tuesday Morning

The guy speaking about Q-systems this morning was Pinhas (not Punhas)
Grossman. I wish I were there - thanks for the synopses!

Posted by: Scott Morrison on April 4, 2007 6:29 AM | Permalink | PGP Sig | Reply to this

### Re: Oberwolfach CFT, Tuesday Morning

Thanks for catching that! I’ve fixed it.

Posted by: urs on April 4, 2007 9:15 AM | Permalink | Reply to this
Read the post On Roberts and Ruzzi's Connections over Posets
Weblog: The n-Category Café
Excerpt: On J. Roberts and G. Ruzzi's concept of G-bundles with connection over posets and its relation to analogous notions discussed at length at the n-Cafe.
Tracked: August 16, 2007 11:15 AM

### Bressler et al

This was the only reference I could find to Bressler here at the cafe

Has anyone look at the very categorical/stacky
Bressler,Gorokhovsky,Nest and Tsygan:
Deformations of algebroid stacks?
arXiv:0810.0030