The n-Category Café

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, …