### Wednesday at the Streetfest II

#### Posted by Guest

Hi - I’ve decided to join in the fun and write a little about Michael Mueger’s talk in the Streetfest. I’ve always been a fan of Michael’s work on modular categories, so it was fun to see what he’s been up to lately. My report here will be quite sketchy because he blasted us with so much information that after a little while I decided to just listen rather than take notes.

John Baez

In case you’re wondering, a modular tensor category - or “modular category”, for short - is a kind of structure that one gets from the conformal blocks of a rational conformal field theory. This has recently been proved quite generally in two different frameworks: Michael Mueger and coauthors proved it using algebraic quantum field theory:

Multi-interval Subfactors and Modularity of Representations in Conformal Field Theory

while Huang proved it in some more algebraic setup. I think this paper by Mueger could be a good place to learn about this stuff:

Conformal Field Theory and Doplicher-Roberts Reconstruction

Anyway, you get modular categories from rational conformal field theories. Or, you can get them as the categories of (nice) representations of quantum groups at a root of unity. Or, you can get them by starting with the group algebra of a finite group, taking its “quantum double” a la Drinfeld, which is a Hopf algebra, and then taking the category of representations of that!

On the other hand, topologists like modular categories because Reshetikhin, Turaev and others showed how to use them to build 3d topological quantum field theories.

We can summarize all this by saying that modular categories serve as a stepping stone when it comes to generalizing and making rigorous Witten’s famous paper on Chern-Simons theory and the Jones polynomial, in which he got certain 3d TQFTs (namely Chern-Simons theories) from certain rational conformal field theories (namely Wess-Zumino-Witten models).

Anyway, mathematicians are now interested in modular categories as objects in their own right, and Mueger has done a lot of cool work on this.

Perhaps I should break down and say what a modular category is. A category is “premodular” if it’s braided monoidal category that’s spherical, abelian, semisimple and C-linear, with finitely many simple objects, where the unit object is simple. A premodular category is “modular” if it has trivial “center” and its “dimension” is nonzero. Here the “center” is the full subcategory consisting of objects x for which the double braiding ${B}_{x,y}{B}_{y,x}$ is the identity for all objects y. The “dimension” is the sum of the squares of the dimensions of all the simple objects.

You should probably skip that last paragraph unless you already sort of understand this stuff!

Anyway, here’s a sample of the kind of thing Mueger has proved:

There’s a way to take the tensor product of modular categories and get a new one, and he’s shown that every modular category is a tensor product of “prime” ones - I believe in a unique way.

Also, for every full premodular category C sitting inside a modular category M he defines the “commutant” C’ to be the full subcategory consisting of objects x whose double braiding with all objects y in C is the identity. He then shows that C” = C and that the dimension of C times the dimension of C’ is the dimension of M. I think he also shows that C tensored with C’ is M, which then feeds into the result I mentioned about prime factorization.

This sort of result makes the structure theory of modular tensor categories even better behaved than that of finite groups! You can read more about this kind of thing here:

On the Structure of Modular Categories

Physicists might be more interested in Mueger’s work on orbifold theories: if there’s a group G acting on a modular category, he can define a new modular category which is a subtle kind of “quotient” of the original one. In the case of modular categories coming from conformal field theory, this captures the orbifold construction! See

Conformal Orbifold Theories and Braided Crossed G-Categories

for details.

Cool stuff! I wish you were down here, Urs.

John Baez

## loop groups and vN algebras

I have been looking at p. 8 of Müger’s paper that you mentioned, as well as at Wassermann’s work (that he and Stolz&Teichner cite a lot)

A. Wassermann

Operator Algebras and Conformal Field Theory III:Fusion of positive energy representations of $\mathrm{LSU}(N)$ using bounded operators

math.QA/9806031

I have to better understand this. But apparently something like

seems to be true, or so. This must be a simple statement, but I seem to be missing some points in Wassermann’s discussion.

Anyway, once this is understood the relation of Stolz&Teichner’s enriched elliptic objects to a ${\mathcal{P}}_{1}\mathrm{Spin}(n)$-2-connection should be clearer (to me, that is, perhaps it’s already clear to you). Probably what they have is more like an associated ${\mathcal{P}}_{1}\mathrm{Spin}(n)$-2-(vector bundle) than a principal 2-bundle.

Possibly you have been telling me similar things before, but I was reminded of them only now that I saw Krüger’s definition 3.7.