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I was reading about realizations of the "Fibonacci" fusion ring $X \otimes X = X \oplus 1$ in Fusion Categories of Rank 2 by Victor Ostrik. Apparently, there are two of them and they arise in various ways:

  • integer-spin representations of integrable $\widehat{sl}_2$-modules of level 3
  • the minimal model $\mathcal{M}(2,5)$ of the Virasoro algebra (central charge c = -22/5)
  • representations of $U_q(sl_2)$ with $q = e^{\pi i /5}, e^{3\pi i / 5}$.

In any of these cases, how is the Fibonacci category realized?

I would like to understand each of these specific categorifications of the Fibonacci fusion ring. Can someone here explain the basics of integrable $\widehat{sl}_2$-modules or about the $\mathcal{M}(2,5)$ minimal model from Conformal Field Theory? I would also like to learn about $U_q(sl_2)$, though it's probably written in many texts.

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I don't understand your question. What does "how is it realized" mean? Usually you realize an abstractly dscribed object by providing a construction, but then you've already answered your own question by providing several constructions. –  Noah Snyder Dec 30 '09 at 21:42
Ostrik's paper explains this category pretty well in the abstract; it's the constructions themselves I do not understand. In other words, where can I learn the basics about \hat{sl}_2 modules or M(2,5)? It would be even better if someone explained them here. –  john mangual Dec 31 '09 at 1:05
Thanks for clarifying what you are after. –  Chris Schommer-Pries Dec 31 '09 at 2:02
You would like to learn about $U_q(sl_2)$ "even though" it is written in many texts? I think what you mean is, you'd like a mathoverflow summary of those texts. With all due respect, I hope that no one obliges you, because a good book such as Kassel is the right way to learn the material. The rest of your question, a specific understanding of the Fibonacci category in its two Galois conjugate forms, is better. –  Greg Kuperberg Dec 31 '09 at 2:29
It sounds like the theory of planar algebras arises naturally in attempts to understand all these physical models. It was probably unrealistic to assume it can all be explained here. These book suggestions really help. –  john mangual Dec 31 '09 at 16:11
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2 Answers

Unfortunately all three of those realizations are the sort of thing you need to read a book about not a MO post. I agree with Greg that Kassel's book is a great place to start for the quantum group construction (I don't know the other two constructions well, presumably for the affine algebra construction you'd want to start with Kac's book?).

On the other hand there is an easier to explain elementary diagrammatic description. As usual with diagram categories you only construct a full subcategory and then you'll need to take the additive and idempotent completions to get an abelian category.

Consider the Temperley-Lieb subcategory, whose objects are indexed by integers and whose morphisms m->n are given by linear combinations of planar diagrams of nonintersecting arcs with m boundary points at the bottom and n boundary points at the top modulo a single relation that a circle can be removed for a multiplicative factor of either the golden ratio or its conjugate. Composition is stacking, tensor product is disjoint union. There's an explicit 4-strand projection (called a Jones-Wenzl idempotent) here that has the property that any way you close it off you get zero. Kill that idempotent. Now look at the "even part" i.e. the full subcategory whose objects are even integers. This is your category. Its simple objects are the 0 and 2-strand Jones-Wenzl idempotents.

There's another way to think of this example. First checkboard shade the regions of all your even diagrams so that they're unshaded on the outside. Then collapse all the dark regions to lines. What you end up with now has half as many boundary points and is allowed to have internal 3-valent and 1-valent vertices. It's easy to see that they satisfy an I=H relation and a relation allowing absorbing vertices. This gives a construction of the Fibonacci category using the Yamada polynomial relations (I think to get the usual Yamada polynomial on the nose here you want to actually throw in a bunch of JW2s everywhere but its six of one half dozen the other).

Finally there's a slightly different diagram description given in the appendix of one of my papers with Emily Peters and Scott Morrison. In our notation there the Fibonacci category is (the additive and idempotent completion of) the tadpole planar algebra T_2.

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The Virasoro minimal model $\mathcal{M}(2,5)$ (or in some conventions also $\mathcal{M}(5,2)$ is the conformal field theory which describes the critical behaviour of the Lee-Yang edge singularity. It is described, for example, in Conformal Field Theory, by di Francesco, Mathieu and Sénéchal; albeit the description of the Lee-Yang singularity itself is perhaps a little too physicsy. Still their treatment of minimal models should be amenable to mathematicians without prior exposure to physics.

At any rate, googling Lee-Yang edge singularity might reveal other sources easier to digest. In general it is the Verlinde formula which relates the fusion ring and the Virasoro characters, and at least for the case of the Lee-Yang singularity, these can be related in turn to Temperley-Lieb algebras and Ocneanu path algebras on a suitable graph. Some details appear in this paper.

The relation between the Virasoro minimal models and the representations of $\widehat{sl}_2$ goes by the name of the coset construction in the Physics conformal field theory literature or also Drinfeld-Sokolov reduction. This procedure gives a cohomology theory (a version of semi-infinite cohomology for a nilpotent subalgebra) which produces Virasoro modules from $\widehat{sl}_2$ modules. Relevant words to google are W- algebras, Casimir algebras,... Of course here we are dealing with the simplest case of $\widehat{sl}_2$ and Virasoro, which is the tip of a very large iceberg. The case of the Lee-Yang edge singularity is simple enough that it appears in many papers as an example from which to understand more general constructions.

I know less about the quantum group story, but this paper of Gaberdiel might be a good starting point.

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