What is known about arbitrary subfactors of integer index?

Question

Let $N\subset M$ be an inclusion of ${\rm II}_1$ factors of finite index, $[M:N]<\infty$. I would be mostly interested in the hyperfinite case, $N\simeq M\simeq R$, but let us just take them arbitrary.

There is an evolving theory about "what can be said about $N\subset M$, in the general case'', which started with Jones' index theorem, in 1983. Well-known results here include the Pimsner-Popa basis and entropy formula, the bimodule interpretation, the planar algebra formalism.

• Question: assuming that we are still in the general case, but with the extra assumption that the index is an integer, $[M,N]\in\mathbb N$, what else can be said about $N\subset M$?

To my knowledge, at least some time ago (5-10 years), the only answer here was just that the Pimsner-Popa basis is a "clean" one, I mean as in standard linear algebra. I was wondering if any advances on this question come from the recent work on subject, in small or arbitrary index I mean, perhaps as some corollaries of the theory developed there (?) I would be interested in any comment/answer here, this is actually a question that I spent some time on, long time ago.

Answer 1

If you also assume finite depth, then there's a hope (it's too vague to call it a conjecture) that all integer index subfactors can be classified "using only finite group theory." That is, if you had a black box which could answer all questions about finite groups and their cohomology you'd be able to understand all finite depth integer index subfactors. The key words to look for to get the flavor of this program are "weakly integral" and "weakly group theoretical" fusion categories.

There are some results in this direction for small integer global index (I'll try to locate them when I'm at a real computer), but if you just assume small index then we only know the answer for index up to and including 5. The integrality assumption doesn't help in the approach we've been using. So index 6 is way out of reach even assuming finite depth.

One could be very optimistic and hope that in infinite depth all integer index subfactors come from (possibly infinite) discrete groups in some suitably general fashion. For example, Bisch-Haagerup subfactors come from certain triples of discrete groups.But I have no idea how one would even try to make "suitably general fashion" precise.

On the negative side, Bisch-Nicoara-Popa have shown that at integral index 6 the classification becomes somewhat wild.

Answer 2

See the issue : Non-“weakly group theoretical” integral fusion categories?

Here is proposed some integral fusion rings, such that the prospective related fusion categories, are integral but not weakly group theoretical.
Currently, we try to build some Kac algebras $\mathbb{A}$, having these fusion rings as their Grothendieck rings. The related subfactors $R^{\mathbb{A}} \subset R$ would be irreducible, depth 2, integer index, but not coming from cohomology and group theory.

You can also read my answer to the issue : Are subfactor planar algebras hard to classify at index 6?