Possible values of the index for subfactor inclusions coming from conformal nets This question is related to Can the minimal index of a subfactor take all values in {4cos^2(pi/n);n=3,4,5,...} u [4,infinity]?
I was wondering what one knows for the special case of conformal nets let's say on the circle. 
For a representation $\pi$ of a conformal net $I\mapsto \mathcal A(I)$ one has a index for the inclusion of type $III_1$ factors: 
$$\pi(\mathcal A(I)) \subset \mathcal \pi(\mathcal A(I'))'$$
where $I$ is any "proper" intervall on the circle.
For the vacuum representation the index is 1 because the inclusion is trivial by Haag duality.
I found that Wassermann showed that the inclusion of $\pi(L_ISU(2)) \subset \pi(L_{I'}SU(2))'$ of positive energy representations at level $\ell$ have index values $\lbrace \sin^2(k \pi/\ell)/ \sin^2(\pi/\ell) \rbrace$. This set contains $4 \cdot \cos^2(\pi/\ell)$ e.g. $k=2$. (btw. I am still looking for the original reference).
Question: Which values can the index take in the set $[4,\infty]$.
 A: I'm not an expert on nets, but these indices are all dimensions of objects in unitary braided tensor categories, right?  You can already use that to get gaps in small dimensions using skein theoretic techniques pioneered by Wenzl in joint work with Kazhdan and then with Tuba (MR1237835 and http://arxiv.org/abs/math/0301142).
To see this worked out explicitly look at:


*

*Longo's "Minimal index and braided subfactors" MR1183606

*Rehren's "On the Range of the Index of Subfactors" MR1359925


For an expository explanation Wenzl's techniques and some other applications of it, you can see Section 3 of one of our papers with Scott and Emily
These techniques are quite difficult extend much further than 6, because we don't know a skein theoretic classification of objects in tensor categories with $X \otimes X \cong A \oplus B \oplus C$.
A: I my unpublished article I have a proof that the values of index for irreducible hyperfinite  subfactors span the interval [8, infinity] , for non-hyperfinite  inclusions I think S.Popa proved that these values span all the real numbers equal or greater than 4.
A: This might be interesting for you.Actually I just proved that the values of index of hyperfinite irreducibe subfactors span the interval [4 , infinity] . Bahman Mashood
A: I am sorry about a wrong statement in the interval [4,8] there are many gaps where the values of index do not exist. 
Bahman
