# 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 $[4,\infty]$.

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Probably a hard question... I would bet that, in the case of acompletely rational conformal net, the statistical dimension of a representation is always a cyclotomic integer (and hence, the index would always be the square of a cyclotomic integer). For conformal nets that are not completely rational, I have no idea what to expect. –  André Henriques Jan 19 '11 at 12:48
Andre's right that in the case of rational nets you get number theoretic obstructions. In addition to cyclotomicity (which gives good gaps arxiv.org/abs/1004.0665) you also get some strong "d-number" obstructions (see arxiv.org/abs/0810.3242 and remember that nets are always braided so you get to use the stronger results in that paper). –  Noah Snyder May 31 '11 at 22:08

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 $X \otimes X \cong A \oplus B \oplus C$.

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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.

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Popa's article is MR1198815 (Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111 (1993), no. 2, 375–405) and indeed shows that you can get all real numbers above 4 as an index of a nonhyperfinite subfactor. There's been a bunch of subsequent papers on this topic by Popa, Shlyakhtenko, Jones, Guionnet, Walker, that shows that any subfactor planar algebra can be realized using free group subfactors. I think this general approach gives Popa's original example just starting with Temperley-Lieb. But none of this is directly relevant to the case of nets. –  Noah Snyder May 31 '11 at 21:50

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

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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

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