Possible values of the index for subfactor inclusions coming from conformal nets - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:32:26Z http://mathoverflow.net/feeds/question/52499 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52499/possible-values-of-the-index-for-subfactor-inclusions-coming-from-conformal-nets Possible values of the index for subfactor inclusions coming from conformal nets Marcel Bischoff 2011-01-19T10:19:47Z 2011-12-28T17:31:21Z <p>This question is related to <a href="http://mathoverflow.net/questions/39968/can-the-minimal-index-of-a-subfactor-take-all-values-in-4cos2pi-nn3-4-5" rel="nofollow">http://mathoverflow.net/questions/39968/can-the-minimal-index-of-a-subfactor-take-all-values-in-4cos2pi-nn3-4-5</a></p> <p>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.</p> <p>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).</p> <p>Question: Which values can the index take in the set $[4,\infty]$.</p> http://mathoverflow.net/questions/52499/possible-values-of-the-index-for-subfactor-inclusions-coming-from-conformal-nets/66571#66571 Answer by Bahman Mashood for Possible values of the index for subfactor inclusions coming from conformal nets Bahman Mashood 2011-05-31T16:32:07Z 2011-05-31T16:32:07Z <p>I my unpublished article I have a proof that the values of index for irreducible hyperfinite subfactors span the interval [8, infinity] , for non-hyperfinite inclusions I think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.</p> http://mathoverflow.net/questions/52499/possible-values-of-the-index-for-subfactor-inclusions-coming-from-conformal-nets/66597#66597 Answer by Noah Snyder for Possible values of the index for subfactor inclusions coming from conformal nets Noah Snyder 2011-05-31T22:04:48Z 2011-05-31T22:04:48Z <p>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 <a href="http://arxiv.org/abs/math/0301142" rel="nofollow">http://arxiv.org/abs/math/0301142</a>).</p> <p>To see this worked out explicitly look at:</p> <ul> <li>Longo's "Minimal index and braided subfactors" MR1183606</li> <li>Rehren's "On the Range of the Index of Subfactors" MR1359925</li> </ul> <p>For an expository explanation Wenzl's techniques and some other applications of it, you can see Section 3 of one of <a href="http://arxiv.org/abs/1003.0022" rel="nofollow">our papers with Scott and Emily</a></p> <p>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 $X \otimes X \cong A \oplus B \oplus C$.</p> http://mathoverflow.net/questions/52499/possible-values-of-the-index-for-subfactor-inclusions-coming-from-conformal-nets/73204#73204 Answer by Bahman for Possible values of the index for subfactor inclusions coming from conformal nets Bahman 2011-08-19T05:54:18Z 2011-08-19T05:54:18Z <p>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</p> http://mathoverflow.net/questions/52499/possible-values-of-the-index-for-subfactor-inclusions-coming-from-conformal-nets/84455#84455 Answer by Bahman Mashood for Possible values of the index for subfactor inclusions coming from conformal nets Bahman Mashood 2011-12-28T17:31:21Z 2011-12-28T17:31:21Z <p>I am sorry about a wrong statement in the interval [4,8] there are many gaps where the values of index do not exist. </p> <p>Bahman</p>