# On the Jones index values for conformal subnets

February 19, 2010

We consider the smallest values taken by the Jones index for an inclusion of
local conformal nets of von Neumann algebras on $S^1$ and show that these values
are quite more restricted than for an arbitrary inclusion of factors. Below 4,
the only non-integer admissible value is $4\cos^2 \pi/10$, which is known to be
attained by a certain coset model. Then no index value is possible in the
interval between 4 and $3 +\sqrt{3}$. The proof of this result based on
$\alpha$-induction arguments. In the case of values below 4 we also give a second
proof of the result. In the course of the latter proof we classify all possible
unitary braiding symmetries on the A D E tensor categories, namely the ones
associated with the even vertices of the $A_n$, $D_{2n}$, $E_6$, $E_8$ Dynkin diagrams.

open access link

Lett. Math. Phys. 92 (2010) 99-108

@article{Carpi:2010nnq,
author = "Carpi, Sebastiano and Kawahigashi, Yasuyuki and Longo,
Roberto",
title = "{On the Jones index values for conformal subnets}",
journal = "Lett. Math. Phys.",
volume = "92",
year = "2010",
pages = "99-108",
doi = "10.1007/s11005-010-0384-6",
eprint = "1002.3710",
archivePrefix = "arXiv",
primaryClass = "math.OA",
SLACcitation = "%%CITATION = ARXIV:1002.3710;%%"
}

Keywords:

*none*