On the Jones index values for conformal subnets

TL;DR

This paper investigates the possible values of the Jones index for conformal subnets, revealing strict restrictions below 4 and classifying related algebraic structures, with implications for conformal field theory models.

Contribution

It establishes new restrictions on Jones index values for conformal nets and classifies unitary braiding symmetries on specific tensor categories.

Findings

Only specific index values below 4 are admissible, including 4cos^2(π/10).

No index values exist between 4 and 3+√3.

Classified all unitary braiding symmetries on A D E tensor categories.

Abstract

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.