Strong additivity of the conformal net of an affine simple Lie algebra

Question

Let $\mathfrak g$ be a complex simple Lie algebra, $l$ be a natural number, and $V=V^l(\hat g)$ be the vertex operator algebra of the affine Lie algebra $\hat{\mathfrak g}$ at level $l$. We know that $V$ can always give rise to a conformal net $\mathcal A_V$ (constructed say by integrating the loop algebra $L_I\mathfrak g$). My question is: Is this conformal net strongly additive?

I know this is true when $\mathfrak g$ is $\mathfrak{sl}(n)$ or $\mathfrak {so}(2n)$. A rigorous proof can be found in Toledano-Laredo's paper "Fusion of Positive Energy Representations of $\text{LSpin}_{2n}$" part 1 section IV.1 (p.76-78) using the Sobolev $\frac 1 2$-norm trick. I see no reason why his proof can not be applied to all other cases, so I think that strong additivity can be proved for conformal nets of any affine simple Lie algebra by making use of this argument. But I didn't see any formal statement claiming this general result. So is there anything wrong when we generalize Toledano's proof to general cases? Do we have strong additivity for all affine simple Lie algebras?

Reference: Fusion of Positive Energy Representations of $\text{LSpin}_{2n}$.

Answer 1

In Section 4.C of my paper https://arxiv.org/pdf/1302.2604.pdf, I discuss various aspects of the loop group conformal nets, with a couple of pointers to the literature.

My understanding is that section IV.1 of Toledano-Laredo's PhD works equally well for all compact Lie groups $G$.

Actually, if you look at the wording that the author uses, he also doesn't seem to restrict to the case $G=Spin(2n)$. For example, his Lemma 1.1.1. starts with "Assume $G$ is simple...".

---

The unpublished paper of A. Wassermann:
Subfactors arising from positive energy representations of some infinite-dimensional groups, available at http://www.staff.science.uu.nl/~henri105/PDF/Wass1.pdf, contains a discussion of strong additivity in Sections 9 and 12.