The balanced structure on the category of representations of a conformal net

TL;DR

This paper proves that the category of representations of a conformal net naturally possesses a balanced tensor structure, with the balance given by the action of the rotation generator, extending the understanding of conformal nets.

Contribution

It establishes a canonical balanced tensor category structure on the representation category of a conformal net, with a simplified proof for the case without group actions.

Findings

The representation category is a balanced W*-tensor category.

The balance is explicitly given by the action of e^{-2πiL_0}.

Future work will extend to group actions on the conformal net.

Abstract

Let $\mathcal{A}$ be a (not necessarily rational) conformal net. We show that the braided $\mathrm{W}^*$-tensor category $\text{Rep}(\mathcal{A})$ of representations of $\mathcal{A}$ is canonically a balanced $\mathrm{W}^*$-tensor category. The balance is given by the action of $e^{-2\pi i L_0}$, where $L_0$ denotes the generator of rotations on $S^1$. In future work, we generalize this result to the larger context of a group acting on $\mathcal{A}$. We provide here a more accessible proof for the case where no group is present.