The balanced structure on the category of representations of a 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 arXiv:2606.03623, 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.