The crossed braided tensor category of twisted and untwisted representations of the Heisenberg conformal net

We show that the category of twisted/untwisted representations of the Heisenberg conformal net $\text{Heis}$ is a continuous Tambara-Yamagami category for the group $\mathbb{R}$. We compute the associators and the $\mathbb{Z}\mathrm{/}2$-crossed braiding. By taking a $\mathbb{Z}\mathrm{/}2$-equivariantization, we obtain an explicit computation of the braided continuous tensor category of representations of the fixed-points conformal net ${\text{Heis}}^{\mathbb{Z}\mathrm{/}2}$. This provides the first explicit computation of a category of representations of a conformal net containing irreducible representations whose tensor product is a direct integral of irreducible representations.