Rational and non-rational two-dimensional conformal field theories arising from lattices

For a (finite-dimensional) real Hilbert space $\mathfrak h$ and an orthogonal projection $p$, we consider the associated Heisenberg Lie algebra and the two-dimensional Heisenberg conformal net. Given an even lattice $Q$ in $\mathfrak h$ with respect to the indefinite bilinear form on $\mathfrak h$ defined by $p$, we construct a two-dimensional conformal net ${\mathcal A}_Q$ extending the Heisenberg conformal net. Moreover, with a certain discreteness assumption on the spectrum of the extension, we show that any two-dimensional extension of the Heisenberg conformal net is of the form ${\mathcal A}_Q$ up to unitary equivalence. We consider explicit examples of even lattices where $\mathfrak h$ is two-dimensional and $p$ is one-dimensional, and we show that the extended net may have completely rational or non-completely rational chiral (i.e. one-dimensional lightray) components, depending on the choice of lattice. In the non-rational case, we exhibit the braided equivalence of a certain subcategory of the representation category of the chiral Heisenberg net corresponding to the two-dimensional lattice extension. Inspired by the charge and braiding structures of these nets, we construct two-dimensional conformal Wightman fields on the same Hilbert spaces. We show that, in some cases, these Wightman fields generate the corresponding extended nets.