2025/06/01 by Adamo, Maria Stella, Giorgetti, Luca, Tanimoto, Yoh
#46L60 #81T05 #81T40 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2506.01008
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 AQ 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 AQ 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.