Theorem 3 of the nLab article "Full field algebra" states that
Theorem 3. Two vertex operator algebras $V$ may appear as the left and right chiral halfs of a full conformal field theory precisely if their modular tensor categories of representations have the same Witt class.
I checked the references therein and searched more, but could not find neither the theorem stated anywhere else, nor the proof. I managed to find out that one can take two copies of the same vertex operator algebra to get a full field algebra in the sense of Huang and Kong, or that there exist unphysical diagonal modular invariants (Davydov2015). Since I am interested in CFTs on the plane, modular invariants are not interesting to me because they are associated with tori?
I am aware of (as in, I know that it exists) the FRS construction, but it only works for rational CFTs and Theorem 3 seems to be more general, unless it's just sloppiness since
"The purpose of the nLab is to provide a public place where people can make notes about stuff. The purpose is not to make polished expositions of material; that is a happy by-product."
Any help will be appreciated. In particular, a little summary of the state of the art would suffice, if Theorem 3 turns out to be very mysterious.