Theorem 3 of the [nLab article "Full field algebra"](https://ncatlab.org/nlab/show/full+field+algebra) states that



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> **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.
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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](http://arxiv.org/abs/1412.8505)). 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](https://ncatlab.org/nlab/show/FRS-theorem+on+rational+2d+CFT), 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.



When two vertex (operator) algebras can be patched-up to a full CFT on a genus 0 surface?
If your VOA $V$ is not rational, then it is quite unlikely that its category of representations is a modular tensor category. That is, you can safely conclude that Theorem 3 contains an unstated assumption that $V$ is rational. At this point, we only know the modular tensor property when $V$ is rational, $C\_2$-cofinite, of CFT type, and self-dual as a $V$-module ([due to Huang](http://arxiv.org/abs/math/0502533)).


