# When two vertex (operator) algebras can be patched-up to a full CFT on a genus 0 surface?

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.

• Note that the notion of a (not necessarily chiral) genus zero CFT is not the same as the notion of a full CFT. Every full CFT yields an example of a genus zero CFT, but there are more genus zero CFTs than full CFTs. Your Theorem 3 does not hold for genus zero CFTs. See the first 10 pages of my course notes for some clarifications about various notions of CFT: staff.science.uu.nl/~henri105/Teaching/CFT-2014.pdf (in there, I called genus zero CFTs "weak CFTs"). – André Henriques Mar 20 '16 at 1:05
• I dropped a note for Urs Schreiber at the nForum to have a look at your question. – Todd Trimble Mar 20 '16 at 1:47
• Theorem 3 is not more general than rational, because to make sense of the Witt class you need rationality. If you want genus 0 then I would say there aren't any restriction on the chiral parts – Marcel Bischoff Mar 20 '16 at 2:41
• This was a statement I took from a talk by Alexei Davydov golem.ph.utexas.edu/category/2010/06/… It seems the relevant published version never materialized. I am taking the statement out of the nLab entry. – Urs Schreiber Mar 21 '16 at 15:36

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).