Readers should be warned that the author is not an expert of CFT and, in fact, not even a novice in physics. What follows should be taken with a hefty pinch of salt.
Given any Riemann surface (as a target manifold) one is able to associate to it an or a super conformal field theory. The word super can just be construed as a -grading of the theory. This is a simplistic version as normally one should also take into account several other parameters like a B-field and so on. Within an SCFT there is a topological sector called a TQFT (topological quantum field theory) which is insensitive to the metric on the target space. There are axiomatic descriptions of this theory due to Atiyah and Segal and in its latest version possibly due to Costello. Roughly an -dimensional TQFT is a functor satisfying a lot of axioms from -manifolds with labelled boundaries (incoming and outgoing) to symmetric monoidal DG (differential graded) categories with some twisting.
Now people with a background on derived categories would like to see the triangulated structure of . Indeed, the homotopy category of a DG category resembles a triangulated category. It seems at this point it is possible to associate two models, namely, and to the theory after Witten. The model leads to a complicated Fukaya category (possibly as an -category) and the model leads to the derived (or DG) category of coherent sheaves on . There is a chiral ring that can be associated to each of these two categories. On the side it is the quantum cohomology ring and on the side it is its Hochschild homology . There is a Frobenius algebra structure on them; on at least when is Calabi-Yau.
The homological mirror symmetry conjecture by Kontsevich would predict the existence of an equivalence between the two categories arising out of the and models. This should naturally give rise to an isomorphism of Frobenius algebras after passing on to their chiral rings.
The construction of CFTs is a hard problem in general. Sometimes it is possible to get one’s hands on to the (possibly local) symmetries of a CFT, which are called chiral algebras and then one might be tempted to construct the CFT out of it. The chiral algebras as mathematical objects have the structure of a VOA (vertex operator algebra) . The construction of CFT from its chiral algebra seems to be possible if the representation category of the chiral algebra has finitely many irreducible objects (communicated by Liang Kong). This can also be phrased in the language of the partition function of the CFT as Gukov and Vafa do. Such a CFT is called rational (abbreviated RCFT). They enjoy some other very desirable properties, which make them particularly interesting. When the target space is a complex torus Gukov and Vafa argue that rationality of the CFT is related to the complex torus having complex multiplication. So in this case the RCFTs are plentiful but in higher dimensions they are supposed to be rather sparse (not dense in the moduli space).
Meng Chen studied the higher dimensional case of the same in her thesis and came up with her own geometric definition of the rationality of a CFT and related it to abelian varieties with large endomorphism rings. It is not clear if her geometric definition of rationality of a CFT is related to the, rather algebraic, notion of rationality presented above.
This is an intriguing connection between physics and number theory. In which direction the information would flow remains to be seen. There are some more interesting connections of this sort relating generating functions of some counting problems of (arithmetic) algebro-geometric nature to the partition functions of CFTs. If the author can weather this storm more postings on them will follow.
Sincere apologies for the inaccuracies and for straying from the main theme of the blog which is arithmetic algebraic geometry. The author just stumbled upon the thesis of Meng Chen recently, got fascinated and wanted to share this new-found knowledge.