Citation: Fuchs, Jürgen, Ingo Runkel, and Christoph Schweigert. "TFT construction of RCFT correlators I: Partition functions." Nuclear Physics B 646.3 (2002): 353-497.
Web: https://arxiv.org/abs/hep-th/0204148
Tags: Modular-tensor-categories, Vertex-operator-algebras, Conformal-field-theory, MTC-reconstruction
This seems like the canonical reference for properly understanding the relationship between modular categories and CFTs. Here's my understanding of the picture. If you take the basic chiral data of a rational conformal field theory (namely, the the fusing and braiding matrices) and package it together in a basis-independent way you naturally arrive at the notion of a modular tensor category - this is what Moore and Seiberg did, so we call this modular category the Moore-Seiberg data. This does not, however, capture the full CFT. To quote the authors: "The Moore--Seiberg data captures the chiral aspects of rational conformal field theory. To arrive at a full conformal field theory with local correlation functions, additional input is required".
The additional input is as follows - the data of a symmetric special Frobenius algebra in the modular category. The point here is as follows. Just like how gapped boundaries of TQFTs fit into a nice Morita theory, conformal boundaries of CFTs fit into a nice Morita theory as well. Instead of picking a Lagrangian algebra you pick a symmetric special Frobenius algebra. Boundary conditions correspond to A-modules and generalized defect lines are A-modules, and in the bulk A is only well-defined up to Morita equivalence. This is part of a larger absolutely monster five-part work. The rest of the parts are as follows:
> Fuchs, Jürgen, Ingo Runkel, and Christoph Schweigert. "TFT construction of RCFT correlators II: Unoriented world sheets." Nuclear Physics B 678.3 (2004): 511-637.
> Fuchs, Jürgen, Ingo Runkel, and Christoph Schweigert. "TFT construction of RCFT correlators: III: simple currents." Nuclear Physics B 694.3 (2004): 277-353.
> Fuchs, Jürgen, Ingo Runkel, and Christoph Schweigert. "TFT construction of RCFT correlators IV:: Structure constants and correlation functions." Nuclear physics B 715.3 (2005): 539-638.
> Fjelstad, Jens, et al. "TFT construction of RCFT correlators V: Proof of modular invariance and factorisation." arXiv preprint hep-th/0503194 (2005).