TFT construction of RCFT correlators V: Proof of modular invariance and factorisation
Jens Fjelstad, Jurgen Fuchs, Ingo Runkel, Christoph Schweigert
TL;DR
The work provides a rigorous, category-theoretic construction of RCFT correlators using a modular tensor category and Frobenius/Jandl algebras, encoding correlators as TFT invariants on 3-manifolds. It proves seminal consistency properties: modular invariance under the relative mapping class group and both boundary and bulk factorisation (including cross-cap constraints for unoriented worlds), enabling reconstruction of general correlators from a finite set of building blocks. The results unify oriented and unoriented theories within a single TFT framework and recover the Cardy case as a special instance, offering a robust, computable approach to RCFT correlators. This framework has significant implications for systematic computation of structure constants and correlation functions in RCFTs from purely categorical data.
Abstract
The correlators of two-dimensional rational conformal field theories that are obtained in the TFT construction of [FRSI,FRSII,FRSIV] are shown to be invariant under the action of the relative modular group and to obey bulk and boundary factorisation constraints. We present results both for conformal field theories defined on oriented surfaces and for theories defined on unoriented surfaces. In the latter case, in particular the so-called cross cap constraint is included.