TFT construction of RCFT correlators I: Partition functions
Jürgen Fuchs, Ingo Runkel, Christoph Schweigert
TL;DR
The paper develops a full RCFT construction from chiral data using symmetric special Frobenius algebras A in a modular tensor category, showing boundary conditions correspond to A-modules and defects to A-A bimodules. By embedding the theory into a 3D TFT, structure constants and torus/annulus partition functions become link invariants, with modular invariance and NIM-rep properties proven. The approach unifies boundary OPE, representation theory, and alpha-induction, and is illustrated via explicit free boson and E7 SU(2)16 modular invariant examples, including the interpretation of left/right chiral algebras as centers of A. It provides a noncommutative-geometric perspective on RCFTs and offers practical tools for computing full CFT data from chiral input. Overall, the framework connects sewing constraints, Morita equivalence, and CFT partition functions through the lens of modular tensor categories and 3D TFT invariants.
Abstract
We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of Moore-Seiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are A-modules, and (generalised) defect lines are A-A-bimodules. The relation with three-dimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in three-manifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIM-rep properties. We suggest that our results can be interpreted in terms of non-commutative geometry over the modular tensor category of Moore-Seiberg data.