TFT construction of RCFT correlators I: Partition functions

TL;DR

This paper develops a framework for rational conformal field theory using Frobenius algebras in modular tensor categories, explicitly computing partition functions and demonstrating their consistency with known physical principles.

Contribution

It introduces a construction of RCFT correlators via symmetric special Frobenius algebras, linking algebraic structures to three-dimensional topological field theory and explicitly calculating partition functions.

Findings

Explicit formulas for torus and annulus partition functions.

Proof of modular invariance and NIM-rep properties.

Connection to non-commutative geometry in the modular tensor category.

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.