TFT construction of RCFT correlators IV: Structure constants and correlation functions

TL;DR

The work develops a universal, TFT-based construction of RCFT correlators by tying the RCFT data to a modular tensor category ${\mathcal C}$ and a symmetric special Frobenius algebra ${A}$ (or a Jandl algebra for unoriented cases). Fundamental correlators are expressed as ribbon-graph invariants in three-manifolds and tied to conformal blocks via holomorphic factorisation, enabling all RCFT correlators to be generated by sewing. A key contribution is the introduction and computation of fusing matrices for ${\mathcal C}_{A}$-modules and ${}_A\mathcal{C}_A$-bimodules, ${\sf G}[A]$ and ${\sf F}[A|A]$, which encode associator data and yield compact closed-form structure constants for boundary, bulk, and defect sectors, including the Cardy case as a consistency check. The framework unifies boundaries, defects, and unoriented surfaces within a Morita-equivalent, model-independent approach, with explicit constructions of ribbon graphs and conformal blocks guiding the calculation of correlators. The results provide a robust foundation for calculating RCFT correlators across oriented and non-oriented world sheets in a purely category-theoretic, topological setting.

Abstract

We compute the fundamental correlation functions in two-dimensional rational conformal field theory, from which all other correlators can be obtained by sewing: the correlators of three bulk fields on the sphere, one bulk and one boundary field on the disk, three boundary fields on the disk, and one bulk field on the cross cap. We also consider conformal defects and calculate the correlators of three defect fields on the sphere and of one defect field on the cross cap. Each of these correlators is presented as the product of a structure constant and the appropriate conformal two- or three-point block. The structure constants are expressed as invariants of ribbon graphs in three-manifolds.