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

TL;DR

This paper computes fundamental correlation functions in two-dimensional rational conformal field theory, providing explicit structure constants and correlators for various field configurations, including defects and cross caps.

Contribution

It introduces a systematic method to derive all correlators from basic three-point functions using ribbon graph invariants in three-manifolds.

Findings

Explicit formulas for bulk, boundary, and defect correlators.

Structure constants expressed as ribbon graph invariants.

Framework applicable to all correlators via sewing procedures.

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.