The three-dimensional origin of the classifying algebra

TL;DR

This paper reveals how the classifying algebra in rational conformal field theory naturally emerges from three-dimensional geometric considerations, linking boundary conditions to ribbon graph invariants.

Contribution

It demonstrates the geometric origin of the classifying algebra from 3D correlator factorization and provides explicit formulas for its structure constants.

Findings

Derived structure constants as ribbon graph invariants

Connected boundary conditions with mapping class group intertwiners

Unveiled geometric interpretation of the classifying algebra

Abstract

It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra. We show how this algebra arises naturally from the three-dimensional geometry of factorization of correlators of bulk fields on the disk. This allows us to derive explicit expressions for the structure constants of the classifying algebra as invariants of ribbon graphs in the three-manifold S^2 x S^1. Our result unravels a precise relation between intertwiners of the action of the mapping class group on spaces of conformal blocks and boundary conditions in rational conformal field theories.