Solitonic sectors, conformal boundary conditions and three-dimensional topological field theory

TL;DR

This paper develops a systematic framework for understanding boundary conditions that break bulk symmetries in rational conformal field theories, using topological methods and solitonic sectors.

Contribution

It introduces an alpha-induction based construction of a fusion ring for boundary fields, linking boundary conditions to topological field theory and solitonic sectors.

Findings

Boundary conditions are expressed via Wilson graphs in three-manifolds.

The fusion ring for boundary fields is constructed with structure constants as annulus coefficients.

Boundary symmetry breaking corresponds to solitonic sectors.

Abstract

The correlation functions of a two-dimensional rational conformal field theory, for an arbitrary number of bulk and boundary fields and arbitrary world sheets can be expressed in terms of Wilson graphs in appropriate three-manifolds. We present a systematic approach to boundary conditions that break bulk symmetries. It is based on the construction, by `alpha-induction', of a fusion ring for the boundary fields. Its structure constants are the annulus coefficients and its 6j-symbols give the OPE of boundary fields. Symmetry breaking boundary conditions correspond to solitonic sectors.