How to add a boundary condition

TL;DR

This paper presents a method to construct boundary conformal field theory (CFT) nets from a given two-dimensional conformal QFT net by using subfactors and local nets, classifying all such boundary nets.

Contribution

It introduces a procedure to add a boundary to a 2D conformal QFT net, producing all locally isomorphic boundary CFT nets via subfactors and local nets.

Findings

All boundary CFT nets arise from the proposed construction.

There are finitely many such boundary nets, and all are classified.

The method involves redefining the C* representation using subfactors.

Abstract

Given a conformal QFT local net of von Neumann algebras B_2 on the two-dimensional Minkowski spacetime with irreducible subnet A\otimes\A, where A is a completely rational net on the left/right light-ray, we show how to consistently add a boundary to B_2: we provide a procedure to construct a Boundary CFT net B of von Neumann algebras on the half-plane x>0, associated with A, and locally isomorphic to B_2. All such locally isomorphic Boundary CFT nets arise in this way. There are only finitely many locally isomorphic Boundary CFT nets and we get them all together. In essence, we show how to directly redefine the C* representation of the restriction of B_2 to the half-plane by means of subfactors and local conformal nets of von Neumann algebras on S^1.