How to Add a Boundary Condition

Given a conformal QFT local net of von Neumann algebras $${\mathcal {B}_2}$$ on the two-dimensional Minkowski spacetime with irreducible subnet $${\mathcal {A} \otimes \mathcal {A}}$$, where $${\mathcal {A}}$$ is a completely rational net on the left/right light-ray, we show how to consistently add a boundary to $${\mathcal {B}_2}$$: we provide a procedure to construct a Boundary CFT net $${\mathcal {B}}$$ of von Neumann algebras on the half-plane x > 0, associated with $${\mathcal {A}}$$, and locally isomorphic to $${\mathcal {B}_2}$$. 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 $${\mathcal {B}_2}$$ to the half-plane by means of subfactors and local conformal nets of von Neumann algebras on S1.