Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case

Understand

Let $\mathcal{A}$ be a completely rational local M\"obius covariant net on $S^1$, which describes a set of chiral observables.

• We show that local M\"obius covariant nets $\mathcal{B}_2$ on 2D Minkowski space which contains $\mathcal{A}$ as chiral left-right symmetry are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category $\mathrm{DHR}(\mathcal{A})$.
• The M\"obius covariant boundary conditions with symmetry $\mathcal{A}$ of such a net $\mathcal{B}_2$ are given by the Q-systems in the Morita equivalence class or by simple objects in the module category modulo automorphisms of the dual category.
• We generalize to reducible boundary conditions.