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

TL;DR

This paper classifies boundary conditions of 2D conformal nets with chiral symmetry using categorical methods, establishing a correspondence with Morita equivalence classes of Q-systems and proving a conjecture relating Rehren's construction to the categorical full center.

Contribution

It introduces a Morita equivalence framework for Q-systems in braided tensor categories and proves the equivalence of Rehren's construction with the categorical full center.

Findings

Classification of boundary conditions via Morita equivalence classes

Proof that Rehren's construction equals the categorical full center

Extension of boundary condition analysis to reducible cases

Abstract

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. To establish this result we define the notion of Morita equivalence for Q-systems (special symmetric $\ast$-Frobenius algebra objects) and non-degenerately braided subfactors. We prove a conjecture by Kong and Runkel, namely that Rehren's construction (generalized Longo-Rehren construction, $\alpha$-induction construction) coincides with the categorical full center. This gives a new view and new results for the study of braided subfactors.