#### DOCUMENTA MATHEMATICA, Vol. 20 (2015), 1137-1184

Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo

Characterization of 2D Rational Local Conformal Nets and Its Boundary Conditions: the Maximal Case

Let $\A$ be a completely rational local Möbius covariant net on $S1$, which describes a set of chiral observables. We show that local Möbius covariant nets $\cB2$ on 2D Minkowski space which contains $\A$ as chiral left-right symmetry are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category $\DHR(\A)$. The Möbius covariant boundary conditions with symmetry $\A$ of such a net $\cB2$ 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.

2010 Mathematics Subject Classification: 81T40, 18D10, 81R15, 46L37

Keywords and Phrases: Conformal Nets, Boundary Conditions, Q-system, Full Center, Subfactors, Modular Tensor Categories.

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