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 $S^{1}$,
which describes a set of chiral observables. We show that local Möbius
covariant nets $\cB_{2}$ 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 $\cB_{2}$ 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.

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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