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

Abstract

Let A be a completely rational local Mobius covariant net on S 1 , which describes a set of chiral observables. We show that local Mobius covari- ant nets B2 on 2D Minkowski space which contain the chiral theory A are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category DHR(A). The Mobius covariant boundary con- ditions with symmetry A of such a net B2 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 con- ditions. To establish this result we define the notion of Morita equiva lence for Q- systems (special symmetric �-Frobenius algebra objects) and non-degenerately braided subfactors. We prove a conjecture by Kong and Runkel, namely that Rehren's construction (generalized Longo-Rehren constru ction,α-induction con- struction) coincides with the categorical full center. Thi s gives a new view and new results for the study of braided subfactors.