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

Marcel Bischoff ; Roberto Longo ; Yasuyuki Kawahigashi

Documenta mathematica, Tome 20 (2015), pp. 1137-1184

Résumé

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 B2​ 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 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 conditions. To establish this result we define the notion of Morita equivalence 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 construction, α-induction construction) coincides with the categorical full center. This gives a new view and new results for the study of braided subfactors.

Détail

DOI : 10.4171/dm/515

Classification : 46L37, 81R15, 81T40
Mots-clés : conformal nets, boundary conditions, Q-system, full center, subfactors, modular tensor categories