Categorical Extensions of Conformal Nets
- PDF / 1,155,771 Bytes
- 77 Pages / 439.37 x 666.142 pts Page_size
- 36 Downloads / 188 Views
Communications in
Mathematical Physics
Categorical Extensions of Conformal Nets Bin Gui Department of Mathematics, Rutgers University, New Brunswick, NJ, USA. E-mail: [email protected]; [email protected] Received: 16 March 2020 / Accepted: 12 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: An important goal in studying the relations between unitary VOAs and conformal nets is to prove the equivalence of their ribbon categories. In this article, we prove this conjecture for many familiar examples. Our main idea is to construct new structures associated to conformal nets: the categorical extensions. Let V be a stronglylocal unitary regular VOA of CFT type, and assume that all V -modules are unitarizable. Then V is associated with a conformal net AV by Carpi et al. (From vertex operator algebras to conformal nets and back, Vol. 254, No. 1213, Memoirs of the American Mathematical Society, 2018). Let Repu (V ) and Repss (AV ) be the braided tensor categories of unitary V -modules and semisimple AV -modules respectively. We show that if one can find enough intertwining operators of V satisfying the strong intertwining property and the strong braiding property, then any unitary V -module Wi can be integrated to an AV -module Hi , and the functor F : Repu (V ) → Repss (AV ), Wi → Hi induces an equivalence of the ribbon categories Repu (V ) − → F(Repu (V )). This, in particular, shows that F(Repu (V )) is a modular tensor category. We apply the above result to all unitary c < 1 Virasoro VOAs (minimal models), many unitary affine VOAs (WZW models), and all even lattice VOAs. In the case of Virasoro VOAs and affine VOAs, one further knows that F(Repu (V )) = Repss (AV ). So we’ve proved the equivalence of the unitary modular tensor categories Repu (V ) Repss (AV ). In the case of lattice VOAs, besides the equivalence of Repu (V ) and F(Repu (V )), we also prove the strong locality of V and the strong integrability of all (unitary) V -modules. This solves a conjecture in Carpi et al. (From vertex operator algebras to conformal nets and back, Vol. 254, No. 1213, Memoirs of the American Mathematical Society, 2018). Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connes Fusion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Conformal nets and their representations . . . . . . . . . . . . . . . .
B. Gui
2.2 Connes fusion Hi (I ) H j (J ) . . . . . . . . 2.3 Actions of conformal nets . . . . . . . . . . . 2.4 Conformal structures . . . . . . . . . . . . . 2.5 Associativity . . . . . . . . . . . . . . . . . . 2.6 C ∗ -Tensor categories . . . . . . . . . . . . . 3. Connes Fusions and Categorical Extensions . . . . 3.1 Categorical extensions . . . . . . . . . . . . 3.2 Connes categorical extensions . . . . . . . . 3.3 Hexagon axioms . . . . . . . . . . . . . . . . 3.4 Uniqueness of tensor categorical structures . . 3.5 Uniqueness of maximal categorical extensions 3.6 Semisimple categorical extensions . . . . . . 4. VOAs
Data Loading...
