Categorical Extensions of Conformal Nets

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 strongly-local unitary regular VOA of CFT type, and assume that all V -modules are unitarizable. Then V is associated with a conformal net A V 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 Rep u ( V ) and Rep ss ( A V ) be the braided tensor categories of unitary V -modules and semisimple A V -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 W i can be integrated to an A V -module H i , and the functor F : Rep u ( V ) → Rep ss ( A V ) , W i ↦ H i induces an equivalence of the ribbon categories Rep u ( V ) → ≃ F ( Rep u ( V ) ) . This, in particular, shows that F ( Rep u ( 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 ( Rep u ( V ) ) = Rep ss ( A V ) . So we’ve proved the equivalence of the unitary modular tensor categories Rep u ( V ) ≃ Rep ss ( A V ) . In the case of lattice VOAs, besides the equivalence of Rep u ( V ) and F ( Rep u ( 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).