Q-Systems and Extensions of Completely Unitary Vertex Operator Algebras

Abstract

Abstract Complete unitarity is a natural condition on a CFT-type regular vertex operator algebra (VOA), which ensures that its modular tensor category (MTC) is unitary. In this paper we show that any CFT-type unitary (conformal) extension $U$ of a completely unitary VOA $V$ is completely unitary. Our method is to relate $U$ with a Q-system $A_U$ in the $C^*$-tensor category $\textrm{Rep}^{\textrm{u}}(V)$ of unitary $V$-modules. We also update the main result of [ 30] to the unitary cases by showing that the tensor category $\textrm{Rep}^{\textrm{u}}(U)$ of unitary $U$-modules is equivalent to the tensor category $\textrm{Rep}^{\textrm{u}}(A_U)$ of unitary $A_U$-modules as unitary MTCs. As an application, we obtain infinitely many new (regular and) completely unitary VOAs including all CFT-type $c<1$ unitary VOAs. We also show that the latter are in one-to-one correspondence with the (irreducible) conformal nets of the same central charge $c$, the classification of which is given by [ 29].