We prove a general mirror duality theorem for a subalgebra U of a simple conformal vertex algebra A and its commutant V = ComA(U). Specifically, we assume that A ≌ ⊕i ∈ IUi ⊗ Vi as a U ⊗ V-module, where the U-modules Ui are simple and distinct and are objects of a semisimple braided ribbon category of U-modules, and the V-modules Vi are semisimple and contained in a (not necessarily rigid) braided tensor category of V-modules. We also assume U = ComA(V). Under these conditions, we construct a braid-reversed tensor equivalence \(\tau :{{\cal U}_A} \to {{\cal V}_A}\) , where \({{\cal U}_A}\) is the semisimple category of U-modules with simple objects Ui, i ∈ I, and \({{\cal V}_A}\) is the category of V-modules whose objects are finite direct sums of Vi. In particular, the V-modules Vi are simple and distinct, and \({{\cal V}_A}\) is a rigid tensor category. As an application, we find a rigid semisimple tensor subcategory of modules for the Virasoro algebra at central charge 13 + 6p + 6p−1, p ∈ ℤ≽2, which is braided tensor equivalent to an abelian 3-cocycle twist of the category of finite-dimensional \(\mathfrak{sl_{2}}\) -modules. Consequently, the Virasoro vertex operator algebra at central charge 13 + 6p + 6p−1 is the \({\rm{PS}}{{\rm{L}}_2}\left(\mathbb{C}\right)\) -fixed-point subalgebra of a simple conformal vertex algebra \({\cal W}\left({-p} \right)\) , analogous to the realization of the Virasoro vertex operator algebra at central charge 13 − 6p − 6p−1 as the \({\rm{PS}}{{\rm{L}}_2}\left(\mathbb{C}\right)\) -fixed-point subalgebra of the triplet algebra \({\cal W}\left( p \right)\) .
