A general mirror equivalence theorem for coset vertex operator algebras

We prove a general mirror duality theorem for a subalgebra U of a simple conformal vertex algebra A and its commutant V = Com(A)(U). Specifically, we assume that A congruent to circle plus(i is an element of I) U-i circle times V-i as a U circle times V-module, where the U-modules U-i are simple and distinct and are objects of a semisimple braided ribbon category of U-modules, and the V-modules V-i are semisimple and contained in a (not necessarily rigid) braided tensor category of V-modules. We also assume U = Com(A)(V). Under these conditions, we construct a braid-reversed tensor equivalence tau : U-A -> V-A, where U-A is the semisimple category of U-modules with simple objects U-i, i is an element of I, and V-A is the category of V-modules whose objects are finite direct sums of V-i. In particular, the V-modules V-i are simple and distinct, and 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 is an element of Z(>= 2), which is braided tensor equivalent to an abelian 3-cocycle twist of the category of finite-dimensional sl(2)-modules. Consequently, the Virasoro vertex operator algebra at central charge 13 + 6p + 6p(-1) is the PSL2(C)-fixed-point subalgebra of a simple conformal vertex algebra W(-p), analogous to the realization of the Virasoro vertex operator algebra at central charge 13- 6p- 6p(-1) as the PSL2(C)-fixed-point subalgebra of the triplet algebra W(p).