On Quantum Galois Theory

The goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the “operator content of orbifold models.” To explain what this might mean, and to clarify the title of the paper, we will assume that the reader is familiar with the algebraic formulation of 2-dimensional CFT in the guise of vertex operator algebras (VOA), see [B], [FLM] and [DM] for more information on this point. In the paper [DVVV], several ideas are proposed concerning the structure of a holomorphic orbifold. In other words, if V is a holomorphic VOA and if G is a finite group of automorphisms of V, then the sub VOA V G of G-invariants is itself a VOA and the subject of [DVVV] is very much concerned with speculation on the nature of the V -modules. It turns out to be more useful − at least for purpose of inductive proofs − to take V to be a simple VOA. We will then see that V G is also simple whenever G is a finite group of automorphisms of V. One consequence of our main results is the following: