The goals of the present paper are to develop the foundations of the theory of abstract orbifold models in conformal field theory – that is to say, the theory of twisted representations of vertex operator algebras. If \(V\) is a vertex operator algebra and \(G\) a (finite) group of automorphisms of \(V\), then the space of \(G\)-invariants \(V^G\) is itself a vertex operator algebra, and one wishes to understand the structure of various module categories for \(V^G\). For background we refer the reader to [C. Dong andG. Mason, Vertex operator algebras and Moonshine, a survey, Adv. Stud. Pure Math. 24, 101-136 (1996;Zbl 0861.17018)]. One of the main features of orbifold theory is the use of twisted modules, or twisted sectors, which are analogues of ordinary modules over a vertex operator algebra but which satisfy a ‘twisted’ version of the main axiom for modules, namely the Jacobi identity.
In the present paper, the authors construct an associative algebra \(A_g(V)\) associated to a vertex operator algebra \(V\) and an automorphism \(g\) of \(V\) of finite order which enjoys the following property: there is a natural bijection between (isomorphism classes of) simple \(A_g(V)\)-modules and simple admissible \(g\)-twisted \(V\)-modules (admissible (twisted) modules are essentially modules for a vertex operator algebra in which some of the axioms are relaxed). More generally, a certain pair of functors between the corresponding module categories are constructed and studied, and it is shown that restriction of the functors to simple objects realizes the above bijection. To some extent, this result allows one to reduce questions about twisted representations of vertex operator algebras to questions about the associative algebra \(A_g(V)\), which are often easier to answer. These results generalize some basic theorems of Zhu [Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Am. Math. Soc. 9, 237-302 (1996;Zbl 0854.17034)], who studied the case in which \(g\) is the identity.
The authors give a number of applications to their main result, the most important of which is concerned with the case of so-called \(g\)-rational vertex operator algebras. Here, one assumes that each admissible \(g\)-twisted module is completely reducible. It is shown that with this assumption, the algebra \(A_g(V)\) is finite-dimensional and semisimple, and that \(V\) has only a finite number of simple \(g\)-twisted modules. Moreover the two functors induce categorical equivalences between the categories of \(A_g(V)\)-modules and admissible \(g\)-twisted modules, and between the categories of finite-dimensional \(A_g(V)\)-modules and \(g\)-twisted modules.

Reviewer: Chongying Dong (Santz Cruz)