Modular categories and orbifold models II

TL;DR

This paper extends the understanding of orbifold models in conformal field theory by describing the representation categories of fixed point algebras in non-holomorphic cases using tensor categories, building on previous holomorphic results.

Contribution

It provides a partial answer to the construction of representation categories of fixed point algebras for non-holomorphic vertex operator algebras, emphasizing tensor category methods.

Findings

Representation category of V^G determined by twisted V-modules and G-action

Avoids VOA techniques, using tensor categories instead

Partial extension of holomorphic case results to non-holomorphic cases

Abstract

This is a continuation of the paper "Modular tensor categories and orbifold theories", arXiv:math.QA/0104242. It discusses orbifold models of conformal filed theory, or, in mathematical language, question of constructing the category of representations of the fixed point algebra $V^G$ for a given vertex operator algebra $V$ with an action of a finite group $G$. The previous paper gave a proof of well-known conjecture of Dijkgraaf-Vafa-Verlinde-Verlinde giving a complete answer to this question in the holomorphic case (when $V$ has a unique simple module, $V$ itself) under the assumption that categories of rrepresentations of $V$, $V^G$ are modular tensor categories. In the current paper, we give a partial answer in non-holomorphic case. In particular, we show that the category of representations of $V^G$ is completely determined by the category of twisted $V$-modules together with the action of $G$ on this category. Our approach is based on describing representations of $V$, $V^G$ and relation between them in terms of tensor categories and avoids using the technique of VOAs as much as possible.