Modular categories and orbifold models II

TL;DR

The paper reformulates orbifold models of vertex operator algebras in tensor-categorical terms by introducing a commutative algebra $A$ in a modular tensor category $\mathcal{C}$ with a finite group $G$ acting by automorphisms and $A^G=\mathbf{1}$. It proves that the category of representations of the fixed-point algebra $\mathcal{V}^G$ is completely determined by the category of twisted $V$-modules together with the $G$-action, via the equivalence $\mathcal{C}\simeq (\mathrm{Rep}\,A)^G$ and $\mathcal{C}_1\simeq (\mathrm{Rep}^0\,A)^G$, and constructs a braided structure on $(\mathrm{Rep}\,A)^G$ with a $G$-crossed/Drinfeld-double flavor. The twisted sector is analyzed under modularity, yielding a decomposition $\mathrm{Rep}\,A=\bigoplus_{g\in G}\mathrm{Rep}_gA$ and a braiding given by $\sigma_{X,Y}$ involving $\sum_g \delta_g\otimes g$, reproducing the holomorphic case as $\mathrm{Rep}\,D^{\omega}(G)$. However, vertical reconstruction (recovering $\mathcal{C}$ from $\mathcal{C}_1$) remains challenging in the non-holomorphic setting, as illustrated by quantum-group examples, indicating limits of reconstruction from the untwisted data alone.

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.

Figures (1)

• Figure 1: Proof of Theorem \ref{['t:dbraiding']}

Theorems & Definitions (45)

• Definition 1.2
• Lemma 1.3
• Definition 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Example 3.1