Modular categories and orbifold models

TL;DR

The paper addresses how to describe the quotient of a modular tensor category by a finite group action, focusing on the holomorphic VOA setting. It develops a category-theoretic framework using a $\mathcal{C}$-algebra $A$ and the functor $\Phi: \mathrm{Rep}\,G\to \mathcal{C}$ to relate orbifold models to the Drinfeld double $D(G)$, introducing twisted sectors via $g$-twisted modules and a constructed object $\tilde{A}$ with a $D(G)$-action. The main results prove that, under suitable conditions (trivial twist, faithful action, holomorphicity, modularity), the representation category $\mathrm{Rep}\,\mathcal{V}^G$ is equivalent to $\mathrm{Rep}\,D(G)$ (and to $\mathrm{Rep}\,D(G,H)$ more generally), with explicit decompositions and functors connecting the orbifold data to Drinfeld doubles. This bridges orbifold conformal field theories, vertex operator algebras, and modular tensor categories, offering a rigorous category-theoretic realization of DVVV-type predictions and detailed descriptions of twisted and untwisted sectors in the $D(G)$ framework.

Abstract

In this paper, we try to answer the following question: given a modular tensor category $\A$ with an action of a compact group $G$, is it possible to describe in a suitable sense the ``quotient'' category $\A/G$? We give a full answer in the case when $\A=\vec$ is the category of vector spaces; in this case, $\vec/G$ turns out to be the category of representation of Drinfeld's double $D(G)$. This should be considered as category theory analog of topological identity ${pt}//G=BG$. This implies a conjecture of Dijkgraaf, Vafa, E. Verlinde and H. Verlinde regarding so-called orbifold conformal field theories: if $\V$ is a vertex operator algebra which has a unique irreducible module, $\V$ itself, and $G$ is a compact group of automorphisms of $\V$, and some not too restricitive technical conditions are satisfied, then $G$ is finite, and the category of representations of the algebra of invariants, $\V^G$, is equivalent as a tensor category to the category of representations of Drinfeld's double $D(G)$. We also get some partial results in the non-holomorphic case, i.e. when $\V$ has more than one simple module.

Figures (12)

• Figure 1: Definition of $I$
• Figure 2: Proof of $IJ=\mathop{\mathrm{id}}\nolimits$
• Figure 3: Definition of $f\colon\Phi(V_\lambda^*)\otimes \Phi(V_\lambda\otimes V_\lambda^*)\to \Phi(V_\lambda^*)$
• Figure 4: Proof of $JI=\mathop{\mathrm{id}}\nolimits$
• Figure 5: Definition of $g$-twisted module

Theorems & Definitions (76)

• Theorem
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Remark 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• proof