Reconstruction and Local Extensions for Twisted Group Doubles, and Permutation Orbifolds

TL;DR

Problem: determine when a modular tensor category from a twisted group double $D_\omega(G)$ can be realized as Rep$(\mathcal{A})$ or Mod$(\mathcal{V})$ of a completely rational conformal net or VOA, and classify conformal extensions. Approach: introduce a groupoid-bundle framework, compute alpha-induction for type 1 extensions, and apply 8-term restriction-inflation cohomology to control twists; prove reconstruction for all $\omega$ and solvable $G$ (with assumptions for VOAs). Contributions: (i) classification of type 1 module categories (Theorem 1), (ii) identification of modular data as twisted doubles for local modules, (iii) permutation orbifold twists depending on central charge mod 24 (Theorem 2), (iv) reconstruction (Theorem 3) and solvable-Group VOA realizations (Theorem 4) with explicit branching rules and examples. Significance: provides the first nontrivial reconstruction instances, clarifies how orbifold twists arise from CFT data, and yields concrete tools for constructing and analyzing MTCs from holomorphic nets and VOAs.

Abstract

We prove the first nontrivial reconstruction theorem for modular tensor categories: the category associated to any twisted Drinfeld double of any finite group, can be realised as the representation category of a completely rational conformal net. We also show that any twisted double of a solvable group is the category of modules of a completely rational vertex operator algebra. In the process of doing this, we identify the 3-cocycle twist for permutation orbifolds of holomorphic conformal nets: unexpectedly, it can be nontrivial, and depends on the value of the central charge modulo 24. In addition, we determine the branching coefficients of all possible local (conformal) extensions of any finite group orbifold of holomorphic conformal nets, and identify their modular tensor categories. All statements also apply to vertex operator algebras, provided the conjecture holds that finite group orbifolds of holomorphic VOAs are rational, with a category of modules given by a twisted group double.