Reconstruction and local extensions for twisted group doubles, and permutation orbifolds

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.

我们证明了模范张量范畴的第一个非平凡重构定理：与任何有限群的任意扭曲Drinfeld 双相关的范畴，可以被实现为一个完全有理共形网的表示范畴。我们还表明，可解群的任何扭曲双是一个完全有理顶点算子代数的模范畴。在这个过程中，我们确定了全纯共形网的置换轨形的3 - 上循环扭曲：出乎意料的是，它可以是非平凡的，并且取决于中心荷模24的值。此外，我们确定了全纯共形网的任何有限群轨形的所有可能的局部（共形）扩张的分支系数，并确定了它们的模范张量范畴。如果猜想成立，即全纯顶点算子代数的有限群轨形是有理的，且其模范畴由一个扭曲群双给出，那么所有陈述也适用于顶点算子代数。