Reconstruction and local extensions for twisted group doubles, and permutation orbifolds
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.
39 pages
Operator algebra methods applied to problems in quantum theory, High Energy Physics - Theory, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Mathematics - Operator Algebras, \(K\)-theory and operator algebras (including cyclic theory), FOS: Physical sciences, Fusion categories, modular tensor categories, modular functors, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Axiomatic quantum field theory; operator algebras, Finite-dimensional groups and algebras motivated by physics and their representations, 18D10 (primary), 17B69, 81T40 (secondary), High Energy Physics - Theory (hep-th), Mathematics - Quantum Algebra, FOS: Mathematics, Quantum Algebra (math.QA), QA, Operator Algebras (math.OA)
Operator algebra methods applied to problems in quantum theory, High Energy Physics - Theory, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Mathematics - Operator Algebras, \(K\)-theory and operator algebras (including cyclic theory), FOS: Physical sciences, Fusion categories, modular tensor categories, modular functors, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Axiomatic quantum field theory; operator algebras, Finite-dimensional groups and algebras motivated by physics and their representations, 18D10 (primary), 17B69, 81T40 (secondary), High Energy Physics - Theory (hep-th), Mathematics - Quantum Algebra, FOS: Mathematics, Quantum Algebra (math.QA), QA, Operator Algebras (math.OA)
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