Conformal Orbifold Theories and Braided Crossed G-Categories

2004/03/31 by Michael Müger, Michael Mueger · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic structures and combinatorial models #Derived category #Equivariant map #Functor #Homotopy and Cohomology in Algebraic Topology #Invariant (physics) #Mathematics #Orbifold #Pure mathematics #Quotient #Subcategory #Triangulated category #math.CT #math.OA #math.QA #msc:18D10 #msc:81T40

paper · pdf · doi:10.1007/s00220-005-1291-z

published as Commun. Math. Phys. 260, 727-762 (2005) · Already online on the site of Commun. Math. Phys. Remark 4.16 corrected for the record. latex2e, approx. 38 pages, requires diagrams.tex

openalex publication_date 2005/02/12 · arxiv created 2005/03/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category GLoc A of twisted representations. This category is a braided crossed G-category in the sense of Turaev. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT -> modular category -> 3-manifold invariant. We then study the relation between GLoc A and the braided (in the usual sense) representation category Rep A^G of the orbifold theory A^G. We prove the equivalence Rep A^G = (GLoc A)^G, which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep A^G. In the opposite direction we have GLoc A = Rep A^G \rtimes S, where S ⊂ Rep A^G is the full subcategory of representations of A^G contained in the vacuum representation of A, and \rtimes refers to the Galois extensions of braided tensor categories of [44,48]. If A is completely rational and G is finite we prove that A has g-twisted representations for every g in G. In the holomorphic case (where Rep A = Vect_C) this allows to classify the possible categories GLoc A and to clarify the role of the twisted quantum doubles D^ω(G) in this context, as will be done in a sequel. We conclude with some remarks on non-holomorphic orbifolds and surprising counterexamples concerning permutation orbifolds.

Citations

• Operator algebras and conformal field theory
• Homotopy field theory in dimension 3 and crossed group-categories
• On G--equivariant modular categories
• Superselection Theory for Subsystems
• Spin Statistics and CPT for Solitons
• On the Structure of Modular Categories
• Classical and quantum conformal field theory
• Modular structure and duality in conformal quantum field theory
• Spherical Categories
• Categories for the Working Mathematician
• On q-analog of McKay correspondence and ADE classification of sl^(2) conformal field theories

Cited by

• Twisted representations of conformal nets and crossed balanced tensor categories
• Balanced tensor categories of representations of fixed-points conformal nets