On the tensor structure of modules for compact orbifold vertex operator algebras
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Mathematische Zeitschrift
On the tensor structure of modules for compact orbifold vertex operator algebras Robert McRae1 Received: 3 December 2018 / Accepted: 18 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Suppose V G is the fixed-point vertex operator subalgebra of a compact group G acting on a simple abelian intertwining algebra V . We show that if all irreducible V G -modules contained in V live in some braided tensor category of V G -modules, then they generate a tensor subcategory equivalent to the category Rep G of finite-dimensional representations of G, with associativity and braiding isomorphisms modified by the abelian 3-cocycle defining the abelian intertwining algebra structure on V . Additionally, we show that if the fusion rules for the irreducible V G -modules contained in V agree with the dimensions of spaces of intertwiners among G-modules, then the irreducibles contained in V already generate a braided tensor category of V G -modules. These results do not require rigidity on any tensor category of V G -modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known; for example they apply when V G is C2 -cofinite but not necessarily rational. When V G is both C2 -cofinite and rational and V is a vertex operator algebra, we use the equivalence between Rep G and the corresponding subcategory of V G -modules to show that V is also rational. As another application, we show that a certain category of modules for the Virasoro algebra at central charge 1 admits a braided tensor category structure equivalent to Rep SU (2), up to modification by an abelian 3-cocycle. Keywords Vertex operator algebras · Compact Lie groups · Braided tensor categories · Virasoro algebra Mathematics Subject Classification Primary 17B69; Secondary 18D10 · 20C35 · 81R10
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Vertex tensor categories . . . . . . . . . . . . . . . . . . 2.2 Group-module categories modified by abelian 3-cocycles
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Robert McRae [email protected] Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA
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R. McRae 2.3 Abelian intertwining algebras and automorphisms . . . . . . . 3 From G-modules to V G -modules . . . . . . . . . . . . . . . . . . 3.1 Schur-Weyl duality for abelian intertwining algebras . . . . . . 3.2 The functor . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Equal fusion rules implies vertex tensor category structure . . 4.2 Vertex tensor category structure implies a braided equivalence 4.3 Examples and applications . . . . . . . . . . . . . . . . . . . Ref
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