Twisted Modules and G -equivariantization in Logarithmic Conformal Field Theory
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Communications in
Mathematical Physics
Twisted Modules and G -equivariantization in Logarithmic Conformal Field Theory Robert McRae Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China. E-mail: [email protected] Received: 27 April 2020 / Accepted: 24 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic tensor category theory of Huang, Lepowsky, and Zhang. In this paper, we study not-necessarilysemisimple or rigid braided tensor categories C of modules for the fixed-point vertex operator subalgebra V G of a vertex operator (super)algebra V with finite automorphism group G. The main results are that every V G -module in C with a unital and associative V -action is a direct sum of g-twisted V -modules for possibly several g ∈ G, that the category of all such twisted V -modules is a braided G-crossed (super)category, and that the G-equivariantization of this braided G-crossed (super)category is braided tensor equivalent to the original category C of V G -modules. This generalizes results of Kirillov and Müger proved using rigidity and semisimplicity. We also apply the main results to the orbifold rationality problem: whether V G is strongly rational if V is strongly rational. We show that V G is indeed strongly rational if V is strongly rational, G is any finite automorphism group, and V G is C2 -cofinite. Contents 1. 2.
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Braided G-crossed Supercategories . . . . . . . . . . . . . 2.1 Superalgebra objects in supercategories . . . . . . . . 2.2 Braided G-crossed supercategories of twisted modules 2.3 G-equivariantization . . . . . . . . . . . . . . . . . . The Main Categorical Theorem . . . . . . . . . . . . . . . Twisted Modules for Vertex Operator Superalgebras . . . . 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . .
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R. McRae
4.2 General theorems . . . . . . . . . . . . . 4.3 Z/2Z-equivariantization for superalgebras 4.4 Application to orbifold rationality . . . . A. Proof of Theorem 2.10 . . . . . . . . . . . . . B. Details for Theorem 3.3 . . . . . . . . . . . .
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1. Introduction Orbifolding is a way to produce new conformal field theories from old ones. Mathematically, a two-dimensional (chiral) conformal field theory can be treated as the representation theory of its chiral a
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