Twisted modules and G-equivariantization in logarithmic conformal field theory

Robert McRae Tsinghua University

Category Theory Mathematical Physics Quantum Algebra Representation Theory mathscidoc:2204.04004

Communications in Mathematical Physics, 383, (3), 1939-2019, 2021.5
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-necessarily-semisimple 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 C_2-cofinite.
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  title={Twisted modules and G-equivariantization in logarithmic conformal field theory},
  author={Robert McRae},
  booktitle={Communications in Mathematical Physics},
Robert McRae. Twisted modules and G-equivariantization in logarithmic conformal field theory. 2021. Vol. 383. In Communications in Mathematical Physics. pp.1939-2019.
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