Twisted modules and $G$-equivariantization in logarithmic conformal field theory

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-necessarily-semisimple or rigid braided tensor categories $\mathcal{C}$ 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 $\mathcal{C}$ with a unital and associative $V$-action is a direct sum of $g$-twisted $V$-modules for possibly several $g\in G$, 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 $\mathcal{C}$ 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.

arXiv comment56 pages, updated contact information and minor edits in this version

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Cited by in corpus (16)

• Tensor categories for vertex operator superalgebra extensions
• On the tensor structure of modules for compact orbifold vertex operator algebras
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• Direct limit completions of vertex tensor categories
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• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• On semisimplicity of module categories for finite non-zero index vertex operator subalgebras
• Orbifolds and minimal modular extensions
• $N=1$ super Virasoro tensor categories
• Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules
• Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras
• Genus-zero Permutation-twisted Conformal Blocks for Tensor Product Vertex Operator Algebras: The Tensor-factorizable Case
• Modular ${\mathbb{Z}}_2$-Crossed Tambara-Yamagami-like Categories for Even Groups
• Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras II: Convergence of Sewing and Higher Genus Pseudo-$q$-traces
• S-matrix in orbifold theory
• Representation theory of vertex operator algebras and orbifold conformal field theory