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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 $gin G$, 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 M{u}ger 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.
The main purpose of this paper is a mathematical construction of a non-perturbative deformation of a two-dimensional conformal field theory. We introduce a notion of a full vertex algebra which formulates a compact two-dimensional conformal field the
We continue the program of constructing (pre)modular tensor categories from 3-manifolds first initiated by Cho-Gang-Kim using $M$ theory in physics and then mathematically studied by Cui-Qiu-Wang. An important structure involved is a collection of ce
The idea of summing over all intermediate states that is central for implementing locality in quantum systems can be realized by coend constructions. In the concrete case of systems of conformal blocks for a certain class of conformal vertex algebras
We introduce and study a category $text{Fin}$ of modules of the Borel subalgebra of a quantum affine algebra $U_qmathfrak{g}$, where the commutative algebra of Drinfeld generators $h_{i,r}$, corresponding to Cartan currents, has finitely many charact
The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of any rational