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We construct two associative algebras from a vertex operator algebra $V$ anda general automorphism $g$ of $V$. The first, called $g$-twisted zero-modealgebra, is a subquotient of what we call $g$-twisted universal envelopingalgebra of $V$. These algebras are generalizations of the correspondingalgebras introduced and studied by Frenkel-Zhu and Nagatomo-Tsuchiya in the(untwisted) case that $g$ is the identity. The other is a generalization of the$g$-twisted version of Zhu's algebra for suitable $g$-twisted modulesconstructed by Dong-Li-Mason when the order of $g$ is finite. We are mainlyinterested in $g$-twisted $V$-modules introduced by the first author in thecase that $g$ is of infinite order and does not act on $V$ semisimply. In thiscase, twisted vertex operators in general involve the logarithm of thevariable. We construct functors between categories of suitable modules forthese associative algebras and categories of suitable (logarithmic) $g$-twisted$V$-modules. Using these functors, we prove that the $g$-twisted zero-modealgebra and the $g$-twisted generalization of Zhu's algebra are in factisomorphic.

Associative algebras for (logarithmic) twisted modules for a vertex operator algebra