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

We construct two associative algebras from a vertex operator algebra V and a general automorphism g of V. The first, called a g-twisted zero-mode algebra, is a subquotient of what we call a g-twisted universal enveloping algebra of V. These algebras are generalizations of the corresponding algebras 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 modules constructed by Dong-Li-Mason when the order of g is finite. We are mainly interested in g-twisted V-modules introduced by the first author in the case that g is of infinite order and does not act on V semisimply. In this case, twisted vertex operators in general involve the logarithm of the variable. We construct functors between categories of suitable modules for these associative algebras and categories of suitable (logarithmic) g-twisted V-modules. Using these functors, we prove that the g-twisted zero-mode algebra and the g-twisted generalization of Zhu’s algebra are in fact isomorphic.