Study of representation theory of fixed point vertex subalgebras

(1) For a vertex algebra V, I define the notion of a V-module with logarithmic terms, which is a natural generalization of a V-module. Moreover I construct the corresponding Zhu algebras. Namely, I establish a one-to-one correspondence between the simple V-modules with logarithmic terms and the simple left modules over the Zhu algebra. This correspondence is a natural generalization of some results by Zhu.
(2) I construct examples of modules with logarithmic terms for lattice vertex algebras. Since lattice vertex algebras have many interesting vertex subalgebras, these examples are also modules with logarithmic terms over such vertex algebras. For example, we have modules with logarithmic terms over some Virasoro vertex algebras and Heisenberg vertex algebras.