Twisted logarithmic modules of free field algebras

TL;DR

This paper constructs explicit examples of twisted modules for free field vertex algebras under non-semisimple automorphisms, involving logarithmic fields and non-semisimple Virasoro actions, with applications to symplectic fermions, free fermions, and $eta ext{-} ext{gamma}$ systems.

Contribution

It provides explicit constructions of twisted modules with logarithmic fields for free field vertex algebras under non-semisimple automorphisms, expanding understanding of such modules.

Findings

Explicit realization of twisted modules as Fock spaces.

Determination of Virasoro action in each case.

Examples include symplectic fermions, free fermions, and $eta ext{-} ext{gamma}$ systems.

Abstract

Given a non-semisimple automorphism $\varphi$ of a vertex algebra $V$, the fields in a $\varphi$-twisted $V$-module involve the logarithm of the formal variable, and the action of the Virasoro operator $L_0$ on such module is not semisimple. We construct examples of such modules and realize them explicitly as Fock spaces when $V$ is generated by free fields. Specifically, we consider the cases of symplectic fermions (odd superbosons), free fermions, and $\beta\gamma$-system (even superfermions). In each case, we determine the action of the Virasoro algebra.