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Twisted logarithmic modules of free field algebras

Given a non-semisimple automorphism $φ$ of a vertex algebra $V$, the fields in a $φ$-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 $βγ$-system (even superfermions). In each case, we determine the action of the Virasoro algebra.