Generators, spanning sets and existence of twisted modules for a grading-restricted vertex (super)algebra

Abstract

For a grading-restricted vertex superalgebra $V$ and an automorphism $g$ of $V$, we give a linearly independent set of generators of the universal lower-bounded generalized $g$-twisted $V$-module ${\hat{M}}_B^{[g]}$ constructed by the author in \cite{H-const-twisted-mod}. We prove that there exist irreducible lower-bounded generalized $g$-twisted $V$-modules by showing that there exists a maximal proper submodule of ${\hat{M}}_B^{[g]}$ for a one-dimensional space $M$. We then give several spanning sets of ${\hat{M}}_B^{[g]}$ and discuss the relations among elements of the spanning sets. Assuming that $V$ is a Möbius vertex superalgebra (to make sure that lowest weights make sense) and that $P(V)$ (the set of all numbers of the form $\Re (\alpha )\in [0,1)$ for $α\in \C$ such that $e^{2\pi i\alpha }$ is an eigenvalue of $g$) has no accumulation point in $\mathbb{R}$ (to make sure that irreducible lower-bounded generalized $g$-twisted $V$-modules have lowest weights). Under suitable additional conditions, which hold when the twisted zero-mode algebra or the twisted Zhu's algebra is finite dimensional, we prove that there exists an irreducible grading-restricted generalized $g$-twisted $V$-module, which is in fact an irreducible ordinary $g$-twisted $V$-module when $g$ is of finite order. We also prove that every lower-bounded generalized module with an action of $g$ for the fixed-point subalgebra $V^g$ of $V$ under $g$ can be extended to a lower-bounded generalized $g$-twisted $V$-module.

41 pages. One section reviewing the construction of lower-bounded $g$-twisted generalized $V$-module is added. Several references and sentences referring to them are added

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