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 ${\widehat{M}}^{[g]}_{B}$ constructed by the author in Huang (Commun Math Phys 377:909–945 (2020)). We prove that there exist irreducible lower-bounded generalized g-twisted V-modules by showing that there exists a maximal proper submodule of ${\widehat{M}}^{[g]}_{B}$ for a one-dimensional space M. We then give several spanning sets of ${\widehat{M}}^{[g]}_{B}$ 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 $\mathfrak {R}(\alpha )\in [0, 1)$ for $\alpha \in \mathbb {C}$ 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.