Generators, spanning sets and existence of twisted modules for a grading-restricted vertex (super)algebra
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Generators, spanning sets and existence of twisted modules for a grading-restricted vertex (super)algebra Yi-Zhi Huang1
© Springer Nature Switzerland AG 2020
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] constructed by the author in Huang (Commun Math Phys g-twisted V -module M B 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 [g] for a one-dimensional space M. We then give several spanning sets of M [g] M B 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 (α) ∈ [0, 1) for α ∈ C such that e2πiα is an eigenvalue of g) has no accumulation point in 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 gradingrestricted 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. Mathematics Subject Classification 17B69 · 81T40
1 Introduction In the representation theory of vertex operator algebras and orbifold conformal field theory, the existence of twisted modules associated to an automorphism of a vertex operator algebra has been an explicitly stated conjecture since mid 1990s. While there
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Yi-Zhi Huang [email protected] Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA 0123456789().: V,-vol
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exists at least one module for a vertex operator algebra (the vertex operator algebra itself), it is not obvious at all why there must be a twisted module for a general vertex operator algebra. Assuming that the vertex operator algebra is simple and C2 -cofinite and the automorphism is of finite order, Dong et al. [9] proved the existence of an irreducible twisted module. But no progress has been made in the general case for more than twenty years. Recently, mathematicians and physicists have discovered that some classes of vertex operator algebras that are not C2 -cofinite have a very rich and exciting representation theory. For example, vertex operator algebras associated to affine Lie algebras at admissible levels are not C2 -cofinite. But the category of ordinary modules for such a vertex operator algebra still has finitely many irreducible modules and every such module is completely reducible (see [1] for the co
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