Principal subspaces of twisted modules for certain lattice vertex operator algebras

Abstract

This is the third in a series of papers studying the vertex-algebraic structure of principal subspaces of twisted modules for lattice vertex operator algebras. We focus primarily on lattices L whose Gram matrix contains only non-negative entries. We develop further ideas originally presented by Calinescu, Lepowsky, and Milas to find presentations (generators and relations) of the principal subspace of a certain natural twisted module for the vertex operator algebra V L . We then use these presentations to construct exact sequences involving this principal subspace, which give a set of recursions satisfied by the multigraded dimension of the principal subspace and allow us to find the multigraded dimension of the principal subspace. and D(1) 4 , Glas. Mat. Ser. III 46, 66 (2011), 49-70.

Principal subspaces of basic modules for twisted affine Lie algebras, q-series multisums, and Nandi’s identities

“…Vertex-algebraic background. We recall certain details relevant to this work from the vertex-algebraic constructions found in the works of Calinescu, Lepowsky, Milas, Penn, Webb, and the fourth author [12,14,35,36,37]. Suppose (15)…”

“…where 1 T is a vector of lowest conformal weight in V T L (see Section 3 of [37] for this notation) and q −wt1 T is introduced to ensure that all powers of q are nonnegative integers. Following Penn, Webb, and the fourth author [35,36,37], we have that for 1 ⩽ i ⩽ d: (18)…”

“…. , l d are the sizes of the orbits of various simple roots (this folding is defined generally by Penn, Webb, and the fourth author [37]). The matrices A[ν] are symmetrized Cartan matrices of types B, C, F, G, and, in the case A…”

“…Principal subspaces and q-series multisums Meanwhile, Penn, Webb, and the fourth author [37] constructed a principal subspace of a twisted module for a lattice vertex operator algebra whose graded dimension is ( 6)…”

Linked Partition Ideals, Directed Graphs and $q$-Multi-Summations

“…It is worth pointing out that the recurrence (46) and its relations to sumlike generating functions have connections with the theory of vertex operator algebras, especially in the context of principal subspaces of modules. For one recent example, see (4.71)-(4.81) in [21].…”