Building vertex algebras from parts

Abstract

Given a collection of modules of a vertex algebra parametrized by an abelian group, together with one dimensional spaces of composable intertwining operators, we assign a canonical element of the cohomology of an Eilenberg-Mac Lane space. This element describes the obstruction to locality, as the vanishing of this element is equivalent to the existence of a vertex algebra structure with multiplication given by our intertwining operators, and given existence, the structure is unique up to isomorphism. The homological obstruction reduces to an "evenness" problem that naturally vanishes for 2-divisible groups, so simple currents organized into odd order abelian groups always produce vertex algebras. Furthermore, in cases most relevant to conformal field theory (i.e., when we have well-behaved contragradients and tensor products), we obtain our spaces of intertwining operators naturally, and the evenness obstruction reduces to the question of whether the contragradient bilinear form on certain order 2 currents is symmetric or skew-symmetric. We show that if we are given a simple regular VOA and integral-weight modules parametrized by a group of even units in the fusion ring, then the direct sum admits the structure of a simple regular VOA, called the simple current extension, and this structure is unique up to isomorphism.

(v3) 38 pages, final version. arXiv admin note: substantial text overlap with arXiv:1208.6254

References in corpus (7)

• Braided tensor categories and extensions of vertex operator algebras
• Construction and Classification of Holomorphic Vertex Operator Algebras
• Simple current extensions beyond semi-simplicity
• Logarithmic tensor product theory for generalized modules for a conformal vertex algebra, Part I
• Generalized Moonshine IV: Monstrous Lie algebras
• Genera of Vertex Operator Algebras and three dimensional Topological Quantum Field Theories
• Logarithmic tensor product theory for generalized modules for a conformal vertex algebra

Cited by in corpus (12)

• Tensor categories for vertex operator superalgebra extensions
• Schur-Weyl Duality for Heisenberg Cosets
• Simple current extensions beyond semi-simplicity
• $\mathcal{W}$-algebras with non-admissible levels and the Deligne exceptional series
• Supersymmetry and the Suzuki chain
• Genus of vertex algebras and mass formula
• On the Genus of the Moonshine Module
• Parafermion vertex operator algebras and W-algebras
• Lattices, Vertex Algebras and Modular Categories
• The classification of chiral WZW models by $H_{+}^4(BG,\mathbb{Z})$
• Simple current extensions of tensor products of vertex operator algebras
• Modular linear differential equations of fourth order and minimal $\mathcal{W}$-algebras