Sphere Packings, Lattices and Groups

Distance-Regular Graphs

Introduction to lie algebras and representation theory

We develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens' classification of $V_1$-structures of meromorphic conformal field theories of central charge 24 is a theorem on vertex operator algebras. Finally we use these results to construct some new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds.

Construction and classification of holomorphic vertex operator algebras

Let V be a simple VOA and consider a representation category of V that is a vertex tensor category in the sense of Huang-Lepowsky. In particular, this category is a braided tensor category. Let J be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension. We prove that $V\oplus J$ is either a VOA or a super VOA. If the representation category of V is in addition ribbon, then the categorical dimension of J decides this parity question. Combining with Carnahan's work, we extend this result to simple currents of arbitrary order. Our next result is a simple sufficient criterion for lifting indecomposable objects that only depends on conformal dimensions. Several examples of simple current extensions that are $C_2$-cofinite and non-rational are then given and induced modules listed.

Simple current extensions beyond semi-simplicity

This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism.

Classification of holomorphic framed vertex operator algebras of central charge 24

In 1993, Schellekens obtained a list of possible 71 Lie algebras of holomorphic vertex operator algebras with central charge 24. However, not all cases are known to exist. The aim of this article is to construct new holomorphic vertex operator algebras using the theory of framed vertex operator algebras and to determine the Lie algebra structures of their weight one subspaces. In particular, we study holomorphic framed vertex operator algebras associated to subcodes of the triply even codes $\RM(1,4)^3$ and $\RM(1,4)\oplus \EuD(d_{16}^+)$ of length 48. These vertex operator algebras correspond to the holomorphic simple current extensions of the lattice type vertex operator algebras $(V_{\sqrt{2}E_8}^+)^{\otimes 3}$ and $V_{\sqrt{2}E_8}^+\otimes V_{\sqrt{2}D_{16}^+}^+$. We determine such extensions using a quadratic space structure on the set of all irreducible modules $R(W)$ of $W$ when $W= (V_{\sqrt{2}E_8}^+)^{\otimes 3}$ or $V_{\sqrt{2}E_8}^+\otimes V_{\sqrt{2}D_{16}^+}^+$. As our main results, we construct seven new holomorphic vertex operator algebras of central charge 24 in Schellekens' list and obtain a complete list of all Lie algebra structures associated to the weight one subspaces of holomorphic framed vertex operator algebras of central charge 24.

Quadratic spaces and holomorphic framed vertex operator algebras of central charge 24

By applying Miyamoto's $\mathbb{Z}_{3}$-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices and their automorphisms of order 3, we construct holomorphic vertex operator algebras of central charge 24 whose Lie algebras of the weight one spaces are of types $A_{2,3}^6$, $E_{6,3}G_{2,1}^{3}$, and $A_{5,3}D_{4,3}A_{1,1}^{3}$, which correspond to No.6, No.17, and No.32 on Schellekens' list, respectively.

Application of a $\mathbb{Z}_{3}$-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices

In this article, we construct three new holomorphic vertex operator algebras of central charge 24 using the ${\mathbb{Z}_{2}}$-orbifold construction associated to inner automorphisms. Their weight one subspaces have the Lie algebra structures D7,3A3,1G2,1, E7,3A5,1, and ${A_{8,3}A_{2,1}^2}$. In addition, we discuss the constructions of holomorphic vertex operator algebras with Lie algebras A5,6C2,3A1,2 and ${D_{6,5}A_{1,1}^2}$ from holomorphic vertex operator algebras with Lie algebras C5,3G2,2A1,1 and ${A_{4,5}^2}$, respectively.

Hiroki Shimakura

Papers

Classification of holomorphic framed vertex operator algebras of central charge 24

Quadratic spaces and holomorphic framed vertex operator algebras of central charge 24

Application of a $\mathbb{Z}_{3}$-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices

Quadratic spaces and holomorphic framed vertex operator algebras of central charge 24

Orbifold Construction of Holomorphic Vertex Operator Algebras Associated to Inner Automorphisms