Fusion rules for the free bosonic orbifold vertex operator algebra

2A-orbifold construction and the baby-monster vertex operator superalgebra

In this article we prove that the full automorphism group of the baby-monster vertex operator superalgebra constructed by Hoehn is isomorphic to 2xB, where B is the baby-monster sporadic finite simple group and determine irreducible modules for the baby-monster vertex operator algebra. Our result has many corollaries. In particular, we can prove that the Z_2-orbifold construction with respect to a 2A-involution of the Monster applied to the moonshine vertex operator algebra yields the moonshine vertex operator algebra itself again.

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

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.

Orbifolds of lattice vertex algebras under an isometry of order two

Every isometry $σ$ of a positive-definite even lattice $Q$ can be lifted to an automorphism of the lattice vertex algebra $V_Q$. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the $σ$-invariant subalgebra $V_Q^σ$ of $V_Q$, known as an orbifold. In the case when $σ$ is an isometry of $Q$ of order two, we classify the irreducible modules of the orbifold vertex algebra $V_Q^σ$ and identify them as submodules of twisted or untwisted $V_Q$-modules. The examples where $Q$ is a root lattice and $σ$ is a Dynkin diagram automorphism are pr…

Reverse orbifold construction and uniqueness of holomorphic vertex operator algebras

In this article, we develop a general technique for proving the uniqueness of holomorphic vertex operator algebras based on the orbifold construction and its "reverse" process. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge $24$ is uniquely determined by its weight one Lie algebra if the Lie algebra has the type $E_{6,3}G_{2,1}^3$, $A_{2,3}^6$ or $A_{5,3}D_{4,3}A_{1,1}^3$.

Orbifold construction of holomorphic vertex operator algebras associated to inner automorphisms

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 has the Lie algebra structures $D_{7,3}A_{3,1}G_{2,1}$, $E_{7,3}A_{5,1}$, and $A_{8,3}A_{2,1}^2$. In addition, we discuss the constructions of holomorphic vertex operator algebras with Lie algebras $A_{5,6}C_{2,3}A_{1,2}$ and $D_{6,5}A_{1,1}^2$ from holomorphic vertex operator algebras with Lie algebras $C_{5,3}G_{2,2}A_{1,1}$ and $A_{4,5}^2$, respectively.

Cyclic orbifolds of lattice vertex operator algebras having group like fusions

Let $L$ be an even (positive definite) lattice and $g\in O(L)$. In this article, we prove that the orbifold vertex operator algebra $V_{L}^{\hat{g}}$ has group-like fusion if and only if $g$ acts trivially on the discriminant group $\mathcal{D}(L)=L^*/L$ (or equivalently $(1-g)L^*<L$). We also determine their fusion rings and the corresponding quadratic space structures when $g$ is fixed point free on $L$. By applying our method to some coinvariant sublattices of the Leech lattice $Λ$, we prove a conjecture proposed by G. Höhn. In addition, we also discuss a construction of certain holomorphic…

Fourvolutions and automorphism groups of orbifold lattice vertex operator algebras

Let $L$ be an even positive definite lattice with no roots, i.e., $L(2)=\{x\in L\mid (x|x)=2\}=\emptyset$. Let $g\in O(L)$ be an isometry of order $4$ such that $g^2=-1$ on $L$. In this article, we determine the full automorphism group of the orbifold vertex operator algebra $V_L^{\hat{g}}$. As our main result, we show that $Aut(V_L^{\hat{g}})$ is isomorphic to $N_{Aut(V_L)}(\langle \hat{g}\rangle)/ \langle\hat{g}\rangle $ unless $L\cong \sqrt{2}E_8$ or $BW_{16}$.

Orbifold construction and Lorentzian construction of Leech lattice vertex operator algebra

We generalize Conway-Sloane's constructions of the Leech lattice from Niemeier lattices using Lorentzian lattice to holomorphic vertex operator algebras (VOA) of central charge 24. It provides a tool for analyzing the structures and relations among holomorphic VOAs related by orbifold construction. In particular, we are able to get some useful information about certain lattice subVOAs associated with the Cartan subalgebra of the weight one Lie algebra. We also obtain a relatively elementary proof that any strongly regular holomorphic VOA of central charge $24$ with a non-trivial weight one sub…

$C_2$-cofiniteness of 2-cyclic permutation orbifold models

In this article, we consider permutation orbifold models of $C_2$-cofinite vertex operator algebras of CFT type. We show the $C_2$-cofiniteness of the 2-cyclic permutation orbifold model $(V\otimes V)^{S_2}$ for an arbitrary $C_2$-cofinite simple vertex operator algebra $V$ of CFT type. We also give a proof of the $C_2$-cofiniteness of a $\Z_2$-orbifold model $V_L^+$ of the lattice vertex operator algebra $V_L$ associated with a rank one positive definite even lattice $L$ by using our result and the $C_2$-cofiniteness of $V_L$.

Extensions of tensor products of ${\mathbb Z}_p$-orbifold models of the lattice vertex operator algebra $V_{\sqrt{2}A_{p-1}}$

Let $p$ be an odd prime and let $\widehatσ$ be an order $p$ automorphism of $V_{\sqrt{2}A_{p-1}}$ which is a lift of a $p$-cycle in the Weyl group ${\rm Weyl}(A_{p-1})\cong {\mathfrak S}_p$. We study a certain extension $V$ of a tensor product of finitely many copies of the orbifold model $V_{\sqrt{2}A_{p-1}}^{\langle \widehatσ \rangle}$ and give a criterion for $V$ that every irreducible $V$-module is a simple current.

Representations of $\mathbb{Z}_{2}$-orbifold of the parafermion vertex operator algebra $K(sl_2,k)$

In this paper, the irreducible modules for the $\mathbb{Z}_{2}$-orbifold vertex operator subalgebra of the parafermion vertex operator algebra associated to the irreducible highest weight modules for the affine Kac-Moody algebra $A_1^{(1)}$ of level $k$ are classified and constructed.

Fusion products of twisted modules in permutation orbifolds

Let $V$ be a vertex operator algebra, $k$ a positive integer and $σ$ a permutation automorphism of the vertex operator algebra $V^{\otimes k}$. In this paper, we determine the fusion product of any $V^{\otimes k}$-module with any $σ$-twisted $V^{\otimes k}$-module.

$\Z_3$-orbifold construction of the Moonshine vertex operator algebra and some maximal $3$-local subgroups of the Monster

In this article, we describe some maximal $3$-local subgroups of the Monster simple group using vertex operator algebras (VOA). We first study the holomorphic vertex operator algebra obtained by applying the orbifold construction to the Leech lattice vertex operator algebra and a lift of a fixed-point free isometry of order $3$ of the Leech lattice. We also consider some of its special subVOAs and study their stabilizer subgroups using the symmetries of the subVOAs. It turns out that these stabilizer subgroups are $3$-local subgroups of its full automorphism group. As one of our main results, …

Fusion rules for $\mathbb{Z}_{2}$-orbifolds of affine and parafermion vertex operator algebras

This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra $K(sl_2,k)$ associated to the integrable highest weight modules for the affine Kac-Moody algebra $A_1^{(1)}$ is the building block of the general parafermion vertex operator $K(\mathfrak{g},k)$ for any finite dimensional simple Lie algebra $\mathfrak{g}$ and any positive integer $k$. We first classify the irreducible modules of $\mathbb{Z}_{2}$-orbifold of the simple affine vertex operator algebra of type $A_1^{(1)}$ and determine their fusion rul…

Finiteness and orbifold Vertex Operator Algebras

In this paper, I investigate the ascending chain condition of right ideals in the case of vertex operator algebras satisfying a finiteness and/or a simplicity condition. Possible applications to the study of finiteness of orbifold VOAs is discussed.

$\mathbb{Z}_2$-orbifold construction associated with $(-1)$-isometry and uniqueness of holomorphic vertex operator algebras of central charge 24

The vertex operator algebra structure of a strongly regular holomorphic vertex operator algebra $V$ of central charge $24$ is proved to be uniquely determined by the Lie algebra structure of its weight one space $V_1$ if $V_1$ is a Lie algebra of the type $A_{1,4}^{12}$, $B_{2,2}^6$, $B_{3,2}^4$, $B_{4,2}^3$, $B_{6,2}^2$, $B_{12,2}$, $D_{4,2}^2B_{2,1}^4$, $D_{8,2}B_{4,1}^2$, $A_{3,2}^4A_{1,1}^4$, $D_{5,2}^2A_{3,1}^2$, $D_{9,2}A_{7,1}$, $C_{4,1}^4$ or $D_{6,2}B_{3,1}^2C_{4,1}$.

Representations and fusion rules for the orbifold vertex operator algebras $L_{\widehat{\frak{sl}_2}}(k,0)^{\mathbb{Z}_3}$

For the cyclic group $\mathbb{Z}_3$ and positive integer $k$, we study the representations of the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$. All the irreducible modules for $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are classified and constructed explicitly. Quantum dimensions and fusion rules for the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are completely determined.

置换轨道和简单电流扩张

Permutation orbifolds and simple current extensions

本文在顶点算子（超）代数框架下研究置换轨道和简单电流扩张，通过张量范畴方法和简单电流扩张建立可解自同构群的诱导理论，并分类不可约扭模。

Bosonic vertex representations of the toroidal superalgebras in type D(m,n)

In this paper, vertex representations of the 2-toroidal Lie superalgebras of type $D(m, n)$ are constructed using both bosonic fields and vertex operators based on their loop algebraic presentation.

由与 Leech 格相关的余不变格产生的某些轨道折叠顶点算子代数的自同构群

Automorphism groups of certain orbifold vertex operator algebras arising from coinvariant lattices associated with the Leech lattice

本文确定了由 Leech 格在共轭类 3C、5C、11A 和 23A 中的余不变格产生的轨道折叠顶点算子代数的自同构群，这些代数出现在 Lam 和 Shimakura 的分类中。

仿射顶点算子超代数 $L_{\widehat{osp(1|2)}}(k,0)$ 的轨形理论

Orbifold theory of the affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(k,0)$

本文开展仿射顶点算子超代数的轨形理论研究，完成单仿射顶点算子超代数 $L_{\widehat{osp(1|2)}}(k,0)$ 轨形的不可约模分类并确定其融合规则。

Permutation Orbifolds of the Heisenberg Vertex Algebra $\mathcal{H}(3)$

We study the $S_3$-orbifold of a rank three Heisenberg vertex algebras in terms of generators and relations. By using invariant theory we prove that the orbifold algebra has a minimal strongly generating set of vectors whose conformal weights are $1,2,3,4,5,6^2$ (two generators of degree $6$). The structure of the cyclic $\mathbb{Z}_3$-oribifold is determined by similar methods. We also study modular properties of characters of modules for these vertex algebras.

Torus n-Point Functions for $\mathbb{R}$-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds

We consider genus one n-point functions for a vertex operator superalgebra with a real grading. We compute all n-point functions for rank one and rank two fermion vertex operator superalgebras. In the rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orbifold n-point functions are given and we describe a generalization of Fay's trisecant identity for elliptic functions.

A Z_2-orbifold model of the symplectic fermionic vertex operator superalgebra

We construct an irrational C_2-cofinite vertex operator algebra associatted to a finite dimensional vector space with a nondegenerate skew-symmetric bilinear form. We also classify its equivalence classes of irreducible modules and determine its automorphism group.

Commutant of $\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(4,0)$ in the cyclic permutation orbifold of $\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(1,0)^{\otimes 4}$

We study the commutant of the vertex operator algebra $\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(4,0)$ in the cyclic permutation orbifold model $(\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(1,0)^{\otimes 4})^τ$ with $τ=(1\,2\,3\,4)$. It is shown that the commutant is isomorphic to a ${\mathbb Z}_2\times{\mathbb Z}_2$-orbifold model of a tensor product of two lattice type vertex operator algebras of rank one.

Permutation orbifolds of Virasoro vertex algebras and $W$-algebras

We study permutation orbifolds of the $2$-fold and $3$-fold tensor product for the Virasoro vertex algebra $\mathcal{V}_c$ of central charge $c$. In particular, we show that for all but finitely many central charges $\left(\mathcal{V}_c^{\otimes 3}\right)^{\mathbb{Z}_3}$ is a $W$-algebra of type $(2, 4, 5, 6^3 , 7, 8^3 , 9^3 , 10^2 )$. We also study orbifolds of their simple quotients and obtain new realizations of certain rational affine $W$-algebras associated to a principal nilpotent element. Further analysis of permutation orbifolds of the celebrated $(2,5)$-minimal vertex algebra $\mathca…

格顶点算子代数的完全不动点自由等距与循环轨形

Completely fixed point free isometry and cyclic orbifold of lattice vertex operator algebras

本文研究格顶点算子代数的循环轨形，证明在特定无不动点等距条件下，额外自同构的存在迫使等距阶为素数或格为 Leech 格及其余不变子格。

Modularity in orbifold theory for vertex operator superalgebras

This paper is about the orbifold theory for vertex operator superalgebras. Given a vertex operator superalgebra V and a finite automorphism group G of V, we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C_2-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C_2-cofinite and g-rational for any g in G.

Fusion rules for the charge conjugation orbifold

We completely determine the fusion rules for the vertex operator algebra $V_L^+$ for a rank one even lattice $L$.

仿费米子顶点算子代数 $K(osp(1|2),k)$ 的轨形表示

Representations of the orbifold of parafermion vertex operator algebra $K(osp(1|2),k)$

针对任意正整数水平 $k$，论文研究 $K(osp(1|2),k)$ 的轨形理论，并完成其轨形不可约模的分类。

A Cyclic Orbifold Theory for Holomorphic Vertex Operator Algebras and Applications

In this thesis we develop an orbifold theory for a finite, cyclic group $G$ acting on a suitably regular, holomorphic vertex operator algebra $V$. To this end we describe the fusion algebra of the fixed-point vertex operator subalgebra $V^G$ and show that $V^G$ has group-like fusion. Then we solve the extension problem for vertex operator algebras with group-like fusion. We use these results to construct five new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds, contributing to the classification of the $V_1$-structures of suitably regular, holomorphic vertex oper…

The relationship between skew group algebras and orbifold theory

Let V be a simple vertex operator algebra and let G be a finite automorphism group of V. In [DY], it was shown that any irreducible V-module is a completely reducible V^G-module where V^G is the G-invariant sub-vertex operator algebra of V. In this paper, we give an alternative proof of this fact using the theory of skew group algebras. We also extend this result to any irreducible g-twisted V-module when g is in the center of G and V is a g-rational vertex operator algebra.

The automorphism group of the Z_2-orbifold of the Barnes-Wall lattice vertex operator algebra of central charge 32

In this article, we prove that the full automorphism group of the Z_2-orbifold of the Barnes-Wall lattice vertex operator algebra of central charge 32 has the shape $2^{27}.E_6(2)$. In order to identify the group structure, we introduce a graph structure on the Griess algebra and show that it is a rank 3 graph associated to $E_6(2)$.

The Large $N$ Limit of Orbifold Vertex Operator Algebras

We investigate the large $N$ limit of permutation orbifolds of vertex operator algebras. To this end, we introduce the notion of nested oligomorphic permutation orbifolds and discuss under which conditions their fixed point VOAs converge. We show that if this limit exists, then it has the structure of a vertex algebra. Finally, we give an example based on $\mathrm{GL}(N,q)$ for which the fixed point VOA limit is also the limit of the full permutation orbifold VOA.

The charge conjugation orbifold $V_{Zα}^{+}$ is rational when $<α,α>/2$ is prime

We prove that the vertex operator algebra $V_{Zα}^{+}$ is rational if $<α,α>/2$ is a prime integer.

Two Large Examples in Orbifold Theory: Abelian Orbifolds and the Charge Conjugation Orbifold on su(n)

Recently the operator algebra and twisted vertex operator equations were given for each sector of all WZW orbifolds, and a set of twisted KZ equations for the WZW permutation orbifolds were worked out as a large example. In this companion paper we report two further large examples of this development. In the first example we solve the twisted vertex operator equations in an abelian limit to obtain the twisted vertex operators and correlators of a large class of abelian orbifolds. In the second example, the twisted vertex operator equations are applied to obtain a set of twisted KZ equations fo…

S-matrix in orbifold theory

The restricted $S$-matrix of $V^G$ is determined for any regular vertex operator algebra $V$ and finite automorphism group $G$ of $V.$ As an application, the $S$-matrices for cyclic permutation orbifolds of prime orders are computed.

Orbifolds of Lattice Vertex Operator Algebras at $d=48$ and $d=72$

Motivated by the notion of extremal vertex operator algebras, we investigate cyclic orbifolds of vertex operator algebras coming from extremal even self-dual lattices in $d=48$ and $d=72$. In this way we construct about one hundred new examples of holomorphic VOAs with a small number of low weight states.

Permutation orbifolds and associative algebras

Let $V$ be a vertex operator algebra and $g=\left(1\ 2\ \cdots k\right)$ be a $k$-cycle which is viewed as an automorphism of the vertex operator algebra $V^{\otimes k}$. It is proved that Dong-Li-Mason's associated associative algebra $A_{g}\left(V^{\otimes k}\right)$ is isomorphic to Zhu's algebra $A\left(V\right)$ explicitly. This result recovers a previous result that there is a one-to-one correspondence between irreducible $g$-twisted $V^{\otimes k}$-modules and irreducible $V$-modules.

Unitary forms for holomorphic vertex operator algebras of central charge $24$

We prove that all holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one subspaces are unitary. The main method is to use the orbifold construction of a holomorphic VOA $V$ of central charge $24$ directly from a Niemeier lattice VOA $V_N$. We show that it is possible to extend the unitary form for the lattice VOA $V_N$ to the holomorphic VOA $V$ by using the orbifold construction and some information of the automorphism group $\mathrm{Aut}(V)$.

Representation theory of vertex operator algebras and orbifold conformal field theory

We discuss some basic problems and conjectures in a program to construct general orbifold conformal field theories using the representation theory of vertex operator algebras. We first review a program to construct conformal field theories. We also clarify some misunderstandings on vertex operator algebras, modular functors and intertwining operator algebras. Then we discuss some basic open problems and conjectures in mathematical orbifold conformal field theory. Generalized twisted modules and their variants, their constructions and some existence results are reviewed. Twisted intertwining op…

$C_2$-cofiniteness of orbifold models for finite groups

We prove that if $V$ is a $C_2$-cofinite simple vertex operator algebra of CFT-type with a nonsingular invariant bilinear form and its an automorphism group $G$ is finite, then an orbifold model $V^G$ is also $C_2$-cofinite.

A $\Z_3$-orbifold theory of lattice vertex operator algebra and $\Z_3$-orbifold constructions

Let $V$ be a simple VOA of CFT-type satisfying $V'\cong V$ and $σ$ a finite automorphism of $V$. We prove that if all $V$-modules are completely reducible and a fixed point subVOA $V^σ$ is $C_2$-cofinite, then all $V^σ$-modules are completely reducible and every simple $V^σ$-module appears in some twisted or ordinary $V$-modules as a $V^σ$-submodule. We also prove that $V_L^σ$ is $C_2$-cofinite for any lattice VOA $V_L$ and $σ\in \Aut(V_L)$ lifted from any triality automorphism of $L$. Using these results, we present two $Z_3$-orbifold constructions as examples. One is the moonshine VOA $V^{\n…

格顶点算子代数的循环轨道折叠的额外自同构

Extra automorphisms of cyclic orbifolds of lattice vertex operator algebras

本文研究了无根偶格顶点算子代数在奇素数阶无不动点等距提升下的循环轨道折叠自同构群，证明了额外自同构存在的充要条件是该格由 Construction B 构造或同构于 Leech 格的余不变格。

Congruence Property in Orbifold Theory

Let $V$ be a rational, selfdual, $C_2$-cofinite vertex operator algebra of CFT type, and $G$ a finite automorphism group of $V.$ It is proved that the kernel of the representation of the modular group on twisted conformal blocks associated to $V$ and $G$ is a congruence subgroup. In particular, the $q$-character of each irreducible twisted module is a modular function on the same congruence subgroup. In the case $V$ is the Frenkel-Lepowsky-Meurman's moonshine vertex operator algebra and $G$ is the monster simple group, the generalized McKay-Thompson series associated to any commuting pair in t…

Modularity of trace functions in orbifold theory for Z-graded vertex operator superalgebras

We study the trace functions in orbiford theory for Z-graded vertex operator superalgebras and obtain a modular invariance result. More precisely, let V be a C_2-cofinite Z-graded vertex operator superalgebra and G a finite automorphism group of V. Then for any commuting pairs (g,h) in G, the hσ-trace functions associated to the simple g-twisted V-modules are holomorphic in the upper half plane where σis the canonical involution on V coming from the superspace structure of V. If V is further g-rational for every g n G, the trace unctions afford a representation for the full modular group SL(2,…

C_2-cofiniteness of cyclic-orbifold models

We prove an orbifold conjecture for a solvable automorphism group. Namely, we show that if V is a C_2-cofinite simple vertex operator algebra and G is a finite solvable automorphism group of V, then the fixed point vertex operator subalgebra V^G is also C_2-cofinite. This offers a mathematically rigorous background to orbifold theories of finite type with solvable automorphism groups.

$S_3$-Permutation Orbifolds of Virasoro Vertex Algebras

In this paper, a continuation of \cite{MPS}, we investigate the $S_3$-orbifold subalgebra of $(\mathcal{V}_c)^{\otimes 3}$, that is, we consider the $S_3$-fixed point vertex subalgebra of the tensor product of three copies of the universal Virasoro vertex operator algebras $\mathcal{V}_c$. Our main result is construction of a minimal, strong set of generators of this subalgebra for any generic values of $c$. More precisely, we show that this vertex algebra is of type $(2,4,6^2,8^2,9,10^2,11,12^3)$. We also investigate two prominent examples of simple $S_3$-orbifold algebras corresponding to ce…

Cyclic permutations of lattice vertex operator algebras

The irreducible modules of the 2-cycle permutation orbifold models of lattice vertex operator algebras of rank 1 are classified, the quantum dimensions of irreducible modules and the fusion rules are determined.

顶点算子超代数的置换轨形与结合代数

Permutation orbifolds of vertex operator superalgebra and associative algebras

本文研究顶点算子超代数张量积的k-循环置换轨形，构造了对应扭Zhu代数的显式同构，复现了扭模一一对应结论，为不动点子代数Zhu代数研究提供工具。

Generalized Twisted Quantum Doubles of a Finite Group and Rational Orbifolds

In previous work the authors introduced a new class of modular quasi-Hopf algebras $D^ω(G, A)$ associated to a finite group $G$, a central subgroup $A$, and a $3$-cocycle $ω\in Z^3(G, C^x)$. In the present paper we propose a description of the class of orbifold models of rational vertex operator algebras whose module category is tensor equivalent to $D^ω(G, A)$-mod. The paper includes background on quasi-Hopf algebras and a discussion of some relevant orbifolds.

对尼梅依尔晶格顶点算子代数的系统轨道构造

Systematic Orbifold Constructions of Schellekens' Vertex Operator Algebras from Niemeier Lattices

本文系统构造了所有70个中心荷为24且具有非零权重一空间的强理性整体顶点算子代数，通过尼梅依尔晶格顶点算子代数及其短自守变换的轨道构造，并证明其由权重一空间的李代数结构唯一确定。

Orbifold theory for vertex algebras and Galois correspondence

Let $V$ be a simple vertex algebra of countable dimension, $G$ be a finite automorphism group of $V$ and $σ$ be a central element of $G$. Assume that ${\cal S}$ is a finite set of inequivalent irreducible $σ$-twisted $V$-modules such that ${\cal S}$ is invariant under the action of $G$. Then there is a finite dimensional semisimple associative algebra ${\cal A}_α(G,{\cal S})$ for a suitable $2$-cocycle $α$ naturally determined by the $G$-action on ${\cal S}$ such that $({\cal A}_α(G,{\cal S}),V^G)$ form a dual pair on the sum $\cal M$ of $σ$-twisted $V$-modules in ${\cal S}$ in the sense that …

Non-Abelian Orbifolds of Lattice Vertex Operator Algebras

We construct orbifolds of holomorphic lattice Vertex Operator Algebras for non-Abelian finite automorphism groups $G$. To this end, we construct twisted modules for automorphisms $g$ together with the projective representation of the centralizer of $g$ on the twisted module. This allows us to extract the irreducible modules of the fixed point VOA $V^G$, and to compute their characters and modular transformation properties. We then construct holomorphic VOAs by adjoining such modules to $V^G$. Applying these methods to extremal lattices in $d=48$ and $d=72$, we construct more than fifty new hol…