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。
摘要：
本文研究无根偶格对应的顶点算子代数在奇素数阶 $p$ 无不动点等距提升下的循环轨道折叠的自同构群。我们证明，当且仅当无根偶格能通过 Construction B 从 $\mathbb{Z}_p$ 上的码构造，或同构于与某阶 $p$ 等距相关的 Leech 格的余不变格时，该循环轨道折叠才包含不由格顶点算子代数自同构诱导的额外自同构。
英文摘要：
In this article, we study the automorphism group of the cyclic orbifold of a vertex operator algebra associated with a rootless even lattice for a lift of a fixed-point free isometry of odd prime order $p$. We prove that such a cyclic orbifold contains extra automorphisms, not induced from automorphisms of the lattice vertex operator algebra, if and only if the rootless even lattice can be constructed by Construction B from a code over $\mathbb{Z}_p$ or is isometric to the coinvariant lattice of the Leech lattice associated with a certain isometry of order $p$.
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 $E_6(2)$.
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.
We classify, up to conjugation, all automorphisms of Niemeier lattices to which we can apply Miyamoto's orbifold construction. Using this classification, we prove that the VOAs obtained in [M] and [SS] are all of holomorphic non-lattice VOAs which we can obtain by applying the $\mathbb{Z}_3$-orbifold construction to a Niemeier lattice and its automorphism.
We construct solitons in affine orbifold nets associated with outer automorphisms, and we show that our construction gives all the twisted representations of the fixed point subnet. This allows us to settle a number of questions concerning such orbifold constructions.
We determine the automorphism groups of the cyclic orbifold vertex operator algebras associated with coinvariant lattices of isometries of the Leech lattice in the conjugacy classes $4C,6E,6G,8E$ and $10F$. As a consequence, we have determined the automorphism groups of all the $10$ vertex operator algebras in [Hö], which are useful to analyze holomorphic vertex operator algebras of central charge $24$.
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 $BW_{16}$.
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 $V_L$.
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.
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.
Using a result of Longo and Xu, we show that the anomaly arising from a cyclic permutation orbifold of order 3 of a holomorphic conformal net $\mathcal A$ with central charge $c=8k$ depends on the "gravitational anomaly" $k\pmod 3$. In particular, the conjecture that holomorphic permutation orbifolds are non-anomalous and therefore a stronger conjecture of Müger about braided crossed $S_n$-categories arising from permutation orbifolds of completely rational conformal nets are wrong. More general, we show that cyclic permutations of order $n$ are non-anomalous if and only if $3\nmid n$ or $24|c…
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…
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…
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…
In this article, we determine the automorphism groups of $14$ holomorphic vertex operator algebras of central charge $24$ obtained by applying the $\mathbb{Z}_2$-orbifold construction to the Niemeier lattice vertex operator algebras and lifts of the $-1$-isometries.
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.
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…
Let $V$ be a simple vertex operator superalgebra and $G$ a finite automorphism group of $V$ containing the canonical automorphism $σ$ such that $V^G$ is regular. It is proved that every irreducible $V^G$-module occurs in an irreducible $g$-twisted $V$-module for some $g\in G$ and the irreducible $V^G$-modules are classified. Moreover, the quantum dimensions of irreducible $V^G$-modules are determined, a global dimension formula for $V$ in terms of twisted modules is obtained and a super quantum Galois theory is established. In addition, the $S$-matrix of $V^G$ is computed
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.
We extend the Dong-Mason theorem on the irreducibility of modules for orbifold vertex algebras from [C. Dong, G. Mason, Duke Math. J. 86 (1997)] 305-321] for the category of weak modules. Let $V$ be a vertex operator algebra, $g$ an automorphism of order $p$. Let $W$ be an irreducible weak $V$--module such that $W,W\circ g,\dots,W\circ g^{p-1}$ are inequivalent irreducible modules. We prove that $W$ is an irreducible weak $V^{\left\langle g\right\rangle }$-module. This result can be applied on irreducible modules of certain Lie algebra $\mathfrak L$ such that $W,W\circ g,\dots,W\circ g^{p-1}$ …
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…
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.
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.
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.
Starting with a vertex algebra $V$, a finite group $G$ of automorphisms of $V$, and a suitable collection of twisted $V$--modules, we construct (twisted) $D$--modules on the stack of pointed $G$--covers, introduced by Jarvis, Kaufmann, and Kimura. The fibers of these sheaves are spaces of orbifold conformal blocks defined in joint work with Edward Frenkel. The key ingredient is a $G$--equivariant version of the Virasoro uniformization theorem.
We calculate the twisted Hochschild and cyclic homology (in the sense of Kustermans, Murphy and Tuset) of all Podles quantum spheres relative to arbitary automorphisms. Our calculations are based on a free resolution due to Masuda, Nakagami and Watanabe. The dimension drop in Hochschild homology can be overcome by twisting by automorphisms induced from the canonical modular automorphism associated to the Haar state on quantum SU(2). We specialize our results to the standard quantum sphere, and identify the class in twisted cyclic cohomology of the 2-cocycle discovered by Schmuedgen and Wagner …
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.
Let V be a simple vertex operator algebra and G a finite automorphism group of V such that V^G is regular. It is proved that every irreducible V^G-module occurs in an irreducible g-twisted V-module for some g in G. Moreover, the quantum dimensions of each irreducible V^G-module are determined and a global dimension formula for V in terms of twisted modules is obtained.
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, …
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.
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.
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.
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.
The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of rational orbifold conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise numbers of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlations functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, und…
We present a systematic study of the orbifolds of the rank $n$ symplectic fermion algebra $\mathcal{A}(n)$, which has full automorphism group $Sp(2n)$. First, we show that $\mathcal{A}(n)^{Sp(2n)}$ and $\mathcal{A}(n)^{GL(n)}$ are $\mathcal{W}$-algebras of type $\mathcal{W}(2,4,\dots, 2n)$ and $\mathcal{W}(2,3,\dots, 2n+1)$, respectively. Using these results, we find minimal strong finite generating sets for $\mathcal{A}(mn)^{Sp(2n)}$ and $\mathcal{A}(mn)^{GL(n)}$ for all $m,n\geq 1$. We compute the characters of the irreducible representations of $\mathcal{A}(mn)^{Sp(2n)\times SO(m)}$ and $\m…
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.
