C_2 cofiniteness of vertex operator algebra V_L^+ when L is a rank one lattice

Rationality, Regularity, and C_2-cofiniteness

We demonstrate that, for CFT vertex operator algebras, C_2-cofiniteness and rationality is equivalent to regularity. In addition, we show that, for C_2-cofinite vertex operators algebras, irreducible weak modules are ordinary modules and C_2-cofinite, and V_L^+ are C_2-cofinite.

C_1-cofiniteness and Fusion Products of Vertex Operator Algebras

Let V be a vertex operator algebra. We prove that if U and W are C_1-cofinite {\mathbb N}-gradable V-modules, then a fusion product U\boxtimes W is well-defined and also a C_1-cofinite {\mathbb N}-gradable V-module, where the fusion product is defined by (logarithmic) intertwining operators. This is also true for C_2-cofinite {\mathbb N}-gradable modules.

A theory of tensor products for vertex operator algebra satsifying C_2-cofiniteness

We reformed the tensor product theory of vertex operator algebras developed by Huang and Lepowsky so that we could apply it to all vertex operator algebras satisfying C_2-cofiniteness. We also showed that the tensor product theory develops naturally if we include not only ordinary modules, but also weak modules with a composition series of finite length (we call it an Artin module). In particular, we don't assume the semisimplicity of the weight operator L(0). Actually, without the assumption of rationality, a C_2-cofiniteness on V is enough to obtain the existence of a tensor product of two A…

Zhu's algebra, C_2-algebra and C_2-cofiniteness of parafermion vertex operator algebras

We determine Zhu's algebra and C_2-algebra of parafermion vertex operator algebras for sl_2. Moreover, we prove the C_2-cofiniteness of parafermion vertex operator algebras for any finite dimensional simple Lie algebras.

Borcherds 李代数与 moonshine 型顶点算子代数的 $C_2$-余有限性

Borcherds's Lie algebra and $C_2$-cofiniteness of vertex operator algebras of moonshine type

本文研究 moonshine 型顶点算子代数的 $C_2$-余有限性，精确刻画中心荷为 24 时 Zhu Poisson 代数的分次结构及 $C_2(V)$ 的显式形式。

在顶点算子代数理论中扭曲融合积的有限性

Cofiniteness for Twisted Fusion Products in Vertex Operator Algebra Theory

本文研究了在顶点算子代数中两个交换有限阶自同构下的扭曲模的有限性问题，证明了在特定条件下扭曲融合积也是C1-有限的，并应用此结果建立了有限融合规则。

$C_{1}$-余有限性与顶点张量范畴

$C_{1}$-cofiniteness and vertex tensor categories

本文推广对数张量范畴理论至模范畴不封闭情形，证明$C_{1}$-余有限广义模的交织算子满足结合性，并赋予该模范畴自然的顶点张量范畴结构。

Module category and $C_2$-cofiniteness of affine vertex operator superalgebras

In this paper, we investigate the Lie algebra structures of weight one subspaces of $C_2$-cofinite vertex operator superalgebras. We also show that for any positive integer $k$, vertex operator superalgebras $L_{sl(1|n+1)}(k,0)$ and $L_{osp(2|2n)}(k,0)$ have infinitely many irreducible admissible modules. As a consequence, we give a proof of the fact that $L_{\mathfrak g}(k,0)$ is $C_2$-cofinite if and only if $\mathfrak g$ is either a simple Lie algebra, or $\mathfrak g=osp(1|2n)$, and $k$ is a nonnegative integer. As an application, we show that $L_{G(3)}(1,0)$ is a vertex operator superalge…

Rationality and $C_2$-cofiniteness of certain diagonal coset vertex operator algebras

In this paper, it is shown that the diagonal coset vertex operator algebra $C(L_{\mathfrak{g}}(k+2,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(2,0))$ is rational and $C_2$-cofinite in case $\mathfrak{g}=so(2n), n\geq 3$ and $k$ is an admissible number for $\hat{\mathfrak{g}}$. It is also shown that the diagonal coset vertex operator algebra $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ is rational and $C_2$-cofinite in case $k$ is an admissible number for $\hat{sl_2}$. Furthermore, irreducible modules of $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ are classified in c…

$C_2$-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.

Abelianizing vertex algebras

To every vertex algebra $V$ we associate a canonical decreasing sequence of subspaces and prove that the associated graded vector space $gr(V)$ is naturally a vertex Poisson algebra, in particular a commutative vertex algebra. We establish a relation between this sequence and the sequence $C_{n}$ introduced by Zhu. By using the (classical) algebra $gr(V)$, we prove that for any vertex algebra $V$, $C_{2}$-cofiniteness implies $C_{n}$-cofiniteness for all $n\ge 2$. We further use $gr(V)$ to study generating subspaces of certain types for lower truncated $Z$-graded vertex algebras.

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.

$C_2$-余有限性与二十面体轨道簇 $V_{L_2}^{A_5}$ 的有理性

$C_2$-Cofiniteness and Rationality of the Icosahedral Orbifold $V_{L_2}^{A_5}$

本文证明二十面体群 $A_5$ 作用于秩一根格顶点算子代数 $V_{L_2}$ 的不动点子代数 $V_{L_2}^{A_5}$ 是强有理的，即满足 $C_2$-余有限性且有理。

Nonmeromorphic operator product expansion and C_2-cofiniteness for a family of W-algebras

We prove the existence and associativity of the nonmeromorphic operator product expansion for an infinite family of vertex operator algebras, the triplet W-algebras, using results from P(z)-tensor product theory. While doing this, we also show that all these vertex operator algebras are C_2-cofinite.

Representations of Vertex Operator Algebras $V_{L_2}^{S_4}$ and $V_{L_2}^{A_5}$

$C_2$ cofiniteness and rationality of $V_{L_2}^{S_4}$ are obtained, and irreducible $V_{L_2}^{S_4}$-modules are classified. With the assumption of rationality and $C_2$ cofiniteness, irreducible $V_{L_2}^{A_5}$-modules are determined. Also, quantum dimensions of these irreducible modules are calculated.

The fixed point subalgebra of a lattice vertex operator algebra by an automorphism of order three

We study the subalgebra of the lattice vertex operator algebra $V_{\sqrt{2}A_2}$ consisting of the fixed points of an automorphism which is induced from an order 3 isometry of the root lattice $A_2$. We classify the simple modules for the subalgebra. The rationality and the $C_2$-cofiniteness are also established.

Fixed point subalgebras of lattice vertex operator algebras by an automorphism of order three

We study the fixed point subalgebra of a certain class of lattice vertex operator algebras by an automorphism of order 3, which is a lift of a fixed-point-free isometry of the underlying lattice. We classify the irreducible modules for the subalgebra. Moreover, the rationality and the $C_2$-cofiniteness of the subalgebra are established. Our result contains the case of the vertex operator algebra associated with the Leech lattice.

Ordered spanning sets for vertex operator algebras and their modules

Moonshine relates three fundamental mathematical objects: the Monster sporadic simple group, the modular function j, and the moonshine module vertex operator algebra. Examining the relationship between modular functions and the representation theory of vertex operator algebras reveals rich structure. In particular, C2-cofiniteness (also called Zhu's finiteness condition) implies the existence of finite generating sets and Poincare-Birkhoff-Witt-like spanning sets for vertex operator algebras and their modules. These spanning sets feature desirable ordering restrictions, e.g., a difference-one …

Representations of the vertex operator algebra V_{L_{2}}^{A_{4}}

The rationality and C_2-cofiniteness of the orbifold vertex operator algebra V_{L_{2}}^{A_{4}} are established and all the irreducible modules are constructed and classified. This is part of classification of rational vertex operator algebras with c=1.

Remarks on Mathieu-Zhao Subspaces of Commutative Associative Algebras and Vertex Algebras

We introduce a notion of Mathieu-Zhao subspaces of vertex algebras. Among other things, we show that for a vertex algebra $V$ and its subspace $M$ that contains $C_2(V)$, $M$ is a Mathieu-Zhao subspace of $V$ if and only if the quotient space $M/C_2(V)$ is a Mathieu-Zhao subspace of a commutative associative algebra $V/C_2(V)$. As a result, one can study the famous Jacobian conjecture in terms of Mathieu-Zhao subspaces of vertex algebras. In addition, for a $CFT$-type vertex operator algebra $V$ that satisfies the $C_2$-cofiniteness condition, we classify all Mathieu-Zhao subspaces $M$ that co…

Flatness and Semi-Rigidity of Vertex Operator Algebras

In vertex operator algebra theories, most of the general theorems are proved under the assumptions of rationality and C_2-cofiniteness. In this paper, we obtain several general theorems without the assumption of rationality so that we can use them for proving rationality of given C_2-cofinite vertex operator algebras. For example, we apply them to orbifold models and show that if g is a finite automorphism of a rational C_2-cofinite vertex operator algebra T of CFT-type with T'\cong T and a fixed point subVOA T^g is C_2-cofinite, then T^g is also rational.

Cofiniteness conditions, projective covers and the logarithmic tensor product theory

We construct projective covers of irreducible V-modules in the category of grading-restricted generalized V-modules when V is a vertex operator algebra satisfying the following conditions: 1. V is C_{1}-cofinite in the sense of Li. 2. There exists a positive integer N such that the differences between the real parts of the lowest conformal weights of irreducible V-modules are bounded by N and such that the associative algebra A_{N}(V) is finite dimensional. This result shows that the category of grading-restricted generalized V-modules is a finite abelian category over C. Using the existence o…

递减滤过、$C_{2}$-代数与扭曲模

Decreasing filtrations, $C_{2}$-algebra and twisted modules

本文研究顶点代数$C_{2}$-代数与扭曲模的关系，通过构造递减滤过并证明相关分次空间为顶点Poisson代数的扭曲模，证明了$C_{2}$-余有限性蕴含$C_{n}$-余有限性，并应用于研究生成子空间。

Modularity of Relatively Rational Vertex Algebras and Fusion Rules of Principal Affine W-Algebras

We study modularity of the characters of a vertex (super)algebra equipped with a family of conformal structures. Along the way we introduce the notions of rationality and cofiniteness relative to such a family. We apply the results to determine modular transformations of trace functions on admissible modules over affine Kac-Moody algebras and, via BRST reduction, trace functions on minimal series representations of principal affine W-algebras.

$C_2$-余有限顶点算子代数的$C_n$-余有限扭曲模

$C_n$-cofinite twisted modules for $C_2$-cofinite vertex operator algebras

该研究针对带一般自同构的顶点算子代数，引入了弱扭曲模的$C_n$-余有限性概念，并证明$C_2$-余有限的CFT型顶点算子代数的所有有限生成弱扭曲模均满足$C_n$-余有限性。

C_2-cofinite W-algebras and their logarithmic representations

We discuss our recent results on the representation theory of $\mathcal{W}$--algebras relevant to Logarithmic Conformal Field Theory. First we explain some general constructions of $\mathcal{W}$-algebras coming from screening operators. Then we review the results on $C_2$--cofiniteness, the structure of Zhu's algebras, and the existence of logarithmic modules for triplet vertex algebras. We propose some conjectures and open problems which put the theory of triplet vertex algebras into a broader context. New realizations of logarithmic modules for $\mathcal{W}$-algebras defined via screenings a…

ADE subalgebras of the triplet vertex algebra W(p): A-series

Motivated by \cite{am1}, for every finite subgroup $Γ\subset PSL(2,\mathbb{C})$ we investigate the fixed point subalgebra $\triplet^Γ$ of the triplet vertex $\mathcal {W}(p)$, of central charge $1-\frac{6(p-1)^{2}}{p}$, $p\geq2$. This part deals with the $A$-series in the ADE classification of finite subgroups of $PSL(2,\mathbb{C})$. First, we prove the $C_2$-cofiniteness of the $A_m$-fixed subalgebra $\triplet^{A_m}$. Then we construct a family of $\am$-modules, which are expected to form a complete set of irreps. As a strong support to our conjecture, we prove modular invariance of (generali…

酉全顶点算子代数的Osterwalder-Schrader公理

Osterwalder-Schrader axioms for unitary full vertex operator algebras

本文为酉全顶点算子代数建立多项式能量界与谱密度条件，并证明其关联函数满足共形版Osterwalder-Schrader公理，同时验证Heisenberg VOA的一族全扩张满足这些假设。

一些简单仿射VOA的变形刚性

Deformation rigidity of some simple affine VOAs

本文证明了某些简单仿射VOA在特定条件下具有变形刚性，并推翻了$C_2$-余有限性和理性作为必要条件的假设。

On the extended W-algebra of type sl_2 at positive rational level

The extended W-algebra of type sl_2 at positive rational level, denoted by M_{p_+,p_-}, is a vertex operator algebra that was originally proposed in [1]. This vertex operator algebra is an extension of the minimal model vertex operator algebra and plays the role of symmetry algebra for certain logarithmic conformal field theories. We give a construction of M_{p_+,p_-} in terms of screening operators and use this construction to prove that M_{p_+,p_-} satisfies Zhu's c_2-cofiniteness condition, calculate the structure of the zero mode algebra (also known as Zhu's algebra) and classify all simpl…

Schur-Weyl Duality for Heisenberg Cosets

Let $V$ be a simple vertex operator algebra containing a rank $n$ Heisenberg vertex algebra $H$ and let $C=\text{Com}\left( {H}, {V}\right)$ be the coset of ${H}$ in ${V}$. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of ${C}$ are found and every simple ${C}$-module is shown to be contained in at least one ${V}$-module. A corollary of this is that if ${V}$ is rational and $C_2$-cofi…

Classification of irreducible modules for the vertex operator algebra M(1)^+, II: higher rank

The vertex operator algebra M(1)^+ is the fixed point set of free bosonic vertex operator algebra M(1) under the -1 automorphism. All irreducible modules for M(1)^+ are classified in this paper for all ranks.

Representations of vertex operator algebra V_L^+ for rank one lattice L

We classify the irreducible modules for the fixed point vertex operator subalgebra V_L^+ of the vertex operator algebra V_L associated to a positive definite even lattice of rank 1 under the automorphism lifted from the -1 isometry of L.

Rationality of vertex operator algebra V_L^+: higher rank

The lattice vertex operator V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from -1 isometry of L. It is established that the fixed point vertex operator algebra V_L^+ is rational.

Rationality of the vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of arbitrary rank

In this paper we prove that the vertex algebra $V_L^+$ is rational if $L$ is a negative definite even lattice of finite rank, or if $L$ is a non-degenerate even lattice of a finite rank that is neither positive definite nor negative definite. In particular, for such even lattices $L$, we show that the Zhu algebras of the vertex algebras $V_L^+$ are semisimple. This extends the result of Abe which establishes the rationality of $V_L^+$ when $L$ is a positive definite even lattice of finite rank.

Classification of irreducible modules of the vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of an arbitrary rank

In this paper, we first classify all irreducible modules of the vertex algebra $V_L^+$ when $L$ is a negative definite even lattice of arbitrary rank. In particular, we show that any irreducible $V_L^+$-module is isomorphic to a submodule of an irreducible twisted $V_L$-module. We then extend this result to a vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of finite rank.

ADE subalgebras of the triplet vertex algebra W(p): D_m-series

We are continuing our study of ADE-orbifold subalgebras of the triplet vertex algebra W(p). This part deals with the dihedral series. First, subject to a certain constant term identity, we classify all irreducible modules for the vertex algebra $\bar{M(1)} ^+$, the $\Z_2$--orbifold of the singlet vertex algebra $\bar{M(1)}$. Then we classify irreducible modules and determine Zhu's and $C_2$--algebra for the vertex algebra $\triplet ^{D_2}$. A general method for construction of twisted $\triplet$--modules is also introduced. We also discuss classification of twisted $\bar{M(1)}$--modules includ…

顶点算子代数的内部Hom的有限性与构造

Finiteness and Construction of Internal Hom for Vertex Operator Algebras

本文构造了受限V-模的广义模H(W^1,W^2)，证明在适当条件下它实现内部Hom对象，并建立与Li模及Du-Huang对偶积的同构，在C1-余有限条件下研究其有限性。

正则 $\mathbb{N}$-分次顶点算子代数中带 Gorenstein $V_0$ 的仿射与格结构

Affine and Lattice Structures in Regular $\mathbb{N}$-Graded Vertex Operator Algebras with Gorenstein $V_0$

本文研究缺少CFT型条件时正则N-分次顶点算子代数中仿射与格结构的保持，证明在若干假设下存在共形嵌入的格顶点算子代数，并指出正则性不足以保证半单性。

Level-Rank Duality for Vertex Operator Algebras of types B and D

For the simple Lie algebra $ \frak{so}_m$, we study the commutant vertex operator algebra of $ L_{\hat{\frak{so}}_{m}}(n,0)$ in the $n$-fold tensor product $ L_{\hat{\frak{so}}_{m}}(1,0)^{\otimes n}$. It turns out that this commutant vertex operator algebra can be realized as a fixed point subalgebra of $L_{\hat{\frak{so}}_{n}}(m,0)$ (or its simple current extension) associated with a certain abelian group. This result may be viewed as a version of level-rank duality.

与秩24奇幺模格相关的多数顶点超代数具有N=4超共形结构

Most vertex superalgebras associated to an odd unimodular lattice of rank 24 have an N=4 superconformal structure

本文证明秩24奇幺模格对应的273个顶点超代数中至少267个含有中心荷6的N=4超共形子代数，方法为研究特定秩6格的嵌入。

Rational Vertex Operator Algebras and the Effective Central Charge

We establish that the Lie algebra of weight one states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank is bounded above by the effective central charge. We show that lattice vertex operator algebras may be characterized by the equalities of the effective central charge, the Lie rank and the central charge, and in particular holomorphic lattice theories may be characterized among all holomorphic vertex operator algebras by the equality of the Lie rank and the central charge.

A generalized Kac-Moody algebra of rank 14

We construct a vertex algebra of central charge 26 from a lattice orbifold vertex operator algebra of central charge 12. The BRST-cohomology group of this vertex algebra is a new generalized Kac-Moody algebra of rank 14. We determine its root space multiplicities and a set of simple roots.

Twisted Heisenberg-Virasoro vertex operator algebra

In this paper, we study a new kind of vertex operator algebra related to the twisted Heisenberg-Virasoro algebra, which we call the twisted Heisenberg-Virasoro vertex operator algebra, and its modules. Specifically, we present some results concerning the relationship between the restricted module categories of twisted Heisenberg-Virasoro algebras of rank one and rank two and several different kinds of module categories of their corresponding vertex algebras. We also study fully the structures of the twisted Heisenberg-Virasoro vertex operator algebra, give a characterization of it as a tensor …

Simple weak modules for the fixed point subalgebra of the Heisenberg vertex operator algebra of rank $1$ by an automorphism of order $2$ and Whittaker vectors

Let $M(1)$ be the vertex operator algebra with the Virasoro element $ω$ associated to the Heisenberg algebra of rank $1$ and let $M(1)^{+}$ be the subalgebra of $M(1)$ consisting of the fixed points of an automorphism of $M(1)$ of order $2$. We classify the simple weak $M(1)^{+}$-modules with a non-zero element $w$ such that for some integer $s\geq 2$, $ω_i w\in{\mathbb C}w$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $ω_{s}w\in{\mathbb C}^{\times}w$, and $ω_i w=0$ for all $i>s$. The result says that any such simple weak $M(1)^{+}$-module is isomorphic to some simple weak $M(1)…

Rank one lattice type vertex operator algebras and their automorphism groups, II: E-series

Let L be the A_1 root lattice and G a finite subgroup of Aut(V_L), where $V_L$ is the associated lattice VOA (in this case, Aut(V) is isomorphic to PSL(2,\Bbb C)). The fixed point subVOA, V^G was studied in q-alg/9710017, which finds a set of generators and determines the automorphism group when G is cyclic (from the "A-series") or dihedral (from the "D-series"). In the present article, we obtain analogous results for the remaining possibilities for G, that it belong to the "E-series": G\cong Alt_4, Alt_5 or Sym_4.

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.

Classification of irreducible modules for the vertex operator algebra M(1)^+

We classify the irreducible modules for the fixed point vertex operator subebra of the rank 1 free bosonic VOA under the -1 automorphism.

Holomorphic Vertex Operator Algebras of Small Central Charges

We provide a rigorous mathematical foundation to the study of strongly rational, holomorphic vertex operator algebras V of central charge c = 8, 16 and 24 initiated by Schellekens. If c = 8 or 16 we show that V is isomorphic to a lattice theory corresponding to a rank c even, self-dual lattice. If c = 24 we prove, among other things, that either V is isomorphic to a lattice theory corresponding to a Niemeier lattice or the Leech lattice, or else the Lie algebra on the weight one subspace V_1 is semisimple (possibly 0) of Lie rank less than 24.

The 3-permutation orbifold of a lattice vertex operator algebra

Irreducible modules of the 3-permutation orbifold of a rank one lattice vertex operator algebra are listed explicitly. Fusion rules are determined by using the quantum dimensions. The $S$-matrix is also given.

Bosonic ghostbusting -- The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion

The rank 1 bosonic ghost vertex algebra, also known as the $βγ$ ghosts, symplectic bosons or Weyl vertex algebra, is a simple example of a conformal field theory which is neither rational, nor $C_2$-cofinite. We identify a module category, denoted category $\mathscr{F}$, which satisfies three necessary conditions coming from conformal field theory considerations: closure under restricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second of these conditions, with the other two already being known. Further, we show that category $\ma…

中心荷为8的Haagerup顶点算子代数

The Haagerup VOA at $c=8$

本文构造了中心荷为8的酉强有理顶点算子代数$U$，其与Haagerup双代数存在ribbon等价，通过多种方法完成论证，权1李代数水平为1和39。

2-permutations of lattice vertex operator algebras: Higher rank

The fusion rules of the 2-permutation orbifold of an arbitrary lattice vertex operator algebra are determined by using the theory of quantum dimension.

A characterization of the rational vertex operator algebra $V_{\Zα}^{+}$}: II

A characterization of vertex operator algebra $V_L^+$ for any rank one positive definite even lattice $L$ is given in terms of dimensions of homogeneous subspaces of small weights. This result reduces the classification of rational vertex operator algebras of central charge 1 to the characterization of three vertex operator algebras in the $E$-series of central charge one.