A spanning set for VOA modules

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.

On the structure of framed vertex operator algebras and their pointwise frame stabilizers

In this paper, we study the structure of a general framed vertex operator algebra. We show that the structure codes (C,D) of a framed VOA V satisfy certain duality conditions. As a consequence, we prove that every framed VOA is a simple current extension of the associated binary code VOA V_C. This result would give a prospect on the classification of framed vertex operator algebras. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in this pointwise stabilizer, which are of order 1, 2 or 4. The 4A-twisted sector and the 4A-twisted orbifold t…

Orbifold Zhu theory associated to intertwining operators

We study intertwining operators among the twisted modules for rational VOAs and show the modular invariance property of the space spanned by the trace functions concerning to some representations of rational orbifold VOAs. Our result generalizes the results obtained by Dong-Li-Mason and Miyamoto, which are basically extensions of Zhu's celebrated works.

泛型 Virasoro VOA 与量子群的模范畴

Module Categories of the Generic Virasoro VOA and Quantum Groups

本文通过直接比较张量范畴结构，证明了泛型 Virasoro VOA 的模范畴与量子群 $\mathcal{U}_q(\mathfrak{sl}_2)$ 的有限维 I 型模范畴之间的 ribbon 等价，并进一步研究了 $C_1$-cofinite 模范畴的相应等价关系。

3-State Potts model and automorphism of vertex operator algebra of order 3

We define an automorphism of VOA of order 3 by using a sub VOA isomorphic to a direct sum of 3-state Potts models $L(\ff,0)$ and an its module $L(\ff,3)$. This automorphism is a 3A element of the monster simple group if $V$ is the moonshine VOA $V^{\natural}$.

Representations of the orbifold VOAS $L_{\hat{\frak{sl}_2}}(k,0)^K$ and the commutant VOAS $C_{{L_{\hat{\mathfrak{so}_m}}(1,0)}^{\otimes 3}}({L_{\hat{\mathfrak{so}_m}}(3,0)})$

For the Klein group $K$, $k\in\mathbb{Z}_{\geqslant 1}$ and $m\in\mathbb{Z}_{\geqslant 4}$, we study the representations of the orbifold vertex operator algebra $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and the commutant vertex operator algebra of $L_{\hat{\mathfrak{so}_m}}(3,0)$ in $L_{\hat{\mathfrak{so}_m}}(1,0)^{\otimes 3}$ which can be realized as the orbifold vertex operator subalgebra $L_{\hat{\mathfrak{sl}_2}}(2m,0)^{K}$ or its extension. All the irreducible modules for $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and $C_{{L_{\hat{\mathfrak{so}_m}}(1,0)}^{\otimes 3}}({L_{\hat{\mathfrak{so}_m}}(3,0)}…

Modular invariance of vertex operator algebras satisfying C_2-cofiniteness

We show that C_2-cofiniteness is enough to prove a modular invariance property of vertex operator algebras without assuming the semisimplicity of Zhu algebra. For example, if a VOA V=\oplus_{m=0}^{\infty}V_m is C_2-cofinite, then the space spanned by generalized characters of V-modules is invariant under the action of SL_2(\Z). In this case, the central charge and conformal weights are all rational numbers. Namely, a VOA satisfying C_2-cofiniteness is a rational conformal field theory in a sense. We also show that C_2-cofiniteness is equivalent to the condition that every weak module is an \N-…

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…

C-Graded Vertex Algebras and Conformal Flow

We consider C-graded vertex algebras, which are vertex algebras V with a C-grading such that V is an admissible V-module generated by 'lowest weight vectors'. We show that such vertex algebras have a 'good' representation theory in the sense that there is a Zhu algebra A(V) and a bijection between simple admissible V-modules and simple A(V)-modules. We also consider pseudo vertex operator algebras, which are C-graded vertex algebras with a conformal vector such that the homogeneous subspaces of V are generalized eigenspaces for L(0); essentially, these are VOAs that lack any semisimplicity or …

带点顶点算子代数的轨形 I

Orbifolds of Pointed Vertex Operator Algebras I

本文系统研究带点顶点算子代数的轨形与张量范畴论的相互作用，证明了 Dijkgraaf-Witten 猜想，实现了任意幂零群作为格 VOA 自同构，推广了 Tambara-Yamagami 范畴，并构造了广泛的辫化交叉融合范畴和模张量范畴。

The Triplet Vertex Operator Algebra W(p) and the Restricted Quantum Group at Root of Unity

We prove the abelian category of the modules over triplet VOA W(p) is category equivalent to the abelian category of the modules over quantum algebra of type sl_2 at root of unity.

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…

酉顶点算子代数模的融合积的几何正性

Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules

本文通过几何刻画融合积上 Hermite 型的正性，证明不可约可酉化融合积必正定，并将其用于循环轨形与 simple current 扩张的完全酉性。

A modular invariance on the theta functions defined on vertex operator algebras

We introduce theta-functions of VOA-modules and show that the space spanned by them has a modular invariance property.

Zhu's algebra, the C_2 algebra, and twisted modules

In his landmark paper, Zhu associated two associative algebras to a vertex operator algebra: what are now called Zhu's algebra and the C_2-algebra. The former has a nice interpretation in terms of the representation theory of the VOA, while the latter only serves as a finiteness condition. In this paper we undertake first steps to unravel the interpretation of the C_2 vector space. In particular we suggest that it seems and controls the twisted representations of the VOA.

编织张量范畴的3d镜像对称性

3d mirror symmetry of braided tensor categories

本文利用边界VOA及其非半单模范畴，研究阿贝尔3d N=4理论A/B扭转中的线算子，并证明镜像理论间的编织张量等价。

顶点算子代数及其模的亏格 $g$ Zhu 递归

Genus $g$ Zhu Recursion for Vertex Operator Algebras and Their Modules

本文在 Schottky 单值化下建立 VOA 及其模的亏格 $g$ Zhu 递归，用于关联函数、Ward 恒等式、Heisenberg VOA 结构及有理格广义 VOA 配分函数。

3d Mirror Symmetry and the $βγ$ VOA

We study the simplest example of mirror symmetry for 3d $\mathcal N=4$ SUSY gauge theories: the A-twist of a free hypermultiplet and the B-twist of SQED. We particularly focus on the category of line operators in each theory. Using the work of Costello-Gaiotto, we define these categories as appropriate categories of modules for the boundary vertex operator algebras present in each theory. For the A-twist of a free hyper, this will be a certain category of modules for the $βγ$ VOA, properly containing the category previously studied by Allen-Wood. Applying the work of Creutzig-Kanade-McRae and …

单位ary顶点算子代数的扩展比较与共形网络

Comparison of Extensions of Unitary Vertex Operator Algebras and Conformal Nets

本文比较了单位ary顶点算子代数的扩展，证明了其共形网络的自然同构及强可积性。

顶点算子代数的有限诱导函子

Finite induction functor for vertex operator algebras

本文为顶点算子代数的嵌入引入有限诱导函子，证明其满足 Frobenius 互反律等性质，并明确计算了标准子代数和抛物型子代数上不可约模的有限诱导。

First cohomologies of affine, Virasoro and lattice vertex operator algebras

In this paper we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated to a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; (iii) the lattice VOA associated to a positive definite even lattice. We prove that in all these cases, the first cohomology $H^1(V, W)$ are given by the zero-mode derivations when $W$ is any $V$-module with an $\N$-grading (not necessarily by the operator $L(0)$). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. For…

自对偶顶点算子超代数与宝贝怪兽 (Self-dual Vertex Operator Super Algebras and the Baby Monster)

Selbstduale Vertexoperatorsuperalgebren und das Babymonster (Self-dual Vertex Operator Super Algebras and the Baby Monster)

本文研究自对偶顶点算子超代数，构造中心荷23.5的较短月光模，宝贝怪兽群作用其上，并引入极值SVOAs概念，给出存在性结果。

Lifts of automorphisms of vertex operator algebras in simple current extensions

In this article, we study isomorphisms between simple current extensions of a simple VOA. For example, we classify the isomorphism classes of simple current extensions of some VOAs. Moreover, we consider the same simple current extension and describe the normalizer of the abelian automorphism group associated with this extension. In particular, we regard the moonshine module as simple current extensions of five subVOAs V_L^+ for 2-elementary totally even lattices L, and describe corresponding five normalizers of elementary abelian 2-group in the automorphism group of the moonshine module in te…

Classification of some vertex operator algebras of rank 3

We discuss the classification of strongly regular vertex operator algebras (VOAs) with exactly three simple modules whose character vector satisfies a monic modular linear differential equation with irreducible monodromy. Our Main Theorem provides a classification of all such VOAs in the form of one infinite family of affine VOAs, one individual affine algebra and two Virasoro algebras, together with a family of eleven exceptional character vectors and associated data that we call the $U$-series. We prove that there are at least $15$ VOAs in the $U$-series occurring as commutants in a Schellek…

Flatness of Tensor Products and Semi-Rigidity for $C_2$-cofinite Vertex Operator Algebras II

Let V be a simple C_2-cofinite VOA of CFT-type and we assume that there is a simple module U such that \Hom_V(U\boxtimes V',V)\not=0 where V' is a restricted dual of V. As the author has shown, an S-transformation S(Ψ_V) of a trace function Ψ_V on V corresponding \begin{pmatrix}0&-1\cr 1&0\end{pmatrix} may contain pseudo-trace functions. Our assumption in this paper is that no pseudo-trace functions appear in S(Ψ_V). Under these assumptions, we prove that every V-module W is semirigid and Ψ_W appears in S(Ψ_V). As a corollary of our main theorem, a fixed point subVOA of a rational VOA of CFT-t…

扭Zhu代数与双模的Noetherian性

Noetherianity of twisted Zhu algebra and bimodules

本文证明了一大类自然顶点算子代数及其模的Zhu代数、双模及其$g$-扭类似具有Noetherian性，为应用非交换代数几何方法研究其表示理论与融合规则提供了可能。

谱流、扭曲模与拟光滑顶点代数的 MLDE

Spectral flow, twisted modules and MLDE of quasi-lisse vertex algebras

本文计算了容许级下 $\mathbb{Z}_2$-扭曲模的融合规则，为拟光滑顶点代数扭曲模的正规化特征标推导了一系列 MLDE，并基于谱流自同构获得了非容许级模的特征标。

顶点算子代数的经典Yang-Baxter方程及其算子形式

Classical Yang-Baxter equation for vertex operator algebras and its operator forms

本文为顶点算子代数引入经典Yang-Baxter方程的类比VOYBE，并通过相对Rota-Baxter算子给出其算子形式刻画，从而推广并恢复李代数中的经典对应关系。

$K$-theory associated to vertex operator algebras

We introduce two $K$-theories, one for vector bundles whose fibers are modules of vertex operator algebras, another for vector bundles whose fibers are modules of associative algebras. We verify the cohomological properties of these $K$-theories, and construct a natural homomorphism from the VOA K-theory to the associative algebra K-theory.

On Z_2-twisted representation of vertex operator superalgebras and the Ising model SVOA

We investigate a general theory of the Z_2-twisted representations of vertex operator superalgebras. Certain one-to-one correspondence theorems are established. We also give an explicit realization of the Ising model SVOA and its Z_2-twisted modules. As an application, we obtain the Gerald Hoehn's Babymonster SVOA VB and its Z_2-twisted module VB_{tw} from the moonshine VOA V^\nat by cutting off the Ising models. It is also shown in this paper that Aut (VB) is finite.

顶点算子代数与拓扑扭曲的陈-西蒙斯-物质理论

Vertex Operator Algebras and Topologically Twisted Chern-Simons-Matter Theories

本文研究拓扑扭曲的陈-西蒙斯-物质理论，提出用边界顶点算子代数描述三维线算子范畴，并将拓扑A-扭曲推广到更广泛的非N=4理论，给出容许单态VOA的具体例子。

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 …

Ordered spanning sets for quasimodules for Mobius vertex algebras

Quasimodules for vertex algebras are generalizations of modules for vertex algebras. These new objects arise from a generalization of locality for fields. Quasimodules tie together module theory and twisted module theory, and both twisted and untwisted modules feature Poincare-Birkhoff-Witt-like spanning sets. This paper generalizes these spanning set results to quasimodules for certain Mobius vertex algebras. In particular this paper presents two spanning sets, one featuring a difference-zero ordering restriction on modes and another featuring a difference-one condition.

Relaxed and logarithmic modules of $\widehat{\mathfrak{sl}_3}$

In [8], the affine vertex algebra $L_k(\mathfrak{sl}_2)$ is realized as a subalgebra of the vertex algebra $Vir_c \otimes Π(0)$, where $Vir_c$ is a simple Virasoro vertex algebra and $Π(0)$ is a half-lattice vertex algebra. Moreover, all $L_k(\mathfrak{sl}_2)$--modules (including, modules in the category $KL_k$, relaxed highest weight modules and logarithmic modules) are realized as $Vir_c \otimes Π(0)$--modules. A natural question is the generalization of this construction in higher rank. In the current paper, we study the case $\mathfrak{g}= \mathfrak{sl}_3$ and present realization of the VO…

Tensor product of Vertex operator algebras

Let $V_1 \otimes V_2$ be a tensor product of VOAs. Using Zhu theory we discuss the theory of representations of V (associative algebra, modules and fusion rules). We prove that this theory is more or less the same as representation theory of tensor product of the associative algebras.

与洛伦兹格点和Narain CFT相关的非手征顶点算子代数

Non-Chiral Vertex Operator Algebra Associated To Lorentzian Lattices And Narain CFTs

该研究定义非手征顶点算子代数，构造偶整洛伦兹格点对应的非手征VOA及其不可约模，得到相关模不变非手征CFT的模空间，证明Narain CFT是其例子并给出手征代数与配分函数分解。

通过逆量子哈密顿约化研究 Nappi-Witten 顶点算子代数

Nappi-Witten vertex operator algebra via inverse Quantum Hamiltonian Reduction

本文利用逆量子哈密顿约化将 Nappi-Witten VOA 嵌入 Heisenberg-Virasoro VOA 与类格点顶点代数的张量积，实现全部松弛最高权模并构造一族对数模。

$C_2$-共有限顶点算子代数I的解析共形块：传播与对偶融合积

Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras I: Propagation and Dual Fusion Products

本文是系列论文的第一篇，针对不一定有理的$C_2$-共有限顶点算子代数，通过推广共形块的传播，证明了对偶融合积的存在性。

Vertex operator algebra arising from the minimal series M(3,p) and monomial basis

We study a vertex operator algebra (VOA) V related to the M(3,p) Virasoro minimal series. This VOA reduces in the simplest case p=4 to the level two integrable vacuum module of $\hat{sl}_2$. On V there is an action of a commutative current a(z), which is an analog of the current e(z) of $\hat{sl}_2$. Our main concern is the subspace W generated by this action from the highest weight vector of V. Using the Fourier components of a(z), we present a monomial basis of W and a semi-infinite monomial basis of V. We also give a Gordon type formula for their characters.

Galilean $W_3$ algebra

Galilean $W_3$ vertex operator algebra $\mathcal GW_3(c_L,c_M)$ is constructed as a universal enveloping vertex algebra of certain non-linear Lie conformal algebra. It is proved that this algebra is simple by using determinant formula of the vacuum module. Reducibility criterion for Verma modules is given, and the existence of subsingular vectors demonstrated. Free field realisation of $\mathcal GW_3(c_L,c_M)$ and its highest weight modules is obtained within a rank 4 lattice VOA.

Flatness of Tensor Products and Semi-Rigidity for C_2-cofinite Vertex Operator Algebras I

We study properties of a C_2-cofinite vertex operator algebra of CFT type. If it is also rational and V'\cong V, then the rigidity of the tensor category of modules has been proved by Huang. When we treat an irrational C_2-cofinite VOA, the rigidity is too strong, because it is almost equivalent to be rational as we see. We introduce a natural weaker condition "semi-rigidity". Under this condition, we prove the following results. For a projective cover P of a V-module V and a finitely generated V-module M, the projective cover of M is a direct summand of the tensor product P\boxtimes M defined…

Vertex operator algebras associated to modular invariant representations for $A_1 ^{(1)}$

We investigate vertex operator algebras $L(k,0)$ associated with modular-invariant representations for an affine Lie algebra $A_1 ^{(1)}$ , where k is 'admissible' rational number. We show that VOA $L(k,0)$ is rational in the category $\cal O$ and find all irreducible representations in the category of weight modules.

Grothendieck-Verdier范畴中的交换代数、刚性与顶点算子代数

Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras

研究Grothendieck-Verdier范畴中交换代数的模范畴及其局部子范畴的刚性继承问题，给出刚性判定条件，应用于顶点算子代数扩张的强有理性证明，降低相关证明难度。

On global generation of vector bundles on the moduli space of curves from representations of vertex operator algebras

We consider global generation of sheaves of coinvariants on the moduli space of curves given by simple modules over certain vertex operator algebras, extending results for affine VOAs at integrable levels on stable pointed rational curves. Examples where global generation fails, and further evidence of positivity are given.

顶点算子超代数的Kazhdan-Lusztig对应关系：从阿贝尔规范理论的拓扑B扭中获得

Kazhdan-Lusztig Correspondence for Vertex Operator Superalgebras from Abelian Gauge Theories

本文通过拓扑B扭的阿贝尔规范理论边界VOA，证明了顶点算子超代数的Kazhdan-Lusztig对应关系，并将线缺陷视为Hopf代数模块，同时验证了Semikhatov-Tipunin猜想的一个案例。

On Infinite Order Simple Current Extensions of Vertex Operator Algebras

We construct a direct sum completion $\mathcal{C}_{\oplus}$ of a given braided monoidal category $\mathcal{C}$ which allows for the rigorous treatment of infinite order simple current extensions of vertex operator algebras as seen in \cite{CKL}. As an example, we construct the vertex operator algebra $V_L$ associated to an even lattice $L$ as an infinite order simple current extension of the Heisenberg VOA and recover the structure of its module category through categorical considerations.

Virasoro 框架及其对 E_8 格型顶点算子代数的稳定化子

Virasoro frames and their Stabilizers for the E_8 lattice type Vertex Operator Algebra

本文研究 E_8 格型顶点算子代数的 Virasoro 框架稳定化子，证明其自同构群作用恰有五个轨道，并给出稳定化子群结构的范畴化描述。

Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 2 standard modules for $D_4^{(1)}$

Let $\gtl$ be an affine Lie algebra of type $D_{\ell}^{(1)}$ and $L(Λ)$ its standard module with a highest weight vector $v_Λ$. For a given $\Z$-gradation $\gtl = \gtl_{-1} + \gtl_0 + \gtl_1$, we define Feigin-Stoyanovsky's type subspace as $$W(Λ) = U(\gtl_1) \cdot v_Λ.$$ By using vertex operator relations for standard modules we reduce the Ponicaré-Brikhoff-Witt spanning set of $W(Λ)$ to a basis and prove its linear independence by using Dong-Lepowsky intertwining operators.

Generators, spanning sets and existence of twisted modules for a grading-restricted vertex (super)algebra

For a grading-restricted vertex superalgebra $V$ and an automorphism $g$ of $V$, we give a linearly independent set of generators of the universal lower-bounded generalized $g$-twisted $V$-module $\widehat{M}^{[g]}_{B}$ constructed by the author in \cite{H-const-twisted-mod}. We prove that there exist irreducible lower-bounded generalized $g$-twisted $V$-modules by showing that there exists a maximal proper submodule of $\widehat{M}^{[g]}_{B}$ for a one-dimensional space $M$. We then give several spanning sets of $\widehat{M}^{[g]}_{B}$ and discuss the relations among elements of the spanning …

关于亚纯开弦顶点代数左模的扩张，I

On the extensions of the left modules for a meromorphic open-string vertex algebra, I

本文研究亚纯开弦顶点代数左模的扩张，证明其与第一上同调类一一对应，并给出Ext^1维数的上界及一个Virasoro VOA的例子。

有理c=1顶点算子代数与顶点算子超代数的分类

Classification of Rational $c=1$ Vertex Operator Algebras and Vertex Operator Superalgebras

该文严格分类了有理c=1顶点算子代数与顶点算子超代数，证实其为秩1整格格VO(S)A或其有限群自同构的orbifold，为后续完整CFT分类奠定基础。

为 $D_4^{(1)}$ 构造标准模的生成集

Generating sets of standard modules for $D_4^{(1)}$

本文提出一种方法，通过顶点算子关系构造 $D_4^{(1)}$ 标准模的生成集，并将其与极限过程联系起来。

三元组顶点算子代数的Yoneda代数

Yoneda algebras of the triplet vertex operator algebra

本文以三元组顶点算子代数W(p)为例，通过确定Ext箭图、Morita等价基本代数和Yoneda代数，比较其可容许模范畴、Zhu代数模范畴和C2-代数模范畴，并揭示这些范畴的无限全局维数及Koszul性质。

利用 $C_{2\ell}^{(1)}$ 证明 $C_\nell^{(1)}$ 的线性无关性

Linear independence for $C_\ell^{(1)}$ by using $C_{2\ell}^{(1)}$

本研究通过建立标准$C_\ell^{(1)}$-模$L(k\Lambda_0)$的组合生成集与$C_{2\ell}^{(1)}$-模的Feigin-Stoyanovsky型子空间组合基的联系，证明该生成集的线性无关性，子空间基的无关性借助顶点算子代数的简单流与缠结算子证得。