On VOAs Associated to Jordan Algebras of Type $C$

秩二抛物型顶点算子代数与nil理想中的nilpotency

Rank-two parabolic-type VOAs and nilpotency of nil ideals

本文研究秩二抛物型子顶点算子代数的分类及其不可约模，发现Zhu代数提供了非nilpotent的nil理想例子，并证明其简单商为满足强单位性质的cofinite irrational VOA。

Energy bounds condition for intertwining operators of type $B$, $C$, and $G_2$ unitary affine vertex operator algebras

The energy bounds condition for intertwining operators of unitary rational vertex operator algebras (VOAs) was studied, first by A.Wassermann for type $A$ affine VOAs, and later by T.Loke for $c<1$ Virasoro VOAs, and by V.Toledano-Laredo for type $D$ affine VOAs. In this paper, we extend their results to affine VOAs of type $B$, $C$, and $G_2$. As a consequence, the modular tensor categories of these unitary vertex operator algebras are unitary.

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)}…

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。

VOAs generated by two conformal vectors whose $τ$-involutions generate $S_3$

We determined the inner products of two conformal vectors with central charge 1/2 whose τ-involutions generates S_3 if none of τ-involutions are trivial. We also see that a subVA generated by such conformal vectors is a VOA with central charge 1/2+20/21 or $4/5+6/7 and has a Griess algebra of dimension three or four, respectively.

关于仿射顶点算子代数的强幺元性质

On the Strong Unital Property for the Affine VOAs

研究仿射VOA的强幺元性质，通过为\(L_{\widehat{\mathfrak{sl}_2}}(1,0)\)构造强幺元，证明\(\mathfrak{sl}_2\)通用仿射VOA在非临界水平\(k\neq -2\)时模过渡代数非强幺元。

W algebra, Cosets and VOA for 4d N = 2 SCFT from M5 branes

We identify vertex operator algebras (VOAs) of a class of Argyres-Douglas (AD) matters with two types of non-abelian flavor symmetries. They are the $W$ algebra defined using nilpotent orbit with partition $[q^m,1^s]$. Gauging above AD matters, we can find VOAs for more general $\mathcal{N}=2$ SCFTs engineered from 6d $(2,0)$ theories. For example, the VOA for general $(A_{N-1}, A_{k-1})$ theory is found as the coset of a collection of above $W$ algebras. Various new interesting properties of 2d VOAs such as level-rank duality, conformal embedding, collapsing levels, coset constructions for kn…

由 σ 型 Ising 向量生成的 OZ 型顶点算子代数的分类

The classification of vertex operator algebras of OZ-type generated by Ising vectors of $σ$-type

本文分类由 σ 型 Ising 向量生成的 OZ 型 VOA，并据此证明其单性、有理性、C₂ 余有限性、酉性及紧实形式的存在性。

细化与顶点算子代数相关的扭曲双模

Refining twisted bimodules associated to VOAs

本文证明顶点算子代数扭曲双模定义中的子空间和可简化为单一子空间，简化了Dong-Jiang构造。

Vertex Algebras for S-duality

We define new deformable families of vertex operator algebras $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ VOAs are equipped with two $\mathfrak{g}$ affine vertex subalgebras whose levels are related by the S-duality operation. They compose accordingly under a natural convolution operation an…

Character Vectors of Strongly Regular Vertex Operator Algebras

We summarize interactions between vertex operator algebras and number theory through the lens of Zhu theory. The paper begins by recalling basic facts on vertex operator algebras (VOAs) and modular forms, and then explains Zhu's theorem on characters of VOAs in a slightly new form. We then axiomatize the desirable properties of modular forms that have played a role in Zhu's theorem and related classification results of VOAs. After this we summarize known classification results in rank two, emphasizing the geometric theory of vector-valued modular forms as a means for simplifying the discussion…

单仿射 VOA $L_k(sl_3)$ 和 $W$-代数 $W_k(sl_3,f)$ 的结合簇

Associated varieties of simple affine VOAs $L_k(sl_3)$ and $W$-algebras $W_k(sl_3,f)$

本文证明了 $V^k(sl_n)$ 极大理想的生成条件，并确定了非容许级下 $L_k(sl_3)$ 及对应 $W$-代数 $W_k(sl_3,f)$ 的结合簇。

VOAs labelled by complex reflection groups and 4d SCFTs

We define and study a class of $\mathcal{N}=2$ vertex operator algebras $\mathcal{W}_{\mathcal{\mathsf{G}}}$ labelled by complex reflection groups. They are extensions of the $\mathcal{N}=2$ super Virasoro algebra obtained by introducing additional generators, in correspondence with the invariants of the complex reflection group $\mathcal{\mathsf{G}}$. If $\mathcal{\mathsf{G}}$ is a Coxeter group, the $\mathcal{N}=2$ super Virasoro algebra enhances to the (small) $\mathcal{N}=4$ superconformal algebra. With the exception of $\mathcal{\mathsf{G}} = \mathbb{Z}_2$, which corresponds to just the $…

作为仿射顶点算子代数共形扩张的 W - 代数

W-algebras as conformal extensions of affine VOAs

研究从仿射顶点代数到顶点超代数的顶点算子超代数同态，给出其为共形及满射的准则，应用于 W - 代数和 W - 超代数，得到多种坍缩或扩张的例子，特别是非允许层级单 W - 代数坍缩为允许层级仿射顶点代数的例子。

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…

On Correlation Functions of Vertex Operator Algebras Associated to Jordan Algebras

In this paper we study certain vertex operator algebras associated to Jordan algebras and compute the correlation function of basic fields

顶点代数与交换代数

Vertex Algebras and Commutative Algebras

本文回顾了顶点算子代数（VOA）理论的发展，重点介绍了过去15年基于VOA与交换代数联系的进展，包括利用函子将VOA问题转化为交换代数问题，解决Gaiotto和Rapčák猜想，并概述了拓扑VOA理论及其在BRST上同调和共形弦理论中的应用。

Vertex operator algebras associated to modified regular representations of the Virasoro algebra

We give an abstract construction, based on the Belavin-Polyakov-Zamolodchikov equations, of a family of vertex operator algebras of rank $26$ associated to the modified regular representations of the Virasoro algebra. The vertex operators are obtained from the tensor products of intertwining operators for a pair of Virasoro algebras. We explicitly determine the structure coefficients that yield the axioms of VOAs. In the process of our construction, we obtain new hypergeometric identities.

Simple Conformal Algebras Generated by Jordan Algebras

In this article, we first give a short introduction to conformal algebras. Then we present three families of simple conformal algebras finite growth generated by simple Jordan algebras of types A, B, C.

Deformation of central charges, vertex operator algebras whose Griess algebras are Jordan algebras

If a vertex operator algebra $V=\oplus_{n=0}^{\infty}V_n$ satisfies $\dim V_0=1, V_1=0$, then $V_2$ has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set $Sym_d(\C)$ of symmetric matrices of degree $d$ becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, central charges influence the properties of vertex operator algebras. In this paper, we construct vertex operator algebras with central charge $c$ and its Griess algebra is isom…

仿射李代数 VOA 的奇异量子子群与扩张 —— 第一部分

Exotic quantum subgroups and extensions of affine Lie algebra VOAs -- part I

本文研究仿射李代数 VOA 的扩张问题，证明对于任意单李代数 g 存在界 K(g) 使得当级 k>K(g) 时仅有简单流扩张，并据此完成了 sl(4)、sl(5) 在所有级以及 sl(n) 在级 k≤5 时的扩张分类。

关于 $C_2$-共有限顶点算子代数的一点函数：伪迹与投射模的迹空间

One-point functions for $C_2$-cofinite VOAs: pseudo-traces and trace spaces of projective modules

通过将环面上的一点函数空间与顶点算子代数表示范畴中投射对象的迹对象关联，利用Arike-Nagatomo伪迹证明Gainutdinov-Runkel映射的满射性，并在分离共形权模$\mathbb{Z}$条件下证明单射性。

三维 $\mathcal{N}=4$ 秩零 SCFT 的边界顶点代数

Boundary vertex algebras for 3d $\mathcal{N}=4$ rank-0 SCFTs

研究拓扑扭变的三维N=4秩零SCFT的边界顶点算子代数，提出基于N=2微观描述的构造技术，应用于新发现的T_r族，得到仿射VOA L_r(osp(1|2))，并发现与最小模型M(2,5)的等级-秩对偶。

格顶点算子代数的 Borel 型子代数

Borel-type subalgebras of the lattice vertex operator algebra

该研究提出格 VOA 的圆锥型、Borel 型与抛物型子 VOA，并对最低秩非平凡 Borel 型例子显式给出其 Zhu 代数的生成元—关系表示。

From VOAs to short star products in SCFT

We build a bridge between two algebraic structures in SCFT: a VOA in the Schur sector of 4d $\mathcal{N}=2$ theories and an associative algebra in the Higgs sector of 3d $\mathcal{N}=4$. The natural setting is a 4d $\mathcal{N}=2$ SCFT placed on $S^3\times S^1$: by sending the radius of $S^1$ to zero, we recover the 3d $\mathcal{N}=4$ theory, and the corresponding VOA on the torus degenerates to the associative algebra on the circle. We prove that: 1) the Higgs branch operators remain in the cohomology; 2) all the Schur operators of the non-Higgs type are lifted by line operators wrapped on th…

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。

关于源自秩一4D SCFT的非容许水平上的若干简单轨形仿射VOA

On some simple orbifold affine VOAs at non-admissible level arising from rank one 4D SCFTs

本文研究源自秩一4D SCFT的非容许水平仿射顶点代数，分类$L_{-2}(G_2)$与$L_{-2}(B_3)$的不可约最高权模，并证明前者的伴生簇是$L_{-2}(D_4)$伴生簇在三次对称群下的轨形。

Classification of extremal vertex operator algebras with two simple modules

In recent work, Wang and the third author defined a class of 'extremal' vertex operator algebras (VOAs), consisting of those with at least two simple modules and conformal dimensions as large as possible for the central charge. In this article we show that there are exactly 15 character vectors of extremal VOAs with two simple modules. All but one of the 15 character vectors is realized by a previously known VOA. The last character vector is realized by a new VOA with central charge 33.

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.

仿射 $\mathcal{W}$-代数与来自三维 $\mathcal N=4$ 非阿贝尔箭图规范理论的 Miura 映射

Affine $\mathcal{W}$-algebras and Miura maps from 3d $\mathcal N=4$ non-Abelian quiver gauge theories

本文研究三维 $\mathcal{N}=4$ 线性箭图规范理论 H-扭转得到的顶点算子代数，揭示其与希格斯分支及仿射 $\mathcal{W}$-代数的关系，并提出箭图约化算法以系统构造新自由场实现。

Representation theory in chiral conformal field theory: from fields to observables

This article develops new techniques for understanding the relationship between the three different mathematical formulations of two-dimensional chiral conformal field theory: conformal nets (axiomatizing local observables), vertex operator algebras (axiomatizing fields), and Segal CFTs. It builds upon previous work which introduced a geometric interpolation procedure for constructing conformal nets from VOAs via Segal CFT, simultaneously relating all three frameworks. In this article, we extend this construction to study the relationship between the representation theory of conformal nets and…

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…

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。

Intertwining operators and vector-valued modular forms for minimal models

Using the language of vertex operator algebras (VOAs) and vector-valued modular forms we study the modular group representations and spaces of 1-point functions associated to intertwining operators for Virasoro minimal model VOAs. We examine all representations of dimension less than four associated to irreducible modules for minimal models, and determine when the kernel of these representations is a congruence or noncongruence subgroup of the modular group. Arithmetic criteria are given on the indexing of the irreducible modules for minimal models that imply the associated modular group repre…

费米子扩展的W-代数 via 3d N=4 超对称 gauge 理论与边界

Fermionic extensions of $W$-algebras via 3d $\mathcal{N}=4$ gauge theories with a boundary

研究3d H-twisted N=4超对称 gauge 理论与边界相关的顶点算子代数性质，发现其为费米子扩展的W-代数。

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。

Q系统与完全单位元顶点算子代数的扩展

Q-systems and extensions of completely unitary vertex operator algebras

本文研究了完全单位元顶点算子代数的扩展与Q系统的关系，并证明了单位元扩展的完全单位元性质。

Differential calculus on Jordan algebra and Jordan modules

Having in mind applications to particle physics we develop the differential calculus over Jordan algebras and the theory of connections on Jordan modules. In particular we focus on differential calculus over the exceptional Jordan algebra and provide a complete characterization of the theory of connections for free Jordan modules.

C*-代数上多重线性映射的Jordan范数与Grothendieck不等式

Jordan norms for multilinear maps on C*-algebras and Grothendieck's inequalities

该研究提出C*-代数上多重线性映射完全有界范数的推广形式Jordan范数，通过映射分解定义，并证明C*-代数对上的有界双线性型均为Jordan有界，且其Jordan范数不超过原范数的两倍。

Novikov-Jordan algebras

Algebras with identity $(a\star b)\star (c\star d) -(a\star d)\star(c\star b)$ $=(a,b,c)\star d-(a,d,c)\star b$ are studied. Novikov algebras under Jordan multiplication and Leibniz dual algebras satisfy this identity. If algebra with such identity has unit, then it is associative and commutative.

对数顶点算子代数的特征标与环面链环的着色不变量

Characters of logarithmic vertex operator algebras and coloured invariants of torus links

本文证明了 $(1,p)$ 单态与三重态顶点算子代数的 $\mathfrak{sl}_r$ 特征标可由特定环面链环的着色 $\mathfrak{sl}_r$ Jones 不变量取极限得到，建立了顶点代数与量子拓扑不变量之间的联系。