Yi-Zhi Huang, Antun Milas
This is Part I of a series of papers constructing intertwining operator superalgebras and vertex tensor categories associated to the superconformal minimal models and other related models. In this paper, we construct the intertwining operator superalgebras and vertex tensor categories in the N=1 case.
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We construct the intertwining operator superalgebras and vertex tensor categories for the N=2 superconformal unitary minimal models and other related models.
Superconformal change of variables formulas for N=1 Neveu-Schwarz vertex operator superalgebras are presented for general invertible superconformal changes of variables. Using the underlying worldsheet supergeometry of propagating superstrings, geometric proofs of the change of variables formulas are given for the case of convergent superconformal changes of variables. More general formal algebraic proofs of the change of variables formulas in the case of formal superconformal changes of variables are then given. Finally, isomorphic families of N=1 NS-VOSAs are derived from the superconformal …
The notion of "N = 2 vertex superalgebra with two odd formal variables" is presented, the main axiom being a Jacobi identity with odd formal variables in which an N=2 superconformal shift is incorporated into the usual Jacobi identity for a vertex superalgebra. It is shown that as a consequence of these axioms, the N=2 vertex superalgebra is naturally a representation of the Lie algebra isomorphic to the three-dimensional algebra of superderivations with basis consisting of the usual conformal operator and the two N=2 superconformal operators. The notion of N=2 Neveu-Schwarz vertex operator su…
In this paper we construct Bernstein--Gelfand--Gelfand type resolution of simple highest weight modules over the simple $\mathcal{N}=2$ vertex operator superalgebra of central charge $c_{p,p'}=3\left(1-\frac{2p'}{p}\right)$ by means of the Kazama--Suzuki coset construction. As an application, we compute the twisted Zhu algebras of the simple $\mathcal{N}=2$ vertex operator superalgebra. We also compute the Frenkel--Zhu bimodule structure associated with a certain simple highest weight module of central charge $c_{3,2}=-1$.
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。
In vertex algebra theory, fusion rules are described as the dimension of the vector space of intertwining operators between three irreducible modules. We describe fusion rules in the category of weight modules for the Weyl vertex algebra. This way we confirm the conjecture on fusion rules based on the Verlinde algebra. We explicitly construct intertwining operators appearing in the formula for fusion rules. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules for the affine Lie superalgebra $\widehat{gl(1 \vert 1)}$.
We study several families of vertex operator superalgebras from a jet (super)scheme point of view. We provide new examples of vertex algebras which are "chiralizations" of their Zhu's Poisson algebras $R_V$. Our examples come from affine $C_\ell^{(1)}$-series vertex algebras ($\ell \geq 1$), certain $N=1$ superconformal vertex algebras, Feigin-Stoyanovsky principal subspaces, Feigin-Stoyanovsky type subspaces, graph vertex algebras $W_Γ$, and extended Virasoro vertex algebra. We also give a counterexample to the chiralization property for the $N=2$ superconformal vertex algebra of central char…
Level-one representations of the quantum affine superalgebra $U_q[\hat{gl(N|N)}]$ associated to the appropriate non-standard system of simple roots and $q$-vertex operators (intertwining operators) associated with the level-one modules are constructed explicitly in terms of free bosonic fields.
Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras
研究简单$\mathfrak{sl}(2)$和$\mathcal{N}=2$超共形顶点算子代数上权模，证明其权模范畴刚性及融合积分解公式成立，通过积分屏蔽流开发新技术，给出$\mathfrak{sl}(2)$对数交织算子显式公式。
We introduce the notion of N=1 supergeometric vertex operator superalgebra motivated by the worldsheet geometry underlying genus-zero, two-dimensional, holomorphic N=1 superconformal field theory. We then show, assuming the convergence of certain projective factors, that the category of such objects is isomorphic to the category of N=1 Neveu-Schwarz vertex operator superalgebras.
The $\mathbb{Z}/2\mathbb{Z}$--graded intertwining operators are introduced. We study these operators in the case of ``degenerate'' N=1 minimal models, with the central charge $c=3/2$. The corresponding fusion ring is isomorphic to the Grothendieck ring for the Lie superalgebra ${\mathfrak osp}(1|2)$. We also discus multiplicity--2 fusion rules and logarithmic intertwiners.
Local Systems of Vertex Operators, Vertex Superalgebras and Modules
本文借助于向量空间的“局部顶点算子系”的概念，给出了向量空间的自同态环的概念在顶点算子代数和超代数中的一个类比.首先证明了（超）向量空间$M$上的任何局部顶点算子系都具有以$M$为模的自然顶点（超）代数结构.然后证明了对于顶点（算子）超代数$V$，给出一个$V$ -模$M$等价于给出一个从$V$到$M$上某个局部顶点算子系的顶点（算子）超代数同态.作为应用，我们证明了某些著名的无限维李代数或李超代数的某些最低权模具有自然顶点算子超代数结构。证明了仿射代数的标准模所对应的顶点算子超代数的合理性。我们还通过引入广义交织算子的概念，给出了李代数从一个模到另一个模的线性同态空间的概念的类似物。证明了顶点算子超代数$V$的从一个模$M^{1}$到另一个模$M^{2}$的广义交织算子空间$G(M^{1},M^{2})$是广义$V$ -模.进一步证明了对于固定顶点算子超代数$V$和
The quantum SUSY N=1 hierarchy based on $sl(2|1)^{(2)}$ twisted affine superalgebra is considered. The construction of the corresponding Baxter's Q-operators and fusion relations is outlined. The relation with the superconformal field theory is discussed.
The algebraic structure on the subspace of the quasi-primary vectors given by the projection of the (n) products of a conformal superalgebra is formulated. As an application the complete list of simple physical conformal superalgebras is given. The list contains a one-parameter family of superconformal algebras with 4 supercharges that is simple for general values.
We determine fusion rules (dimensions of the space of intertwining operators) among simple modules for the vertex operator algebra obtained as an even part of the symplectic fermionic vertex operator superalgebra. By using these fusion rules we show that the fusion algebra of this vertex operator algebra is isomorphic to the group algebra of the Klein four group over Z.
From vertex operator superalgebras to graded-local conformal nets and back
通过引入强分级局部顶点算子超代数概念，将顶点算子代数与共形网的对应推广至超代数情形，并证明了格VOSA、超Moonshine VOSA及短Moonshine VOSA的强分级局部性。
In this note we prove that the vertex operator superalgebras associated to the unitary representations of the N=2 superconformal algebra are rational.
Yang-Baxter relations symmetric with respect to the ortho-symplectic superalgebras are studied. We start from the formulation of graded algebras and the linear superspace carrying the vector (fundamental) representation of the ortho-symplectic supergroup. On this basis we study the analogy of the Yang-Baxter operators considered earlier for the cases of orthogonal and symplectic symmetries: the vector (fundamental) R matrix, the L operator defining the Yangian algebra and its first and second order evaluations. We investigate the condition for L(u) in the case of the truncated expansion in inv…
Based on the Kazama-Suzuki type coset construction and its inverse coset between the subregular $\mathcal{W}$-algebras for $\mathfrak{sl}_n$ and the principal $\mathcal{W}$-superalgebras for $\mathfrak{sl}_{1|n}$, we prove weight-wise linear equivalences of their representation categories. Our main results are then improvements of these correspondences incorporating the monoidal structures. Firstly, in the rational case, we obtain the classification of simple modules and their fusion rules via simple current extensions from their Heisenberg cosets. Secondly, beyond the rational case, we use ce…
We compute the modular transformation formula of the characters for a certain family of (finitely or uncountably many) simple modules over the simple $\mathcal{N}=2$ vertex operator superalgebra of central charge $c_{p,p'}=3\left(1-\frac{2p'}{p}\right),$ where $(p,p')$ is a pair of coprime positive integers such that $p\geq2$. When $p'=1$, the formula coincides with that of the $\mathcal{N}=2$ unitary minimal series found by F. Ravanini and S.-K. Yang. In addition, we study the properties of the corresponding "modular $S$-matrix", which is no longer a matrix if $p'\geq2$.
We prove that the family of non-linear $W$-algebras $SW(3/2,2)$ which are extensions of the $N=1$ superconformal algebra by a primary supercurrent of conformal weight $2$ can be realized as a quantum Hamiltonian reduction of the Lie superalgebra $osp(3|2)$. In consequence we obtain an explicit free field realization of the algebra in terms of the screening operators. At central charge $c=12$ the $SW(3/2,2)$ superconformal algebra corresponds to the superconformal algebra associated to sigma models based on eight-dimensional manifolds with special holonomy $Spin(7)$, i.e., the Shatashvili-Vafa …
We prove some conjectures about vertex algebras which emerge in gauge theory constructions associated to the geometric Langlands program. In particular, we present the conjectural kernel vertex algebra for the $S T^2 S$ duality transformation in $SU(2)$ gauge theory. We find a surprising coincidence, which gives a powerful hint about the nature of the corresponding duality wall. Concretely, we determine the branching rules for the small $N=4$ superconformal algebra at central charge $-9$ as well as for the generic large $N=4$ superconformal algebra at central charge $-6$. Moreover we obtain th…
We shall first present an explicit realization of the simple $N=4$ superconformal vertex algebra $L_{c} ^{N=4}$ with central charge $c=-9$. This vertex superalgebra is realized inside of the $ b c βγ$ system and contains a subalgebra isomorphic to the simple affine vertex algebra $L_{A_1} (- \tfrac{3}{2} Λ_0)$. Then we construct a functor from the category of $L_{c} ^{N=4}$--modules with $c=-9$ to the category of modules for the admissible affine vertex algebra $L_{A_{2} } (-\tfrac{3}{2} Λ_0)$. By using this construction we construct a family of weight and logarithmic modules for $L_{c} ^{N=4}…
Motivated by recent progress on the correspondence between string theory on anti-de Sitter space and conformal field theory, we address the question of constructing space-time N extended superconformal algebras on the boundary of AdS_3. Based on a free field realization of an affine SL(2|N/2) current superalgebra residing on the world sheet, we construct explicitly the Virasoro generators and the N supercurrents. N is even. The resulting superconformal algebra has an affine SL(N/2) \otimes U(1) current algebra as an internal subalgebra. Though we do not complete the general superalgebra, we ou…
Let $L_c$ be simple vertex operator superalgebra(SVOA) associated to the vacuum representation of N=2 superconformal algebra with the central charge $c$. Let $c_m = {3m}/{m+2}$. We classify all irreducible modules for the SVOA $L_{c_m}$. When $m$ is an integer we prove that the set of all unitary representations of N=2 superconformal algebra with the central charge $c_m$ provides all irreducible $L_{c_m}$-modules. When $m \notin {\N} $ and $m$ is an admissible rational number we show that irreducible $L_{c_m}$-modules are parameterized with the union of one finite set and union of finitely man…
The new general upper bound mu <= [c/24] + 1 for the minimal weight mu of a self-dual vertex operator superalgebra of central charge c different from 47/2 is proven. For central charges c <= 48, further improved estimates are given and examples of with large minimal weight are discussed. We also study the case of vertex operator superalgebras with N=1 supersymmetry which was first considered by Witten in connection with three-dimensional quantum gravity. The upper bound mu^* <= (1/2)[c/12]+1/2 for the minimal superconformal weight is obtained for c different from 47/2. In addition, we show tha…
A nonmeromorphic extension of the moonshine module vertex operator algebra
本文描述了由Leech格构造的无扭顶点算子代数{}与它的（唯一）不可约扭模的直和上的交换交织代数（在Dong和Lepowsky意义下）的一个自然结构.当我们限制自己的月光模块，我们得到了一个新的和概念性的证明，月光模块具有一个顶点算子代数的自然结构。这个阿贝尔交织代数也包含一个不可约的扭曲模块的月光模块关于明显的对合。此外，它还包含一个顶点算子超代数和这个顶点算子超代数关于对合的扭曲模，对合在偶数子空间上是恒等式，在奇数子空间上是$-1$.还给出了Dixon，Ginsparg和Harvey观察到的超共形结构。
A class of Fock representations of non-central extensions of the super-diffeomorphism algebra in (N+1|M) dimensions is constructed, by superization of the paper [physics/9705040]. The representations act on trajectories in (N|M)-dimensional superspace, the extra dimension being the parameter along the trajectory. The restrictions to various subalgebras are considered. In particular, the centrally extended superconformal algebra is obtained by restriction to the contact superalgebra K(1|1). This shows that one of the basic assumptions in superstring theory (the distinguished nature of the super…
A supergeometric interpretation of vertex operator superalgebras
将Huang关于顶点算子代数的几何解释推广到顶点算子超代数的超几何解释。特别是，几何领域的穿孔和局部解析坐标的指数的导数扩展到几何的超球穿孔，一个给定的自旋结构，和局部超共形坐标的指数的superderivations。引入了超几何顶点算子超代数的概念，并在Grassmann代数上引入了奇形式变量的（超代数）顶点算子超代数的概念.主要结果是：具有给定自旋结构的Grassmann代数上的超几何顶点算子超代数范畴与Grassmann代数上的顶点算子超代数范畴是同构的.此外，两种不同的可能自旋结构之间的超几何对称性自然会引起超代数结构的对称性。
In this paper we continue the study of representation theory of formal distribution Lie superalgebras initiated in q-alg/9706030. We study finite Verma-type conformal modules over the N=2, N=3 and the two N=4 superconformal algebras and also find explicitly all singular vectors in these modules. From our analysis of these modules we obtain a complete list of finite irreducible conformal modules over the N=2, N=3 and the two N=4 superconformal algebras.
Simple smooth modules over the superconformal current algebra
本文通过先分类 Heisenberg-Clifford 代数上的简单光滑模，进而完整分类超共形流代数上的简单光滑模，并给出最高权模与 Whittaker 模的刻画及非张量积型例子。
Modules Over Affine Lie Superalgebras
研究了仿射李超代数${\cal G}$上的模，特别是针对${\cal G}=\hat{OSP(1,2)}$。通过对Verma模的研究，证明了卡茨-穆迪代数中的许多结果都可以推广到Verma模的超情形.其中最重要的是广义Kac-Kazhdan公式和Malikov-Feigin-Fuchs构造，它们给出了仿射Kac-Moody超代数上Verma模中奇异向量的权和显式.我们还考虑了$\hat{OSP(1,2)}$的容许表示到$\hat{SL(2)}\otimes$维拉索罗代数的容许表示的分解，由此得到了环面上的模变换和融合规则。环面上不同的边界条件对应于当前超代数和特征标或超特征标的不同模化，这可能与Neveu-Schwarz或Ramond超共形代数的Hamilton约化有关.最后，通过自由场实现，构造了由Wakimoto模构成的Felder BRST复形。
We present a new and asymmetric N=4 superconformal algebra for arbitrary central charge, thus completing our recent work on its classical analogue with vanishing central charge. Besides the Virasoro generator and 4 supercurrents, the algebra consists of an internal SL(2) \otimes U(1) Kac-Moody algebra in addition to two spin 1/2 fermions and a bosonic scalar. The algebra is shown to be invariant under a linear twist of the generators, except for a unique value of the continuous twist parameter. At this value, the invariance is broken and the algebra collapses to the small N=4 superconformal al…
We obtain the superconformal algebra associated to a sigma model with target a manifold with $G_{2}$ holonomy, i.e., the Shatashvili-Vafa $G_{2}$ algebra as a quantum Hamiltonian reduction of the exceptional Lie superalgebra $D(2,1;α)$ for $α=1$. We produce the complete family of $W$-algebras $SW(\frac{3}{2},\frac{3}{2}, 2)$ (extensions of the $N=1$ superconformal algebra by two primary supercurrents of conformal weight $\frac{3}{2}$ and $2$ respectively) as a quantum Hamiltonian reduction of $D(2,1;α)$. As a corollary we find a free field realization of the Shatashvili-Vafa $G_{2}$ algebra, a…
A Conjecture of Warnaar-Zudilin from Deformations of Lie Superalgebras
通过李超代数形变方法，证明了Warnaar-Zudilin猜想中的$q$-级数恒等式，并填补了Stoyanovsky关于主子空间特征与仿射顶点代数特征等同的空白。
Feigin-Semikhatov conjecture and related topics
本文围绕已证明的Feigin-Semikhatov猜想，综述A型次正则W-代数与主W-超代数在代数、模及含融合规则的缠结算子层面对应关系的最新进展，该猜想是Feigin-Frenkel对偶的变体与相关代数联系的推广。
We give two constructions of grading-restricted vertex (super)algebras. We first give a new construction of a class of grading-restricted vertex (super)algebras originally obtained by Meurman and Primc using a different method. This construction is based on a new definition of vertex operators and a new method. Our second construction is a generalization of the author's construction of the moonshine module vertex operator algebra and a related vertex operator superalgebra. This construction needs properties of intertwining operators formulated and proved by the author.
We consider the extension of the Heisenberg vertex operator algebra by all its irreducible modules. We give an elementary construction for the intertwining vertex operators and show that they satisfy a complex parametrized generalized vertex operator algebra. We illustrate some of our results with the example of integral lattice vertex operator superalgebras.
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