Chongying Dong, Zhongping Zhao
This paper gives an analogue of A_g(V) theory for a vertex operator superalgebra V and an automorphism g of finite order. The relation between the g-twisted V-modules and A_g(V)-modules is established. It is proved that if V is g-rational, then A_g(V) is finite dimensional semisimple associative algebra and there are only finitely many irreducible g-twisted V-modules.
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In this note we study holomorphic integer graded vertex superalgebras. We prove that all such vertex superalgebras of central charge 8 and 16 are purely even. For the case of central charge 24 we prove that the weight-one Lie superalgebra is either zero, of superdimension 24, or else is one of an explicit list of 1332 semisimple Lie superalgebras.
On integral forms for vertex superalgebras associated with affine Lie superalgebras and their modules
本文研究仿射顶点超代数及其模的积分形式，通过构造仿射李超代数通用包络超代数的积分形式，并利用Chevalley基推广Kostant-Garland积分形式理论，进而构造基于仿射李超代数的顶点（算子）超代数及其模的积分形式，并探讨积分形式包含共形向量以及反梯度模中的积分形式。
The Regular Representation of the twisted queer $q$-Schur Superalgebra
本文研究量子 queer 超代数及其扭 queer $q$-Schur 超代数的表示，证明后者正则模可分解为不可约子模直和，从而确立其半单性。
We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to twisted Hopf algebras that are twisted dual. Then, using induction and restriction functors coming from such towers, we obtain a categorification of the twisted Heisenberg double and its Fock space representation. We show that towers of wreath product algebras (in particular, the tower of Sergeev superalgebras) and the tower of nilcoxeter graded superalgebras…
We classify irreducible $σ$-twisted modules for the N=1 super triplet vertex operator superalgebra $\mathcal{SW}(m)$ introduced recently [Adamovic D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of $σ$-twisted modules are also determined. These results, combined with our previous work in the untwisted case, show that the $SL(2,\mathbb{Z})$-closure of the space spanned by irreducible characters, irreducible supercharacters and $σ$-twisted irreducible characters is $(9m+3)$-dimensional. We present strong evidence that this is also the (full) space of g…
Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1 irreducible integrable highest weight representations and classify the finite dimensional irreducible representations of Uq(g). This makes essential use of the Drinfeld realisation for Uq(g), and quantum correspondences between affine Kac-Moody superalgebras, developed in earlier papers.
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.
Orbifold theory of the affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(k,0)$
本文开展仿射顶点算子超代数的轨形理论研究，完成单仿射顶点算子超代数 $L_{\widehat{osp(1|2)}}(k,0)$ 轨形的不可约模分类并确定其融合规则。
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 $\mathbb{Z}/2\mathbb{Z}$--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.
Twisted modules for N=2 supersymmetric vertex operator superalgebras are classified for the vertex operator superalgebra automorphisms which are lifts of a finite automorphism of the N=2 Neveu-Schwarz Lie superalgebra representation. These include the Ramond-twisted sectors and mirror-twisted sectors for N=2 vertex operator superalgebras, as well as twisted modules related to more general "spectral flow" representations of the N=2 Neveu-Schwarz algebra. We present the construct of twisted modules for free N=2 vertex operator superalgebras for all of the N=2 Neveu-Schwarz Lie superalgebra autom…
We study the trace functions in orbiford theory for Z-graded vertex operator superalgebras and obtain a modular invariance result. More precisely, let V be a C_2-cofinite Z-graded vertex operator superalgebra and G a finite automorphism group of V. Then for any commuting pairs (g,h) in G, the hσ-trace functions associated to the simple g-twisted V-modules are holomorphic in the upper half plane where σis the canonical involution on V coming from the superspace structure of V. If V is further g-rational for every g n G, the trace unctions afford a representation for the full modular group SL(2,…
Twisted modules of $\frac{1}{2}\mathbb{Z}$-graded modular vertex superalgebras
本文研究了模1/2Z-graded顶点超代数的g-扭模理论，构建了Zhu代数的扭版本，并建立了简单模之间的对应关系，同时研究了affine Lie超代数和模Neveu-Schwarz顶点超代数的扭模结构。
In this paper, the notion of unitary vertex operator superalgebra is introduced. It is proved that the vertex operator superalgebras associated to the unitary highest weight representations for the Neveu-Schwarz Lie superalgebra, Heisenberg superalgebras and to positive definite integral lattices are unitary vertex operator superalgebras. The unitary structures are then used to study the structures of vertex operator superalgebras, it is proved that any unitary vertex operator superalgebra is a direct sum of strong CFT type unitary simple vertex operator superalgebras. The classification of un…
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.
We derive from the super RS algebra the Drinfeld basis of the twisted quantum affine superalgebra $U_q[osp(2|2)^{(2)}]$ by means of the Gauss decomposition technique. We explicitly construct a nonclassical level-one representation of $U_q[osp(2|2)^{(2)}]$ in terms of two $q$-deformed free boson fields.
Vertex operator superalgebras are studied and various results on rational Vertex operator superalgebras are obtained. In particular, the vertex operator super subalgebras generated by the weight 1/2 and weight 1 subspaces are determined. It is also established that if the even part $V_{\bar 0}$ of a vertex operator superalgebra $V$ is rational, so is $V.$
Twisted Zhu algebras
本文研究自由生成预分次顶点超代数的 $(g,H)$-扭曲 Zhu 代数，证明其 PBW 基、非线性李超代数泛包络实现及 BRST 上同调交换性，并用于计算仿射顶点超代数与仿射 $W$-代数。
We introduce the notion of ``local system of $\Bbb{Z}_{T}$-twisted vertex operators'' on a $\Bbb{Z}_{2}$-graded vector space $M$, generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$ has a vertex superalgebra structure with an automorphism $σ$ of order $T$ with $M$ as a $σ$-twisted module. Then we prove that for a vertex (operator) superalgebra $V$ with an automorphism $σ$ of order $T$, giving a $σ$-twisted $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from…
In the present paper, a class of new simple modules over the $N=1$ Ramond algebra are constructed, which are induced from simple modules over some finite dimensional solvable Lie superalgebras. These new modules are simple restricted modules over the $N=1$ Ramond algebra. Combined with the result in \cite{L1}, a classification of simple weak $ψ$-twisted $\bar W(0,c)$-modules under certain conditions is also given. At last, some examples of simple restricted $N=1$ Ramond modules as various versions of Whittaker modules are presented (classical Whittaker modules were studied in \cite{LPX}).
We construct and classify $(1 \; 2\; \cdots \; k)$-twisted $V^{\otimes k}$-modules for $k$ odd and for $V$ a vertex operator superalgebra. This extends previous results of the author, along with Dong and Mason, classifying all permutation-twisted modules for tensor product vertex operator algebras, to the setting of vertex operator superalgebras for odd order permutations. We show why this construction does not extend to the case of permutations of even order in the superalgebra case and how the construction and classification in the even order case is fundamentally different than that for the…
In this note we prove that the vertex operator superalgebras associated to the unitary representations of the N=2 superconformal algebra are rational.
$(d,σ)$-twisted Affine-Virasoro superalgebras
本文引入了带有偶导子和有限阶自同构的 $(d,σ)$-扭曲 Affine-Virasoro 超代数及其泛中心扩张，并利用权函子和 $A$-覆盖方法完成了对其单尖点模与单拟有限模的分类。
The notions of N=1 Neveu-Schwarz vertex operator superalgebra over a Grassmann algebra and with odd formal variables and of N=1 Neveu-Schwarz vertex operator superalgebra over a Grassmann algebra and without odd formal variables are introduced, and we show that the respective categories of such objects are isomorphic. The weak supercommutativity and weak associativity properties for an N=1 Neveu-Schwarz vertex operator superalgebra with odd formal variables are established, and we show that in the presence of the other axioms, weak supercommutativity and weak associativity are equivalent to th…
Based on the loop-algebraic presentation of 2-toroidal Lie superalgebras, free field representation of toroidal Lie superalgebras of type $A(m, n)$ is constructed using both vertex operators and bosonic fields.
In this paper, vertex representations of the 2-toroidal Lie superalgebras of type $D(m, n)$ are constructed using both bosonic fields and vertex operators based on their loop algebraic presentation.
Semisimplicity of module categories of certain affine vertex operator superalgebras
本文证明了某些仿射顶点算子超代数在共形但非容许水平上的 Kazhdan-Lusztig 范畴是半单的，并由此得到其有限长度广义模范畴具有辫子张量范畴结构。
Using recent results of the author along with Vander Werf, we present the classification and construction of mirror-twisted modules for N=2 supersymmetric vertex operator superalgebras of the form $V \otimes V$ for the signed transposition mirror map automorphism. In particular, we show that the category of such mirror-twisted sectors for $V\otimes V$ is isomorphic to the category of N=1 Ramond sectors for $V$.
Ordinary modules for affine vertex operator superalgebras
本文证明边界容许水平下仿射顶点算子超代数的普通模范畴有限半单，等同于有限长广义模范畴，并进一步证明其为ribbon融合超范畴。
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.
Classification of Self-Dual Vertex Operator Superalgebras of Central Charge at Most 24
本文分类了中心荷为24的自对偶顶点算子超代数，在特定假设下确定共有969个，并利用2-邻域图和简单流扩展构造证明它们均源自Conway群，进而推广至中心荷至多为24的情形。
Drinfeld realisations are constructed for the quantum affine superalgebras of the series ${\rm\mathfrak{osp}}(1|2n)^{(1)}$,${\rm\mathfrak{sl}}(1|2n)^{(2)}$ and ${\rm\mathfrak{osp}}(2|2n)^{(2)}$. By using the realisations, we develop vertex operator representations and classify the finite dimensional irreducible representations for these quantum affine superalgebras.
We construct the intertwining operator superalgebras and vertex tensor categories for the N=2 superconformal unitary minimal models and other related models.
Y(z)-injective vertex superalgebras and Hopf actions
本文研究Y(z)-内射顶点超代数，证明单顶点超代数与具PBW基的顶点超代数属于该类，且内忠实作用于其上的有限维Hopf代数必为群代数，将不动点子代数研究归约为群作用情形。
We construct and classify $(1 \; 2 \; \cdots \; k)$-twisted $V^{\otimes k}$-modules for $k$ even and $V$ a vertex operator superalgebra. In particular, we show that the category of weak $(1 \; 2 \; \cdots \; k)$-twisted $V^{\otimes k}$-modules for $k$ even is isomorphic to the category of weak parity-twisted $V$-modules. This result shows that in the case of a cyclic permutation of even order, the construction and classification of permutation-twisted modules for tensor product vertex operator superalgebras is fundamentally different than in the case of a cyclic permutation of odd order, as pr…
We give an analogue for vertex operator algebras and superalgebras of the notion of endomorphism ring of a vector space by means of a notion of ``local system of vertex operators'' for a (super) vector space. We first prove that any local system of vertex operators on a (super) vector space $M$ has a natural vertex (super)algebra structure with $M$ as a module. Then we prove that for a vertex (operator) superalgebra $V$, giving a $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of vertex operators on $M$. As applications, we pro…
We conjecture that the category of permutation-twisted modules for a multi-fold tensor product vertex operator superalgebra and a cyclic permutation of even order is isomorphic to the category of parity-twisted modules for the underlying vertex operator superalgebra. This conjecture is based on our observations of the cyclic permutation-twisted modules for free fermions as we discuss in this work, as well as previous work of the first author constructing and classifying permutation-twisted modules for tensor product vertex operator superalgebras and a permutation of odd order. In addition, we …
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…
In this article we prove that the full automorphism group of the baby-monster vertex operator superalgebra constructed by Hoehn is isomorphic to 2xB, where B is the baby-monster sporadic finite simple group and determine irreducible modules for the baby-monster vertex operator algebra. Our result has many corollaries. In particular, we can prove that the Z_2-orbifold construction with respect to a 2A-involution of the Monster applied to the moonshine vertex operator algebra yields the moonshine vertex operator algebra itself again.
After giving some definitions for vertex operator SUPERalgebras and their modules, we construct an associative algebra corresponding to any vertex operator superalgebra, such that the representations of the vertex operator algebra are in one-to-one correspondence with those of the corresponding associative algebra. A way is presented to decribe the fusion rules for the vertex operator superalgebras via modules of the associative algebra. The above are generalizations of Zhu's constructions for vertex operator algebras. Then we deal in detail with vertex operator superalgebras corresponding to …
Huang's geometric interpretation of vertex operator algebras is extended to a supergeometric interpretation of vertex operator superalgebras. In particular, the geometry of spheres with punctures and local analytic coordinates in terms of exponentials of derivations is extended to the geometry of superspheres with punctures, a given spin structure, and local superconformal coordinates in terms of exponentials of superderivations. The notion of supergeometric vertex operator superalgebra is introduced, and in addition, the notion of (superalgebraic) vertex operator superalgebra over a Grassmann…
It is proved that g-rationality of a vertex operator superalgebra V=V_{\bar0}+V_{\bar1} for all g in G imply rationality of V^G, and also imply that each irreducible V^G-module is a submodule of an irreducible g-twisted V-module for some g in G, where G is any finite abelian subgroup of Aut(V). We also prove that for any finite solvable G, rationality of V^G implies g-rationality of V for any g in G.
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。
We calculate twisted denominator identities of the fake monster superalgebra and use them to construct new examples of supersymmetric generalized Kac-Moody superalgebras. Their denominator identities give new infinite product identities.
The shadow V' of a self-dual vertex operator superalgebra V is defined as the direct sum of the irreducible modules of its even vertex operator subalgebra $V_{(0)}$ not contained in $V=V_{(0)}+V_{(1)}$. We describe the self-dual ``very nice'' unitary rational vertex operator superalgebras V of rank c whose shadows have the largest possible minimal weights c/8 or c/8-1. The results are analogous to and imply the corresponding results for self-dual binary codes and lattices.
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…
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.
This paper is to study vertex operator superalgebras which are strongly generated by their weight-$2$ and weight-$\frac{3}{2}$ homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra $V$ is simple, then $V_{(2)}$ has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and $V_{(\frac{3}{2})}$ is naturally a $V_{(2)}$-module equipped with a $V_{(2)}$-valued symmetric bilinear form and a non-degenerate ($\mathbb{C}$-valued) symmetric bilinear form, satisfying a set of conditions. O…
Rationality of vertex operator superalgebras with rational conformal weights
本文证明了容许级别下的仿射顶点代数及特定仿射顶点算子超代数、极小 $W$-代数关于一族 Virasoro 元的有理性，并得出弱模范畴特定子范畴半单的结论。
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…
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.
A cohomology theory of supercommutative algebras and grading-restricted vertex superalgebras
本文推广黄一智的相关理论，构建分次限制顶点超代数的上同调理论，同时补充此前未明确给出的超交换结合代数上同调理论构建，为顶点超代数形变等后续研究奠基。
The minimal irreducible representations of $U_q[gl(m|n)]$, i.e. those irreducible representations that are also irreducible under $U_q[osp(m|n)]$ are investigated and shown to be affinizable to give irreducible representations of the twisted quantum affine superalgebra $U_q[gl(m|n)^{(2)}]$. The $U_q[osp(m|n)]$ invariant R-matrices corresponding to the tensor product of any two minimal representations are constructed, thus extending our twisted tensor product graph method to the supersymmetric case. These give new solutions to the spectral-dependent graded Yang-Baxter equation arising from $U_q…
We construct the screening currents of the quantum superalgebra $U_q(\hat{sl}(N|1))$ for an arbitrary level $k \neq -N+1$. We show that these screening currents commute with the superalgebra modulo total difference. We propose bosonizations of the vertex operators by using the screening currents. We check that these vertex operators are the intertwiners among the Fock-Wakimoto representation and the typical representation for rank $N \leq 4$.
In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations.
Vertex Superalgebras for Hypertoric Varieties and 3d Abelian Gauge Theories
通过构造光滑超环面簇上的顶点算子超代数层，证明了该超代数的仿射簇恢复奇异超环面簇，验证了3d Higgs分支猜想，并揭示了与Kuwabara构造的顶点算子超代数的关系。
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.
In this paper, we prove the categories of lower bounded twisted modules of positive integer levels for simple vertex operator algebras associated with affine Lie algebras and general automorphisms are semisimple, using the twisted generalization of Zhu's algebra for these vertex operator algebras, constructed in \cite{HY}. We also show that the category of lower bounded twisted modules for a general automorphism is equivalent to the category of lower bounded twisted modules for the corresponding diagram automorphism.
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 …
Classical freeness of orthosymplectic affine vertex superalgebras
本文证明了简单仿射顶点超代数 $L_n(\mathfrak{osp}_{m|2r})$ 在满足 $-\frac{m}{2}+r+n+1>0$ 时具有经典自由性，推广了顶点代数经典自由性理论。
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