Toshiyuki Abe
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.
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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。
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.
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,…
We examine in detail the Jacobi-Trudi characters over the ortho-symplectic Lie superalgebras spo(2|2m+1) and spo(2n|3). We furthermore relate them to Serganova's notion of Euler characters.
The character formula of any finite dimensional irreducible module $L_λ$ for Lie superalgebra $\mathfrak{osp}(n|2)$ is computed. As a by-product, the decomposition of tensor module $L_λ\otimes \mathbb{C}^{n|2}$, where $\mathbb{C}^{n|2}$ is the natural representation, is obtained.
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.
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.
We introduce a semisimple tensor category $\mc{O}^{int}_q(m|n)$ of modules over an quantum ortho-symplectic superalgebra. It is a natural counterpart of the category of finitely dominated integrable modules over the quantum classical (super) algebra of type $B_{m+n}$, $C_{m+n}$, $D_{m+n}$ or $B(0,m+n)$ from a viewpoint of super duality. We classify the irreducible modules in $\mc{O}^{int}_q(m|n)$ and show that an irreducible module in $\mc{O}^{int}_q(m|n)$ has a unique crystal base in case of type $B$ and $C$. An explicit description of the crystal graph is given in terms of a new combinatoria…
We present a systematic study of the orbifolds of the rank $n$ symplectic fermion algebra $\mathcal{A}(n)$, which has full automorphism group $Sp(2n)$. First, we show that $\mathcal{A}(n)^{Sp(2n)}$ and $\mathcal{A}(n)^{GL(n)}$ are $\mathcal{W}$-algebras of type $\mathcal{W}(2,4,\dots, 2n)$ and $\mathcal{W}(2,3,\dots, 2n+1)$, respectively. Using these results, we find minimal strong finite generating sets for $\mathcal{A}(mn)^{Sp(2n)}$ and $\mathcal{A}(mn)^{GL(n)}$ for all $m,n\geq 1$. We compute the characters of the irreducible representations of $\mathcal{A}(mn)^{Sp(2n)\times SO(m)}$ and $\m…
Permutation orbifolds of vertex operator superalgebra and associative algebras
本文研究顶点算子超代数张量积的k-循环置换轨形，构造了对应扭Zhu代数的显式同构，复现了扭模一一对应结论，为不动点子代数Zhu代数研究提供工具。
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…
Representations of the orbifold of parafermion vertex operator algebra $K(osp(1|2),k)$
针对任意正整数水平 $k$，论文研究 $K(osp(1|2),k)$ 的轨形理论，并完成其轨形不可约模的分类。
On irreducibility of modules of Whittaker type: twisted modules and nonabelian orbifolds
本文将Dong-Mason不可约性定理推广到顶点算子超代数的非阿贝尔轨形，证明了中心元扭模在满足一定非同构条件下的不可约性，并应用于多种顶点代数的Whittaker型模。
We prove the additive version of the conjecture proposed by Ginzburg and Kaledin. This conjecture states that if X/G is an orbifold modeled on a quotient of a smooth affine symplectic variety X (over C) by a finite group G\subset Aut(X) and A is a G-stable quantum algebra of functions on X then the graded vector space HH(A^G) of the Hochschild cohomology of the algebra A^G of invariants is isomorphic to the graded vector space H_{CR}(X/G)((h)) of the Chen-Ruan (stringy) cohomology of the orbifold X/G.
We give a method to construct pseudo-trace functions for vertex operator algebras satisfying Zhu's finiteness condition not through higher Zhu's algebras and apply our method to the Z_2-orbifold model associated with d-pairs of symplectic fermions. For d=1, we determine the dimension of the space of one-point functions. For d>1, we construct 2^{2d-1}+3 linearly independent one-point functions and study their values at the vacuum vector.
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。
We introduce a modified quantum enveloping algebra as well as a (modified) covering quantum algebra for the ortho-symplectic Lie superalgebra osp(1|2). Then we formulate and compute the corresponding canonical bases, and relate them to the counterpart for sl(2). This provides a first example of canonical basis for quantum superalgebras.
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.
We define and compute the continuous orbifold partition function and a generating function for all $n$-point correlation functions for the rank two free fermion vertex operator superalgebra on a genus two Riemann surface formed by self-sewing a torus. The partition function is proportional to an infinite dimensional determinant with entries arising from torus Szego kernel and the generating function is proportional to a finite determinant of genus two Szego kernels. These results follow from an explicit analysis of all torus $n$-point correlation functions for intertwiners of the irreducible m…
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…
Permutation orbifolds and simple current extensions
本文在顶点算子（超）代数框架下研究置换轨道和简单电流扩张，通过张量范畴方法和简单电流扩张建立可解自同构群的诱导理论，并分类不可约扭模。
Topological Quantum Mechanics on Orbifolds and Orbifold Index
本文通过研究辛轨道折叠上的拓扑量子力学模型，构造了量子HKR映射的轨道折叠版本，并利用精确半经典近似给出了轨道折叠代数指标的几何与量子场论解释。
We define the partition and $n$-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by sewing two tori together. For the free fermion vertex operator superalgebra we obtain a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus Szegö kernels. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. Using the bosonized formalism, a new genus two Jacobi product identi…
Let $(X,ω)$ be a symplectic orbifold which is locally like the quotient of a $\mathbb{Z}_2$ action on $\reals^n$. Let $A^{((\hbar))}_X$ be a deformation quantization of $X$ constructed via the standard Fedosov method with characteristic class being $ω$. In this paper, we construct a universal deformation of the algebra $A^{((\hbar))}_X$ parametrized by codimension 2 components of the associated inertia orbifold $\widetilde{X}$. This partially confirms a conjecture of Dolgushev and Etingof in the case of $\mathbb{Z}_2$ orbifolds. To do so, we generalize the interpretation of Moyal star-product …
We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the $K$-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold…
Let $Γ$ be a finite group acting faithfully and linearly on a vector space $V$. Let $T(V)$ ($S(V)$) be the tensor (symmetric) algebra associated to $V$ which has a natural $Γ$ action. We study generalized quadratic relations on the tensor algebra $T(V)\rtimes Γ$. We prove that the quotient algebras of $T(V)\rtimes Γ$ by such relations satisfy PBW property. Such quotient algebras can be viewed as quantizations of linear or constant Poisson structures on $S(V)\rtimes Γ$, and are natural generalizations of symplectic reflection algebras.
We study Yang-Baxter equations with orthosymplectic supersymmetry. We extend a new approach of the construction of the spinor and metaplectic $\hat{\cal R}$-operators with orthogonal and symplectic symmetries to the supersymmetric case of orthosymplectic symmetry. In this approach the orthosymplectic $\hat{\cal R}$-operator is given by the ratio of two operator valued Euler Gamma-functions. We illustrate this approach by calculating such $\hat{\cal R}$ operators in explicit form for special cases of the $osp(n|2m)$ algebra, in particular for a few low-rank cases. We also propose a novel, simpl…
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