Rationality of unitary N=2 vertex operator superalgebras

Positive Energy Unitary Irreducible Representations of the Superalgebras osp(1|2n,R)

We give the classification of the positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n,R).

Vertex Operator Superalgebras and Their Representations

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 …

Simple current extensions of vertex operator algebras by unitary modules

In this paper, a condition making vertex operator superalgebras to be unitary is determined and an analogue of conformal spin-statistics theorem in conformal field theory is proved. As an application of these results, it is proved that under some assumptions there exist vertex operator algebra structures on the direct sum of simple current unitary modules of unitary vertex operator algebras.

Local Systems of Vertex Operators, Vertex Superalgebras and Modules

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…

Intertwining operator superalgebras and vertex tensor categories for superconformal algebras, II

We construct the intertwining operator superalgebras and vertex tensor categories for the N=2 superconformal unitary minimal models and other related models.

Self-dual Vertex Operator Superalgebras with Shadows of large minimal weight

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.

极小W-代数导出的共形网

Conformal nets from minimal W-algebras

本文证明了所有幺正极小W-代数的强分级局部性，从而产生不可约分级局部共形网，并基于此提出了统一构造以获得包含N=3,4超Virasoro网在内的新共形网，同时证明了完全有理猜想对N=0至4超Virasoro代数成立。

中心荷至多为24的自对偶顶点算子超代数的分类

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。

The quantum superalgebra $U_q[osp(1/2n)]$: deformed para-Bose operators and root of unity representations

We recall the relation between the Lie superalgebra $osp(1/2n)$ and para-Bose operators. The quantum superalgebra $U_q[osp(1/2n)]$, defined as usual in terms of its Chevalley generators, is shown to be isomorphic to an associative algebra generated by so-called pre-oscillator operators satisfying a number of relations. From these relations, and the analogue with the non-deformed case, one can interpret these pre-oscillator operators as deformed para-Bose operators. Some consequences for $U_q[osp(1/2n)]$ (Cartan-Weyl basis, Poincaré-Birkhoff-Witt basis) and its Hopf subalgebra $U_q[gl(n)]$ are …

Unitary Representations of $W$ Infinity Algebras

We study the irreducible unitary highest weight representations, which are obtained from free field realizations, of $W$ infinity algebras ($W_{\infty}$, $W_{1+\infty}$, $W_{\infty}^{1,1}$, $W_{\infty}^M$, $W_{1+\infty}^N$, $W_{\infty}^{M,N}$) with central charges ($2$, $1$, $3$, $2M$, $N$, $2M+N$). The characters of these representations are computed. We construct a new extended superalgebra $W_{\infty}^{M,N}$, whose bosonic sector is $W_{\infty}^M\oplus W_{1+\infty}^N$. Its representations obtained from a free field realization with central charge $2M+N$, are classified into two classes: con…

单位ary N=(2,2) 全顶点代数的上同调环与镜像对称

Cohomology ring of unitary $N=(2,2)$ full vertex algebra and mirror symmetry

本文研究了N=(2,2)超对称共形场论的上同调环、Hodge数及Witten指数，通过构造泛化全顶点算子超代数，探讨了上同调环的Poincaré对称、T对称及Frobenius代数结构，并讨论了镜像对称与Hodge理论的关系。

Jacobson generators of the quantum superalgebra $U_q[sl(n+1|m)]$ and Fock representations

As an alternative to Chevalley generators, we introduce Jacobson generators for the quantum superalgebra $U_q[sl(n+1|m)]$. The expressions of all Cartan-Weyl elements of $U_q[sl(n+1|m)]$ in terms of these Jacobson generators become very simple. We determine and prove certain triple relations between the Jacobson generators, necessary for a complete set of supercommutation relations between the Cartan-Weyl elements. Fock representations are defined, and a substantial part of this paper is devoted to the computation of the action of Jacobson generators on basis vectors of these Fock spaces. It i…

Characters of the Positive Energy UIRs of D=4 Conformal Supersymmetry

We give character formulae for the positive energy unitary irreducible representations of the N-extended D=4 conformal superalgebras su(2,2/N). Using these we also derive decompositions of long superfields as they descend to the unitarity threshold. These results are also applicable to irreps of the complex Lie superalgebras sl(4/N). Our derivations use results from the representation theory of su(2,2/N) developed already in the 80s.

中心荷不超过24的全纯顶点算子超代数的幺正性与强分次局部性

Unitarity and strong graded locality of holomorphic vertex operator superalgebras with central charge at most 24

本文研究中心荷不超过24的良好全纯VOSA的幺正性与强分次局部性，排除特定假设赝副本与例外情形，证明多数此类VOSA满足相关性质，可生成对应共形网。

On Exceptional Vertex Operator (Super) Algebras

We consider exceptional vertex operator algebras and vertex operator superalgebras with the property that particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendents of the vacuum. We show that the genus one partition function and characters for simple ordinary modules must satisfy modular linear differential equations. We show the rationality of the central charge and module lowest weights, modularity of solutions, the dimension of each graded space is a rational function of the central charge and that the lowest weight primaries generat…

Deformed Clifford Cl_q(n|m) and orthosymplectic U_q[osp(2n+1|2m)] superalgebras and their root of unity representations

It is shown that the Clifford superalgebra Cl(n|m) generated by m pairs of Bose operators (odd elements) anticommuting with n pairs of Fermi operators (even elements) can be deformed to Cl_q(n|m) such that the latter is a homomorphic image of the quantum superalgebra U_q[osp(2n+1|2m)]. The Fock space F(n|m) of Cl_q(n|m) is constructed. At q being a root of unity (q=\exp ({iπl/k})) q-bosons (and q-fermions) are operators acting in a finite-dimensional subspace F_{l/k}(n|m) of F(n|m). Each F_{l/k}(n|m) is turned through the above mentioned homomorphism into an irreducible (root of unity) U_q[osp…

W-algebras as coset vertex algebras

We prove the long-standing conjecture on the coset construction of the minimal series principal $W$-algebras of $ADE$ types in full generality. We do this by first establishing Feigin's conjecture on the coset realization of the universal principal $W$-algebras, which are not necessarily simple. As consequences, the unitarity of the "discrete series" of principal $W$-algebras is established, a second coset realization of rational and unitary $W$-algebras of type $A$ and $D$ are given and the rationality of Kazama-Suzuki coset vertex superalgebras is derived.

Representations of N=2 superconformal vertex algebra

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…

范畴扩展中的无界场算子

Unbounded field operators in categorical extensions of conformal nets

本文证明了VOA张量范畴与conformal net张量范畴的等价性，并给出了conformal net完全理性的新证明。

$C^*$-张量范畴中的单倍代数与 Schellekens 列表

Haploid algebras in $C^*$-tensor categories and the Schellekens list

本文利用 $C^*$-张量范畴中单倍代数与 Q-系统的刚性等价，证明 Schellekens 列表前 70 个全纯 VOA 的幺正性与强局域性，并分类低中心荷超 Virasoro 扩张。

紧李群对偶的直极限完备化中的代数对象与c=1顶点算子代数的分类

Algebra objects in direct limit completions of compact Lie group duals and the classification of $c=1$ vertex operator algebras

该研究通过直极限完备化的代数对象分类，结合A₅-轨道模型的强合理性假设，完成了中心电荷c=1的强合理顶点算子代数的完整分类。