Q-systems and extensions of completely unitary vertex operator algebras
Bin Gui
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。
我们关联了完全单位元顶点算子代数的扩展与（可交换的）Q系统。作为应用，我们证明了任何完全单位元顶点算子代数的单位元扩展都是完全单位元的。
We relate extensions of completely unitary VOAs and (commutative) Q-systems. As an application, we show that any unitary extension of a completely unitary VOA is completely unitary.
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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.
Non-unitary Wightman CFTs and non-unitary vertex algebras
本文建立了带拟初级生成族的 Möbius 顶点代数与单位圆上不必幺正的 Möbius 协变 Wightman CFT 之间最一般的范畴等价，并证明非幺正 Reeh–Schlieder 定理。
This is the first part in a two-part series of papers constructing a unitary structure for the modular tensor category (MTC) associated to a unitary rational vertex operator algebra (VOA).
We prove an equivalence between the following notions: (i) unitary Möbius vertex algebras, and (ii) Wightman conformal field theories on the circle (with finite-dimensional conformal weight spaces) satisfying an additional condition that we call uniformly bounded order. Reading this equivalence in one direction, we obtain new analytic and operator-theoretic information about vertex operators. In the other direction we characterize OPEs of Wightman fields and show they satisfy the axioms of a vertex algebra. As an application we establish new results linking unitary vertex operator algebras wit…
This is the second half of a two-part series studying tensor categories of unitary vertex operator algebras from a unitary point of view.
We provide a complete classification of unitary subalgebras of even rank-one lattice vertex operator algebras. As a consequence of the correspondence between vertex operator algebras and conformal nets, we also obtain a complete classification of conformal subnets of even rank-one lattice conformal nets.
Unitary vertex operator algebras are introduced and studied. It is proved that most well-known rational vertex operator algebras are unitary. The classification of unitary vertex operator algebras with central charge c less than or equal to 1 is also discussed.
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…
In this note we prove that the vertex operator superalgebras associated to the unitary representations of the N=2 superconformal algebra are rational.
Energy bounds for vertex operator algebra extensions
本文研究简单酉顶点算子代数在何种条件下具有能量有界性，通过紧轨道形和指数能量有界加有限直和分解两个充分条件，证明正则能量有界子代数可推出全代数能量有界，并应用于半单李代数的仿射顶点算子代数扩张。
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.
Osterwalder-Schrader axioms for unitary full vertex operator algebras
本文为酉全顶点算子代数建立多项式能量界与谱密度条件，并证明其关联函数满足共形版Osterwalder-Schrader公理，同时验证Heisenberg VOA的一族全扩张满足这些假设。
We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra V a conformal net A_V acting on the Hilbert space completion of V and prove that the isomorphism class of A_V does not depend on the choice of the scalar product on V. We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex …
We prove that all holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one subspaces are unitary. The main method is to use the orbifold construction of a holomorphic VOA $V$ of central charge $24$ directly from a Niemeier lattice VOA $V_N$. We show that it is possible to extend the unitary form for the lattice VOA $V_N$ to the holomorphic VOA $V$ by using the orbifold construction and some information of the automorphism group $\mathrm{Aut}(V)$.
Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules
本文通过几何刻画融合积上 Hermite 型的正性，证明不可约可酉化融合积必正定，并将其用于循环轨形与 simple current 扩张的完全酉性。
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。
Higher-dimensional full vertex algebras and systems of conformally covariant correlation functions
本文为紧致d维共形场论构造欧几里得关联函数系统，引入共形d-顶点代数并证明其存在性，且对d≥2给出广义自由标量场的例子，涵盖幺正与非幺正情形。
A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method.
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。
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。
This is a continuation of a previous study of quantum vertex algebras of Zamolodchikov-Faddeev type. In this paper, we focus our attention on the special case associated to diagonal unitary rational quantum Yang-Baxter operators. We prove that the associated weak quantum vertex algebras, if not zero, are irreducible quantum vertex algebras with a normal basis in a certain sense.
A unitary vertex operator algebra arising from the 3C-algebra
本文通过3C代数的共轭表示证明了特定顶点算子代数的单位性，并推导了其模的融合规则。
Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category $\mathcal{C}$ and a central charge $c$. A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category $\mathcal{C}$, there exists a unitary chiral conformal field theory $V$ so that its modular tensor category $\mathcal{C}_V$ is $\mathcal{C}$. In this paper, we initiate a mathematical program in and around this conjecture. We define a cla…
On the semi-infinite cohomology of graded-unitary vertex algebras
本文研究了具有手性量子动量图的分级酉顶点代数的相对半无穷上同调，证明其具有类似紧致Kähler流形微分形式的结构且本身分级酉，为源自超对称量子场论的顶点算子代数确立了分级酉性，并建立了类似Lefschetz作用的USp(2)作用及形式性结果。
We construct the intertwining operator superalgebras and vertex tensor categories for the N=2 superconformal unitary minimal models and other related models.
The extensions of the affine vertex operator algebras $L_{sl_2}(k,0)$ and the preunitary vertex operator algebras with central charges $c<1$ are classified. In particular, the unitary vertex operator algebras with central charges $c<1$ are classified.
Rationality of holomorphic vertex operator algebras
证明CFT型酉单全纯顶点算子代数必为有理的，即其所有N-可分级模均为V的直和。
Fusion rules among irreducible modules of the free bosonic orbifold vertex operator algebra are completely determined.
Graded Unitarity in the SCFT/VOA Correspondence
本文提出分级单位性概念，用于描述四维超对称场论对应下的顶点代数性质，探讨其分类及兼容性条件。
Reconstruction of vertex algebras in even higher dimensions
本文研究偶数高维顶点代数的重构问题，发现在共形李代数幺正作用等自然条件下，可从低维逆过程重构高维顶点代数。
Defining relations for minimal unitary quantum affine W-algebras
本文证明了泛最小量子仿射 W-代数的酉最高权模下降到单商，给出酉单最小量子仿射 W-代数的定义关系与不可约正能量模列表，并分类了相关单仿射顶点代数的不可约最高权模。
In two-dimensional conformal field theory (CFT) the building blocks are given by chiral CFTs, i.e.~CFTs on the unit circle (compactified light-ray). They are generated by quantum fields depending on one light-ray coordinate only. There are two mathematical formulations of chiral CFT, the one based on vertex operator algebras (VOAs) and the one based on conformal nets. We describe some recent results which, for first time, gives a general construction of conformal nets from (unitary) VOAs.
Completely fixed point free isometry and cyclic orbifold of lattice vertex operator algebras
本文研究格顶点算子代数的循环轨形，证明在特定无不动点等距条件下，额外自同构的存在迫使等距阶为素数或格为 Leech 格及其余不变子格。
Every conformal net has an associated unitary VOA
本文证明每个共形网都有相关的酉顶点算子代数，并给出其表示产生酉模的条件，推进了酉VOA与共形网等价猜想的研究。
We explore new connections between the fields and local observables in two dimensional chiral conformal field theory. We show that in a broad class of examples, the von Neumann algebras of local observables (a conformal net) can be obtained from the fields (a unitary vertex operator algebra) via a continuous geometric interpolation procedure involving Graeme Segal's functorial definition of conformal field theory. In particular, we construct conformal nets from these unitary vertex operator algebras by showing that 'geometrically mollified' versions of the fields yield bounded, local operators…
The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo-unitary algebras su(p,q) of the Cartan series A_l and the pseudo-unitary algebras u(p,q), are completely determined and classified for arbitrary p,q. In addition to the su(p,q) and u({p,q}) algebras, whose second cohomology group is well known to be trivial, each family includes many non-semisimple algebras; their central extensions, which are explicitly given, can be classified into three types as far as their properties under contraction are involved. A …
Given a holomorphic $C_2$-cofinite vertex operator algebra $V$ with graded dimension $j-744$, Borcherds's proof of the monstrous moonshine conjecture implies any finite order automorphism of $V$ has graded trace given by a "completely replicable function", and by work of Cummins and Gannon, these functions are principal moduli of genus zero modular groups. The action of the monster simple group on the monster vertex operator algebra produces 171 such functions, known as the monstrous moonshine functions. We show that 154 of the 157 non-monstrous completely replicable functions cannot possibly …
The fusion rules for the vertex operator algebras M(1)^+ (of any rank) and V_L^+ (for any positive definite even lattice L) are determined completely.
In this article, we completely determine the isomorphism classes of lattice vertex operator algebras and the vertex operator subalgebras fixed by a lift of the -1-isometry of the lattice. We also provide similar results for certain even lattices associated with doubly-even binary codes.
We study strongly graded vertex algebras and their strongly graded modules, which are conformal vertex algebras and their modules with a second, compatible grading by an abelian group satisfying certain grading restriction conditions. We consider a tensor product of strongly graded vertex algebras and its tensor product strongly graded modules. We prove that a tensor product of strongly graded irreducible modules for a tensor product of strongly graded vertex algebras is irreducible, and that such irreducible modules, up to equivalence, exhaust certain naturally defined strongly graded irreduc…
We prove the Verlinde conjecture in the following general form: Let V be a simple vertex operator algebra satisfying the following conditions: (i) The homogeneous subspaces of V of weights less than 0 are 0, the homogeneous subspace of V of weight 0 is spanned by the vacuum and V' is isomorphic to V as a V-module. (ii) Every weak V-module gradable by nonnegative integers is completely reducible. (iii) V is C_2-cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Then the matrices …
The lattice vertex operator algebra $V_L$ associated to a positive definite even lattice $L$ has an automorphism of order 2 lifted from -1-isometry of $L$. We prove that for the fixed point vertex operator algebra $V_L^+$, any $\Z_{\geq0}$-graded weak modules is completely reducible.
For the Klein group $K$ and a positive integr $k$, irreducible modules of the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{K}$ have been classified and constructed in \cite{JWa}. In this paper, we determine completely the fusion rules of $L_{\widehat{\frak{sl}_2}}(k,0)^{K}$.
Towards a classification of graded unitary ${\mathcal W}_3$ algebras
本文研究四维单位性对从四维通过SCFT/VOA对应得到的${\mathcal W}_3$顶点代数的约束，证明除了$(3,q+4)$极小模型外，其他中央荷值不兼容四维单位性。
This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism.
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.
We introduce a new approach that allows us to determine the structure of Zhu's algebra for certain vertex operator (super)algebras which admit horizontal $\mathbb{Z} $-grading. By using this method and an earlier description of Zhu's algebra for the singlet W-algebra, we completely describe the structure of Zhu's algebra for the triplet vertex algebra W(p). As a consequence, we prove that Zhu's algebra A(W(p)) and the related Poisson algebra P(W(p)) have the same dimension. We also completely describe Zhu's algebras for the N=1 triplet vertex operator superalgebra SW(m). Moreover, we obtain si…
Let $V$ be a simple vertex operator algebra which admits the continuous, faithful action of a compact Lie group $G$ of automorphisms. We establish a Schur-Weyl type duality between the unitary, irreducible modules for $G$ and the irreducible modules for $V^G$ which are contained in $V$ where $V^G$ is the space of $G$-invariants of $V.$ We also prove a concomitant Galois correspondence between vertex operator subalgebras of $V$ which contain $V^G$ and closed Lie subgroups of $G$ in the case that $G$ is abelian.
We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a {\em tree algebra}. Using the Riemann-Hilbert correspondence, we further prove that a vertex tensor category in the sense of Huang and Lepowsky gives rise to a tree algebra over $\C$. We also show that the chiral WZW model of a simply connected simple compact Lie group gives rise to a tree algebra over $\Q$.
Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $θ$ the automorphism of $V_{L}$ induced from the $-1$-isometry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $θ$. We show that every weak $V_{L}^{+}$-module is completely reducible.
A class of vertex operator algebras generated by Virasoro vectors
本文研究一类由简单 Virasoro 向量生成的简单 OZ 型顶点算子代数，证明其由 Griess 代数唯一确定，给出了自同构群及为酉代数的必要条件。
For the cyclic group $\mathbb{Z}_3$ and positive integer $k$, we study the representations of the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$. All the irreducible modules for $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are classified and constructed explicitly. Quantum dimensions and fusion rules for the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are completely determined.
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…
Let $L((n-\tfrac 3 2)Λ_0)$, $n \in \Bbb N$, be a vertex operator algebra associated to the irreducible highest weight module $L((n-\tfrac 3 2)Λ_0)$ for a symplectic affine Lie algebra. We find a complete set of irreducible modules for $L((n-\tfrac 3 2)Λ_0)$ and show that every module for $L((n-\tfrac 3 2)Λ_0)$ from the category $\Cal O$ is completely reducible.
In this paper, for every one-dimensional formal group $F$ we formulate and study a notion of vertex $F$-algebra and a notion of $ϕ$-coordinated module for a vertex $F$-algebra where $ϕ$ is what we call an associate of $F$. In the case that $F$ is the additive formal group, vertex $F$-algebras are exactly ordinary vertex algebras. We give a canonical isomorphism between the category of vertex $F$-algebras and the category of ordinary vertex algebras. Meanwhile, for every formal group we completely determine its associates. We also study $ϕ$-coordinated modules for a general vertex $\Z$-graded a…
It is proved that if any Z-graded weak module for vertex operator algebra V is completely reducible, then V is rational and C_2-cofinite. That is, V is regular. This gives a natural characterization of regular vertex operator algebras.
Let $V$ be a simple VOA of CFT-type satisfying $V'\cong V$ and $σ$ a finite automorphism of $V$. We prove that if all $V$-modules are completely reducible and a fixed point subVOA $V^σ$ is $C_2$-cofinite, then all $V^σ$-modules are completely reducible and every simple $V^σ$-module appears in some twisted or ordinary $V$-modules as a $V^σ$-submodule. We also prove that $V_L^σ$ is $C_2$-cofinite for any lattice VOA $V_L$ and $σ\in \Aut(V_L)$ lifted from any triality automorphism of $L$. Using these results, we present two $Z_3$-orbifold constructions as examples. One is the moonshine VOA $V^{\n…
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.
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