Yan-Jun Chu, Fang Huang, Zhu-Jun Zheng
Analogue to commutants in the theory of associative algebras, one can construct a new subalgebra of vertex algebra known as a vertex algebra commutant. In this paper, for the adjoint representation $V$ of Lie algebra $sl(2,\C)$, we describe a commutant of $βγ$- System $S(V)$ by giving its generators, moreover, we get a new Howe pair of vertex algebras.
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Let $V$ be a vertex operator subalgebra of $U$. Assume that $U$, $V$, and its commutant $V^c$ in $U$ are CFT-type, self-dual, and regular VOAs. Assume also that the double commutant $V^{cc}$ equals $V$. We prove that any intertwining operator of $V$ is a compression of intertwining operators of $U$.
We study the commutant $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$ of $L_{\widehat{\frak{sl}}_{2}}(n,0)$ in the vertex operator algebra $L_{\widehat{\frak{sl}}_{2}}(1,0)^{\otimes n}$, for $n\geq 2$. The main results include a complete classification of all irreducible $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$-modules and a proof that $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$ is a rational vertex operator algebra. As a consequence, every irreducible $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$-module arises from the coset construction as conjectured in \cite{LS}.
Given a vertex algebra $\mathcal{V}$ and a subalgebra $\mathcal{A}\subset \mathcal{V}$, the commutant $\text{Com}(\mathcal{A},\mathcal{V})$ is the subalgebra of $\mathcal{V}$ which commutes with all elements of $\mathcal{A}$. This construction is analogous to the ordinary commutant in the theory of associative algebras, and is important in physics in the construction of coset conformal field theories. When $\mathcal{A}$ is an affine vertex algebra, $\text{Com}(\mathcal{A},\mathcal{V})$ is closely related to rings of invariant functions on arc spaces. We find strong finite generating sets for a…
On the commutant of the principal subalgebra in the $A_1$ lattice vertex algebra
本文研究 $A_1$ 格顶点算子代数 $V_{A_1}$ 的主子代数 $W$ 的交换子 $C$，给出其无穷多个元素的显式极小强生成元集，证明 $C$ 非有限生成，并确定其 Zhu 泊松代数结构及伴随簇，同时验证 $W$ 与 $C$ 构成对偶对。
For any vertex algebra V and any subalgebra A of V, there is a new subalgebra of V known as the commutant of A in V. This construction was introduced by Frenkel-Zhu, and is a generalization of an earlier construction due to Kac-Peterson and Goddard-Kent-Olive known as the coset construction. In this paper, we interpret the commutant as a vertex algebra notion of invariant theory. We present an approach to describing commutant algebras in an appropriate category of vertex algebras by reducing the problem to a question in commutative algebra. We give an interesting example of a Howe pair (ie, a …
We study the commutant of the vertex operator algebra $\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(4,0)$ in the cyclic permutation orbifold model $(\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(1,0)^{\otimes 4})^τ$ with $τ=(1\,2\,3\,4)$. It is shown that the commutant is isomorphic to a ${\mathbb Z}_2\times{\mathbb Z}_2$-orbifold model of a tensor product of two lattice type vertex operator algebras of rank one.
In this article we study and obtain a classification of Ising vectors in vertex operator algebras associated to binary codes and $\sqrt{2}$ times root lattices, where an Ising vector is a conformal vector with central charge 1/2 generating a simple Virasoro sub VOA. Then we apply our results to study certain commutant subalgebras related to root systems. We completely classify all Ising vectors in such commutant subalgebras and determine their full automorphism groups.
For the Klein group $K$, $k\in\mathbb{Z}_{\geqslant 1}$ and $m\in\mathbb{Z}_{\geqslant 4}$, we study the representations of the orbifold vertex operator algebra $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and the commutant vertex operator algebra of $L_{\hat{\mathfrak{so}_m}}(3,0)$ in $L_{\hat{\mathfrak{so}_m}}(1,0)^{\otimes 3}$ which can be realized as the orbifold vertex operator subalgebra $L_{\hat{\mathfrak{sl}_2}}(2m,0)^{K}$ or its extension. All the irreducible modules for $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and $C_{{L_{\hat{\mathfrak{so}_m}}(1,0)}^{\otimes 3}}({L_{\hat{\mathfrak{so}_m}}(3,0)}…
The bcβγ-system W of rank 3 has an action of the affine vertex algebra V_0(sl_2), and the commutant vertex algebra C =Com(V_0(sl_2), W) contains copies of V_{-3/2}(sl_2) and Odake's algebra O. Odake's algebra is an extension of the N=2 superconformal algebra with c=9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V_{-3/2}(sl_2) and O form a Howe pair (i.e., a pair of mutual commutants) inside C. More generally, any finite-dimensional representation of a Lie algebra g gives rise to a similar Howe pair, and this example corres…
For the simple Lie algebra $ \frak{so}_m$, we study the commutant vertex operator algebra of $ L_{\hat{\frak{so}}_{m}}(n,0)$ in the $n$-fold tensor product $ L_{\hat{\frak{so}}_{m}}(1,0)^{\otimes n}$. It turns out that this commutant vertex operator algebra can be realized as a fixed point subalgebra of $L_{\hat{\frak{so}}_{n}}(m,0)$ (or its simple current extension) associated with a certain abelian group. This result may be viewed as a version of level-rank duality.
For n\geq 2, there is a free field realization of the affine vertex superalgebra A associated to psl(n|n) at critical level inside the bcβγsystem W of rank n^2. We show that the commutant C=Com(A,W) is purely bosonic and is freely generated by n+1 fields. We identify the Zhu algebra of C with the ring of invariant differential operators on the space of n\times n matrices under SL_n \times SL_n, and we classify the irreducible, admissible C-modules with finite dimensional graded pieces. For n\leq 4, C is isomorphic to the W_n^{(2)}-algebra at critical level, and we conjecture that this holds fo…
We give a coset realization of the vertex operator algebra $M(1)^+$ with central charge $\ell$. We realize $M(1)^+$ as a commutant of certain affine vertex algebras of level -1 in the vertex algebra $L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}Λ_0) \otimes L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}Λ_0)$. We show that the simple vertex algebra $L_{C_{\ell} ^{(1)}}(-Λ_0)$ can be (conformally) embedded into $L_{A_{2 \ell -1} ^{(1)}} (-Λ_0)$ and find the corresponding decomposition. We also study certain coset subalgebras inside $L_{C_{\ell} ^{(1)}}(-Λ_0)$.
The notion of {\it free} generalized vertex algebras is introduced. It is equivalent to the notion of {\it generalized principal subspaces} associated with lattices which are not necessarily integral. Combinatorial bases and the characters of the free generalized vertex algebras are given. As an application, the commutants of principal subspaces are described by using generalized principal subspaces.
In this paper, we study a new kind of vertex operator algebra related to the twisted Heisenberg-Virasoro algebra, which we call the twisted Heisenberg-Virasoro vertex operator algebra, and its modules. Specifically, we present some results concerning the relationship between the restricted module categories of twisted Heisenberg-Virasoro algebras of rank one and rank two and several different kinds of module categories of their corresponding vertex algebras. We also study fully the structures of the twisted Heisenberg-Virasoro vertex operator algebra, give a characterization of it as a tensor …
We prove an $\text{SL}_2 (\mathbb{Z})$-invariance property of multivariable trace functions on modules for a regular VOA. Applying this result, we provide a proof of the inversion transformation formula for Siegel theta series. As another application, we show that if $V$ is a regular VOA containing a regular subVOA $U$ whose commutant $U^c$ is regular and satisfies $(U^c)^c =U$, then all simple $U$-modules appear in some simple $V$-module.
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。
We discuss the classification of strongly regular vertex operator algebras (VOAs) with exactly three simple modules whose character vector satisfies a monic modular linear differential equation with irreducible monodromy. Our Main Theorem provides a classification of all such VOAs in the form of one infinite family of affine VOAs, one individual affine algebra and two Virasoro algebras, together with a family of eleven exceptional character vectors and associated data that we call the $U$-series. We prove that there are at least $15$ VOAs in the $U$-series occurring as commutants in a Schellek…
E7 part: In this paper, we study McKay's E7 observation on the Baby Monster. By investigating so called derived c=7/10 Virasoro vectors, we show that there is a natural correspondence between dihedral subgroups of the Baby Monster and certain subalgebras of the Baby Monster vertex operator algebra which are constructed by the nodes of the affine E7 diagram. This allows us to reinterpret McKay's E7 observation via the theory of vertex operator algebras. For a class of vertex operator algebras including the Moonshine module, we will show that the product of two Miyamoto involutions associated to…
Tensor category of $\mathbb{Z}_2$-orbifold of Heisenberg vertex operator algebra and its applications
本文证明Heisenberg顶点算子代数的Z₂-orbifold M(1)^+ 的有限长度模的张量范畴是顶点和braided张量范畴，并探讨其在简单affine顶点代数的泛化模中的应用。
We study McKay's observation on the Monster simple group, which relates the 2A-involutions of the Monster simple group to the extended E_8 diagram, using the theory of vertex operator algebras (VOAs). We first consider the sublattices L of the E_8 lattice obtained by removing one node from the extended E_8 diagram at each time. We then construct a certain coset (or commutant) subalgebra U associated with L in the lattice VOA V_{\sqrt{2}E_8}. There are two natural conformal vectors of central charge 1/2 in U such that their inner product is exactly the value predicted by Conway. The Griess alge…
Attached to a vector space V is a vertex algebra S(V) known as the beta-gamma system or algebra of chiral differential operators on V. It is analogous to the Weyl algebra D(V), and is related to D(V) via the Zhu functor. If G is a connected Lie group with Lie algebra g, and V is a linear G-representation, there is an action of the corresponding affine algebra on S(V). The invariant space S(V)^{g[t]} is a commutant subalgebra of S(V), and plays the role of the classical invariant ring D(V)^G. When G is an abelian Lie group acting diagonally on V, we find a finite set of generators for S(V)^{g[t…
Quantization of parafermion vertex algebras
针对有限维单李代数对应的仿射李代数顶点代数，本文构造其量子化形式作为单量子仿射顶点代数的\u27e6\u27e6\frak g\u27e7\u27e7-进量子顶点子代数，并证明该量子仿射顶点代数含与量子格顶点代数同构的子代数且二者满足双交换子性质。
A general mirror equivalence theorem for coset vertex operator algebras
本研究针对单共形顶点代数的子代数及其交换子证明了一般镜像对偶定理，构造了逆编织张量等价，并应用于Virasoro代数得到相关不动点子代数结论。
Equivalence Relations on Vertex Operator Algebras, I: Genus
本文研究顶点算子代数的体亏格与双曲亏格两种等价关系，证明双曲亏格更精细，提出c=32全纯顶点算子代数分类计划与数量下界，并猜想Siegel-Weil恒等式。
On rationality for $C_2$-cofinite vertex operator algebras
本文证明了 $C_2$-cofinite 顶点算子代数的理性性，并应用该结果证明了相关 $W$-代数的强理性以及共集理性问题的转化。
We extend a quantized skew Howe duality result for Type $\mathbf{A}$ algebras to orthogonal types via a seesaw. We develop an operator commutant version of the First Fundamental Theorem of invariant theory for $U_q(\mathfrak{so}_n)$ using a double centralizer property inside a quantized Clifford algebra. We obtain a multiplicity-free decomposition of tensor powers of the $U_q(\mathfrak{so}_{2n})$ spin representation by explicitly computing joint highest weights with respect to an action of $U_q(\mathfrak{so}_{2n}) \otimes U_q'(\mathfrak{so}_m)$. Clifford algebras are an essential feature of ou…
We develop an operator commutant version of the First Fundamental Theorem of invariant theory for the general linear quantum group $U_q(\mathfrak{gl}_n)$ by using a double centralizer property inside a quantized Clifford algebra. In particular, we show that $U_q(\mathfrak{gl}_m)$ generates the centralizer of the $U_q(\mathfrak{gl}_n)$-action on the tensor product of braided exterior algebras $\bigwedge_q(\mathbb{C}^n)^{\otimes m}$. We obtain a multiplicity-free decomposition of the $U_q(\mathfrak{gl}_n) \otimes U_q(\mathfrak{gl}_m)$-module $\bigwedge_q(\mathbb{C}^n)^{\otimes m} \cong \bigwedge…
A commutative associative algebra $A$ over ${\mathbb C}$ with a derivation is one of the simplest examples of a vertex algebra. However, the differences between the modules for $A$ as a vertex algebra and the modules for $A$ as an associative algebra are not well understood. In this paper, I give the classification of finite-dimensional indecomposable untwisted or twisted modules for the polynomial ring in one variable over ${\mathbb C}$ as a vertex algebra.
Let V be a simple vertex operator algebra and G a finite automorphism group. Then there is a natural right G-action on the set of all inequivalent irreducible V-modules. Let S be a finite set of inequivalent irreducible V-modules which is closed under the action of G. There is a finite dimensional semisimple associative algebra A_α(G,S) for a suitable 2-cocycle αnaturally determined by the G-action on S such that A_α(G,S) and the vertex operator algebra V^G form a dual pair on the sum of V-modules in S in the sense of Howe. In particular, every irreducible V-module is completely reducible V^G-…
This paper illustrates the combinatorial approach to vertex algebra - study of vertex algebras presented by generators and relations. A necessary ingredient of this method is the notion of free vertex algebra. Borcherds \cite{bor} was the first to note that free vertex algebras do not exist in general. The reason for this is that vertex algebras do not form a variety of algebras, because the locality axiom (see sec 2 below) is not an identity. However, a certain subcategory of vertex algebras, obtained by restricting the order of locality of generators, has a universal object, which we call th…
The admissible modules for $\hat{sl}_2$ are studied from the point of view of vertex operator algebra. If $l$ is rational such that $l+2={p\over q}$ for some coprime positive integers $p\ge 2$ and $q$, Kac and Wakimoto found finitely many distinguished irreducible representations for $\hat{sl}_2$, called admissible representations. In this paper we prove that the vertex operator algebra $L(l,0)$ associated to irreducible highest weight representation of $l$ is not rational if $l$ is not a positive integer. However if we change the Virasoro algebra in certain way, $L(l,0)$ becomes a rational ve…
In 1993, Schellekens obtained a list of possible 71 Lie algebras of holomorphic vertex operator algebras with central charge 24. However, not all cases are known to exist. The aim of this article is to construct new holomorphic vertex operator algebras using the theory of framed vertex operator algebras and to determine the Lie algebra structures of their weight one subspaces. In particular, we study holomorphic framed vertex operator algebras associated to subcodes of the triply even codes $\RM(1,4)^3$ and $\RM(1,4)\oplus \EuD(d_{16}^+)$ of length 48. These vertex operator algebras correspond…
Three examples of free field constructions for the vertex operators of the elliptic quantum group ${\cal A}_{q,p}(\hat{sl}_2)$ are obtained. Two of these (for $p^{1/2}=\pm q^{3/2},p^{1/2}=-q^2$) are based on representation theories of the deformed Virasoro algebra, which correspond to the level 4 and level 2 $Z$-algebra of Lepowsky and Wilson. The third one ($p^{1/2}=q^{3}$) is constructed over a tensor product of a bosonic and a fermionic Fock spaces. The algebraic structure at $p^{1/2}=q^{3}$, however, is not related to the deformed Virasoro algebra. Using these free field constructions, an …
Why do the symmetries of the monster vertex algebra form a finite simple group?
本文利用顶点算子代数新进展与经典群论，证明了怪物顶点代数的自同构群是有限单群，从而为 Frenkel-Lepowsky-Meurman 构造提供了更直接的论证。
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