Toshiyuki Abe
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
It is shown that a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V) is semisimple and each irreducible admissible V-module is ordinary. A contravariant form on a Verma type admissible V-module is constructed and the radical is exactly the maximal proper submodule. As an application the rationality of V_L^+ for any positive definite even lattice is obtained.
We study the vertex algebras associated with modular invariant representations of affine Kac-Moody algebras at fractional levels, whose simple highest weight modules are classified by Joseph's characteristic varieties. We show that an irreducible highest weight representation of a non-twisted affine Kac-Moody algebra at an admissible level k is a module over the associated simple affine vertex algebra if and only if it is an admissible representation whose integral root system is isomorphic to that of the vertex algebra itself. This in particular proves the conjecture of Adamovic and Milas on …
In this paper we prove that the vertex algebra $V_L^+$ is rational if $L$ is a negative definite even lattice of finite rank, or if $L$ is a non-degenerate even lattice of a finite rank that is neither positive definite nor negative definite. In particular, for such even lattices $L$, we show that the Zhu algebras of the vertex algebras $V_L^+$ are semisimple. This extends the result of Abe which establishes the rationality of $V_L^+$ when $L$ is a positive definite even lattice of finite rank.
We prove the conjecture of Kac-Wakimoto on the rationality of exceptional W-algebras for the first non-trivial series, namely, for the Bershadsky-Polyakov vertex algebras $W_3^{(2)}$ at level $k=p/2-3$ with $p=3,5,7,...$. This gives new examples of rational conformal field theories.
This paper studies the twisted representations of vertex operator algebras. Let V be a vertex operator algebra and g an automorphism of V of finite order T. For any m,n in (1/T)Z_+, an A_{g,n}(V)-A_{g,m}(V)-bimodule A_{g,n,m}(V) is constructed. The collection of these bimodules determines any admissible g-twisted V-module completely. A Verma type admissible g-twisted V-module is constructed naturally from any A_{g,m}(V)-module. Furthermore, it is shown with the help of bimodule theory that a simple vertex operator algebra V is g-rational if and only if its twisted associative algebra A_g(V) is…
In this note we prove that the vertex operator superalgebras associated to the unitary representations of the N=2 superconformal algebra are rational.
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.
The lattice vertex operator V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from -1 isometry of L. It is established that the fixed point vertex operator algebra V_L^+ is rational.
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。
In this paper we prove the rationality of the capped vertex function with descendents for arbitrary Nakajima quiver varieties with generic stability conditions. We generalise the proof given by Smirnov to the general case, which requires to use techniques of tautological classes rather than the fixed-point basis. This result confirms that the "monodromy" of the capped vertex function is trivial, which gives a strong constraint for the monodromy of the capping operators. We will also provide a GIT wall-crossing formula for the capped vertex function in terms of the quantum difference operators …
We prove the rationality of all the minimal series principal W-algebras discovered by Frenkel, Kac and Wakimoto in 1992, thereby giving a new family of rational and C_2-cofinite vertex operator algebras. A key ingredient in our proof is the study of Zhu's algebra of simple W-algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu's algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu's algebra of all the admissible affine vertex algebras as well.
We demonstrate that, for CFT vertex operator algebras, C_2-cofiniteness and rationality is equivalent to regularity. In addition, we show that, for C_2-cofinite vertex operators algebras, irreducible weak modules are ordinary modules and C_2-cofinite, and V_L^+ are C_2-cofinite.
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。
The rationality of the parafermion vertex operator algebra associated to any finite dimensional simple Lie algebra and any nonnegative integer is established and the irreducible modules are determined.
In this paper, it is shown that the diagonal coset vertex operator algebra $C(L_{\mathfrak{g}}(k+2,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(2,0))$ is rational and $C_2$-cofinite in case $\mathfrak{g}=so(2n), n\geq 3$ and $k$ is an admissible number for $\hat{\mathfrak{g}}$. It is also shown that the diagonal coset vertex operator algebra $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ is rational and $C_2$-cofinite in case $k$ is an admissible number for $\hat{sl_2}$. Furthermore, irreducible modules of $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ are classified in c…
In vertex operator algebra theories, most of the general theorems are proved under the assumptions of rationality and C_2-cofiniteness. In this paper, we obtain several general theorems without the assumption of rationality so that we can use them for proving rationality of given C_2-cofinite vertex operator algebras. For example, we apply them to orbifold models and show that if g is a finite automorphism of a rational C_2-cofinite vertex operator algebra T of CFT-type with T'\cong T and a fixed point subVOA T^g is C_2-cofinite, then T^g is also rational.
Using the Zhu algebra for a certain category of $\mathbb{C}$-graded vertex algebras $V$, we prove that if $V$ is finitely $Ω$-generated and satisfies suitable grading conditions, then $V$ is rational, i.e. has semi-simple representation theory, with one dimensional level zero Zhu algebra. Here $Ω$ denotes the vectors in $V$ that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal element $ω_μ$ parameterized by $μ\in \mathbb{C}$, and prove that for certain non-integer values of $μ$, these vertex algebras, wh…
The rationality and C_2-cofiniteness of the orbifold vertex operator algebra V_{L_{2}}^{A_{4}} are established and all the irreducible modules are constructed and classified. This is part of classification of rational vertex operator algebras with c=1.
This paper is an exposition of the representation theory of vertex operator algebras in terms of associative algebras A_n(V) and their bimodules. A new result on the rationality is given. That is, a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V) is a semisimple associative algebra and each irreducible admissible $V$-module is ordinary.
The vertex operator algebras and modules associated to the highest weight modules for the Virasoro algebra over an arbitrary field F whose characteristic is not equal to 2 are studied. The irreducible modules of vertex operator algebra L(1/2,0)_F are classified. The rationality of L(1/2,0)_F is established if the characteristic of F is different from 7.
$C_2$ cofiniteness and rationality of $V_{L_2}^{S_4}$ are obtained, and irreducible $V_{L_2}^{S_4}$-modules are classified. With the assumption of rationality and $C_2$ cofiniteness, irreducible $V_{L_2}^{A_5}$-modules are determined. Also, quantum dimensions of these irreducible modules are calculated.
It was shown by Abe, Buhl and Dong that the vertex algebra $V_L^+$ and its irreducible weak modules satisfy the $C_2$-cofiniteness condition when $L$ is a positive definite even lattice. In this paper, we extend their results by showing that the vertex algebra $V_L^+$ and its irreducible weak modules are $C_2$-cofinite when $L$ is a negative definite even lattice and when $L$ is a non-degenerate even lattice that is neither negative definite nor positive definite.
A notion of meromorphic open-string vertex algebra is introduced. A meromorphic open-string vertex algebra is an open-string vertex algebra in the sense of Kong and the author satisfying additional rationality (or meromorphicity) conditions for vertex operators. The vertex operator map for a meromorphic open-string vertex algebra satisfies rationality and associativity but in general does not satisfy the Jacobi identity, commutativity, the commutator formula, the skew-symmetry or even the associator formula. Given a vector space \mathfrak{h}, we construct a meromorphic open-string vertex algeb…
We study representations of the meromorphic open-string vertex algebra (MOSVAs hereafter) defined in [H3], a noncommutative generalization of vertex (operator) algebra. We start by recalling the definition of a MOSVA $V$ and left $V$-modules in [H3]. Then we define right $V$-modules and $V$-bimodules that reflect the noncommutative nature of $V$. When $V$ satisfies a condition on the order of poles of the correlation function (which we call pole-order condition), we prove that the rationality of products of two vertex operators implies the rationality of products of any numbers of vertex opera…
We study modularity of the characters of a vertex (super)algebra equipped with a family of conformal structures. Along the way we introduce the notions of rationality and cofiniteness relative to such a family. We apply the results to determine modular transformations of trace functions on admissible modules over affine Kac-Moody algebras and, via BRST reduction, trace functions on minimal series representations of principal affine W-algebras.
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.
First, we prove the Kac-Wakimoto conjecture on modular invariance of characters of exceptional affine W-algebras. In fact more generally we prove modular invariance of characters of all lisse W-algebras obtained through Hamiltonian reduction of admissible affine vertex algebras. Second, we prove the rationality of a large subclass of these W-algebras, which includes all exceptional W-algebras of type A and lisse subregular W-algebras in simply laced types. Third, for the latter cases we compute S-matrices and fusion rules. Our results provide the first examples of rational W-algebras associate…
$C_2$-Cofiniteness and Rationality of the Icosahedral Orbifold $V_{L_2}^{A_5}$
本文证明二十面体群 $A_5$ 作用于秩一根格顶点算子代数 $V_{L_2}$ 的不动点子代数 $V_{L_2}^{A_5}$ 是强有理的，即满足 $C_2$-余有限性且有理。
We prove a generalized rationality property and a new identity that we call the ``Jacobi identity'' for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. Together with associativity and commutativity for intertwining operators proved by the author in \cite{H2} and \cite{H3}, the results of the present paper solve completely the problem of finding a natural purely al…
We study the subalgebra of the lattice vertex operator algebra $V_{\sqrt{2}A_2}$ consisting of the fixed points of an automorphism which is induced from an order 3 isometry of the root lattice $A_2$. We classify the simple modules for the subalgebra. The rationality and the $C_2$-cofiniteness are also established.
We reformed the tensor product theory of vertex operator algebras developed by Huang and Lepowsky so that we could apply it to all vertex operator algebras satisfying C_2-cofiniteness. We also showed that the tensor product theory develops naturally if we include not only ordinary modules, but also weak modules with a composition series of finite length (we call it an Artin module). In particular, we don't assume the semisimplicity of the weight operator L(0). Actually, without the assumption of rationality, a C_2-cofiniteness on V is enough to obtain the existence of a tensor product of two A…
We study the fixed point subalgebra of a certain class of lattice vertex operator algebras by an automorphism of order 3, which is a lift of a fixed-point-free isometry of the underlying lattice. We classify the irreducible modules for the subalgebra. Moreover, the rationality and the $C_2$-cofiniteness of the subalgebra are established. Our result contains the case of the vertex operator algebra associated with the Leech lattice.
We study $\N$-graded $ϕ$-coordinated modules for a general quantum vertex algebra $V$ of a certain type in terms of an associative algebra $\widetilde{A}(V)$ introduced by Y.-Z. Huang. Among the main results, we establish a bijection between the set of equivalence classes of irreducible $\N$-graded $ϕ$-coordinated $V$-modules and the set of isomorphism classes of irreducible $\widetilde{A}(V)$-modules. We also show that for a vertex operator algebra, rationality, regularity, and fusion rules are independent of the choice of the conformal vector.
Some of the consequences that follow from the C_2 condition of Zhu are analysed. In particular it is shown that every conformal field theory satisfying the C_2 condition has only finitely many n-point functions, and this result is used to prove a version of a conjecture of Nahm, namely that every representation of such a conformal field theory is quasirational. We also show that every such vertex operator algebra is a finite W-algebra, and we give a direct proof of the convergence of its characters as well as the finiteness of the fusion rules.
Modular Zhu algebra theory and Virasoro vertex algebras
该研究利用高阶朱代数和模转换代数发展顶点代数表示理论，给出正特征域上莫比乌斯顶点代数合理性条件，证明特定离散系列维拉索罗顶点算子代数的合理性，还研究了特征$p>2$时简单商的全纯性情况。
Twisted associative algebras associated to vertex algebras
本文构造了与顶点代数相关且不依赖共形结构的扭曲结合代数序列，并证明顶点算子代数的 $g$-有理性、$g$-正则性与扭曲融合规则独立于共形向量的选取。
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 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.
Let L be a positive definite even lattice and V_L^+ be the fixed points of the lattice VOA V_L associated to L under an automorphism of V_L lifting the -1 isometry of L. For any positive rank, the full automotphism group of V_L^+ is determined if L does not have vectors of norm 2 or 4. For any L of rank 2, a set of generators and the full automorphism group of V_L^+ are determined.
The irreducible modules for the fixed point vertex operator subalgebra V_L^+ of the vertex operator algebra V_L associated to an arbitrary positive definite even lattice L under the automorphism lifted from the -1 isometry of L are classified.
We classify the irreducible modules for the fixed point vertex operator subalgebra V_L^+ of the vertex operator algebra V_L associated to a positive definite even lattice of rank 1 under the automorphism lifted from the -1 isometry of L.
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。
In this paper we build an abstract description of vertex algebras from their basic axioms. Starting with Borcherds' notion of a vertex group, we naturally construct a family of multilinear singular maps parameterised by trees. These singular maps are defined in a way which focusses on the relations of singularities to their inputs. In particular we show that this description of a vertex algebra allows us to present generalised notions of rationality, commutativity and associativity as natural consequences of the definition. Finally, we show that for a certain choice of vertex group, axiomatic …
We consider how a vertex operator algebra can be extended to an abelian intertwining algebra by a family of weak twisted modules which are {\em simple currents} associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except for $E_8$) and to get various extensions of the vertex operator algebras associated with integrable representations.
A characterization of vertex operator algebra $V_L^+$ for any rank one positive definite even lattice $L$ is given in terms of dimensions of homogeneous subspaces of small weights. This result reduces the classification of rational vertex operator algebras of central charge 1 to the characterization of three vertex operator algebras in the $E$-series of central charge one.
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…
Let L be a rank one positive definite even lattice. We prove that a vertex operator algebra (VOA) V_L^+ satisfies the C_2 condition. Here, V_L^+ is a fixed point sub-VOA of the VOA V_L associated with the automorphism lifted from the -1 isometry of L.
A regular vertex operator algebra is a vertex operator algebra such that any weak module (without grading) is a direct sum of ordinary irreducible modules. In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the moonshine module vertex operator algebra $V^{\natural},$ the vertex operator algebras $L(l,0)$ associated with the integrable representations of affine algebras of level $l,$ the vertex operator algebras $L(c_{p,q},0)$ associated with irreducible highest weight representations for the discrete series of the Viraso…
We study minimal generating subspaces of "weak PBW type" for vertex operator algebras that satisfy a certain condition on the grading, and we develop a procedure for finding such subspaces. As applications, we derive some results on generalized modules and produce a minimal generating space of weak PBW type for the vertex operator algebra $V_L$ associated with any positive-definite even lattice $L$.
We study and classify systems of certain screening operators arising in a generalized vertex operator algebra, or more generally an abelian intertwining algebra with an associated vertex operator (super)algebra. Screening pairs arising from weight one primary vectors acting commutatively on a lattice vertex operator algebra (the vacuum module) are classified into four general types, one type of which has been shown to play an important role in the construction and study of certain important families of $\mathcal{W}$-vertex algebras. These types of screening pairs we go on to study in detail th…
It is proved that Zhu's algebra for vertex operator algebra associated to a positive-definite even lattice of rank one is a finite-dimensional semiprimitive quotient algebra of certain associative algebra introduced by Smith. Zhu's algebra for vertex operator algebra associated to any positive-definite even lattice is also calculated and is related to a generalization of Smith's algebra.
This paper consists of two parts: (1) Using a Z[1/2]-form of Virasoro vertex operator algebra L(1/2,0) with central charge 1/2, we obtain a modular vertex operator algebra over any field F of finite characteristic different from 2. We determine the generators and classify the irreducible modules for this vertex operator algebra. (2) We investigate modular framed vertex operator algebras. In particular, the rationality of modular framed vertex operator algebras is established. For a modular code vertex operator algebra, the irreducible modules are constructed and classified. Moreover, a Z[1/2]-…
Given a finite-dimensional reductive Lie algebra $\mathfrak{g}$ equipped with a nondegenerate, invariant, symmetric bilinear form $B$, let $V^k(\mathfrak{g},B)$ denote the universal affine vertex algebra associated to $\mathfrak{g}$ and $B$ at level $k$. Let $\mathcal{A}^k$ be a vertex (super)algebra admitting a homomorphism $V^k(\mathfrak{g},B)\rightarrow \mathcal{A}^k$. Under some technical conditions on $\mathcal{A}^k$, we characterize the coset $\text{Com}(V^k(\mathfrak{g},B),\mathcal{A}^k)$ for generic values of $k$. We establish the strong finite generation of this coset in full generali…
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 paper, we first classify all irreducible modules of the vertex algebra $V_L^+$ when $L$ is a negative definite even lattice of arbitrary rank. In particular, we show that any irreducible $V_L^+$-module is isomorphic to a submodule of an irreducible twisted $V_L$-module. We then extend this result to a vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of finite rank.
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 …
The 3-transposition groups that act on a vertex operator algebra in the way described by Miyamoto are classified under the assumption that the group is centerfree and the VOA carries a positive-definite invariant Hermitian form. This generalizes and refines the result of Kitazume and Miyamoto. Application to a similar but different situation is also considered in part by a slight generalization of the argument.
We apply an idea of framed vertex operator algebras to a construction of local conformal nets of (injective type III_1) factors on the circle corresponding to various lattice vertex operator algebras and their twisted orbifolds. In particular, we give a local conformal net corresponding to the moonshine vertex operator algebras of Frenkel-Lepowsky-Meurman. Its central charge is 24, it has a trivial representation theory in the sense that the vacuum sector is the only irreducible DHR sector, its vacuum character is the modular invariant J-function and its automorphism group (the gauge group) is…
For a positive-definite, even, integral lattice $L$, the lattice vertex operator algebra $V_L$ is known to be rational and $C_2$-cofinite, and thus the fusion products of its modules always exist. The fusion product of two untwisted irreducible $V_L$-modules is well-known, namely $V_{L+λ} \boxtimes_{V_L} V_{L+μ} = V_{L + λ+ μ}$. In this paper, we determine the other two fusion products: $V_{L+λ} \boxtimes_{V_L} V_L^{T_χ}$ and $V_L^{T_{χ_1}} \boxtimes_{V_L} V_L^{T_{χ_2}}$.
Let L be a positive definite even lattice of rank one and V_L^+ be the fixed points of the lattice VOA V_L associated to L under an automorphism of V_L lifting the -1$ isometry of L. A set of generators and the full automorphism group of V_L^+ are determined.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。