T. Abe, G. Buhl, C. Dong
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.
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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…
$C_2$-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 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…
It is proved that the regularity of parafermion vertex operator algebras associated to integrable highest weight modules for affine Kac-Moody algebra A_1^{(1)} implies the C_2-cofiniteness of parafermion vertex operator algebras associated to integrable highest weight modules for any affine Kac-Moody algebra. In particular, the parafermion vertex operator algebra associated to an integrable highest weight module of small level for any affine Kac-Moody algebra is C_2-cofinite and has only finitely many irreducible modules. Also, the parafermion vertex operator algebras with level 1 are determin…
$C_2$ 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.
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.
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.
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.
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 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.
Deformation rigidity of some simple affine VOAs
本文证明了某些简单仿射VOA在特定条件下具有变形刚性，并推翻了$C_2$-余有限性和理性作为必要条件的假设。
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.
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。
The rank 1 bosonic ghost vertex algebra, also known as the $βγ$ ghosts, symplectic bosons or Weyl vertex algebra, is a simple example of a conformal field theory which is neither rational, nor $C_2$-cofinite. We identify a module category, denoted category $\mathscr{F}$, which satisfies three necessary conditions coming from conformal field theory considerations: closure under restricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second of these conditions, with the other two already being known. Further, we show that category $\ma…
Affine and Lattice Structures in Regular $\mathbb{N}$-Graded Vertex Operator Algebras with Gorenstein $V_0$
本文研究缺少CFT型条件时正则N-分次顶点算子代数中仿射与格结构的保持，证明在若干假设下存在共形嵌入的格顶点算子代数，并指出正则性不足以保证半单性。
Vertex operator algebra and parenthesized braid operad
研究顶点算子代数在构型空间上的共形块与括号化辫子操作子的兼容性，证明C1-余有限模构成伪辫子范畴，无需有理或C2-余有限假设。
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。
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