Modular invariance of characters of vertex operator algebras
Modular invariance of characters of vertex operator algebras
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DOI：
10.1090/s0894-0347-96-00182-8
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发表时间：
1995
影响因子：
3.9
通讯作者：
Yongchang Zhu
中科院分区：
文献类型：
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作者：
Yongchang Zhu
与有限维李代数不同，无限维李代数理论的一个显著特征是某些表示的模不变性。已知[Fr]， [KP]对于给定仿射李代数，在模群SL2(Z)的通常作用下，由具有固定水平的可积最高权模的特征所张成的线性空间是不变的。对于Virasoro代数的最小级数，在[Ca]和[IZ]中也观察到类似的结果。在这两种情况下，我们都使用显式字符公式来证明模不变性。仿射李代数的特征公式在[K]中计算，而Virasoro代数的特征公式本质上包含在[FF]中；参见[R]的显式计算。无限维李代数与模群之间的这种神秘联系可以用二维共形场论来解释。仿射李代数和Virasoro代数的最高权模产生了共形场论。特别是与可积最高模和最小级数相关的共形场论是有理的。在相应的共形场论中，这些模的性质被理解为环面上配分函数的全纯部分。从这个角度来看，模群SL2(Z)的作用是显而易见的。在共形场论的研究中，物理学家提出了手性代数的概念（参见[MS]）。另外，为了将Monster零星群实现为具有一定代数结构的对称群，在[FLM1]中构造了Monster零星群的无限维梯度表示，即Moonshine模块。这种代数结构后来在[Bo]中被发现，并被称为顶点代数；给出了顶点算子代数的第一个公理。在[FLM2]中证明了Moonshine模是一个顶点算子代数，Monster群是它的自同构群。值得注意的是，Moonshine模的特征也是一个模函数，即j(τ)−744。结果表明，顶点算子代数可以看作是物理文献中手性代数的一个严格的数学定义。并期望一对同构顶点算子代数及其表示（对应于全纯扇区和反全纯扇区）是建立某类型共形场论所需要的基本对象。
In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual action of the modular group SL2(Z). The similar result for the minimal series of the Virasoro algebra is observed in [Ca] and [IZ]. In both cases one uses the explicit character formulas to prove the modular invariance. The character formula for the affine Lie algebra is computed in [K], and the character formula for the Virasoro algebra is essentially contained in [FF]; see [R] for an explicit computation. This mysterious connection between the infinite dimensional Lie algebras and the modular group can be explained by the two dimensional conformal field theory. The highest weight modules of affine Lie algebras and the Virasoro algebra give rise to conformal field theories. In particular, the conformal field theories associated to the integrable highest modules and minimal series are rational. The characters of these modules are understood to be the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. From this point of view, the role of the modular group SL2(Z) is manifest. In the study of conformal field theory, physicists arrived at the notion of chiral algebras (see e.g. [MS]). Independently, in the attempt to realize the Monster sporadic group as a symmetry group of certain algebraic structure, an infinite dimensional graded representation of the Monster sporadic group, the so called Moonshine module, was constructed in [FLM1]. This algebraic structure was later found in [Bo] and called the vertex algebra; the first axioms of vertex operator algebras were formulated in that paper. The proof that the Moonshine module is a vertex operator algebra and the Monster group acts as its automorphism group was given in [FLM2]. Notably the character of the Moonshine module is also a modular function, namely j(τ) − 744. It turns out that the vertex operator algebra can be regarded as a rigorous mathematical definition of the chiral algebras in the physical literature. And it is expected that a pair of isomorphic vertex operator algebras and their representations (corresponding to the holomorphic and antiholomorphic sectors) are the basic objects needed to build a conformal field theory of a certain type.