同调群上的弦配分函数、Hilbert格式和仿射李代数表示
String Partition Functions, Hilbert Schemes, and Affine Lie Algebra Representations on Homology Groups
摘要：
本文简要介绍了无限维李代数、顶点算子代数和点的Hilbert格式的最高权表示，以及它们在超共形量子力学和超弦模型的椭圆族中的物理应用。所有这些概念的共同联系和许多例子中考虑的文件是要找到一个非常重要的特点理论的无限维李代数：模块化的性质的字符（生成函数）的某些表示。对于相应的共形场论，最高权模的特征标表示环面上配分函数的全纯部分。本文讨论了单模群（和模群）和双曲几何的（Selberg型）Ruelle谱函数在椭圆亏格和相关的$q$ -级数计算中的作用.对于数学家来说，椭圆形属通常与空间的新数学不变量相关联，而对于物理学家来说，椭圆形属是单圈弦配分函数（因此它们适用于例如拓扑卡西米尔效应计算）。我们证明了椭圆亏格可以方便地转化为乘积表达式，从而可以继承适当的多阶李代数的同调性质。
英文摘要：
This review paper contains a concise introduction to highest weight representations of infinite dimensional Lie algebras, vertex operator algebras and Hilbert schemes of points, together with their physical applications to elliptic genera of superconformal quantum mechanics and superstring models. The common link of all these concepts and of the many examples considered in the paper is to be found in a very important feature of the theory of infinite dimensional Lie algebras: the modular properties of the characters (generating functions) of certain representations. The characters of the highest weight modules represent the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. We discuss the role of the unimodular (and modular) groups and the (Selberg-type) Ruelle spectral functions of hyperbolic geometry in the calculation of elliptic genera and associated $q$-series. For mathematicians, elliptic genera are commonly associated to new mathematical invariants for spaces, while for physicists elliptic genera are one-loop string partition function (therefore they are applicable, for instance, to topological Casimir effect calculations). We show that elliptic genera can be conveniently transformed into product expressions which can then inherit the homology properties of appropriate polygraded Lie algebras.
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