$S$ -函数、双曲几何的谱函数和顶点算子及其在Weyl和正交群不变量结构中的应用
$S$-Functions, Spectral Functions of Hyperbolic Geometry, and Vertex Operators with Applications to Structure for Weyl and Orthogonal Group Invariants
摘要：
本文分析了量子同调不变量（$\mathfrak{sl}_N$链同调的庞加莱多项式）。在适当拓扑空间的同调维数精确已知的情况下，基于欧拉-庞加莱公式的Kovanov-Rozansky型同调的计算过程可以明显简化。我们用双曲几何的对称函数和谱函数来表示经典群的不可约张量表示的形式特征。在Labastida-Mariño-Ooguri-Vafa猜想的基础上，我们得到了Chern-Simons配分函数关于Ruelle谱函数的无穷乘积形式的表示（考虑了纽结、非纽结和链环的情况）.我们还导出了正交Chern-Simons配分函数的无穷积公式，并分析了无穷积结构的奇异性和对称性。
英文摘要：
In this paper we analyze the quantum homological invariants (the Poincaré polynomials of the $\mathfrak{sl}_N$ link homology). In the case when the dimensions of homologies of appropriate topological spaces are precisely known, the procedure of the calculation of the Kovanov-Rozansky type homology, based on the Euler-Poincaré formula can be appreciably simplified. We express the formal character of the irreducible tensor representation of the classical groups in terms of the symmetric and spectral functions of hyperbolic geometry. On the basis of Labastida-Mariño-Ooguri-Vafa conjecture, we derive a representation of the Chern-Simons partition function in the form of an infinite product in terms of the Ruelle spectral functions (the cases of a knot, unknot, and links have been considered). We also derive an infinite-product formula for the orthogonal Chern-Simons partition functions and analyze the singularities and the symmetry properties of the infinite-product structures.
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