$S$-Functions, Spectral Functions of Hyperbolic Geometry, and Vertex Operators with Applications to Structure for Weyl and Orthogonal Group Invariants

TL;DR

This paper connects hyperbolic spectral functions with quantum invariants of links, providing simplified calculations for homologies, expressing group characters via hyperbolic functions, and deriving infinite-product formulas for Chern-Simons partition functions.

Contribution

It introduces new formulas linking hyperbolic spectral functions to quantum invariants and simplifies homology calculations based on known topological data.

Findings

Expressed irreducible tensor characters using hyperbolic spectral functions

Derived infinite-product formulas for Chern-Simons partition functions

Analyzed singularities and symmetries of the spectral structures

Abstract

In this paper we analyze the quantum homological invariants (the Poincar\'e 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-Poincar\'e 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-Mari\~{n}o-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.