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

Abstract

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.

arXiv comment33 pages. arXiv admin note: text overlap with arXiv:1006.4266, arXiv:1012.2636, arXiv:1310.2981, arXiv:1207.5569, arXiv:0910.0083 by other authors

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• Vector Generation Functions, q-Spectral Functions of Hyperbolic Geometry and Vertex Operators for Quantum Affine Algebras
• Moduli space of logarithmic states in critical massive gravities
• Chern-Simons Invariants on Hyperbolic Manifolds and Topological Quantum Field Theories
• The Duistermaat-Heckman formula with application to circle actions and Poincaré $q$-polynomials in twisted equivariant K-theory