Multipartite Generating Functions and Infinite Products for Quantum Invariants

TL;DR

This paper develops a mathematical framework connecting multipartite generating functions, Bell polynomials, and Ruelle spectral functions to derive infinite-product formulas for quantum invariants like Chern-Simons partition functions and knot invariants.

Contribution

It introduces a novel approach linking hyperbolic geometry, spectral functions, and quantum invariants through infinite-product formulas and modular properties.

Findings

Derived infinite-product formula for Chern-Simons partition functions

Connected spectral functions with knot invariants construction

Demonstrated modular properties in the infinite-product structure

Abstract

We show that multipartite generation functions can be written in terms of the Bell polynomials (known as Fa\`a di Bruno's formula) and the Ruelle spectral functions, whose spectrum is encoded in the Patterson-Selberg function of the hyperbolic three-geometry. We derive an infinite-product formula for the Chern-Simons partition functions and analyze appropriate q-series which leads to the construction of knot invariants. With the help of the Ruelle spectral functions symmetric and modular properties in infinite-product structure can be described.