Hilbert schemes, Verma modules and spectral functions of hyperbolic geometry with application to quantum invariants

TL;DR

This paper explores spectral functions, algebraic modules, and topological invariants in hyperbolic geometry, linking them to quantum invariants and analyzing their implications in mathematical physics.

Contribution

It introduces new connections between spectral functions, Verma modules, and quantum invariants, providing novel insights into hyperbolic geometry and topological quantum field theories.

Findings

Analysis of Ruelle spectral functions in hyperbolic geometry

Relation between Chern-Simons invariants and spectral functions

Description of exponential action in Chern-Simons theory

Abstract

In this article we exploit Ruelle-type spectral functions and analyze the Verma module over Virasoro algebra, boson-fermion correspondence, the analytic torsion, the Chern-Simons and $\eta$ invariants, as well as the generation function associated to dimensions of the Hochschild homology of the crossed product $\mathbb{C}[S_n]\ltimes \mathcal{A}^{\otimes n}$ ($\mathcal{A}$ is the $q$-Weyl algebra). After analysing the Chern-Simons and $\eta$ invariants of Dirac operators by using irreducible $SU(n)$-flat connections on locally symmetric manifolds of non-positive section curvature, we describe the exponential action for the Chern-Simons theory.