Chern-Simons Invariants on Hyperbolic Manifolds and Topological Quantum Field Theories

TL;DR

This paper derives formulas for the Chern-Simons invariant on negatively curved hyperbolic 3-manifolds, linking it to Selberg functions, R-torsion, and quantum invariants, with implications for topological quantum field theories.

Contribution

It introduces new formulas for Chern-Simons invariants on hyperbolic manifolds and connects them to Selberg functions and quantum invariants, expanding understanding of topological quantum field theories.

Findings

Formulas for Chern-Simons invariants on hyperbolic 3-manifolds.

Connection between Selberg functions, R-torsion, and Dirac operators.

Representation of quantum partition functions as infinite products with symmetry properties.

Abstract

We derive formulas for the classical Chern-Simons invariant of irreducible $SU(n)$-flat connections on negatively curved locally symmetric three-manifolds. We determine the condition for which the theory remains consistent (with basic physical principles). We show that a connection between holomorphic values of Selberg-type functions at point zero, associated with R-torsion of the flat bundle, and twisted Dirac operators acting on negatively curved manifolds, can be interpreted by means of the Chern-Simons invariant. On the basis of Labastida-Marino-Ooguri-Vafa conjecture we analyze a representation of the Chern-Simons quantum partition function (as a generating series of quantum group invariants) in the form of an infinite product weighted by S-functions and Selberg-type functions. We consider the case of links and a knot and use the Rogers approach to discover certain symmetry and modular form identities.