Chern–Simons invariants on hyperbolic manifolds and topological quantum field theories

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 the Labastida–Mariño–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.