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 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.

19 pages

References in corpus (4)

• Partition Functions for Quantum Gravity, Black Holes, Elliptic Genera and Lie Algebra Homologies
• Plethysms, replicated Schur functions and series, with applications to vertex operators
• $S$-Functions, Spectral Functions of Hyperbolic Geometry, and Vertex Operators with Applications to Structure for Weyl and Orthogonal Group Invariants
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Cited by in corpus (2)

• Moduli space of logarithmic states in critical massive gravities
• Hilbert schemes, Verma modules and spectral functions of hyperbolic geometry with application to quantum invariants