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.
Highlights
On the other hand, the Chern–Simons partition function is a generating series of quantum group invariants weighted by S-functions
On the basis of the Labastida–Mariño–Ooguri– Vafa conjecture we analyze a representation of the Chern– Simons quantum partition function in the form of an infinite product weighted by S-functions and Selberg-type functions
Recall that the Chern–Simons theory has been conjectured to be equivalent to a topological string theory 1/N expansion in physics
Summary
Chern character allows one to map the analytical Dirac index in terms of K-theory classes into a topological index which can be expressed in terms of cohomological characteristic classes. Citing [8], the appropriate class of Riemannian manifolds for which a result of this type can be expected is that of non-positively curved locally symmetric manifolds, while the class of self-adjoint operators whose eta invariants are interesting to compute is that of Dirac-type operators, even with additional coefficients in locally flat bundles. It is one of the purpose of this paper.
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