Semiclassical approximation for Chern-Simons theory and 3-hyperbolic invariants

Abstract

The invariant integration method for Chern-Simons theory defined on the compact hyperbolic manifold Γ\H 3 is verified in the semiclassical approximation. The semiclassical limit for the partition function is presented. We discuss briefly L 2 − analytic torsion and the eta invariant of Atiyah-Patodi-Singer for compact hyperbolic 3-manifolds.

On Partition Functions of Hyperbolic Three-Geometry and Associated Hilbert Schemes

“…Let us next introduce some well-known functions and their modular properties under the action of SL(2, Z). The special cases associated with (13), (14) are (see [20]):…”

Expository remarks on three-dimensional gravity and hyperbolic invariants

“…The Chern-Simons invariant of X Γ = Γ\H 3 can be derived from the index of the Dirac operator. The following formula for the U (n)−Chern-Simons invariant of an irreducible flat connection on the real hyperbolic three-manifolds holds [8]: The analytic torsion for manifold X Γ has been calculated (in the presence of non-vanishing Betti numbers b i ≡ b i (X Γ )) in [9], and it is given by…”

Complex Topological Invariants of Three-Hyperbolic Manifolds

“…For a closed oriented hyperbolic three-manifolds of the form X Γ = Γ\H 3 and for acyclic χ the L 2 −analytic torsion gets the form [30,17,18]: [τ an (X Γ )] 2 = R χ (0), where R χ (s) is the Ruelle function. A Ruelle type zeta function for s large can be defined as the product over prime closed geodesics γ of factors det(I − ξ(γ)e −s (γ) ), where (γ) is the length of γ, and can be continued meromorphically to the entire complex plane C [22].…”

“…The analytic torsion for manifold X Γ has been calculated (in the presence of non-vanishing Betti numbers b i ≡ b i (X Γ ) in [17,18], and it is given by…”