The renormalization of volume and Chern-Simons invariant for hyperbolic 3-manifolds

TL;DR

This paper develops a method to renormalize the Chern-Simons invariant for hyperbolic 3-manifolds, revealing its asymptotic behavior and connection to complex volume involving torsion and mean curvature.

Contribution

It introduces a new renormalization approach for the Chern-Simons invariant using asymptotics along equidistance foliations, linking it to complex volume and geometric invariants.

Findings

Asymptotic expansion of the Chern-Simons invariant for hyperbolic 3-manifolds.

Identification of an exponentially divergent term involving torsion 2-form.

Definition of a complex-valued quantity combining mean curvature and torsion on embedded surfaces.

Abstract

We renormalize the Chern-Simons invariant for convex-cocompact hyperbolic 3-manifolds by finding the asymptotics along an equidistance foliation. We prove that the metric Chern-Simons invariant has an exponentially divergent term given by the integral of the torsion 2-form with respect to a Weitzenb\"ock connection. This produces the asymptotics of hyperbolic volume plus the metric Chern-Simons invariant, which is often called complex volume. The leading coefficient of the asymptotics introduces a complex-valued quantity consisting of mean curvature and torsion 2-form, which is defined on smooth surfaces embedded in a Riemann-Cartan 3-manifold.