The volume and Chern-Simons invariant of a representation

Abstract

We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic $\mathrm{PSL}(2,C)$-representation of a tame $3$-manifold. If the representation is the geometric representation of a hyperbolic $3$-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal triangulation with no use of additional combinatorial topology. In particular, the Chern-Simons invariant is computed just as easily as the volume