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A hyperbolic 3-manifold M carries a flat PSL(2;C)-connection whose Chern-Simons invariant has been much studied since the early 1980's.  For example, its real part is the volume of M.  Explicit formulas in terms of a triangulation involve the dilogarithm.  In joint work with Andy Neitzke we use 3-dimensional spectral network to abelianize the computation of complex Chern-Simons invariants. The locality of the Chern-Simons invariant, expressed in the language of topological field theory, plays an important role.  The dilogarithm arises from a novel construction involving Chern-Simons invariants of flat C*-connections over a 2-torus.

GT Program Seminar: Complex Chern-Simons Invariants Of 3-Manifolds Via Abelianization