3d spectral networks and classical Chern-Simons theory

Abstract

We define the notion of spectral network on manifolds of dimension $\le 3$. For a manifold $X$ equipped with a spectral network, we construct equivalences between Chern-Simons invariants of flat $\mathrm{SL}(2,\mathbb{C})$-bundles over $X$ and Chern-Simons invariants of flat ${\mathbb{C}}^{\times }$-bundles over ramified double covers $\tilde{X}$. Applications include a new viewpoint on dilogarithmic formulas for Chern-Simons invariants of flat $\mathrm{SL}(2,\mathbb{C})$-bundles over triangulated 3-manifolds, and an explicit description of Chern-Simons lines of flat $\mathrm{SL}(2,\mathbb{C})$-bundles over triangulated surfaces. Our constructions heavily exploit the locality of Chern-Simons invariants, expressed in the language of extended (invertible) topological field theory.

97 pages, 27 figures