3d spectral networks and classical Chern-Simons theory

TL;DR

This paper introduces spectral networks on 3-manifolds, establishing equivalences between Chern-Simons invariants of different bundle types, with applications to dilogarithmic formulas and explicit descriptions over triangulated surfaces.

Contribution

It defines spectral networks on low-dimensional manifolds and constructs equivalences between Chern-Simons invariants of flat bundles, providing new insights and explicit descriptions.

Findings

Equivalence between Chern-Simons invariants of ${ m SL}(2,{f C})$ and ${f C}^ imes$-bundles

New viewpoint on dilogarithmic formulas for 3-manifolds

Explicit description of Chern-Simons lines over triangulated surfaces

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 $\widetilde X$. 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.