Surveys in Differential Geometry

Volume 26 (2021)

3d spectral networks and classical Chern–Simons theory

Pages: 51 – 155

DOI: https://dx.doi.org/10.4310/SDG.2021.v26.n1.a4

Authors

Daniel S. Freed (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Andrew Neitzke (Department of Mathematics, Yale University, New Haven, Connecticut, U.S.A.)

Abstract

$\def\SL{\mathrm{SL}_2 \mathbb{C}}$We define the notion of spectral network on manifolds of dimension $\leq 3$. For a manifold $X$ equipped with a spectral network, we construct equivalences between Chern–Simons invariants of flat $\SL$-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 $\SL$-bundles over triangulated $3$-manifolds, and an explicit description of Chern–Simons lines of flat $\SL$-bundles over triangulated surfaces. Our constructions heavily exploit the locality of Chern–Simons invariants, expressed in the language of extended (invertible) topological field theory.

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To Chern, who taught us all

This material is based upon work supported by the National Science Foundation under Grant Numbers DMS-1611957, DMS-1711692, DMS-2005286, DMS-2005312, and by the Simons Fellowship in Mathematics. This work was initiated at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.

Published 22 January 2024