Spectral Networks And Non-Abelianization For Reductive Groups

Non-abelianization was introduced in [16] as a way to study the moduli space of local systems of n-dimensional vector spaces on a Riemann surface X. This thesis, which is based on the forthcoming paper [23], explains how to generalize non-abelianization to the setting of G-local systems, for any reductive Lie group G. The main tool used to achieve this goal is a graph on X called a spectral network. These graphs have been introduced in [16] for groups of type A, and extended in [27] to groups of type ADE. We construct spectral networks for all reductive G, using a branched cover of X called a cameral cover, which is, in general, different from the spectral cover used in previous work on the subject. Our framework emphasizes the relationship between spectral networks and the trajectories of quadratic differentials, which provides a strategy to prove genericity results about spectral networks. Finally, we show how to associate, in an equivariant fashion, unipotent automorphisms called Stokes factors to edges of a spectral network. We define non-abelianization as a "cut and reglue" construction: we cut along the spectral network and reglue using the Stokes factors. Our construction, unlike the one in [16], does not rely on choices of trivializations for the local systems or for the branched cover.