Spectral Networks and Non-abelianization

TL;DR

This paper extends the non-abelianization process to all reductive algebraic groups, connecting moduli spaces of local systems via spectral networks and quadratic differentials, enriching the geometric understanding of these structures.

Contribution

It generalizes the non-abelianization map to arbitrary reductive groups and interprets spectral networks through quadratic differentials, broadening the scope of previous work.

Findings

Generalization of non-abelianization to all reductive groups

Spectral networks related to quadratic differentials

Descriptions of generic spectral network behavior

Abstract

We generalize the non-abelianization of Gaiotto-Moore-Neitzke from the case of $SL(n)$ and $GL(n)$ to arbitrary reductive algebraic groups. This gives a map between a moduli space of certain $N$-shifted weakly $W$-equivariant $T$-local systems on an open subset of a cameral cover $\tilde{X}\rightarrow X$ to the moduli space of $G$-local systems on a punctured Riemann surface $X$. For classical groups, we give interpretations of these moduli spaces using spectral covers. Non-abelianization uses a set of lines on the Riemann surface $X$ called a spectral network, defined using a point in the Hitchin base. We show that these lines are related to trajectories of quadratic differentials on quotients of $\tilde{X}$. We use this to describe some of the generic behaviour of lines in a spectral network.