Spectral Networks and Non-abelianization

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 $\tilde{X}\to X$ 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.

104 pages

References in corpus (5)

• Higher Topos Theory
• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces
• On the monodromy of the deformed cubic oscillator
• Asymptotic Geometry of the Moduli Space of Parabolic $SL(2,\mathbb{C})$-Higgs Bundles