A geometric recipe for twisted superpotentials

We give a pedagogical introduction to spectral networks and abelianization, as well as their relevance to $\mathcal{N}=2$ supersymmetric field theories in four dimensions. Motivated by a conjecture of Nekrasov-Rosly-Shatashvili, we detail a geometric recipe for computing the effective twisted superpotential for $\mathcal{N}=2$ field theories of class $\mathcal{S}$ as a generating function of the brane of opers, with respect to the spectral coordinates found from abelianization. We present two new examples, the simplest Argyres-Douglas theory and the pure $SU(2)$ gauge theory, while we conjecture the $\epsilon$-expansion of the effective twisted superpotential for the $E_6$ Minahan-Nemeschansky theory.
Comment: 102 pages, 42 figures; v2: references added