A geometric recipe for twisted superpotentials

We give a pedagogical introduction to spectral networks and abelianization, as well as their relevance to $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ N = 2 field theories of class $$ \mathcal{S} $$ 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 E-expansion of the effective twisted superpotential for the E6 Minahan-Nemeschansky theory.