A geometric recipe for twisted superpotentials

TL;DR

This paper introduces a geometric method to compute effective twisted superpotentials in certain supersymmetric theories using spectral networks and abelianization, providing new examples and conjectures.

Contribution

It presents a novel geometric recipe for calculating twisted superpotentials as generating functions of branes of opers, expanding the computational tools in supersymmetric field theories.

Findings

Derived explicit superpotentials for Argyres-Douglas and pure SU(2) theories.

Conjectured the epsilon-expansion for E6 Minahan-Nemeschansky theory.

Connected spectral networks with effective superpotential computations.

Abstract

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.