The rank-2 classification problem III: curves with additional automorphisms

This is the third in a series of papers which outlines an approach to the classification of $\mathcal{N}{=}2$ superconformal field theories at rank 2 via the study of their Coulomb branch geometries. Here we use the fact that the encoding of a Coulomb branch geometry as a Seiberg-Witten curve and 1-form enjoys a large reparametrisation invariance. While there is always a unique way to fix this invariance such that the curve and 1-form are single-valued over the Coulomb branch -- the "canonical frame" of the curve used in the first two papers in this series -- there are other useful frames in which the curve is single-valued but the 1-form is allowed to be multi-valued. In these frames, which we call "automorphism frames", the 1-form is periodic up to an automorphism twist. We argue that the multi-valuedness of the automorphism frame can simplify the computational complexity of finding new consistent scale invariant solutions. We demonstrate this in an example by using the automorphism frame to construct for the first time a genus 2 Seiberg-Witten curve for the $\mathcal{N}{=}4$ SU(3) superYang-Mills theory, a solution that is hard to find by other approaches.