Genus 2 Seiberg-Witten curves for rank 2 \( \mathcal{N} \) =4 superYang-Mills theories

We determine new genus 2 Seiberg-Witten curves for four dimensional rank 2 absolute $\mathcal{N}$ \( \mathcal{N} \) =4 superYang-Mills theories using the automorphism twist approach. The conformal manifolds of these curves agree with those predicted by S-duality orbits of global structures, and we use this to identify which of the two S-duality orbits of the $\mathfrak{so}\left(5\right)\simeq \mathfrak{sp}\left(4\right)$ \( \mathfrak{so}(5)\simeq \mathfrak{sp}(4) \) superYang-Mills theory the genus-2 curve corresponds to. We also compare the curves to earlier constructions of Seiberg-Witten curves for these theories as spectral curves of integrable systems. These spectral curves have genus greater than the rank, and so only give a Coulomb branch geometry upon projection to a sublattice of the homology lattice of the curves. We show how to determine the correct sublattice projection, and find that the integrable system curves do not apply to our theories.