Elliptic loci of SU(3) vacua

The space of vacua of many four-dimensional, 𝒩=2 supersymmetric gauge theories can famously be identified with a family of complex curves. For gauge group SU(2), this gives a fully explicit description of the low-energy effective theory in terms of an elliptic curve and associated modular fundamental domain. The two-dimensional space of vacua for gauge group SU(3) parametrizes an intricate family of genus two curves. We analyse this family using the so-called Rosenhain form for these curves. We demonstrate that two natural one-dimensional subloci of the space of SU(3) vacua, 𝓔_u and 𝓔_v, each parametrize a family of elliptic curves. For these elliptic loci, we describe the order parameters and fundamental domains explicitly. The locus 𝓔_u contains the points where mutually local dyons become massless and is a fundamental domain for a classical congruence subgroup. Moreover, the locus 𝓔_v contains the superconformal Argyres–Douglas points and is a fundamental domain for a Fricke group.