Elliptic Loci of SU(3) Vacua

Abstract

The space of vacua of many four-dimensional, $\mathcal{N}=2$ 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 analyze 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, ${\mathcal{E}}_u$ and ${\mathcal{E}}_v$, each parametrize a family of elliptic curves. For these elliptic loci, we describe the order parameters and fundamental domains explicitly. The locus ${\mathcal{E}}_u$ contains the points where mutually local dyons become massless, and is a fundamental domain for a classical congruence subgroup. Moreover, the locus ${\mathcal{E}}_v$ contains the superconformal Argyres-Douglas points, and is a fundamental domain for a Fricke group.

39 pages + Appendices, 5 figures, v2: minor changes and extended discussion on automorphisms, v3: minor changes, published version

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