Elliptic Loci of SU(3) Vacua
Johannes Aspman, Elias Furrer, Jan Manschot

TL;DR
This paper explores specific elliptic subloci within the complex family of genus two curves describing the vacua of $SU(3)$ supersymmetric gauge theories, explicitly characterizing their order parameters and fundamental domains.
Contribution
It identifies and analyzes two elliptic subloci of the $SU(3)$ vacua space, describing their properties and associated modular groups in detail.
Findings
The sublocus $\\mathcal{E}_u$ is a fundamental domain for a classical congruence subgroup.
The sublocus $\mathcal{E}_v$ contains Argyres-Douglas points and is a fundamental domain for a Fricke group.
Explicit descriptions of order parameters and fundamental domains for these loci.
Abstract
The space of vacua of many four-dimensional, supersymmetric gauge theories can famously be identified with a family of complex curves. For gauge group , 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 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 vacua, and , each parametrize a family of elliptic curves. For these elliptic loci, we describe the order parameters and fundamental domains explicitly. The locus contains the points where mutually local dyons become massless, and is a fundamental domain for…
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