Elliptic Loci of SU(3) Vacua

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, $\mathcal{N}=2$ 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.