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 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, \({\mathcal {E}}_u\) 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.

可以用一系列复杂曲线来识别许多四维\（{{mathcal {N}} = 2 \）超对称规范理论的真空空间。对于量规组SU（2），这根据椭圆曲线和相关的模块化基本域对低能效理论进行了完全明确的描述。量规组SU（3）的二维真空空间参数化了两个曲线属的复杂族。我们使用所谓的Rosenhain形式对这些曲线进行分析。我们证明了SU（3）vacua空间的两个自然一维子位置，\（{\ mathcal {E}} _ u \）和\（{{mathcal {E}} _ v \），每个参数都会构成一族椭圆曲线。对于这些椭圆基因座，我们明确描述了阶数参数和基本域。轨迹\（{\ mathcal {E}} _ U \）包含其中相互当地dyons成为无质量的点，是一个经典的一致性分组的基本领域。此外，轨迹\（{\ mathcal {E}} _ v \）包含超共形的Argyres–Douglas点，并且是Fricke组的基本域。