We introduce unsupervised machine learning techniques in order to identify
toric phases of 4d N=1 supersymmetric gauge theories corresponding to the same
toric Calabi-Yau 3-fold. These 4d N=1 supersymmetric gauge theories are
worldvolume theories of a D3-brane probing a toric Calabi-Yau 3-fold and are
realized in terms of a Type IIB brane configuration known as a brane tiling. It
corresponds to the skeleton graph of the coamoeba projection of the mirror
curve associated to the toric Calabi-Yau 3-fold. When we vary the complex
structure moduli of the mirror Calabi-Yau 3-fold, the coamoeba and the
corresponding brane tilings change their shape, giving rise to different toric
phases related by Seiberg duality. We illustrate that by employing techniques
such as principal component analysis (PCA) and t-distributed stochastic
neighbor embedding (t-SNE), we can project the space of coamoeba labelled by
complex structure moduli down to a lower dimensional phase space with phase
boundaries corresponding to Seiberg duality. In this work, we illustrate this
technique by obtaining a 2-dimensional phase diagram for brane tilings
corresponding to the cone over the zeroth Hirzebruch surface F0.

通过应用无监督机器学习技术，我们识别出与同一拓扑 Calabi-Yau 3 - 褶对应的 4d N=1 超对称规范理论的扭曲相。这些 4d N=1 超对称规范理论是探测扭曲 Calabi-Yau 3 - 褶的 D3 - 膜的世界卷理论，通过一种被称为膜铺砌的 Type IIB 膜构型来实现。它对应于与扭曲 Calabi-Yau 3 - 褶关联的镜像曲线的余有壳图。当我们变动镜像 Calabi-Yau 3 - 褶的复结构调制度时，余有壳和相应的膜铺砌会改变形状，产生由 Seiberg 二重性相关的不同的扭曲相。通过采用主成分分析（PCA）和 t - 分布随机近邻嵌入（t-SNE）等技术，我们可以将由复结构调制度标记的余有壳空间投影到具有相应 Seiberg 二重性相界的低维相空间上。在此工作中，我们通过获取与零 Hirzebruch 曲面 F0 对应的膜铺砌的二维相图来说明这种技术。