Unsupervised machine learning techniques for exploring tropical coamoeba, brane tilings and Seiberg duality
Rak-Kyeong Seong
Abstract
We introduce unsupervised machine learning techniques in order to identify toric phases of $4d$ $\mathcal{N}=1$ supersymmetric gauge theories corresponding to the same toric Calabi-Yau 3-fold. These $4d$ $\mathcal{N}=1$ supersymmetric gauge theories are world volume 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 and $t$-distributed stochastic neighbor embedding, we can project the space of coamoeba labeled 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 ${F}_{0}$.