Generative AI for Brane Configurations, Tropical Coamoeba and 4d N=1 Quiver Gauge Theories

We introduce a generative AI model to obtain Type IIB brane configurations that realize toric phases of a family of 4d N=1 supersymmetric gauge theories. These 4d N=1 quiver gauge theories are worldvolume theories of a D3-brane probing a toric Calabi-Yau 3-fold. The Type IIB brane configurations that realize this family of 4d N=1 theories are known as brane tilings and are given by the tropical coamoeba projection of the mirror curve associated with the toric Calabi-Yau 3-fold. The shape of the mirror curve and its coamoeba projection, as well as the corresponding Type IIB brane configuration and the toric phase of the 4d N=1 theory, all depend on the complex structure moduli parameterizing the mirror curve. We train a generative AI model, a conditional variational autoencoder (CVAE), that takes a choice of complex structure moduli as input and generates the corresponding tropical coamoeba. This enables us not only to obtain a high-resolution representation of the entire phase space for a family of brane tilings corresponding to the same toric Calabi-Yau 3-fold, but also to continuously track the movements of the mirror curve and individual branes in the corresponding Type IIB brane configurations during phase transitions associated with Seiberg duality.