Machine Learning Toric Duality in Brane Tilings

TL;DR

A fully connected neural network is trained to identify classes of Seiberg dual theories realised on orbifolds of the conifold and achieves remarkably accurate results; namely, upon fixing a choice of Kasteleyn matrix representative, the regressor achieves a mean absolute error of 0.021.

Abstract

We apply a variety of machine learning methods to the study of Seiberg duality within 4d $\mathcal{N}=1$ quantum field theories arising on the worldvolumes of D3-branes probing toric Calabi-Yau 3-folds. Such theories admit an elegant description in terms of bipartite tessellations of the torus known as brane tilings or dimer models. An intricate network of infrared dualities interconnects the space of such theories and partitions it into universality classes, the prediction and classification of which is a problem that naturally lends itself to a machine learning investigation. In this paper, we address a preliminary set of such enquiries. We begin by training a fully connected neural network to identify classes of Seiberg dual theories realised on $\mathbb{Z}_m\times\mathbb{Z}_n$ orbifolds of the conifold and achieve $R^2=0.988$. Then, we evaluate various notions of robustness of our methods against perturbations of the space of theories under investigation, and discuss these results in terms of the nature of the neural network's learning. Finally, we employ a more sophisticated residual architecture to classify the toric phase space of the $Y^{6,0}$ theories, and to predict the individual gauged linear $σ$-model multiplicities in toric diagrams thereof. In spite of the non-trivial nature of this task, we achieve remarkably accurate results; namely, upon fixing a choice of Kasteleyn matrix representative, the regressor achieves a mean absolute error of $0.021$. We also discuss how the performance is affected by relaxing these assumptions.

Figures (13)

• Figure 1: The dimer model capturing the Klebanov-Witten theory, presented both as (a) a tiling of $T^2$ and (b) an infinite tessellation of $\mathbb{R}^2$. The fundamental domain spanned by the torus is drawn as a white outline.
• Figure 2: The brane tiling corresponding to the $\mathbf{dP}_{\mathbf{1}}$ theory drawn on (a) the 2-torus and (b) the infinite plane.
• Figure 3: The four toric phases on the third del Pezzo surface, $\textbf{dP}_{\boldsymbol{3}}$. For each phase, the dimer, quiver graph, toric diagram, and superpotential are shown. This figure collects results found in Beasley:2001zpFeng:2002zwFranco:2005rj. In each toric diagram, the multiplicities of the external nodes are all 1; only the multiplicity of the central node is explicitly displayed. We emphasise that, in each case, the dimer encodes all the information contained in the quiver, toric diagram, and superpotential.
• Figure 4: The GLSM multiplicities of a selection of toric phases of $Y^{6,0}$.
• Figure 5: Visual representation of our dataset generation, starting from the values of $m$ and $n$ and ending with a set of tensors, one for each Kasteleyn matrix produced via Seiberg duality.