Metaheuristic Generation of Brane Tilings
Yang-Hui He, Vishnu Jejjala, Tomás S. R. Silva
TL;DR
This work encodes 4d $\mathcal{N}=1$ quiver gauge theories from brane tilings as permutation pairs $(\sigma_B,\sigma_W)$ with $\sigma_F=(\sigma_B\sigma_W)^{-1}$ and uses simulated annealing to search for geometrically consistent tilings by turning the consistency criteria into an optimization problem. The authors define an energy function $Energy(\sigma_B,\sigma_W)=6- (\text{PT-2 Score})-(\text{PT-3a Score})-(\text{PT-3b Score})-(\text{PT-5 Score})-(\text{CONS-1 Score})-(\text{CONS-2 Score})$ and employ a cycle-preserving Move to navigate the large search space $\mathfrak{S}_d\times\mathfrak{S}_d$, enabling efficient discovery of valid tilings. They reproduce known tilings up to PT-1 equivalence and present a new $d=26$ tiling with explicit $\sigma_B,\sigma_W$, a 10-node quiver, and a 16-term superpotential, whose moduli space is a non-compact Calabi–Yau threefold with toric data and Hilbert-series information. Overall, the work demonstrates metaheuristic optimization as a viable addition to building and analyzing gauge theories from brane tilings, expanding the catalog of admissible quivers and their geometric moduli.
Abstract
The combinatorics of dimer models on brane tilings describe a large class of four-dimensional $\mathcal{N}=1$ gauge theories that afford quiver descriptions and have toric moduli spaces. We introduce a combinatorial optimization method leveraging simulated annealing to explicitly construct geometrically consistent brane tilings, providing a proof of concept for efficient generation of gauge theories using metaheuristic techniques. The implementation of this idea recovers known examples and allows us to derive a new brane tiling with $26$ quantum fields, illustrating the potential of metaheuristic techniques as a valuable addition to the toolbox for constructing and analyzing gauge theories from brane tilings.