Consistency and Derangements in Brane Tilings

Understand

Brane tilings describe Lagrangians (vector multiplets, chiral multiplets, and the superpotential) of four dimensional $\mathcal{N}=1$ supersymmetric gauge theories.

• These theories, written in terms of a bipartite graph on a torus, correspond to worldvolume theories on $N$ D$3$-branes probing a toric Calabi-Yau threefold singularity.
• A pair of permutations compactly encapsulates the data necessary to specify a brane tiling.
• We show that geometric consistency for brane tilings, which ensures that the corresponding quantum field theories are well behaved, imposes constraints on the pair of permutations, restricting certain products constructed from the pair to have no one-cycles.