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Vishnu Jejjala, University of the Witwatersrand

Consistency and derangements

Brane tilings describe Lagrangians of 4d $\mathcal{N}=1$ supersymmetric gauge theories. These theories, written in terms of a bipartite graph on a torus, correspond to the worldvolume theories on $D3$-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. Such permutations are known as derangements. We illustrate this formulation of consistency with known brane tilings.