Consistency and Derangements in Brane Tilings

TL;DR

This paper explores the mathematical conditions, specifically derangements, necessary for the geometric consistency of brane tilings, which model supersymmetric gauge theories on D3-branes probing toric Calabi-Yau singularities.

Contribution

It introduces a permutation-based framework to characterize consistency conditions in brane tilings, linking geometric constraints to derangements in permutation pairs.

Findings

Consistency conditions restrict permutation products to have no one-cycles.

Formulation of counting formulas for consistent brane tilings.

Illustration of the permutation approach with known brane tilings.

Abstract

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. Permutations without one-cycles are known as derangements. We illustrate this formulation of consistency with known brane tilings. Counting formulas for consistent brane tilings with an arbitrary number of chiral bifundamental fields are written down in terms of delta functions over symmetric groups.