Brane tilings and reflexive polygons (Q2901184)

The paper reports progresses in the study of \(N=1\) supersymmetric gauge theories defined on D-branes that probe singular non-compact Calabi-Yau 3-folds. The matter content of the 4-dimensional world-volume theories is encoded in a graph known as a quiver. A subset of these theories possesses mesonic moduli spaces which are toric and associated to convex lattice polygons (known as toric diagrams of the Clabi-Yau singularity). Reflexive polygons are associated with mirror-paired Calabi-Yau manifolds and of great interest in string theory. This paper identifies all 30 4-dimensional supersymmetric quiver gauge theories whose moduli space is represented by one of the 16 reflexive polygons. It uses the method of brane tilings which combine the matter content and the superpotential of the quiver theory on a periodic graph on \(\mathbb T^2\). Some of the theories are toric (Seiberg) dual to each other, and this duality corresponds to the duality of the associated reflexive polygons.