Brane Tilings and Reflexive Polygons

TL;DR

The paper classifies 30 four-dimensional ${ m N}=1$ quiver gauge theories whose mesonic moduli spaces correspond to the 16 reflexive polygons, using brane tilings and the forward algorithm to connect quiver data with toric diagrams. By computing mesonic Hilbert series and applying plethystics, it identifies the generator lattices and shows they are dual to the toric diagrams, establishing a generator–toric duality for all phases. A novel specular duality is proposed, swapping external and internal toric points and pairing the 30 theories into duals; this reveals a deeper structure in the moduli spaces beyond standard Seiberg duality. The work provides a comprehensive atlas of generator data, dualities, and Hilbert-series structures across all reflexive-polygon phases, with implications for mirror symmetry and AdS/CFT realizations of toric Calabi–Yau moduli spaces.

Abstract

Reflexive polygons have attracted great interest both in mathematics and in physics. This paper discusses a new aspect of the existing study in the context of quiver gauge theories. These theories are 4d supersymmetric worldvolume theories of D3 branes with toric Calabi-Yau moduli spaces that are conveniently described with brane tilings. We find all 30 theories corresponding to the 16 reflexive polygons, some of the theories being toric (Seiberg) dual to each other. The mesonic generators of the moduli spaces are identified through the Hilbert series. It is shown that the lattice of generators is the dual reflexive polygon of the toric diagram. Thus, the duality forms pairs of quiver gauge theories with the lattice of generators being the toric diagram of the dual and vice versa.

Figures (50)

• Figure 1: The $16$ convex polygons which are reflexive. The polygons have been $GL(2,\mathbb{Z})$ adjusted to reflect the duality under (\ref{['es00_20']}). The green internal points are the origins. $G$ is the area of the polygon with the smallest lattice triangle having normalized area $1$, and $n_G$ is the number of extremal points which are in black. The $4$ polygons with $G=6$ are self-dual. The paired polygons in 8 and 10 are $GL(2,\mathbb{Z})$ equivalent and are each others dual polygon.
• Figure 2: The $16$ reflexive polygons as toric diagrams for $30$ brane tilings. The $16$ polygons have been $GL(2,\mathbb{Z})$ transformed to illustrate the blow down from $\mathbb{C}^3/\mathbb{Z}_{4} \times\mathbb{Z}_{4}~(1,0,3)(0,1,3)$ whose toric diagram contains all 16 reflexive polygons. Each polygon is labelled by $(G|n_p:n_i|n_W)$, where $G$ corresponds to the number of $U(n)$ gauge groups, $n_p$ to the number of GLSM fields with non-zero R-charge (number of extremal points in the toric diagram or just the order of the polygon), $n_i$ to the multiplicity of the single interior point of the toric diagram, and $n_W$ to the number of superpotential terms. A reflexive polygon can correspond to multiple quiver gauge theories which are related by toric (Seiberg) duality and distinguished via $n_i$ and $n_W$.
• Figure 3: The quiver for phase b of the Hirzebruch $\mathbb{F}_0$ model. Vertices $1$ and $3$ share the same incidence information with no matter fields between them. They are combined into a block. All matter fields intersecting the block are colored red and are combined such that a red arrow represents all possible connections from and to all vertices within the block.
• Figure 4: The quiver, toric diagram, and brane tiling of Model 1. The red arrows in the quiver indicate all possible connections between blocks of nodes.
• Figure 5: The quiver, toric diagram, and brane tiling of Model 2.