On the Origin of Toric Diagrams

Abstract

Five-dimensional superconformal field theories ($5d$ SCFTs) can be encoded by Generalized Toric Polygons (GTPs), where external legs of the dual $(p,q)$ five-brane web correspond to $T$-cones. Hanany-Witten transitions act on these geometries by flipping $T$-cones about their apex, thereby naturally endowing the choice of origin in the polygon with physical significance. It was recently conjectured that a suitably graded Hilbert series equals the Ehrhart series of the dual polytope, which, in turn, is an invariant under such mutations. In this paper, we introduce a prescription for assigning scaling dimensions to fields in the toric gauge theory associated with the underlying toric diagram and show that the resulting Hilbert series of the coherent component of the moduli space matches the geometric Hilbert series given by the Ehrhart series of the dual polytope once an origin is specified. We validate our construction through several non-trivial examples, including cases with multiple admissible choices of origin leading to distinct GTPs and brane-web realizations. Our results provide evidence that ordinary brane tilings retain non-trivial information about generalized toric polygons and suggest the existence of a deeper combinatorial structure underlying GTPs.