Toric Duality as Seiberg Duality and Brane Diamonds

TL;DR

The paper argues that toric duality for N=1 theories with toric moduli spaces is a manifestation of Seiberg duality. By combining field theory dualities, brane interval/diamond pictures, and partial resolutions of toric singularities such as $C^3/(Z_3\times Z_3)$, it demonstrates explicit Seiberg-dual pairs for known toric duals and derives new phases. It also develops a quiver-duality framework that mirrors Seiberg duality at the level of adjacency matrices, and compares Seiberg duality with Picard-Lefschetz monodromy, showing both connections and limitations. The work provides both a unifying physical interpretation of toric duality and practical methods to generate additional toric dual phases, including for del Pezzo surfaces. Overall, it reinforces the view that toric dualities reflect deep IR equivalences of gauge theories arising from D-branes at toric singularities.

Abstract

We use field theory and brane diamond techniques to demonstrate that Toric Duality is Seiberg duality for N=1 theories with toric moduli spaces. This resolves the puzzle concerning the physical meaning of Toric Duality as proposed in our earlier work. Furthermore, using this strong connection we arrive at three new phases which can not be thus far obtained by the so-called ``Inverse Algorithm'' applied to partial resolution of C^3/Z_3 x Z_3. The standing proposals of Seiberg duality as diamond duality in the work by Aganagic-Karch-Lüst-Miemiec are strongly supported and new diamond configurations for these singularities are obtained as a byproduct. We also make some remarks about the relationships between Seiberg duality and Picard-Lefschetz monodromy.