D-Brane Gauge Theories from Toric Singularities and Toric Duality

TL;DR

The paper develops a canonical algorithm to derive D-brane world-volume gauge theories from toric singularities by leveraging partial resolutions of Abelian orbifolds. It presents a dual-track framework: a forward procedure that converts gauge theory data into toric data, and an inverse procedure that reconstructs gauge theory data from toric diagrams, using matrices like Δ, Q_t, and K to capture D- and F-term structure. The authors apply the method to toric del Pezzo surfaces and the zeroth Hirzebruch surface, extracting explicit quivers and superpotentials, and they demonstrate that multiple distinct theories can share the same toric moduli space, a phenomenon they term toric duality. The work provides tools to classify IR universality classes of gauge theories arising on D-branes and establishes a bridge between toric geometry and SUSY gauge dynamics with potential extensions to non-Abelian setups. These techniques offer a systematic way to navigate the inverse problem and illuminate the landscape of gauge theories with identical toric descriptions.

Abstract

Via partial resolution of Abelian orbifolds we present an algorithm for extracting a consistent set of gauge theory data for an arbitrary toric variety whose singularity a D-brane probes. As illustrative examples, we tabulate the matter content and superpotential for a D-brane living on the toric del Pezzo surfaces as well as the zeroth Hirzebruch surface. Moreover, we discuss the non-uniqueness of the general problem and present examples of vastly different theories whose moduli spaces are described by the same toric data. Our methods provide new tools for calculating gauge theories which flow to the same universality class in the IR. We shall call it ``Toric Duality.''

Figures (5)

• Figure 1: The toric diagram for the singularity ${{\rm C} }^3/({{\rm Z{}Z} }_2 \times {{\rm Z{}Z} }_2)$ and the quiver diagram for the gauge theory living on a D-brane probing it. We have labelled the nodes of the toric diagram by columns of $G_t$ and those of the quiver, with the gauge groups $U(1)_{\{A,B,C,D\}}$.
• Figure 2: The toric diagram showing the resolution of the ${{\rm C} }^3 / ({{\rm Z{}Z} }_2 \times {{\rm Z{}Z} }_2)$ singularity to the suspended pinch point (SPP). The numbers $i$ at the nodes refer to the $i$-th column of the matrix $G_t$ and physically correspond to the fields $p_i$ in the linear $\sigma$-model.
• Figure 3: The quiver diagram showing the matter content of a D-brane probing the SPP singularity. We have not marked in the chargeless field $\phi$ (what in a non-Abelian theory would become an adjoint) because thus far the toric techniques do not yet know how to handle such adjoints.
• Figure 4: The resolution of the Gorenstein singularity ${{\rm C} }^3/({{\rm Z{}Z} }_3 \times {{\rm Z{}Z} }_3)$ to the three toric del Pezzo surfaces as well as the zeroth Hirzebruch surface. We have labelled explicitly which columns (linear $\sigma$-model fields) are to be associated to each node in the toric diagrams and especially which columns are to be eliminated (fields acquiring non-zero VEV) in the various resolutions. Also, we have labelled the nodes of the parent toric diagram with the coordinates as given in the matrix $G_t$ for ${{\rm C} }^3/({{\rm Z{}Z} }_3 \times {{\rm Z{}Z} }_3)$.
• Figure 5: The quiver diagrams for the matter content of the brane world-volume gauge theory on the 4 toric del Pezzo singularities as well as the zeroth Hirzebruch surface. We have specifically labelled the $U(1)$ gauge groups (A, B, ..) and the bi-fundamentals (1, 2, ..) in accordance with our conventions in presenting the various matrices $Q_t$, $\Delta$ and $K$. As a reference we have also included the quiver for the parent ${{\rm Z{}Z} }_3 \times {{\rm Z{}Z} }_3$ theory.