Symmetries of Toric Duality

TL;DR

This work identifies multiplicity symmetry in GLSM fields as the true origin of toric duality for D-branes probing toric Calabi–Yau singularities. By analyzing ${ m C}^2/{ m Z}_n$ and ${ m C}^3/({ m Z}_k imes{ m Z}_k)$ cases, the author shows that different toric phases arise from varying multiplicities on a single toric diagram, and demonstrates this by explicitly obtaining all known dual phases for $F_0$, $dP_0$–$dP_3$ via partial resolutions. The paper further connects these multiplicities to divisors and monodromy, arguing that permutation symmetries of GLSM fields generate monodromy actions that realize Seiberg duality in the toric setting, and uses flavor and node symmetries to constrain or uniquely determine superpotentials. The results suggest a unifying geometric mechanism behind toric duality, enriching the link between toric geometry, quiver gauge theories, and string-theoretic dualities, with implications for predicting dual phases from a fixed toric diagram. A concrete appendix treats the A-type singularities, clarifying how the general dual cone yields the total count of GLSM fields as $2^n+1$ and underpins the observed multiplicity patterns.

Abstract

This paper serves to elucidate the nature of toric duality dubbed in hep-th/0003085 in the construction for world volume theories of D-branes probing arbitrary toric singularities. This duality will be seen to be due to certain permutation symmetries of multiplicities in the gauged linear sigma model fields. To this symmetry we shall refer as ``multiplicity symmetry.'' We present beautiful combinatorial properties of these multiplicities and rederive all known cases of torically dual theories under this new light. We also initiate an understanding of why such multiplicity symmetry naturally leads to monodromy and Seiberg duality. Furthermore we discuss certain ``flavor'' and ``node'' symmetries of the quiver and superpotential and how they are intimately related to the isometry of the background geometry, as well as how in certain cases complicated superpotentials can be derived by observations of the symmetries alone.

Figures (11)

• Figure 1: Toric diagram of $\hbox{$\,{\rm C}$}^3/({ Z Z}_3 \times { Z Z}_3$), with the GLSM fields labelled explicitly (q.v. toric).
• Figure 2: Quiver diagrams of the two torically dual theories corresponding to the cone over the zeroth Hirzebruch surface $F_0$.
• Figure 3: Toric diagrams of the two torically dual theories corresponding to the cone over the zeroth Hirzebruch surface $F_0$, with the surviving GLSM fields indicated explicitly.
• Figure 4: Toric diagrams of the two torically dual theories corresponding to the cone over the $dP_2$, with the surviving GLSM fields indicated explicitly.
• Figure 5: Toric diagrams of the four torically dual theories corresponding to the cone over the $dP_3$, with the surviving GLSM fields indicated explicitly.