D-brane gauge theories from toric singularities of the form $C^3/Γ$ and $C^4/Γ$

TL;DR

This work investigates how D-brane worldvolume gauge theories can be reconstructed from toric data of singularities, by applying the inverse toric procedure to partial resolutions of $\mathbb{C}^3/(\mathbb{Z}_2\times \mathbb{Z}_2)$, $\mathbb{C}^3/(\mathbb{Z}_2\times \mathbb{Z}_3)$, and $\mathbb{C}^4/(\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2)$. Using the framework of homogeneous coordinates $p_\alpha$, dual cones $\sigma^\vee$, and total charge matrices $Q_t$ (and their reductions $Q_t^{\text{red}}$), the author constructs invariant variables (e.g., $x,y,z,w$ with $xyz=w^2$ and $xyzw=v^2$ in higher cases) and reads off the corresponding gauge groups and matter content, including superpotential structure when chargeless fields are absent. The results show consistency with known D-brane gauge theories for the studied blowups, while noting ambiguities in the presence of chargeless fields that prevent a unique superpotential determination; the study also discusses embeddings of toric diagrams into parent singularities and the implications for IR equivalences of distinct UV theories with the same moduli space. Overall, the work validates the utility of the inverse toric approach for extracting worldvolume gauge theory data from toric geometries and outlines directions for extending the analysis to additional higher-dimensional orbifolds and their resolutions.

Abstract

We discuss examples of D-branes probing toric singularities, and the computation of their world-volume gauge theories from the geometric data of the singularities. We consider several such examples of D-branes on partial resolutions of the orbifolds ${\bf C^3/Z_2\times Z_2}$,${\bf C^3/Z_2\times Z_3}$ and ${\bf C^4/Z_2\times Z_2 \times Z_2}$.

Figures (4)

• Figure 1: Toric diagram showing a) the ${\bf C^3/Z_2\times Z_2}$ orbifold, b) a resolution to the conifold, c) resolution to the ${\bf Z_2}$ orbifold singularity and d) a second resolution to the ${\bf Z_2}$ orbifold singularity. We have also marked the fields that remain dynamical in these resolutions.
• Figure 2: Toric diagram showing a resolution of the ${\bf C^3/Z_2\times Z_3}$ orbifold along with the choice of fields to be resolved.
• Figure 3: Toric diagram showing some resolutions of the ${\bf C^3/Z_2\times Z_3}$ orbifold that we have considered. We have marked the various fields that are chosen to remain dynamical on resolving the parent singularity to these cases.
• Figure 4: Toric data for the orbifold ${\bf C^4/Z_2\times Z_2\times Z_2}$ and two of its blowups to the ${\bf Z_2\times Z_2}$ singularity and the ${\bf Z_2}$ singularity.