D-branes on Three-dimensional Nonabelian Orbifolds

TL;DR

The paper addresses D3-branes on nonabelian orbifolds $\mathbb{C}^3/\Gamma$ with $\Gamma \in \{\Delta(3n^2),\Delta(6n^2)\}$ by deriving explicit quiver gauge theories that encode the worldvolume spectrum. It develops the orbifold projection formalism and, for each family, provides tensor-product rules that fix the matter content and quiver structure in terms of irreducible representations of the groups, including special cases when $n$ is or is not divisible by 3. The main contributions are the explicit gauge groups and multiplicities for all $n$, together with compact tensor-product expressions that generate the quiver diagrams and their arrows, revealing a lattice-like and brane-web–style organization. The work is significant for constructing and understanding $\mathcal{N}=1$ gauge theories on nonabelian orbifolds, with potential implications for AdS/CFT-inspired dualities, brane configurations, and connections to the McKay correspondence.

Abstract

We study D-branes on a three complex dimensional nonabelian orbifold ${\bf C}^3/Γ$ with $Γ$ a finite subgroup of SU(3). We present general formulae necessary to obtain quiver diagrams which represent the gauge group and the spectrum of the D-brane worldvolume theory for dihedral-like subgroups $Δ(3n^2)$ and $Δ(6n^2)$. It is found that the quiver diagrams have a similar structure to webs of branes.

Figures (2)

• Figure 1: Parts of the quiver diagram for ${\bf C}^3/\Delta(3n^2)$.
• Figure 2: Parts of the quiver diagram for ${\bf C}^3/{\bf Z}_n \times {\bf Z}_n$.