Branes at $\C^4/\Ga$ Singularity from Toric Geometry

TL;DR

This paper investigates toric singularities of the form C^4/Γ with finite abelian groups, explicitly constructs charge matrices for partial resolutions, and explores the geometric interpretation of field theory parameters and moduli space.

Contribution

It provides explicit charge matrices for resolutions of C^4/Z_2×Z_2×Z_2 orbifolds and analyzes their geometric and field theory implications.

Findings

Derived three algebraic equations parametrizing the singularity

Connected the geometric parameters to the worldvolume field theory

Analyzed the moduli space of D1 branes at the singularity

Abstract

We study toric singularities of the form of $\C^4/\Ga$ for finite abelian groups $\Ga \subset SU(4)$. In particular, we consider the simplest case $\Ga=\Z_2 \times \Z_2 \times \Z_2$ and find explicitly charge matrices for partial resolutions of this orbifold by extending the method by Morrison and Plesser. We obtain three kinds of algebraic equations, $z_1 z_2 z_3 z_4=z_5^2, z_1 z_2 z_3=z_4^2 z_5 $ and $z_1 z_2 z_5 = z_3 z_4$ where $z_i$'s parametrize $\C^5$. When we put $N$ D1 branes at this singularity, it is known that the field theory on the worldvolume of $N$ D1 branes is T-dual to $2 \times 2 \times 2 $ brane cub model. We analyze geometric interpretation for field theory parameters and moduli space.