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

TL;DR

This work investigates toric singularities of the form ${\mathbb C}^4/\Gamma$ with finite abelian $\Gamma\subset SU(4)$, focusing on the simplest nontrivial case ${\Gamma=\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2}$. By extending Morrison–Plesser’s method, the authors construct explicit charge matrices for partial resolutions and derive three invariant algebraic relations among coordinates in ${\mathbb C}^5$: $${\ z_1 z_2 z_3 z_4 = z_5^2,\quad z_1 z_2 z_3 = z_4^2 z_5,\quad z_1 z_2 z_5 = z_3 z_4 }$$, identifying the toric structure of the singularity. Placing $N$ D1 branes leads, via T-duality, to a $2\times 2\times 2$ brane cub model; the paper analyzes the geometric interpretation of field-theory parameters and shows the moduli space is a toric variety consistent with the quotient singularity. The results connect gauge-theory data with explicit toric constructions and suggest paths to generalize to other abelian groups and to related brane configurations in higher-dimensional Calabi–Yau geometries.

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.