Partial resolution of complex cones over Fano ${\cal{B}}$

TL;DR

This paper uses partial resolution techniques to embed toric data of simpler Calabi-Yau four-folds within more complex Fano cone geometries, aiding in understanding their associated quiver gauge theories.

Contribution

It demonstrates that the toric data of $ ext{C}^4$ and Fano $ ext{P}^3$ can be embedded into Fano ${ m B}$ theories, providing a new approach to derive quiver Chern-Simons theories via higgsing.

Findings

Toric data of $ ext{C}^4$ and $ ext{P}^3$ embedded in Fano ${ m B}$ theories.

Partial resolution justifies derivation of quiver theories from parent Fano theories.

Higgsing matter fields relates different quiver gauge theories.

Abstract

In our recent paper arXiv:1108.2387, we systematized inverse algorithm to obtain quiver gauge theory living on the M2-branes probing the singularities of special kind of Calabi-Yau four-folds which were complex cones over toric Fano $\mathbb{P}^3$, ${\cal{B}}_1$, ${\cal{B}}_2$, ${\cal{B}}_3$. These quiver gauge theories cannot be given a dimer tiling presentation. We use the method of partial resolution to show that the toric data of $\mathbb{C}^4$ and Fano $\mathbb{P}^3$ can be embedded inside the toric data of Fano ${\cal{B}}$ theories. This method indirectly justfies that the two node quiver Chern-Simons theories corresponding to $\mathbb{C}^4$, Fano $\mathbb{P}^3$ and their orbifolds can be obtained by higgsing matter fields of the three node parent quiver corresponding to Fano ${\cal{B}}_1$, ${\cal{B}}_2$, ${\cal{B}}_3$, ${\cal{B}}_4$ three-folds.