Inverse algorithm and M2-brane theories

TL;DR

This paper develops an inverse algorithm to derive quiver gauge theories for M2-branes on toric Calabi-Yau 4-folds, successfully identifying theories for all 18 toric Fano 3-folds, including those previously inaccessible.

Contribution

It introduces a systematic inverse algorithm to construct quiver gauge theories from toric data, covering all toric Fano 3-folds, including cases lacking dimer tiling representations.

Findings

Successfully obtained quiver theories for all 18 toric Fano 3-folds.

Identified that some quiver theories lack dimer tiling presentations.

Extended the understanding of M2-brane gauge theories on complex cones over toric Fano 3-folds.

Abstract

Recent paper arXiv:1103.0553 studied the quiver gauge theories on coincident $M2$ branes on a singular toric Calabi-Yau 4-folds which are complex cone over toric Fano 3-folds. There are 18 toric Fano manifolds but only 14 toric Fano were obtained from the forward algorithm. We attempt to systematize the inverse algorithm which helps in obtaining quiver gauge theories on $M2$-branes from the toric data of the Calabi-Yau 4-folds. In particular, we obtain quiver gauge theories on coincident $M2$-branes corresponding to the remaining 4 toric Fano 3-folds. We observe that these quiver gauge theories cannot be given a dimer tiling presentation.