Birational Transformations and 2d (0,2) Quiver Gauge Theories beyond Toric Fano 3-folds

TL;DR

This paper explores how birational transformations relate to 2d (0,2) quiver gauge theories and toric Calabi-Yau 4-folds, revealing equivalence classes and potential for a minimal model program in this context.

Contribution

It identifies a family of birational transformations connecting toric Fano 3-folds and extends these to toric Calabi-Yau 4-folds, linking geometric transformations with gauge theory equivalences.

Findings

Birational transformations correspond to mass deformations in gauge theories.

The transformations preserve the Hilbert series of mesonic moduli spaces.

They categorize Calabi-Yau 4-folds and gauge theories into equivalence classes.

Abstract

We show that a family of birational transformations that relate toric Fano 3-folds defined by reflexive lattice polytopes can be identified with mass deformations of corresponding 2d (0,2) supersymmetric quiver gauge theories. These theories are realized by a Type IIA brane configuration known as brane brick models. We further show that the same family of birational transformations extends to more general toric Calabi-Yau 4-folds, including those defined by non-reflexive toric diagrams. Under these birational transformations, the mesonic moduli spaces of the associated abelian 2d (0,2) supersymmetric gauge theories and brane brick models share the same number of generators and the same Hilbert series when refined only under the U(1)R symmetry. Since these transformations categorize toric Calabi-Yau 4-folds and their corresponding 2d (0,2) supersymmetric gauge theories into non-trivial equivalence classes, we anticipate that our findings will pave the way for a `Minimal Model Program' for quiver gauge theories corresponding to toric Calabi-Yau manifolds.