D2-brane probes of non-toric cDV threefolds via monopole superpotentials

TL;DR

This paper develops a framework for constructing gauge theories on D2-branes probing non-toric cDV threefold singularities, using ADE fibrations and monopole superpotentials, and employs 3d mirror symmetry for analysis.

Contribution

It introduces a novel method to encode cDV singularities via Higgs fields and constructs associated gauge theories, including non-toric and non-resolvable cases.

Findings

Successfully reproduces quiver-collapsing mechanisms.

Provides explicit examples of non-toric cDV singularities.

Utilizes 3d mirror symmetry to analyze gauge theories.

Abstract

We develop a framework to construct worldvolume gauge theories on D2-branes probing compound Du Val (cDV) Calabi-Yau threefold singularities. By viewing these singularities as ADE surface fibrations over the complex $w$-plane, we encode their geometry in a Higgs field $\Phi(w)$. A D2-brane probe perceives $\Phi(w)$ as an $\mathcal{N}=2$ deformation of its 3d $\mathcal{N}=4$ affine Dynkin quiver gauge theory via polynomial and monopole superpotential terms. By exploiting 3d mirror symmetry, we obtain an effective theory that correctly reproduces the quiver-collapsing mechanism known in the mathematical literature. We present several examples, including non-toric and non-resolvable cases.