Sasakian quiver gauge theories and instantons on Calabi-Yau cones

TL;DR

This paper explores the construction of new quiver gauge theories from SU(2)-equivariant reductions on Sasaki-Einstein orbifolds and relates them to instanton moduli spaces, Calabi-Yau cones, and Nakajima quiver varieties.

Contribution

It introduces novel quiver gauge theories based on affine ADE Dynkin diagrams from Sasaki-Einstein orbifolds and connects them to instanton moduli spaces and Nakajima quiver varieties.

Findings

Derived new quiver gauge theories on manifolds from Sasaki-Einstein orbifolds.

Connected vacua of these theories to instanton moduli spaces via Nahm equations.

Linked the theories to Nakajima quiver varieties and D-brane moduli spaces.

Abstract

We consider SU(2)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form $M\times S^3/\Gamma$, where $M$ is a smooth manifold and $S^3/\Gamma$ is a three-dimensional Sasaki-Einstein orbifold. We obtain new quiver gauge theories on $M$ whose quiver bundles are based on the affine ADE Dynkin diagram associated to $\Gamma$. We relate them to those arising through translationally-invariant dimensional reduction over the associated Calabi-Yau cones $C(S^3/\Gamma)$ which are based on McKay quivers and ADHM matrix models, and to those arising through SU(2)-equivariant dimensional reduction over the leaf spaces of the characteristic foliations of $S^3/\Gamma$ which are K\"ahler orbifolds of $\mathbb{C} P^1$ whose quiver bundles are based on the unextended Dynkin diagram corresponding to $\Gamma$. We use Nahm equations to describe the vacua of SU(2)-equivariant quiver gauge theories on the cones as moduli spaces of spherically symmetric instantons. We relate them to the Nakajima quiver varieties which can be realized as Higgs branches of the worldvolume quiver gauge theories on D$p$-branes probing D$(p+4)$-branes which wrap an ALE space, and to the moduli spaces of spherically symmetric solutions in putative non-abelian generalizations of two-dimensional affine Toda field theories.