Sasakian quiver gauge theory on the Aloff-Wallach space $X_{1,1}$

TL;DR

This paper explores the reduction of gauge theories on spaces involving the Aloff-Wallach space $X_{1,1}$, using SU(3)-equivariance and quiver diagrams to construct gauge bundles and analyze instanton solutions.

Contribution

It introduces a novel SU(3)-equivariant framework for gauge theories on $X_{1,1}$ using quiver diagrams and compares these with theories on related spaces, expanding understanding of instantons on Sasaki-Einstein manifolds.

Findings

Constructed quiver bundles from SU(3) weight diagrams.

Analyzed Hermitian Yang-Mills equations on the cone over $X_{1,1}$.

Compared gauge theories on $X_{1,1}$ and $Q_3$.

Abstract

We consider the SU(3)-equivariant dimensional reduction of gauge theories on spaces of the form $M^d \times X_{1,1}$ with d-dimensional Riemannian manifold $M^d$ and the Aloff-Wallach space $X_{1,1}$= SU(3)/U(1) endowed with its Sasaki-Einstein structure. The condition of SU(3)-equivariance of vector bundles, which has already occurred in the studies of Spin(7)-instantons on cones over Aloff-Wallach spaces, is interpreted in terms of quiver diagrams, and we construct the corresponding quiver bundles, using (parts of) the weight diagram of SU(3). We consider three examples thereof explicitly and then compare the results with the quiver gauge theory on $Q_3$ =SU(3)/(U(1) x U(1)), the leaf space underlying the Sasaki-Einstein manifold $X_{1,1}$. Moreover, we study instanton solutions on the metric cone $C(X_{1,1})$ by evaluating the Hermitian Yang-Mills equation. We briefly discuss some features of the moduli space thereof, following the main ideas of a treatment of Hermitian Yang-Mills instantons on cones over generic Sasaki-Einstein manifolds in the literature.