Instantons on Special Holonomy Manifolds

TL;DR

This paper develops a matrix-based ansatz for instantons on special holonomy manifolds, linking instanton equations to matrix models and gradient flows, and explores their relation to quiver gauge theories.

Contribution

It introduces a generalized scalar ansatz for instantons on special holonomy cones, reducing the problem to matrix models and gradient flows, and discusses extensions to Kaehler-Einstein and Calabi-Yau cones.

Findings

Instantons are parameterized by constrained matrix-valued functions.

Instanton equations reduce to matrix model and Newtonian mechanics equations.

Framework enables association of quiver gauge theories with special holonomy manifolds.

Abstract

We consider cones over manifolds admitting real Killing spinors and instanton equations on connections on vector bundles over these manifolds. Such cones are manifolds with special (reduced) holonomy. We generalize the scalar ansatz for a connection proposed by Harland and Nolle in such a way that instantons are parameterized by constrained matrix-valued functions. Our ansatz reduces instanton equations to a matrix model equations which can be further reduced to Newtonian mechanics with particle trajectories obeying first-order gradient flow equations. Generalizations to Kaehler-Einstein manifolds and resolved Calabi-Yau cones are briefly discussed. Our construction allows one to associate quiver gauge theories with special holonomy manifolds.