Explicit Construction of Hermitian Yang-Mills Instantons on Coset Manifolds

TL;DR

This paper develops a systematic method to explicitly construct Hermitian Yang-Mills instantons on six-dimensional coset manifolds using generalized 't Hooft symbols, extending the well-known four-dimensional instanton framework.

Contribution

The paper introduces a novel approach to construct Hermitian Yang-Mills instantons in six dimensions via generalized 't Hooft symbols, linking spin connections and topological invariants.

Findings

Explicit construction of instantons on coset manifolds

Analysis of topological invariants like instanton and Euler numbers

Extension of 't Hooft symbols to six dimensions

Abstract

In four dimensions, 't Hooft symbols offer a compact and powerful framework for describing the self-dual structures fundamental to instanton physics. Extending this to six dimensions, the six-dimensional 't Hooft symbols can be constructed using the isomorphism between the Lorentz group $Spin(6)$ and the unitary group $SU(4)$. We demonstrate that the six-dimensional self-dual structures governed by the Hermitian Yang-Mills equations can be elegantly organized using these generalized 't Hooft symbols. We also present a systematic method for constructing Hermitian Yang-Mills instantons from spin connections on six-dimensional manifolds using the generalized 't Hooft symbols. We provide a thorough analysis of the topological invariants such as instanton and Euler numbers.