Self-Dual Yang-Mills Fields in Eight Dimensions

TL;DR

This paper investigates strongly self-dual Yang-Mills fields in eight-dimensional space, deriving a topological bound related to the first Pontrjagin class, with self-dual solutions achieving this bound.

Contribution

It establishes a new topological bound for self-dual Yang-Mills fields in eight dimensions and characterizes solutions that attain this bound.

Findings

Derived a topological inequality involving the Yang-Mills action and Pontrjagin class.

Identified that strongly self-dual fields realize the lower bound.

Provided conditions characterizing self-dual solutions in 8D.

Abstract

Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fields $F_{\mu \nu}$. We derive a topological bound on ${\bf R}^8$, $\int_{M} ( F,F )^2 \geq k \int_{M} p_1^2$ where $p_1$ is the first Pontrjagin class of the SO(n) Yang-Mills bundle and $k$ is a constant. Strongly self-dual Yang-Mills fields realise the lower bound.