Cohomological Yang-Mills Theory in Eight Dimensions

TL;DR

This paper develops nearly topological Yang-Mills theories on special eight-dimensional manifolds with holonomy groups, linking them to supersymmetric theories and four-dimensional Seiberg-Witten equations.

Contribution

It constructs new topological Yang-Mills models on eight-manifolds with special holonomy, extending the framework to Joyce and Calabi-Yau manifolds and connecting to supersymmetric theories.

Findings

Defined instanton equations using invariant four-forms.

Related the models to eight-dimensional supersymmetric Yang-Mills theory.

Reduced the models to four-dimensional non-abelian Seiberg-Witten equations.

Abstract

We construct nearly topological Yang-Mills theories on eight dimensional manifolds with a special holonomy group. These manifolds are the Joyce manifold with $Spin(7)$ holonomy and the Calabi-Yau manifold with SU(4) holonomy. An invariant closed four form $T_{\mu\nu\rho\sigma}$ on the manifold allows us to define an analogue of the instanton equation, which serves as a topological gauge fixing condition in BRST formalism. The model on the Joyce manifold is related to the eight dimensional supersymmetric Yang-Mills theory. Topological dimensional reduction to four dimensions gives non-abelian Seiberg-Witten equation.