Higher Dimensional Analogues of Donaldson-Witten Theory

TL;DR

This paper develops higher-dimensional topological field theories analogous to Donaldson-Witten theory on special holonomy manifolds, extending the framework to eight, seven, and six dimensions with connections to string theory.

Contribution

It introduces new topological quantum field theories in higher dimensions on manifolds with special holonomy, generalizing Donaldson-Witten theory without the need for twisting.

Findings

Stress tensor is BRST exact for certain metric variations.

Proposes invariants for manifolds with $Spin(7)$, $G_2$, and Calabi-Yau holonomy.

Theories naturally arise from supersymmetric Yang-Mills theory and relate to string theory.

Abstract

We present a Donaldson-Witten type field theory in eight dimensions on manifolds with $Spin(7)$ holonomy. We prove that the stress tensor is BRST exact for metric variations preserving the holonomy and we give the invariants for this class of variations. In six and seven dimensions we propose similar theories on Calabi-Yau threefolds and manifolds of $G_2$ holonomy respectively. We point out that these theories arise by considering supersymmetric Yang-Mills theory defined on such manifolds. The theories are invariant under metric variations preserving the holonomy structure without the need for twisting. This statement is a higher dimensional analogue of the fact that Donaldson-Witten field theory on hyper-K\"ahler 4-manifolds is topological without twisting. Higher dimensional analogues of Floer cohomology are briefly outlined. All of these theories arise naturally within the context of string theory.