Twisted self-duality

We examine a generalization of the usual self-duality equations for Yang-Mills theory when the color space admits a nontrivial involution. This involution allows us to construct a nontrivial twist which may be combined with the Hodge star to form a twisted self-dual curvature. We will construct a simple example of twisted self-duality for su(2) ? su(2) gauge theory along with its explicit solutions, both in Euclidean and Minkowski backgrounds, and then dimensionally reduce from four dimensions to obtain families of nontrivial nonlinear equations in lower dimensions. This twisted self-duality constraint will be shown to arise in E-7 exceptional field theory through a Scherk-Schwarz reduction and we will show how an Eguchi-Hanson gravitational instanton also obeys the twisted self-duality condition.