Gauge theory on nonorientable four-manifolds

In this thesis we generalize the moduli space of anti-self dual connections on some principal bundle over an oriented manifold to the case where the base manifold X is nonorientable. To do this, we work with principal Pin4 bundles P over X. To achieve a meaningful generalization of the Hodge star operator on two forms from the oriented setting we twist the star operator on forms with an involution on the Lie algebra and gain an involution *on the bundle of adjoint valued two forms. A connection is said to be * -ASD if its curvature lies in the (- 1)-eigenspace of the generalized star operator. We investigate the properties of the moduli space Mof * -ASD gauge equivalenceclasses of connections on P. To do this we start with an analysis of Pin4 and give a classification theorem for Pin4 bundles over a four-complex. We proceed to describe the local behaviour of M, calculate its formal dimension and investigate the singularities which arise from reducible connections. Then we define a natural compactification of M and compare Mand its compactifications to the ASD moduli space on some SU2 bundle over the orientation cover of X. We conclude by describing the charge one moduli space on the real projective for space. Finally we proved some theorems about the moduli space over an oriented four- manifold X which is equipped with a smooth orientation preserving action of a finite group. We prove two theorems about the equivariant ASD moduli space for a SU2 or SO3 bundle over X. The first one gives a criterion under which the moduli space contains a non empty fixed set, and the second one gives conditions under which one can avoid reducible ASD connections by a slight equivariant pertubation of the Riemannian metric on X.