Geometry of the Moduli Space of Self-Dual Connections Over the Four-Sphere

A Riemannian metric on a compact four-manifold induces a natural L^2 metric on the corresponding moduli space of (anti-) self-dual connections on a principal G-bundle P. When the bundle structure group G is SU(2) and -c_2(P) = k, Groisser, Parker and others have found explicit formulas for the components of the L^2 metric on the moduli space {cal M}_ {k} when k = 1 and the four-manifold is the sphere S^4 or the complex projective space doubcIP^2. The moduli space {cal M}_1(S ^4) is diffeomorphic to the open five-ball, while {cal M}_1(doubc IP^2) is diffeomorphic to the open cone over doubcIP^2: these moduli spaces have finite volume and diameter with respect to the L^2 metric. Donaldson, Groisser, and Parker have conjectured that the moduli space {cal M}_ {k} has finite volume and diameter with respect to the L^2 metric for any integer k. We consider the case where the four-manifold is the sphere S^4 with its standard round metric, the group G is SU(2), and k = 2. We obtain estimates for the components of the L^2 metric on the moduli space {cal M}_2(S^4) of self-dual SU(2) -connections over the four-sphere, a non-compact 13-dimensional manifold which is homotopic to the Grassman manifold of real 2-planes in IR^5. As an application, we show that the space {cal M}_2(S^4) has finite volume and diameter with respect to the L^2 metric.