Geometry of the Moduli Space of SelfDual Connections Over the FourSphere
Abstract
A Riemannian metric on a compact fourmanifold induces a natural L^2 metric on the corresponding moduli space of (anti) selfdual connections on a principal Gbundle 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 fourmanifold 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 fiveball, 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 fourmanifold 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 selfdual SU(2) connections over the foursphere, a noncompact 13dimensional manifold which is homotopic to the Grassman manifold of real 2planes 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.
 Publication:

Ph.D. Thesis
 Pub Date:
 1992
 Bibcode:
 1992PhDT........62F
 Keywords:

 STABLE BUNDLES;
 Mathematics; Physics: Elementary Particles and High Energy