Symplectic Structures on Gauge Theory

We study certain natural differential forms and their equivariant extensions on the space of connections. These forms are defined using the family local index theorem. When the base manifold is symplectic, they define a family of symplectic forms on the space of connections. We will explain their relationships with the Einstein metric and the stability of vector bundles. These forms also determine primary and secondary characteristic forms (and their higher level generalizations). Received: 27 February 1996 / Accepted: 7 July 1997