Applications of the Ashtekar gravity to four dimensional hyperkähler geometry and Yang-Mills Instantons

Abstract

The Ashtekar-Mason-Newman equations are used to construct the hyperkähler metrics on four dimensional manifolds. These equations are closely related to anti self-dual Yang-Mills equations of the infinite dimensional gauge Lie algebras of all volume preserving vector fields. Several examples of hyperkähler metrics are presented through the reductions of anti self-dual connections. For any gauge group anti self-dual connections on hyperkähler manifolds are constructed using the solutions of both Nahm and Laplace equations.