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

Abstract

The Ashtekar–Jacobson–Smolin–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.