Invariant Sp(1)-instantons on \(S^ 2\times S^ 2\) (Q1093925)

The mathematical exercise of constructing ``instanton'' solutions on the spacetime manifold \(S^ 2\times S^ 2\) is solved in this paper. An instanton is a solution of the self-duality condition for the quaternionic line bundle (or SU(2) gauge potential) on \(S^ 2\times S^ 2\) considered in this paper. The self-duality condition is formulated, and shown to have two types of solution due to the product nature of the manifold. One is the analog of the quaternionic instanton on \(S^ 4\) (ordinary Euclidean spacetime), while the other solution is basically trivial.