Gauged hyperinstantons and monopole equations

The monopole equations in the dual abelian theory of the N = 2 gauge-theory, recently proposed by Witten as a new tool to study topological invariants, are shown to be the simplest elements in a class of instanton equations that follow from the improved topological twist mechanism introduced by the authors in previous papers. When applied to the N = 2 σ-model, this twisting procedure suggested the introduction of the so-called hyperinstantons that are the solutions to an appropriate condition of triholomorphicity imposed on the maps q:m → n from a four-dimensional almost quaternionic world-manifold m to an almost quaternionic target manifold n. When gauging the σ-model by coupling it to the vector multiplet of a gauge group G, one gets instantonic conditions (named by us gauged hyperinstantons) that reduce to the Seiberg-Witten equations for >m = n = R4 and G = U(1). The deformation of the self-duality condition on the gauge-field strength due to the monopole-hyperinstanton is very similar to the deformation of the self-duality condition on the Riemann curvature previously observed by the authors when the hyperinstantons are coupled to topological gravity. In this paper the general form of the hyperinstantonic equations coupled to both gravity and gauge multiplets is presented.