Q -quaternions and q-deformed su(2) instantons

Abstract

We construct (anti-)instanton solutions of a would-be q-deformed su(2) Yang-Mills theory on the quantum Euclidean space Rq4 [the SOq(4)-covariant noncommutative space] by reinterpreting the function algebra on the latter as a q-quaternion bialgebra. Since the (anti-)self-duality equations are covariant under the quantum group of deformed rotations, translations, and scale change, by applying the latter we can generate new solutions from the one centered at the origin and with unit size. We also construct multi-instanton solutions. As they depend on noncommuting parameters playing the roles of “sizes” and “coordinates of the centers” of the instantons, this indicates that the moduli space of a complete theory should be a noncommutative manifold. Similarly, gauge transformations should be allowed to depend on additional noncommutative parameters.