The noncommutative topology of anti-self-dual gauge fields

Through techniques afforded by C∗C∗-algebras and Hilbert modules, we study the topology of spaces which parametrize families of instanton gauge fields on noncommutative Euclidean four-spheres Sσ4. By deforming the ADHM construction of instantons on the classical sphere S4S4, we obtain families of instantons on the quantum sphere which are naturally parametrized by noncommutative topological spaces. Using the internal gauge theory of Sσ4 determined by the inner automorphisms of its function algebra, we find that one may always recover a classical parameter space by making a suitable choice of internal gauge.