Noncommutative instantons come in families

A construction of instantons in the context of noncommutative geometry, in particular SU(2) instantons on a noncommutative 4 sphere, has been recently reported. Firstly, a noncommutative principal fibration A(S 4 θ )↬ A(S 7 θ ) which 'quantizes' the classical SU(2)-Hopf fibration over S 4 , has been constructed in [11] on the toric noncommutative four-sphere S 4 θ The generators of A 4 θ are the entries of a projection p which describes the basic instanton on A 4 θ . That is, p gives a projective module of finite type p [A( S 4 θ )] 4 and a connection ▽ = p ° D on it which has a self-dual curvature and charge 1, in some appropriate sense; this is the basic instanton. In [12] infinitesimal instantons — 'the tangent space to the moduli space' -were constructed using infinitesimal conformal transformations, that is elements in a quantized enveloping algebra U θ ( so (5, 1)). In [10] we looked at a global construction and obtain generic charge 1 instantons by 'quantizing' the action of the Lie groups SL(2, ) and SO(2) on the basic instanton which enter the classical construction [1]. We review all this here.