Noncommutative instantons come in families

Abstract

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(S4θ)↬ A(S7θ) which ‘quantizes’ the classical SU(2)-Hopf fibration over S4, has been constructed in [11] on the toric noncommutative four-sphere S4θ The generators of A4θ are the entries of a projection p which describes the basic instanton on A4θ. That is, p gives a projective module of finite type p[A(S4θ)]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 Uθ(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, ℌ) and SO(2) on the basic instanton which enter the classical construction [1]. We review all this here.