 For example, the dimension of the space of SU2 instantons of rank "k">0 is 8"k"−3.To do this they used the Atiyah–Singer index theorem to calculate the dimension of the tangent space of the moduli space at a point; the tangent space is essentially the space of solutions of an elliptic differential operator, given by the linearization of the non-linear Yang–Mills equations.These moduli spaces were later used by Donaldson to construct his invariants of 4-manifolds.