Exact results for N = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

We provide a contour integral formula for the exact partition function of $$ \mathcal{N} $$ = 2 supersymmetric U(N) gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for U(2) $$ \mathcal{N} $$ = 2∗ theory on $$ {\mathrm{\mathbb{P}}}^2 $$ for all instanton numbers. In the zero mass case, corresponding to the $$ \mathcal{N} $$ = 4 supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a longstanding conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new.