N=2 supersymmetric gauge theory on connected sums of S2 x S2

We construct 4D N = 2 theories on an infinite family of 4D toric manifolds with the topology of connected sums of S-2 x S-2. These theories are constructed through the dimensional reduction along a non -trivial U(1) -fiber of 5D theories on toric Sasaki Einstein manifolds. We discuss the conditions under which such reductions can be carried out and give a partial classification result of the resulting 4D manifolds. We calculate the partition functions of these 4D theories and they involve both instanton and anti-instanton contributions, thus generalizing Pestun's famous result on S-4.