\( \mathcal{N}=2 \) supersymmetric gauge theory on connected sums of S 2 × S 2

We construct 4D $$ \mathcal{N}=2 $$ theories on an infinite family of 4D toric manifolds with the topology of connected sums of S 2 × 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.