Twisting with a Flip (The Art of Pestunization).

We construct N = 2 supersymmetric Yang–Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. It turns out that for every fixed point one can allocate either instanton or anti-instanton contributions to the partition function, and that this is compatible with supersymmetry. The equivariant Donaldson–Witten theory is a special case of our construction. We present a unified treatment of Pestun's calculation on S 4 and equivariant Donaldson–Witten theory by generalizing the notion of self-duality on manifolds with a vector field. We conjecture the full partition function for a N = 2 theory on any 4D manifold with a Killing vector. Using this new notion of self-duality to localize a supersymmetric theory is what we call "Pestunization". [ABSTRACT FROM AUTHOR]