# Twisting with a Flip (the Art of Pestunization)

## TL;DR

This paper develops a unified framework for constructing and analyzing ${
m N}=2$ supersymmetric Yang-Mills theories on 4D manifolds with Killing vectors, generalizing Pestun's localization method and introducing the concept of 'Pestunization' for computing partition functions.

## Contribution

It introduces a new approach to localize supersymmetric theories on 4D manifolds with Killing vectors, extending Pestun's work and proposing a conjecture for the full partition function.

## Key findings

- Unified treatment of Pestun's $S^4$ calculation and equivariant Donaldson-Witten theory.
- Construction of ${
m N}=2$ theories with fixed points allowing instanton/anti-instanton contributions.
- Proposal of a conjecture for the partition function on arbitrary 4D manifolds with Killing vectors.

## Abstract

We construct ${\cal N}=2$ 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 ${\cal N}=2$ 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".