Towards the localization of SUSY gauge theory on a curved space

We consider an N=4 supersymmetric gauge theory on a curved space. We try to generalize Pestun's localization calculation on the four-sphere to a more general class of curved spaces. We calculated the Q-exact term to localize the path-integral, and when it becomes positive definite, we obtain a configuration where the path-integral localizes. We also evaluate the super Yang-Mills action in this configuration.