Partition Function of $N=2$ Gauge Theories on a Squashed $S^4$ with   $SU(2)\times U(1)$ Isometry
  We study $N=2$ supersymmetric gauge theories on a large family of squashed 4-spheres preserving $SU(2)\times U(1)\subset SO(4)$ isometry and determine the conditions under which this background is supersymmetric. We then compute the partition function of the theories by using localization technique. The results indicate that for $N=2$ SUSY, including both vector-multiplets and hypermultiplets, the partition function is independent of the arbitrary squashing functions as well as of the other supergravity background fields. 
Alejandro Cabo-Bizet, Edi Gava, V. I. Giraldo-Rivera, M. Nouman   Muteeb, K.S. Narain
hep-th