Partition function of N=2N=2 gauge theories on a squashed S4S4 with SU(2)×U(1)SU(2)×U(1) isometry

We study N=2N=2 supersymmetric gauge theories on a large family of squashed 4-spheres preserving SU(2)×U(1)⊂SO(4)SU(2)×U(1)⊂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=2N=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.