5D SYM on 3D deformed spheres

We reconsider the relation of superconformal indices of superconformal field theories of class SS with five-dimensional N=2N=2 supersymmetric Yang–Mills theory compactified on the product space of a round three-sphere and a Riemann surface. We formulate the five-dimensional theory in supersymmetric backgrounds preserving N=2N=2 and N=1N=1 supersymmetries and discuss a subtle point in the previous paper concerned with the partial twisting on the Riemann surface. We further compute the partition function by localization of the five-dimensional theory on a squashed three-sphere in N=2N=2 and N=1N=1 supersymmetric backgrounds and on an ellipsoid three-sphere in an N=1N=1 supersymmetric background.