{In this thesis, we focus on the moduli space of ${mathcal N}=2$ supersymmetry gauge theories on four dimensional ellipsoid $S^{4}_{b^{2}}$. We first review Seiberg-Witten theory and the duality between two- and four- dimensional theory on flat space ${mathbb{R}}^{4}$. Furthermore, we also review the construction of SW theory on $S^{4}_{b^{2}}$ and the exact partition function on Coulomb branch by Localization Principle. We add another deformed term ${ f Q}{mathcal V}_{ m Higgs}$ to find out new set of saddle point equations whose solutions include the Higgs branch and generalized instanton-vortex mixed configuration. Evaluating partition function by Residue theorem, we find the factorizable structures of the corresponding codimension 2 surface defects. We also discuss the physical interpretations of non-factorizable structures.}