On Higgs branch localization of Seiberg-Witten theories on an ellipsoid

In this note, we consider so-called Higgs branch localization for four-dimensional N = 2 field theories on a 4D ellipsoid. We find a new set of saddle point equations arising from the additional Higgs branch deformation term, whose solutions include both Higgs branch and BPS instanton-vortex mixed configurations. By evaluating the contour integral, we also demonstrate that the ellipsoid partition almost factorizes into purely b- and b(-1)-dependent parts, using SQCD as an explicit example. We identify various factorized parts with the ellipsoid partition function of two-dimensional N = (2, 2) SQCDA, which is precisely the vortex world volume theory. We also give a physical interpretation for the non-factorizable parts and discuss future directions.