Semichiral fields on S2 and generalized Kähler geometry

We study a class of two-dimensional N=22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=\left(2,2\right) $$\end{document} supersymmetric gauge theories, given by semichiral multiplets coupled to the usual vector multiplet. In the UV, these theories are traditional gauge theories deformed by a gauged Wess-Zumino term. In the IR, they give rise to nonlinear sigma models on noncompact generalized Kähler manifolds, which contain a three-form field H and whose metric is not Kähler. We place these theories on S2 and compute their partition function exactly with localization techniques. We find that the contribution of instantons to the partition function that we define is insensitive to the deformation, and discuss our results from the point of view of the generalized Kähler target space.