Hyperkahler Sigma Model and Field Theory on Gibbons-Hawking Spaces

We describe a novel deformation of the 3-dimensional sigma model with hyperk\"ahler target, which arises naturally from the compactification of a 4-dimensional $\mathcal{N}=2$ theory on a hyperk\"ahler circle bundle (Gibbons-Hawking space). We derive the condition for which the deformed sigma model preserves 4 out of the 8 supercharges. We also study the contribution from a NUT center to the sigma model path integral, and find that supersymmetry implies it is a holomorphic section of a certain holomorphic line bundle over the hyperk\"ahler target. We study explicitly the case where the original 4-dimensional theory is pure $U(1)$ super Yang-Mills, and show that the contribution from a NUT center in this case is simply the Jacobi theta function.