  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.

High Energy Physics - Theory