Field Theory on Gibbons-Hawking Spaces

The IR Lagrangian of an N=2 supersymmetric gauge theory compactified on R^3 x S^1 is a 3d sigma model with a hyperkahler target space. We consider a related problem of constructing the IR Lagrangian of an N=2 theory on a special family of circle fibrations over R^3 -- namely Gibbons-Hawking spaces. We propose that the answer is a novel deformation of the standard 3d hyperkahler sigma model and study the constraints imposed by supersymmetry on such a theory. In particular, supersymmetry implies that the contribution of the NUT center to the sigma model path integral must be a a holomorphic section of a certain holomorphic line bundle over the hyperkahler target space. We illustrate various features of this deformed sigma model using examples of certain Abelian gauge theories on Gibbons-Hawking spaces.