Hyperkähler sigma model and field theory on Gibbons-Hawking spaces

Abstract

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 N = 2 theory on a hyperkahler 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 hyperkahler 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. © The Authors.