Omega-deformed SYM on a Gibbons-Hawking Space

We study an $\mathcal{N}=2$, pure $U(1)$ SYM theory on a Gibbons-Hawking space $\Omega$-deformed using the $U(1)$ isometry. The resultant 3D theory, after an appropriate "Nekrasov-Witten" change of variables, is asymptotically equivalent to the undeformed theory at spatial infinity but differs from it as one approaches the NUT centers which are fixed points under the $U(1)$ action. The 3D theory may be recast in the form of a generalized hyperk\"ahler sigma model introduced in \cite{Dey:2014lja} where the target space is a one-parameter family of hyperk\"ahler spaces. The hyperk\"ahler fibers have a preferred complex structure which for the deformed theory depends on the parameter of $\Omega$-deformation. The metric on the hyperk\"ahler fiber can be reduced to a standard metric on $\mathbb{C} \times T^2$ with the modular parameter of the torus depending explicitly on the $\Omega$-deformation parameter. The contribution of the NUT center to the sigma model path integral, expected to be a holomorphic section of a holomorphic line bundle over the target space on grounds of supersymmetry, turns out to be a Jacobi theta function in terms of certain "deformed" variables.