  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.
