The most physical and understandable definition of Nekrasov's partition function to me uses five-dimensional gauge theories. Namely, any 4d N=2 susy gauge theory has a 5d version with the same matter content, so that compactifying it on a small $S^1$ brings it back to the original 4d theory.

Then we put the theory on the so-called Omega background: it is $\mathbb{R}^4 \times [0,\beta]$, but $(\vec{x},0)$ and $(\vec{x'},\beta)$ are identified by a rotation
$$
\vec x'=\begin{pmatrix}
\cos \beta\epsilon_1 & \sin\beta\epsilon_1 & 0 & 0\\
-\sin \beta\epsilon_1 & \cos\beta\epsilon_1 & 0 & 0\\
0& 0 &\cos \beta\epsilon_2 & \sin\beta\epsilon_2\\
0& 0 &-\sin \beta\epsilon_2 & \cos\beta\epsilon_2
\end{pmatrix}\vec x.
$$

Then we take the limit $\beta\to 0$, keeping $\epsilon_{1,2}$ fixed. (Strictly speaking we also need to add a background $SU(2)_R$ symmetry gauge field, so that some of the susy is preserved.)

Most of what Nekrasov did using his cohomological framework can be seen directly in this higher-dimensional setup. See e.g. Sec. 3.2 of my review article in preparation, available here.

This post imported from StackExchange Physics at 2014-08-23 04:59 (UCT), posted by SE-user Yuji