Motivation for the Deformed Nekrasov Partition Function I have recently been doing research on the AGT Correspondence between the Nekrasov Instanton Partition Function and Louiville Conformal Blocks (http://arxiv.org/abs/0906.3219). When looking at the Nekrasov Partition Function one defines a deformed metric in terms of the "deformation parameters" $\epsilon_1, \epsilon_2$ which seem to define a $SO(4)$ action on a standard Euclidean Metric, breaking translational symmetry. Much of the literature on these functions seems to be in the math department, defining the functions categorically in terms of sheaves and what-not (http://arxiv.org/abs/math/0311058) and even the original paper (http://arxiv.org/abs/hep-th/0206161) approaches the subject from a cohomological perspective.
Is there any obvious physical motivation for looking at partition functions in this strange deformed spacetime? Or should I view it as simply a mathematical manipulation?
 A: 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.
