Motivation for the Deformed Nekrasov Partition Function

Question

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?

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

Comments

mitchell.physics.tamu.edu/Conference/string2010/documents/… ... I think its chief utility lies somewhere in the space between M-theory and SQCD.

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

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The deformation parameters have a meaning in topological string theory, see for example arxiv.org/abs/arXiv:1302.6993 by Antoniadis et al. for a recent perspective.

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

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I think the paper by Nekrasov and Witten gives a nice picture. I don't understand it well enough myself to give an answer but you could take a look at it. arxiv.org/abs/1002.0888

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

Answer 1

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