Seiberg and Witten and Nekrasov managed to completely find the exact partition function of the $\mathcal{N}=2$ SYM theory on $\mathbb{R}^4$. As in $\mathcal{N}=2$ in $\mathcal{N}=1$ the NSZV (Novikov-Shifman-Vainshtein-Zakharov beta function) equation completely determines the evolution of the gauge coupling.

What is different in $\mathcal{N}=1$ theories compared to $\mathcal{N}=2$ theories that give obstructions to obtaining exact results like in the Nekrasov case or Pestun case?

I know that the $\mathcal{N}=1$ theory is chiral and this is one difference already but what other differences are there that make life complicated? Can we define topological twists in these theories? Can we put them in curved manifold without breaking supersymmetry? Is there any hope to obtain something new?


Clearly one difficultly is just the fact that having less supersymmetry means you have less control over the theory and trying to set up the localisation computation can be more difficult. However, one of the other problems is that the $S^4$ partition function for 4d $\mathcal{N}=1$ is an ill-defined quantity. $\log Z_{S^4}$ contains some finite contribution $$ \log Z_{S^4}=\frac{1}{12}K(\bar{\lambda},\lambda)+\dots $$ which, when you have only $\mathcal{N}=1$ supersymmetry, may always be removed by a $\mathcal{N}=1$ preserving counterterm and it is therefore not universal - $Z_{S^4}$ is a regularisation scheme dependent quantity for a generic $\mathcal{N}=1$ theory. On the other hand, for theories with $\mathcal{N}=2$ supersymmetry the contribution cannot be removed by a $\mathcal{N}=2$ preserving counterterm thus $Z_{S^4}$ is well defined and scheme independent for $\mathcal{N}=2$.

I would recommend that you take a look chapter 4 of the lecture notes of Z.Komargodski at http://indico.ictp.it/event/7624/session/19/contribution/84/material/0/0.pdf


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