Partition Functions of $\mathcal{N}=1$ Gauge Theories on $S^2\times {\mathbb{R}}_{\epsilon }^2$ and Duality

Abstract

We compute the partition functions of $\mathcal{N}=1$ gauge theories on $S^2\times {\mathbb{R}}_{\epsilon }^2$ using supersymmetric localization. The path integral reduces to a sum over vortices at the poles of $S^2$ and at the origin of ${\mathbb{R}}_{\epsilon }^2$. The exact partition functions allow us to test Seiberg duality beyond the supersymmetric index. We propose the $\mathcal{N}=1$ partition functions on the $\Omega$-background, and show that the Nekrasov partition functions can be recovered from these building blocks.

51 pages, no figures; v2: discussions added in Sec. 2.4 and Sec. 5, published version

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