Matrix models from localization of five-dimensional supersymmetric noncommutative U(1) gauge theory

Abstract

We study localization of five-dimensional supersymmetric $U(1)$ gauge theory on ${\mathbb{S}}^3\times {\mathbb{R}}_{\theta }^2$ where ${\mathbb{R}}_{\theta }^2$ is a noncommutative (NC) plane. The theory can be isomorphically mapped to three-dimensional supersymmetric $U(N\to \infty )$ gauge theory on ${\mathbb{S}}^3$ using the matrix representation on a separable Hilbert space on which NC fields linearly act. Therefore the NC space ${\mathbb{R}}_{\theta }^2$ allows for a flexible path to derive matrix models via localization from a higher-dimensional supersymmetric NC $U(1)$ gauge theory. The result shows a rich duality between NC $U(1)$ gauge theories and large $N$ matrix models in various dimensions.

50 pages, 2 figures; v2 corrected typos

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