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Gauge theories on noncommutative ${\mathbb C}P^N$ and BPS-like equations

We give the Fock representation of a noncommutative $\mathbb{C}P^N$ and gauge theories on it. The Fock representation is constructed based on star products given by deformation quantization with separation of variables and operators which act on states in the Fock space are explicitly described by functions of inhomogeneous coordinates on ${\mathbb C}P^N$. Using the Fock representation, we are able to discuss the positivity of Yang-Mills type actions and the minimal action principle. Other types of actions including the Chern-Simons term are also investigated. BPS-like equations on noncommutative $\mathbb{C}P^1$ and $\mathbb{C}P^2$ are derived from these actions. There are analogies between BPS-like equations on $\mathbb{C}P^1$ and monopole equations on ${\mathbb R}^3$, and BPS-like equations on $\mathbb{C}P^2$ and instanton equations on ${\mathbb R}^8$. We discuss solutions of these BPS-like equations.