Gauge theories on noncommutative CPN and Bogomol'nyi-Prasad-Sommerfield-like equations

We give the Fock representation of a noncommutative CPN 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 CPN. Using the Fock representation, we are able to discuss the positivity of Yang-Mills type actions and the minimal action principle. Bogomol'nyi-Prasad-Sommerfield (BPS)-like equations on noncommutative CP1 and CP2 are derived from these actions. There are analogies between BPS-like equations on CP1 and mono-pole equations on R-3 and BPS-like equations on CP2 and instanton equations on R-8. We discuss solutions of these BPS-like equations. (C) 2015 AIP Publishing LLC.