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

Abstract

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 monopole equations on ℝ3 and BPS-like equations on ℂP2 and instanton equations on ℝ8. We discuss solutions of these BPS-like equations.