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

Abstract

We give the Fock representation of a noncommutative CP 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 CP 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 CP 1 and CP 2 are derived from these actions. There are analogies between BPS-like equations on CP 1 and monopole equations on R 3 , and BPS-like equations on CP 2 and instanton equations on R 8 . We discuss solutions of these BPS-like equations.

Noncommutative deformations of locally symmetric Kähler manifolds

“…The first one is about the representation of the noncommutative manifolds given by the deformation quantization with separation of variables. In [29], the Fock representation for noncommutative CP N is constructed. Using new star products with separation of variables for Riemann surfaces given in this article, we can make such kind of Fock representation, similarly.…”