Hiroshi Umetsu

Explicit formulas for noncommutative deformations of CPN and CHN

We give explicit expressions of a deformation quantization with separation of variables for CP^N and CH^N. This quantization method is one of the ways to perform a deformation quantization of Kahler manifolds, which is introduced by Karabegov. Star products are obtained as explicit formulae in all order in the noncommutative parameter. We also give the Fock representations of the noncommutative CP^N and CH^N.

Noncommutative CPN and CHN and their physics

We study noncommutative deformation of manifolds by constructing star products. We start from a noncommutative d and discuss more genaral noncommutative manifolds. In general, star products can not be described in concrete expressions without some exceptions. In this article we introduce new examples of noncommutative manifolds with explicit star products. Karabegovs deformation quantization of PN and HN with separation of variables gives explicit calulable star products represented by gamma functions. Using the results of star products between inhomogeneous coordinates, we find creation and anihilation operators and obtain the Fock representation of the noncommutative PN and HN.

Deformation Quantization with Separation of Variables and Gauge Theories

We construct a gauge theory on a noncommutative homogeneous Kahler manifold by using the deformation quantization with separation of variables for Kahler manifolds. A model of noncommutative gauge theory that is connected with an ordinary Yang–Mills theory in the commutative limit is given. As an examples, we review a noncommutative \( \mathbb{C}P^N \) and construct a gauge theory on it. We also give details of the proof showing that the noncommutative \( \mathbb{C}P^N \) constructed in this paper coincides with the one given by Bordemann, Brischle, Emmrich and Waldmann [1].

Noncommutative

We study noncommutative deformation of manifolds by constructing star products. We start from a noncommutative d and discuss more genaral noncommutative manifolds. In general, star products can not be described in concrete expressions without some exceptions. In this article we introduce new examples of noncommutative manifolds with explicit star products. Karabegovs deformation quantization of PN and HN with separation of variables gives explicit calulable star products represented by gamma functions. Using the results of star products between inhomogeneous coordinates, we find creation and anihilation operators and obtain the Fock representation of the noncommutative PN and HN.

CP^{N}

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.

and

We study gauge theories on noncommutative homogeneous Kahler manifolds. To make the noncommutative manifolds, we use the deformation quantization with separation of variables for Kahler manifolds. We construct models of noncommutative gauge theories that are connected with usual Yang-Mills theories in the commutative limits. It is expected that the models connecting to commutative gauge theories are uniquely determined. As examples, we give noncommutative CPN and noncommutative CHN and gauge theories on them. Some kinds of gauge symmetry breaking and topological symmetry breaking by noncommutative deformations are observed by concrete geometrical calculations.

CH^N

We study noncommutative deformation of manifolds by constructing star products. We start from a noncommutative d and discuss more genaral noncommutative manifolds. In general, star products can not be described in concrete expressions without some exceptions. In this article we introduce new examples of noncommutative manifolds with explicit star products. Karabegovs deformation quantization of PN and HN with separation of variables gives explicit calulable star products represented by gamma functions. Using the results of star products between inhomogeneous coordinates, we find creation and anihilation operators and obtain the Fock representation of the noncommutative PN and HN.