Gauge Theories in Noncommutative Homogeneous K\"ahler Manifolds

Understand

We construct a gauge theory on a noncommutative homogeneous K\"ahler manifold, where we employ the deformation quantization with separation of variables for K\"ahler manifolds formulated by Karabegov.

• A key point in this construction is to obtaining vector fields which act as inner derivations for the deformation quantization.
• We show that these vector fields are the only Killing vector fields.
• We give an explicit construction of this gauge theory on noncommutative ${\mathbb C}P^N$ and noncommutative ${\mathbb C}H^N$.