Hermitian-Einstein metrics from noncommutative U (1) instantons

We show that Hermitian-Einstein metrics can be locally constructed by a map from (anti-)self-dual two-forms on Euclidean R4 to symmetric two-tensors introduced in Yang and Salizzoni [Phys. Rev. Lett. 96 201602 (2006); e-print arXiv:hep-th/0512215]. This correspondence is valid not only for a commutative space but also for a noncommutative space. We choose U(1) instantons on a noncommutative C2 as the self-dual two-form, from which we derive a family of Hermitian-Einstein metrics. We also discuss the condition when the metric becomes Kahler.We show that Hermitian-Einstein metrics can be locally constructed by a map from (anti-)self-dual two-forms on Euclidean R4 to symmetric two-tensors introduced in Yang and Salizzoni [Phys. Rev. Lett. 96 201602 (2006); e-print arXiv:hep-th/0512215]. This correspondence is valid not only for a commutative space but also for a noncommutative space. We choose U(1) instantons on a noncommutative C2 as the self-dual two-form, from which we derive a family of Hermitian-Einstein metrics. We also discuss the condition when the metric becomes Kahler.