Hermitian-Einstein metrics from noncommutative U(1) instantons

Abstract

We show that Hermitian-Einstein metrics can be locally constructed by a map from (anti-)self-dual two-forms on Euclidean R 4 to symmetric two-tensors introduced in [1]. This correspondence is valid not only for a commutative space but also for a noncommutative space. We choose U (1) instantons on a noncommutative C 2 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 Kähler.