Hermitian-Einstein metrics from noncommutative $U\left(1 \right)$ instantons

TL;DR

This paper demonstrates a method to construct Hermitian-Einstein metrics from (anti-)self-dual two-forms, applicable to both commutative and noncommutative spaces, using $U(1)$ instantons on noncommutative ${f C}^2$.

Contribution

It introduces a novel correspondence between self-dual forms and Hermitian-Einstein metrics that extends to noncommutative geometries.

Findings

Constructed Hermitian-Einstein metrics from noncommutative $U(1)$ instantons.

Established conditions for the metrics to be Kähler.

Extended the self-dual form to metric correspondence to noncommutative spaces.

Abstract

We show that Hermitian-Einstein metrics can be locally constructed by a map from (anti-)self-dual two-forms on Euclidean ${\mathbb R}^4$ to symmetric two-tensors introduced in "Gravitational instantons from gauge theory," H. S. Yang and M. Salizzoni, Phys. Rev. Lett. (2006) 201602, [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 ${\mathbb C}^2$ 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 K\"ahler.