Hermitian–Einstein Metrics from Non-commutative U(1) Solutions

Abstract

AbstractWe 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” Yang and Salizzoni (Phys Rev Lett 201602, 2006 [hep-th/0512215]). This correspondence is valid not only for a commutative space but also for a noncommutative space. We choose \(U\left (1 \right )\) 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 two-forms are not instantons but they are solutions to the Yang–Mills equations.KeywordsNoncommutative geometryGauge field theoryMathematics Subject Classification (2010)Primary 53Z05; Secondary 83C05