The self-dual classical double copy, and the Eguchi-Hanson instanton

TL;DR

This paper explores the self-dual classical double copy relating gauge and gravity solutions, exemplified by the Eguchi-Hanson instanton, revealing a differential operator framework and properties of the resulting gauge field.

Contribution

It introduces a differential operator formulation of the double copy for self-dual solutions and analyzes the Eguchi-Hanson instanton within this framework.

Findings

The double copy can be formulated using a differential operator for self-dual solutions.

The gauge field from the Eguchi-Hanson instanton is dipole-like and charge-neutral.

The approach provides insights into the structure of classical solutions in gauge and gravity theories.

Abstract

The double copy is a map from non-abelian gauge theories to gravity, that has been demonstrated both for scattering amplitudes and exact classical solutions. In this study, we reconsider the double copy for exact solutions that are self-dual in either the gauge or gravity theory. In this case, one may formulate a general double copy in terms of a certain differential operator, which generates the gauge and gravity solutions from a harmonic function residing in a biadjoint scalar theory. As an illustration, we examine the single copy of the well-known Eguchi-Hanson instanton in gravity. The gauge field thus obtained represents an abelian-like object whose field is dipole-like at large distances, and which has no magnetic or electric charge.