Gravitational instantons with conformally coupled scalar fields

TL;DR

This paper introduces new regular Euclidean solutions in General Relativity with Maxwell and conformally coupled scalar fields, analyzing their topology, invariants, and effects of higher-curvature corrections.

Contribution

It provides novel conformally invariant Eguchi-Hanson and Taub-NUT solutions with explicit topological and action calculations, extending previous solutions.

Findings

Solutions have nontrivial topology characterized by Hirzebruch signature and Euler characteristic.

Asymptotically locally Euclidean limits share global properties despite local differences.

Euclidean on-shell action and charges are finite without boundary counterterms.

Abstract

We present novel regular Euclidean solutions to General Relativity in presence of Maxwell and conformally coupled scalar fields. In particular, we consider metrics of the Eguchi-Hanson and Taub-NUT families to solve the field equations analytically. The solutions have nontrivial topology labeled by the Hirzebruch signature and Euler characteristic that we compute explicitly. We find that, although the solutions are locally inequivalent with the original (anti-)self-dual Eguchi-Hanson metric, their asymptotically locally Euclidean limit leads to the same global properties. We revisit the Taub-NUT solution previously found in the literature, analyze their nuts and bolts structure, and obtain the renormalized Euclidean on-shell action as well as their topological invariants. Additionally, we discuss how the solutions get modified in presence of higher-curvature corrections that respect conformal invariance. In the conformally invariant case, we obtain novel Eguchi-Hanson and Taub-NUT solutions and demonstrate that both Euclidean on-shell action and Noether-Wald charges are finite without any reference to intrinsic boundary counterterms.