Gravitational instantons with conformally coupled scalar fields

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, they have the same global properties in the flat limit. 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.

Automated summary

The paper presents novel regular Euclidean solutions to General Relativity with Maxwell and conformally coupled scalar fields. The authors analyze the metrics of the Eguchi-Hanson and Taub-NUT families and solve the field equations analytically. The solutions have nontrivial topology and the authors compute the Euler characteristic explicitly. They also investigate the modifications to the solutions in the presence of higher-curvature corrections that respect conformal invariance.