Non-conformally Einstein instantons in Conformal Gravity with and without nonlinear matter fields

TL;DR

This paper investigates non-conformally Einstein gravitational instantons within four-dimensional Conformal Gravity (CG), examining both vacuum configurations and those backreacted by nonlinear conformal matter (conformally coupled scalars and ModMax electrodynamics). It first analyzes the Lorentzian Kerr-NUT-AdS extension in CG, computes Noether-Wald charges, and then performs a Euclidean continuation to identify a curve in parameter space where the Weyl tensor becomes (anti-)self-dual, saturating a gravitational BPS bound via the Chern-Pontryagin index. It then introduces nonlinear conformal matter and constructs generalized Taub-NUT-AdS and Eguchi-Hanson instantons, computing their temperatures, partition functions, and charges (showing finiteness due to conformal invariance) and deriving a nonlinear Riegert-type limit. Across all cases, the authors analyze global properties, duality curves, and on-shell actions, highlighting when solutions are not conformally Einstein through the Dunajski-Tod criteria. The results extend the landscape of CG instantons with finite actions and well-defined thermodynamics, offering potential holographic avenues and insights into nonlinear conformal effects in gravity.

Abstract

In this work, we study non-conformally Einstein gravitational instantons in four-dimensional Conformal Gravity, both in vacuum and in the presence of nonlinear conformal matter. First, the one-parameter extension of the Kerr-NUT-AdS metric is analyzed. We obtain their conserved charges by using the Noether-Wald formalism. It turns out that they receive corrections from the linear modes present in Conformal Gravity, which are properly identified. Then, we perform the analytic continuation into the Euclidean section and find the curve in parameter space along which this solution becomes regular and globally (anti)-self-dual. Using the Dunajski-Tod theorem, we show that the metric is not conformally Einstein. Then, the backreaction of nonlinear conformal matter is considered. In particular, we find new gravitational instantons in the presence of conformally coupled scalar fields and ModMax electrodynamics. We compute the partition function and conserved charges, which turn out to be finite by virtue of the conformal invariance of the theory. As a byproduct, we also obtain a generalization of the Riegert metric dressed with nonlinear conformal matter as a particular limit of these instantons. For all cases, we analyze the global properties, the curve in parameter space where the solutions are (anti)-self-dual, and the on-shell Euclidean action, among other features.