Non-Conformally Einstein Gravitational Instantons

• The paper demonstrates that non-conformally Einstein gravitational instantons are complete, Bach-flat solutions that cannot be mapped to Einstein spaces through any conformal rescaling.
• It reveals that the inclusion of linear modes and additional parameters broadens the classical Kerr–NUT–(A)dS landscape to accommodate topologically nontrivial and (anti)-self-dual geometries.
• The work shows that coupling nonlinear conformal matter, such as scalars and ModMax electrodynamics, enriches the moduli space and alters conserved charges in these complex configurations.

Non-conformally Einstein gravitational instantons are complete, regular, four-dimensional Riemannian solutions to higher-derivative (especially conformally invariant) gravitational field equations whose metrics cannot be mapped to Einstein spaces by any conformal rescaling. These instantons, which are solutions to the Bach-flat condition in four-dimensional conformal gravity, exist both in vacuum and in the presence of conformally invariant nonlinear matter, and they exhibit topologically nontrivial, (anti)-self-dual, or generalized (non-Einstein) features, thereby broadening the classical landscape of gravitational instantons far beyond the conformally Einstein sector.

1. Structure of Non-Conformally Einstein Instantons in Conformal Gravity

Four-dimensional Conformal Gravity is defined by the action

$$I_{\mathrm{CG}} = \alpha \int d^4x\,\sqrt{|g|}\; W^{\mu\nu}_{\ \ \lambda\rho} \, W_{\mu\nu}^{\ \ \lambda\rho}$$

where $W_{\mu\nu\lambda\rho}$ is the Weyl tensor. The equations of motion require the vanishing of the Bach tensor $B_{\mu\nu}$: $$B_{\mu\nu} = \nabla^\rho\nabla^\sigma W_{\mu\rho\nu\sigma} + \frac{1}{2} R^{\rho\sigma} W_{\mu\rho\nu\sigma} = 0.$$ All Einstein spaces (with $R_{\mu\nu} \propto g_{\mu\nu}$) are automatically Bach-flat, and any metric conformally related to an Einstein metric remains a solution due to conformal invariance. However, nontrivial Bach-flat metrics can exist that do not lie in the conformal class of any Einstein metric. This is regulated by necessary and sufficient conditions (as given by the Dunajski–Tod theorem): the metric is conformally Einstein if and only if certain curvature functionals built from gradients and contractions of the Weyl tensor vanish identically. Metrics violating these conditions are genuinely non-conformally Einstein.

A canonical example is an extension of the Kerr–NUT–(A)dS family in which the so-called "linear mode" (the coefficient $b$ or $c$ in the metric function) is nonzero. These instantons retain regularity, (anti)-self-duality, and can be globally smooth but fail the Dunajski–Tod constraints, precluding any conformal mapping to Einstein geometries (Corral et al., 13 Oct 2025).

2. Explicit Metrics and Parameter Space Structure

The non-conformally Einstein instantons typically generalize the canonical Kerr–NUT–(A)dS geometry by including additional integration constants $b$ and $c$ in the (Euclidean) metric functions,

$$\begin{aligned} ds^2 = \;& -\frac{\Delta_r}{\Xi^2 \Sigma}[dt + (2n\cos\theta + 2nC - a\sin^2\theta) d\varphi]^2\ & + \frac{\Sigma}{\Delta_r} dr^2 + \frac{\Sigma}{\Delta_\theta} d\theta^2 + \frac{\Delta_\theta\sin^2\theta}{\Xi^2\Sigma}[a\,dt - (r^2 + a^2 + n^2 - 2a n C) d\varphi]^2 \end{aligned}$$

with $\Delta_r$, $\Sigma$, and $\Delta_\theta$ augmented by $b$ and $c$ terms (see (Corral et al., 13 Oct 2025)). Even as the conventional mass parameter $m\to 0$, the solution remains nontrivial due to the persistence of the linear modes.

Upon analytic continuation to the Euclidean sector, regularity and absence of conical singularities are enforced by fixing the periods of the Euclidean time and azimuthal coordinates, yielding smooth gravitational instanton geometries. There exists a curve (a codimension-one subvariety in parameter space) along which the Weyl tensor becomes globally (anti)-self-dual. On this curve, the instantons saturate a BPS bound in conformal gravity, with their action quantized in terms of the Chern–Pontryagin index,

$$I_{\mathrm{CG}} = \pm 16\pi^2 \alpha\, P_1[\mathcal{M}],$$

where $P_1[\mathcal{M}]$ is the topological density integrated over the instanton manifold.

Using the Dunajski–Tod criteria, it is shown that with nonvanishing $b$ (representing the linear or non-Einstein mode), the conditions for conformal Einstein equivalence fail, guaranteeing these instantons are outside the Einstein class.

3. Conserved Charges and Non-Einstein Hair

The presence of linear (non-Einstein) modes $b$ and $c$ modifies the conserved charges derived via the Noether–Wald formalism,

$$q^{\mu\nu} = -4\alpha\left( W^{\mu\nu}_{\ \ \lambda\rho} \nabla^\lambda \xi^\rho + 2 \xi^\lambda C_\lambda^{\ \mu\nu} \right).$$

Here, $\xi^\rho$ is a Killing vector and $C_\lambda^{\ \mu\nu}$ the Cotton tensor. The mass $M$ and angular momentum $J$ receive corrections proportional to $b$ and $c$. In the limit $b,c\to 0$, standard Einstein gravity results are recovered, but otherwise, the solutions support additional "hair." These corrections are physical in conformal gravity and constitute nontrivial charges not present in Einstein gravity (Corral et al., 13 Oct 2025).

4. Interplay with Nonlinear Conformal Matter Fields

Coupling conformal gravity to nonlinear conformal matter expands the space of non-conformally Einstein instantons:

• Conformally coupled scalar: The addition of a scalar with a $|\phi|^4$-self-interaction, suitably conformalized (including a Gauss-Bonnet boundary term), yields deformations of the instanton solutions that preserve Bach-flatness and regularity but again cannot be mapped to Einstein spaces by a Weyl scaling.
• ModMax electrodynamics: The ModMax action,

$$I_{MM} = -\int d^4x\,\sqrt{|g|} [X \cosh\gamma - \sqrt{X^2 - Y^2}\sinh\gamma],$$

with $X = \frac{1}{4} F_{\mu\nu} F^{\mu\nu}$, $Y = \frac{1}{4} \tilde{F}_{\mu\nu} F^{\mu\nu}$, is a nonlinear, conformal, and SO(2)-dual invariant extension of Maxwell’s theory. Nonlinear electromagnetic fields via ModMax can be coupled to these instantons, leading to new dyonic, conformal non-Einstein configurations (e.g., generalizations of Taub-NUT–AdS or Eguchi–Hanson instantons), with dyonic charges tied to the nonlinear matter parameters.

For all such matter-coupled cases, the on-shell Euclidean action and conserved charges remain finite due to conformal invariance and nonlinear matter field properties. Regularity (e.g., at the "nut" or "bolt" fixed points) is enforced by appropriate periodicity relations in the Euclidean section.

5. Curve of (Anti)-Self-Duality and BPS Bound

For both vacuum and matter-coupled solutions, there exists a codimension-one curve in the space of integration constants (e.g., relating mass, linear mode, and NUT parameter) along which the solution is globally (anti)-self-dual. On this locus, the on-shell Euclidean action saturates a BPS-type topological bound,

$$I_{\mathrm{CG}} = \pm 16\pi^2 \alpha\, P_1[\mathcal{M}],$$

with $P_1[\mathcal{M}]$ taking values computed from characteristic classes. This topological saturation is linked to the vanishing of bulk energy-momentum and underlines the special role of these instantons.

Additionally, in the presence of nonlinear matter, logarithmic corrections to the on-shell action may appear, reflecting the underlying matter sector's contribution to the gravitational instanton partition function.

6. Riegert Metric Generalization and Static Limit

In the static, zero–NUT limit ($n\to 0$), the general nonlinear matter–dressed instantons reduce to generalized Riegert metrics of the form

$$ds^2 = \left( \frac{r^2}{\ell^2} + b r + 1 + c \right) dt^2 + \frac{dr^2}{\frac{r^2}{\ell^2} + b r + 1 + c } + r^2 ( d\theta^2 + \sin^2\theta d\varphi^2 )$$

accompanied by a scalar hair parameter $c$ and electric/magnetic charge parameters from the nonlinear electrodynamics sector. The Riegert metric in pure conformal gravity represents a "thermalized vacuum," but with nonlinear matter, it encodes nontrivial field profiles and generalized charges. This generalization demonstrates how nontrivial matter fields endow conformal gravity instantons with new physical content and enrich their moduli spaces.

7. Significance, Topology, and Physical Implications

Non-conformally Einstein instantons in conformal gravity mark a genuinely new sector of nonperturbative gravitational configurations:

• Rich moduli: Parameter spaces are enlarged by additional integration constants not available in the Einstein sector; these parameters control regularity, (anti)-self-duality, and matter hair.
• Topologically quantized action: For globally (anti)-self-dual instantons, the action is topologically quantized, ensuring their role as nonperturbative saddle-points in the gravitational path integral.
• Finiteness and conserved charges: Despite their complexity, all conserved charges and partition functions are finite, and the physical content (including energy and angular momentum) is altered by the non-Einstein modes.
• Nonlinear matter coupling: Conformal matter interactions (scalar and nonlinear electromagnetic) generate further generalizations, yielding instantons with new types of topological and energetic properties. This broadens the landscape of gravitational instantons relevant for quantum gravity and holography.

These findings suggest that the spectrum of gravitational instantons in conformal gravity is not exhausted by conformal rescalings of Einstein metrics, but includes distinct, topologically rich, and physically meaningful non-conformally Einstein configurations, both in vacuum and with nonlinear conformal matter (Corral et al., 13 Oct 2025).