BPS Solutions of 4d Euclidean N=2 Supergravity with Higher Derivative Interactions

We study fully BPS and a broad class of half-BPS stationary configurations of four-dimensional Euclidean N=2 supergravity with higher-derivative interactions. Working within the off-shell conformal supergravity framework of de Wit and Reys (arXiv:1706.04973), we analyse the complete set of Killing spinor equations and obtain the corresponding algebraic and differential constraints. We further derive the Euclidean attractor equations and evaluate the Wald entropy for the fully BPS AdS_2 x S^2 background. For half-BPS stationary configurations, we obtain the generalized stabilization equations expressing all fields in terms of harmonic functions on three-dimensional flat base space, extending the Lorentzian analysis of Cardoso et al (arXiv:hep-th/0009234) to the Euclidean signature. Our results provide a framework for studying supersymmetric saddles and computing the gravitational indices entirely within Euclidean higher-derivative supergravity, without recourse to analytic continuation.

💡 Research Summary

This paper investigates fully BPS and a wide class of half‑BPS stationary configurations in four‑dimensional Euclidean N=2 supergravity with higher‑derivative interactions, using the off‑shell conformal supergravity framework developed by de Wit and Reys (arXiv:1706.04973). The authors begin by reviewing the Euclidean N=2 Weyl multiplet, vector multiplets, and hypermultiplets, emphasizing the symplectic‑Majorana reality conditions that differ from the Lorentzian case and dictate the appropriate analytic continuation rules.

The Killing spinor equations are derived from the Q‑ and S‑supersymmetry variations of the fermions. By imposing suitable superconformal gauge‑fixings, the authors reduce the full set of equations to algebraic and differential constraints on the bosonic fields. For fully BPS solutions (two independent Killing spinors), these constraints force the geometry to be Euclidean AdS₂ × S². The scalar fields of the vector multiplets satisfy Euclidean attractor equations that are the natural continuation of the familiar Lorentzian attractor mechanism. Using Wald’s formula, the authors compute the entropy of this background, showing that higher‑derivative terms (e.g., Weyl‑squared and F⁴ couplings) contribute additive corrections to the area term, exactly as expected from the off‑shell formalism.

For half‑BPS configurations, the authors impose an embedding condition that makes one Killing spinor proportional to the other (up to a gamma‑matrix factor). Assuming a three‑dimensional flat base space, they introduce harmonic functions Hⁱ(𝑥) and Hᵢ(𝑥) on ℝ³, which encode the electric and magnetic charges. All bosonic fields—scalars Xⁱ±, gauge potentials Wⁱ_μ, auxiliary fields Y_{ij}ⁱ, etc.—are expressed in terms of these harmonic functions through generalized stabilization equations. Remarkably, the same functional form persists even when higher‑derivative interactions are present, demonstrating the robustness of the off‑shell approach. This result is the Euclidean analogue of the stabilization equations originally derived by Cardoso, de Wit, and Mohaupt for Lorentzian N=2 Poincaré supergravity (hep‑th/0009234).

The paper emphasizes the relevance of these Euclidean BPS solutions for the computation of supersymmetric gravitational indices. Previously, such indices were defined by analytically continuing Lorentzian black‑hole saddles to Euclidean signature; the present work provides intrinsically Euclidean saddles that preserve supersymmetry without any continuation, thereby offering a more rigorous foundation for index calculations. Moreover, the explicit BPS backgrounds are essential ingredients for applying supersymmetric localization to supergravity, a program that could eventually yield exact results for black‑hole entropy and related observables.

In the concluding sections, the authors outline future directions: extending the analysis to Euclidean N=4 and N=8 supergravities, constructing multi‑center Euclidean solutions, and establishing a detailed dictionary between Lorentzian and Euclidean variables in the presence of higher‑derivative terms. Appendices collect conventions, the explicit form of the Weyl‑squared Lagrangian, and the reality conditions for all fields.

Overall, the work delivers a comprehensive, off‑shell treatment of BPS configurations in Euclidean N=2 supergravity with higher‑derivative couplings, derives the Euclidean attractor mechanism and Wald entropy for the fully supersymmetric AdS₂ × S² background, and formulates generalized stabilization equations for half‑BPS stationary solutions. These results lay the groundwork for intrinsic Euclidean computations of gravitational indices and for supersymmetric localization in higher‑derivative supergravity.