Maximally symmetric nuts in 4d $\mathcal{N}=2$ higher derivative supergravity

We initiate a systematic study of supersymmetric backgrounds in 4d $\mathcal{N}=2$ Euclidean supergravity in the presence of infinite towers of higher derivative corrections. Adopting a Gibbons-Hawking view towards the evaluation of the action in terms of nuts and bolts, we consider the two maximally symmetric vacua $\mathbb{R}^4$ and $\mathbb{H}^4$ (Euclidean AdS$_4$) and their unique supersymmetric deformations with (anti-) self-dual Maxwell tensors corresponding to a single nut at the center. These are the Omega background of Nekrasov-Okounkov, $\Omega\, \mathbb{R}^4$, and its generalization with a cosmological constant of Martelli-Passias-Sparks, denoted $\Omega\, \mathbb{H}^4$ (also known as the gravity dual of the $U(1) \times U(1)$ squashed sphere). We write down the BPS configurations in the superconformal formalism in the presence of vector multiplets and derive the corresponding off- and on-shell actions. Our results provide a rigorous proof for important parts of the conjecture in arXiv:2111.06903 and its holographic corollary in arXiv:2204.02992, which we discuss in detail along with extensions such as the addition of hypermultiplets and the presence of conical defects.