# Euclidean supersymmetric solutions with the self-dual Weyl tensor

## TL;DR

This paper classifies Euclidean supersymmetric solutions with self-dual Weyl tensor in four-dimensional ${
m N}=2$ gauged supergravity, revealing new solutions with enhanced supersymmetry and connecting various known metrics through Killing-Yano tensors.

## Contribution

It demonstrates the existence of additional Killing spinors in certain solutions and unifies different classes of self-dual metrics within a common framework.

## Key findings

- Existence of second Killing spinor in Reissner-Nordström-Taub-NUT-AdS metric
- All solutions reduce to the self-dual Carter family via Killing-Yano tensors
- Identification of conditions for supersymmetry enhancement in these geometries

## Abstract

We explore the Euclidean supersymmetric solutions admitting the self-dual gauge field in the framework of ${\cal N}=2$ minimal gauged supergravity in four dimensions. According to the classification scheme utilizing the spinorial geometry or the bilinears of Killing spinors, the general solution preserves one quarter of supersymmetry and is described by the Przanowski-Tod class with the self-dual Weyl tensor. We demonstrate that there exists an additional Killing spinor, provided the Przanowski-Tod metric admits a Killing vector that commutes with the principal one. The proof proceeds by recasting the metric into another Przanowski-Tod form. This formalism enables us to show that the self-dual Reissner-Nordstr\"om-Taub-NUT-AdS metric possesses a second Killing spinor, which has been missed over many years. We also address the supersymmetry when the Przanowski-Tod space is conformal to each of the self-dual ambi-toric K\"ahler metrics. It turns out that three classes of solutions are all reduced to the self-dual Carter family, by virtue of the nondegenerate Killing-Yano tensor.