Abstract
We explore the Euclidean supersymmetric solutions admitting the self-dual gauge field in the framework of 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–Nordström–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 Kähler 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.
Highlights
The localization technique [1, 2] in supersymmetric gauge theories defined on a curved Riemannian background provides a powerful tool to implement exact nonperturbative calculations, such as the expectation value of a Wilson loop and the partition functions
On account of the G-structure restriction coming from the Killing spinor, the corresponding supersymmetric solutions to the four-dimensional Euclidean gauged supergravity can be classified in a systematic manner [9,10,11]
Ref. [11] analyzed the condition under which the Euclidean supersymmetry is enhanced by focusing on the integrability of the Killing spinor equation. They concluded that the half-supersymmetric self-dual solutions are exhausted by the one for which the Toda equation is separable with respect to the coordinates (x, y) and z (see (4.33) in [11])
Summary
The localization technique [1, 2] in supersymmetric gauge theories defined on a curved Riemannian background provides a powerful tool to implement exact nonperturbative calculations, such as the expectation value of a Wilson loop and the partition functions. The method embraced there was to cast the Plebanski-Demianski metric into two different Przanowski-Tod forms, by making full use of the Killing-Yano tensor found in [18] and the ambi-Kahler property [19]. This was the first work that demonstrated explicitly that the self-dual solution admits the enhancement of super and hidden symmetries, compared to the non-self-dual counterparts. The KillingYano tensor for the self-dual Plebanski-Demianski solution was constructed based on the Petrov-D property This makes it obscure whether the presence of the second Killing spinor for the self-dual metric is generic or specific to the algebraically special solutions.
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