Geometry of Killing spinors in neutral signature

Abstract

We classify the supersymmetric solutions of minimal N = 2 gauged supergravity in four dimensions with neutral signature. They are distinguished according to the sign of the cosmological constant and whether the vector field constructed as a bilinear of the Killing spinor is null or non-null. In neutral signature the bilinear vector field can be spacelike, which is a new feature not arising in Lorentzian signature. In the Λ < 0 non-null case, the canonical form of the metric is described by a fibration over a threedimensional base space that has U(1) holonomy with torsion. We find that a generalized monopole equation determines the twist of the bilinear Killing field, which is reminiscent of an Einstein-Weyl structure. If, moreover, the electromagnetic field strength is self-dual, one gets the Kleinian signature analogue of the Przanowski-Tod class of metrics, namely a pseudo-hermitian spacetime determined by solutions of the continuous Toda equation, conformal to a scalar-flat pseudo-Kähler manifold, and admitting in addition a charged conformal Killing spinor. In the Λ < 0 null case, the supersymmetric solutions define an integrable null Kähler structure. In the Λ > 0 non-null case, the manifold is a fibration over a Lorentzian Gauduchon-Tod base space. Finally, in the Λ > 0 null class, the metric is contained in the Kundt family, and it turns out that the holonomy is reduced to Sim(1) × Sim(1). There appear no self-dual solutions in the null class for either sign of the cosmological constant.

Real Killing spinors in neutral signature

“…However, for Λ > 0, the supercovariant derivative does commute with charge conjugation, and Majorana Killing spinors should exist. Furthermore, one might expect (some of) the geometries found in our work to arise as appropriately tuned limits of Λ > 0 solutions in from [29]. However, this is not the case 1 , and the complete geometric interpretation of the class (ii) solutions remains elusive.…”

“…Finally, we note that the issue of supersymmetry counting and Majorana Killing spinors is also of relevance in connection with the work of [29], in which solutions of the minimal gauged supergravity in neutral signature, and with nonzero cosmological constant Λ, were classified. The classification was undertaken for solutions with Λ < 0 and Λ > 0 separately.…”

Counting the number of Killing vectors in a 3D spacetime

“…For τ 2 (k + 4τ 2 2 ) = 0 with k = −1, the metric describes the 3-dimensional Gödel universe, which is sourced by a dust with a negative cosmological constant (see e.g. [26]).…”

Supersymmetric near-horizon geometry and Einstein-Cartan-Weyl spaces

“…Moreover, under the identifications (59), eqns. (47), (48), (49), (50) and (58) become respectively (14), (39), (40), (38) and (41). Likewise, the generalized monopole equations (37) and (44) coincide.…”

“…As we will see below, using (11), equ. (15) can be rewritten as a generalized monopole equation [36][37][38][39][40].…”