Real Killing spinors in neutral signature

Abstract

Spinorial geometry methods are used to classify solutions admitting Majorana Killing spinors of the minimal 4-dimensional supergravity in neutral signature, with vanishing cosmological constant and a single Maxwell field strength. Two classes of solutions preserving the minimal amount of supersymmetry are found. The first class admits a null-Kähler structure and corresponds to a class of self-dual solutions found by Bryant. The second class admits a null and rotation-free geodesic congruence with respect to which a parallel frame can be chosen. Examples of solutions in the former class are pseudo-hyper-Kähler manifolds; and examples in the latter class include self-dual solutions, as well as a neutral-signature IWP-type solution.

Neutral signature gauged supergravity solutions

“…A systematic classification of solutions in four-dimensional Einstein-Maxwell theory with or without cosmological constant in neutral signature was considered in [24]. In a more recent paper [25], we have used spinorial geometry techniques to classify supersymmetric solutions of minimal four-dimensional Einstein-Maxwell supergravity with neutral signature and without a cosmological constant. Such a theory can be thought of as a truncation of the N = 2 supergravity theory obtained from Hull's M theory [26] via a reduction on CY 3 × S 1 [27].…”

“…Such a theory can be thought of as a truncation of the N = 2 supergravity theory obtained from Hull's M theory [26] via a reduction on CY 3 × S 1 [27]. In the analysis of [25], two orbits for the Majorana Killing spinors were found using appropriately chosen Spin(2, 2) gauge transformations. The solutions for the orbit represented by a chiral Killing spinor were found to correspond to a sub-class of the solutions found [20][21][22].…”

“…In section 2, we introduce the Killing spinor equation of our theory as well as a summary of conventions and results of [25] which are relevant to our subsequent analysis. Relations involving two Spin(2, 2) invariant spinor bilinears are presented and it is demonstrated that only one of the two canonical Killing spinors obtained in [25] survives in the gauged theory. Section 3 contains the analysis of the Killing spinor equations for the remaining non-chiral orbit and the geometric conditions arising from this analysis.…”

“…In this section, we introduce the Killing spinor equation which we shall analyse, and summarize a number of results concerning spinor conventions from [25] which will be particularly JHEP02(2021)030 useful. The Killing spinor equation (KSE) is given by…”

“…For spinors, we adopt the same conventions as introduced in [25]. In particular, there exists a charge conjugation operator C * , [C * , Γ µ ] = 0, and hence if satisfies (2.1) then so does C * .…”