All the timelike supersymmetric solutions of all ungauged d = 4 supergravities

Abstract

We determine the form of all timelike supersymmetric solutions of all N ≥ 2, d = 4 ungauged supergravities, for N ≤ 4 coupled to vector supermultiplets, using the Usp(n,n)-symmetric formulation of Andrianopoli, D'Auria and Ferrara and the spinor-bilinears method, while preserving the global symmetries of the theories all the way.As previously conjectured in the literature, the supersymmetric solutions are always associated to a truncation to an N = 2 theory that may include hypermultiplets, although fields which are eliminated in the truncations can have non-trivial values, as is required by the preservation of the global symmetry of the theories.The solutions are determined by a number of independent functions, harmonic in transverse space, which is twice the number of vector fields of the theory (n). The transverse space is flat if an only if the would-be hyperscalars of the associated N = 2 truncation are trivial.

Geometry of Killing spinors in neutral signature

“…Viewing U = −f (dt + ω) as a 1-form, the differential relation for U yields the governing equation for the base space 1-form ω, 16) where Ω µ measures the twist of the Killing vector,…”

“…al [4] has triggered many new developments in this field, cf. [5][6][7][8][9][10][11][12][13][14][15][16][17] for an (incomplete) list of references. Due to the reduction of the equations of motion to a simpler set of equations on a certain base space of reduced holonomy, over which the full spacetime is fibered, now a huge catalogue of supersymmetric solutions is available, some of which have been missed in the old ansatz-based approach.…”

The FGK formalism for black p-branes in d dimensions

“…The normalizations have been chosen so as to recover the particular cases considered in Refs. [7,1] 3) in the action, where R ΛΣ (φ) is a scalar dependent matrix such that 4) and where 5) and the ansatz should take into account that the same brane can also be magnetically charged with respect to the dual of the (p + 1)-form potentials, which are also (p + 1)-forms, i.e. they can be dyonic.…”

“…[1] have been used in much of the literature on black holes: the attractors values of the scalars on the horizon for a given set of charges of given model or class of models are determined and the entropy of the corresponding extremal black holes is computed without ever having to construct the complete black-hole spacetime metric explicitly. Actually, only in some supergravity theories it is known how to perform this construction (notably, in N = 2, d = 4 supergravity), even if the generic form of the solutions is known, in principle, for all 4-dimensional supergravities [3]. For extremal non-supersymmetric the situation is worse: a systematic procedure to construct the solutions does not exist even for N = 2, d = 4 supergravities, except in trivial cases.…”

Supersymmetric black lenses in five dimensions

“…It turns out that the metric of supersymmetric solutions admitting a timelike bilinear Killing field is described in an adapted coordinate system by an R-bundle over the 4-dimensional hyper-Kähler base space and the governing equations are linear, allowing us to find a number of exact solutions of physical interest. The program undertaken in [16] has replaced the standard ansatz-based approaches and provoked a drastic progress in classifications of BPS solutions with various fractions of supersymmetry in diverse supergravities [21][22][23][24][25][26][27][28][29][30][31][32][33]. An alternative approach referred to as a spinorial geometry also gives us an elegant and powerful method for classifying supergravity solutions (see [34][35][36][37][38][39][40][41] for an incomplete list of references).…”