Classification, geometry and applications of supersymmetric backgrounds

Abstract

We review the remarkable progress that has been made the last 15 years towards the classification of supersymmetric solutions with emphasis on the description of the bilinears and spinorial geometry methods. We describe in detail the geometry of backgrounds of key supergravity theories, which have applications in the context of black holes, string theory, M-theory and the AdS/CFT correspondence unveiling a plethora of existence and uniqueness theorems. Some other aspects of supersymmetric solutions like the Killing superalgebras and the homogeneity theorem are also presented, and the non-existence theorem for certain smooth supergravity flux compactifications is outlined. Amongst the applications described is the proof of the emergence of conformal symmetry near black hole horizons and the classification of warped AdS backgrounds that preserve more than 16 supersymmetries.

On stringy de Sitter spacetimes

“…In particular, we will focus on a class of models arising from type IIB/F-theory, the so called 'axilaton' models 1 [12][13][14][15][16][17]: (1) evade the oft-mentioned no-go theorem 2 [19][20][21][22], (2) form a discretuum owing to their stringy SL(2; Z) monodromy, (3) the overall lengthscale is determined by dimensional transmutation, (4) require g s ∼ O (1), with an effective incorporation of S-duality, and (5) relate two classes of standard supersymmetric string theory solutions [23,24] and [25], and a third, novel class. This resonates with some recent assessments [26], and some features of the recent efforts [27,28]; it reminds of the "T-fold" constructions [29][30][31][32], and qualifies the standard low-energy effective theory limit description as encouraging but incomplete: more of the stringy degrees of freedom must be included, as also advocated recently in the phase-space approach [33][34][35][36][37][38][39][40][41], and also in the different, earlier double field theory approach [42][43][44][45][46].…”

Twisted form hierarchies, Killing-Yano equations and supersymmetric backgrounds

“…In the past fifteen years there has been much progress towards the classification of solutions of supergravity theories that preserve a fraction of the supersymmetry, for a review see [1] and references within. Two main methods have been used for this.…”

“…The gravitino KSE of a supergravity theory is the vanishing condition of the supersymmetry variation of the gravitino evaluated at the vanishing locus of all fermionic fields of the theory. Geometrically the gravitino KSE is a parallel transport equation, D = 0, for the supersymmetry parameter, 1 , with respect to the supercovariant connection, D, which is constructed from the fields of the theory. The supersymmetry variations of the remaining fermions of the theory give rise to algebraic conditions on .…”

Real Killing spinors in neutral signature

“…This is then applied to classifying supergravity solutions by employing gauge transformations in order to express the Killing spinors in simplified canonical forms, which are then used to solve the Killing spinor equations. Such techniques were first used to classify supersymmetric solutions in D=11 supergravity [16], and have also been applied to heterotic and type II supergravity theories [17,18,19,20]; see also the review [22] for a comprehensive description of the applications of spinorial geometry to the classification programme.…”