Supergravity combines the ideas of general relativity and supersymmetry. Loosely speaking, it is a gauge theory of local supersymmetry. Furthermore, supergravity emerges as the low-energy limit of string theory, and understanding its properties is a key to understanding string theory.
Supergravity is the low-energy limit of string theory, which is the most promising candidate theory capable of unifying quantum theory with general relativity. Significant progress has been made in developing techniques for analysing the geometric properties of supersymmetric solutions in higher dimensions. Supersymmetry is a mathematical extension of the standard types of symmetries, such as rotations, and it is a key property of string theory.
Using new spinorial geometry methods, systematic classifications of supersymmetric supergravity solutions have been constructed. The spinorial geometry techniques are particularly useful in analysing solutions with more than the minimal amount of supersymmetry possible. For example, they have been used to either exclude, or prove uniqueness of, certain types of supergravity solutions with very large amounts of supersymmetry.
We are interested in the classifications of anti-de Sitter supergravity solutions, which are particularly important in the context of the AdS/CFT duality. In the case of two-dimensional anti-de Sitter solutions, the topology of the internal space plays a key role in determining the amount of supersymmetry preserved by these solutions. Furthermore, combining this analysis with the homogeneity theorem for highly supersymmetric supergravity solutions, and the constraints obtained from the super-Jacobi identities in the associated superalgebras, produces strict conditions on the types of possible highly supersymmetric anti-de Sitter geometries.
There are many remaining open questions regarding the general classification of supergravity solutions, which are also of particular relevance in the context of black holes.