De Sitter space in supergravity and M theory

Two ways in which de Sitter space can arise in supergravity theories are discussed. In the first, it arises as a solution of a conventional supergravity, in which case it necessarily has no Killing spinors. For example, de Sitter space can arise as a solution of N = 8 gauged supergravities in four or five dimensions. These lift to solutions of 11-dimensional supergravity or D = 10 IIB supergravity which are warped products of de Sitter space and non-compact spaces of negative curvature. In the second way, de Sitter space can arise as a supersymmetric solution of an unconventional supergravity theory, which typically has some kinetic terms with the "wrong" sign; such solutions are invariant under a de Sitter supergroup. These solutions lift to supersymmetric solutions of unconventional supergravities in D = 10 or D = 11, which nonetheless arise as field theory limits of theories that can be obtained from M-theory by timelike T-dualities and related dualities. Brane solutions interpolate between these solutions and at space and lead to a holographic duality between theories in de Sitter vacua and euclidean conformal field theories. Previous results are reviewed and generalised, and discussion is included of Kaluza-Klein theory with non-compact internal spaces, brane and cosmological solutions, and holography on de Sitter spaces and product spaces.