Supersymmetry, duality and holonomy

In this thesis, I study various aspects of solutions to eleven-dimensional supergravity and its descendents. The former is at one corner of the moduli space of M-theory. While it is not clear how to formulate M-theory; it is equally interesting to see how far we can proceed from this low energy window. First of all, various techniques are applied to construct supergravity solutions preserving partial supersymmetry. A seven-dimensional membrane solution in the gauged supergravity is constructed by lifting a self-dual string in six dimensions, and its supersymmetric property is explored in certain detail. Then fractional BPS solutions from Sn x Sn reduction of six and ten-dimensional supergravities are constructed via the method of G-structures. The form of the solutions is totally determined by Laplace equations with specified boundary conditions. Secondly, the concept of duality is realized in two aspects. A certain type of *-theory, obtained from time-like T-dualization of the usual string and M-theory, is studied and its hyperbolic reduction results in de Sitter solutions, which are favored by modern astrophysical observations. Then the mass of R-charged AdS black holes is properly defined with Hamilton-Jacobi counterterms added in the supergravity action, as another illustration of the AdS/CFT correspondence. Finally, the concept of holonomy in Riemannian geometry is generalized to the discussion of generic supersymmetric solutions in various supergravity theories, as an alternate and comparable method for classifying supersymmetric solutions of fractional BPS. The subtlety of higher-order integrability is also discussed at the end.