Manifolds with reduced holonomy in superstring theories

The main results presented in this dissertation are the following - We have shown that in $d=4$ weak hyperkahler torsion structures are the same that hypercomplex structures and the same that the Plebanski-Finley conformally invariant heavens. With the help of this identification we have found the most general local form of an hyperkahler torsion space in four dimensions. We also presented an Ashtekar like formulation for them in which to finding an hyperkahler torsion metric is reduced to solve a quadratic differential system. - It is found the most general form for the target space metric to the moduli space metric of several $(n>1)$ identical matter hypermultiplets for the type-IIA superstrings compactified on a Calabi-Yau threefold, near conifold singularities, even taking into account non-perturbative D-instanton quantum corrections. The metric in consideration is "toric hyperkahler" if we do not take into account the gravitational corrections. - It is constructed a family of toric hyperkahler spaces in eight dimensions. This spaces are lifted to backgrounds of the eleven dimensional supergravity preserving certain ammount of supersymmetry, with and without the presence of fluxes. Several type IIA and IIB backgrounds have been found by reduction along a circle and by use of T-duality rules. - It is constructed a family of $Spin(7)$, $G_2$ and weak $G_2$ holonomy metrics. The result has been lifted to a supergravity solution preserving one supersymmetry. The presence of a toric symmetry allows a reduction to type IIA backgrounds by usual reduction along one of the Killing vectors