Flux compactification geometries and de Sitter vacua in M-theory

We address the issue of flux compactification geometries and moduli stabilization within M-theory compactified on S-1/Z(2) which is also known as heterotic M-theory. After explaining its linear flux background and its constraint, we will examine the issue of moduli stabilization in this background. Our focus will hereby lie on the stabilization of the dilaton which appears geometrized in this theory as the orbifold length modulus. A crucial role in this stabilization is played by open membrane instantons which stretch from each ten-dimensional boundary of the heterotic M-theory to an intermediate M5-brane. The intermediate M5-brane, which is needed to avoid consistency problems with the linear background, gets stabilized at the middle of the S-1/Z(2) orbifold interval. We will see that in the large volume limit positive energy vacua emerge. We will next cover the exact flux compactification background which extends the linear approximative background into the regime where a realistic phenomenology can be addressed. The exact background avoids problems with negative Calabi-Yau volume and represents the eleven-dimensional origin of the popular five-dimensional domain-wall solution. The issue of stabilizing the dilaton (orbifold-length) is examined anew in the exact background which allows for gaugino condensation and open membrane instantons to appear on an equal footing. Once more positive energy vacua with stabilized moduli appear which break supersymmmetry spontaneously through non-vanishing F-terms. It is shown that with a hidden gauge group, considerably smaller than E-8, a promising phenomenology and also new dark matter candidates arise.