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We obtain the vacuum solutions for M-theory compactified on eight-manifolds with non-vanishing four-form flux by analyzing the scalar potential appearing in the threedimensional theory. Many of these vacua are not supersymmetric and yet have a vanishing three-dimensional cosmological constant. We show that in the context of Type IIB compactifications on Calabi-Yau threefolds with fluxes and external brane sources α ′corrections generate a correction to the supergravity potential proportional to the Euler number of the internal manifold which spoils the no-scale structure appearing in the classical potential. This indicates that α ′ -corrections may indeed lead to a stabilization of the radial modulus appearing in these compactifications. April 2002the results of [17] by determining the minima of the potential calculated in [21]. This analysis will also show that the non-supersymmetric vacua found in [17] are classically stable. Our second aim is to investigate the fate of the no-scale structure of the potential if one considers the effect of higher derivative corrections of string theory and M-theory.We confirm the expectation of [5] that the no-scale structure in Type IIB compactifications does not survive in the quantum theory once higher order α ′ -corrections are taken into account. In particular, this implies that breaking supersymmetry via a (0,3)-flux induces a non-vanishing potential. Due to the relationship between Type IIB compactifications with three-form flux and M-theory compactifications with four-form flux [19], [25] a similar result should be valid for the non-supersymmetric fluxes in M-theory [17], which also lead to a vanishing cosmological constant at leading order.This paper is organized as follows. In section 2 we rederive the results of [17] from the scalar potential derived in [21] and show that the non-supersymmetric vacua are classically stable. In section 3 we calculate higher order α ′3 -corrections to the scalar potential computed in [5]. We show that these corrections generate a scalar potential that depends on the Calabi-Yau volume and is proportional to the Euler number of the internal manifold.This spoils the no-scale structure of the classical scalar potential for manifolds with nonvanishing Euler number and suggests that further α ′ -corrections may lead to a stabilization of the radial modulus. Some of the technical details of the computation are relegated to an appendix. (Non)-Supersymmetric Solutions in M-theoryIn this section we derive the non-supersymmetric vacuum solutions with vanishing cosmological constant computed in [17] from the superpotentials found in [19] and [21].We use the notation and conventions of [21]. The Scalar PotentialThe scalar potential of M-theory compactified on a fourfold Y 4 to three dimensions ground fluxes is performed in [22], [23] while the potential for M-theory on G 2 -holonomy manifolds with fluxes has been computed in [24].

We discuss mirror symmetry in generalized Calabi-Yau compactifications of type II string theories with background NS fluxes. Starting from type IIB compactified on CalabiYau threefolds with NS three-form flux we show that the mirror type IIA theory arises from a purely geometrical compactification on a different class of six-manifolds. These mirror manifolds have SU(3) structure and are termed half-flat; they are neither complex nor Ricci-flat and their holonomy group is no longer SU (3). We show that type IIA appropriately compactified on such manifolds gives the correct mirror-symmetric lowenergy effective action.