M Theory , Orientifolds and G - Flux

We study the properties of M and F theory compactifications to three and four dimensions with background fluxes. We provide a simple construction of supersymmetric vacua, including some with orientifold descriptions. These vacua, which have warp factors, typically have fewer moduli than conventional Calabi-Yau compactifications. The mechanism for anomaly cancellation in the orientifold models involves background RR and NS fluxes. We consider in detail an orientifold of K3 × T 2 with background fluxes. After a combination of T and S-dualities, this type IIB orientifold is mapped to a compactification of the SO(32) heterotic string on a non-Kähler space with torsion. The realization that certain string compactifications can be described in multiple ways has provided a significant improvement in our understanding of non-perturbative string dynamics. Among the more interesting compactifications are those with N=1 and N=2 supersymmetry in four dimensions. An intriguing class of such four dimensional vacua are described by compactifying F theory on a Calabi-Yau four-fold M [1]. These vacua are naturally related to compactifications of type IIA string theory and M theory to two and three dimensions, respectively. The strong coupling limit of type IIA compactified on M is well described by the three dimensional theory obtained by compactifying M theory on the same four-fold. If the four-fold admits an elliptic fibration with base B, then we can consider a particular degeneration of this M theory compactification in which the area of the elliptic fiber shrinks to zero. In this limit, M theory on the four-fold goes over to type IIB compactified on the base B with a varying coupling constant. The coupling constant is τ of the elliptic fiber. This four-dimensional type IIB vacuum is known as an F theory compactification. Among the novel features of these vacua is the need to cancel a tadpole anomaly. For this class of compactifications, the anomaly is given by χ/24, where χ is the Euler characteristic of the four-fold. If χ/24 is integral, then the anomaly can be cancelled by placing a sufficient number of spacetime filling branes on points of the compactification manifold [2]. For type IIA, the required branes are strings, while in M and F theory, membranes and D3-branes are required, respectively. However, there is at least one other way of cancelling the anomaly in type IIA or M theory, which is by introducing a background flux for the four-form field strength G [3]. The G-flux …