Abstract
We continue the investigation into non-maximally symmetric compactifications of the heterotic string. In particular, we consider compactifications where the internal space is allowed to depend on two or more external directions. For preservation of supersymmetry, this implies that the internal space must in general be that of a $Spin(7)$ manifold, which leads to a $1/4$-BPS four-dimensional supersymmetric perturbative vacuum breaking all but one supercharge. We find that these solutions allow for internal geometries previously excluded by the domain-wall-type solutions, and hence the resulting four-dimensional superpotential is more generic. In particular, we find an interesting resemblance to the superpotentials that appear in non-geometric flux compactifications of type II string theory. If the vacua are to be used for phenomenological applications, they must be lifted to maximal symmetry by some non-perturbative or higher-order effect.
Highlights
Particular, it is a simple application of Stokes’ theorem to see that a closed flux cannot be used to stabilize moduli in maximally symmetric perturbative compactifications [19]
We consider compactifications where the internal space is allowed to depend on two or more external directions. This implies that the internal space must in general be that of a Spin(7) manifold, which leads to a 1/4-BPS four-dimensional supersymmetric perturbative vacuum breaking all but one supercharge
In this paper we will go further and generalize the perturbative compactification ansatz from the half-flat domain wall case to include a broader class of solutions; in particular, we allow a dependence of the internal geometry on up to two external coordinates
Summary
The bosonic part of the ten-dimensional effective action for heterotic supergravity at lowest order in α′ is given by. Considering theories preserving N = 1 supersymmetry in the effective four-dimensional theory, the Killing spinors reduce the structure group of the internal six compact dimensional manifold down to SU(3). This type of background corresponds to 1/4-BPS states from the point of view of the effective fourdimensional N = 1 supergravity The presence of this spinor on the eight-dimensional geometry will reduce its structure group down to Spin(7), the stability subgroup [40]. As mentioned earlier, the eight-dimensional part of the metric ansatz is further assumed to have a product structure, with two non-compact coordinates and an internal six-dimensional compact manifold This implies the existence of two globally defined oneforms dx and dy corresponding to the non-compact directions.
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