Compactification of superstrings and torsion

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Abstract
The compactification of the 10-dimensional superstring theories down to a supersymmetric 4-dimensional vacuum is reanalyzed by allowing for the possibility of torsion in the 6-dimensional compact manifold. New conditions on the generalized complex compact manifold and metric are derived. The connection and the Riemann and Ricci curvature tensors are obtained in an SU(3) basis. There is no 4-dimensional cosmological term despite a nonvanishing Ricci tensor or scalar curvature in the compact 6-dimensional manifold. However, if the Ricci tensor is required to vanish due to other considerations then torsion must also vanish, yielding a Kähler metric. Solutions to the equations are discussed and some explicit examples of 6-dimensional metrics are provided for both vanishing and nonvanishing torsion. However, we have not yet succeeded in solving for the gauge field with non-nonzero torsion.