On the Compactification of Type IIA String Theory

The ten-dimensional type IIA string effective action with cosmological constant term is dimensionally reduced on a d-dimensional torus to derive lower dimensional effective action. The symmetries of the reduced effective action are examined. It is shown that the resulting six-dimensional theory does not remain invariant under $SO(4,4)$ symmetry whereas the reduced action, in the absence of the cosmological constant respects the symmetry as was shown by Sen and Vafa. New class of black hole solutions are obtained in five and four dimensions in the presence of cosmological constant. For the six-dimensional theory, a four-brane solution is also presented.