Orbifold Branes in the $M_D \times M_{d^+} \times M_{d^-}$
  Compactification of type II string on $S^1/Z_2$ and their cosmological
  applications
  In this paper, we study the implementation of brane worlds in type II string
theory. Starting with the NS/NS sector of type II string, we first compactify
the $(D+d_{+} + d_{-})$-dimensional spacetime, and reduce the corresponding
action to a D-dimensional effective action, where the topologies of $M_{d_{+}}$
and $M_{d_{-}}$ are arbitrary. We further compactify one of the $(D-1)$ spatial
dimensions on an $S^{1}/Z_{2}$ orbifold, and derive the gravitational and
matter field equations both in the bulk and on the branes. Then, we investigate
two key issues in such a setup: (i) the radion stability and radion mass; and
(ii) the localization of gravity, and the corresponding Kaluza-Klein (KK)
modes. We show explicitly that the radion is stable and its mass can be in the
order of $GeV$. In addition, the gravity is localized on the visible brane, and
its spectrum of the gravitational KK towers is discrete and can have a mass gap
of $TeV$, too. The high order Yukawa corrections to the 4-dimensional Newtonian
potential is exponentially suppressed, and can be negligible. Applying such a
setup to cosmology, we obtain explicitly the field equations in the bulk and
the generalized Friedmann equations on the branes.
