  We consider a $\sigma$-model formulation of open string theory in the
presence of D-branes. We perform two-loop computations and discuss
gravitational corrections to Born-Infeld action when branes are non-trivially
embedded in a curved ambient space. In particular for the case of a stack of
$N$ coincident D-branes we analyze couplings of the form
$R_{ijkl}[\Phi^i,\Phi^j][\Phi^k,\Phi^l]$.
