  The mechanism responsible for the extraordinary interface conductivity of
LaAlO$_{3}$ on SrTiO$_{3}$ and its insulator-metal transition remains
controversial. Here, using density functional theory calculations, we establish
a comprehensive and coherent picture that the interplay of electronic
reconstructions, lattice distortions, and surface oxygen vacancies fully
compensates the polarization potential divergence in LaAlO$_{3}$/SrTiO$_{3}$,
explaining naturally the experimental observations under different conditions.
While lattice distortions and a charge redistribution between LaO and AlO$_2$
sub-layers play a dominant role in insulating state, a spontaneous appearance
of 1/4 oxygen vacancies per AlO$_{2}$ sub-layer at the LaAlO$_{3}$ surface
accompanied by 0.5$e^{-}$ charge-transfer into the interface is responsible for
interface conductivity and the discontinuous transition in
LaAlO$_{3}$/SrTiO$_{3}$. Our model also explain properties of
LaAlO$_{3}$/SrTiO$_{3}$ prepared with different growth conditions.
