  We revisit an effective space-time torsion curvature in a second order
formalism, underlying the non-linear $U(1)$ gauge dynamics, of a two form on a
$D_4$-brane in type IIA superstring theory. The formalism incorporates the
significance of a global NS two form into the theory via its perturbative
coupling to a dynamical two form. In particular, we explore the non-linear
gauge dynamics on a $D_4$-brane in presence of a non-trivial background metric.
The fact that the global modes of a NS two form in an open string theory
sources the background metric on a $D_4$-brane may hint at the existence of an
anti $D_4$-brane in the formalism. An effective de Sitter universe is shown to
emerge on a vacuum created pair of $(D{\bar D})_3$-brane by a local two form at
the past horizon with a Big Bang. We obtain a number of 4D de Sitter quantum
black holes, including a Reissner-Nordstrom vacuum, with and without a
propagating torsion. The quantum black holes are shown to be free from
curvature singularity at $r\rightarrow 0$. In a low energy limit, the
nonperturbative correction sourced by a torison may seen to be insignificant.
The quantum black hole undergoes an expansion in the limit and presumably
identifies with the Einstein vacuum. Interestingly our analysis reveals a
plausible quintessence (axion) on an anti $D_3$-brane which may source the dark
energy in a $D_3$-brane universe. Arguably a brane universe moves away from its
anti-brane due to the conjectured repulsive gravity underlying the
quintessence. It leads to a growth in extra fifth dimension between a brane and
an anti-brane which may provide a clue behind an accelerating universe observed
in cosmology.
