  We investigate an effective torsion curvature in a second order formalism
underlying a two form world-volume dynamics in a $D_5$-brane. In particular, we
consider the two form in presence of a background (open string) metric in a
$U(1)$ gauge theory. Interestingly the formalism may be viewed via a
non-coincident pair of $(D{\bar D})_5$-brane with a global NS two form on an
anti brane and a local two form on a brane. The energy-momentum tensor is
computed in the six dimensional CFT. It is shown to source a metric fluctuation
on a vacuum created pair of $(D{\bar D})_4$-brane at a cosmological horizon by
the two form quanta in the gauge theory. The emergent gravity scenario is shown
to describe a low energy (perturbative) string vacuum in $6D$ with a
(non-perturbative) quantum correction by a lower ($p<5$) dimensional $D_p$
brane or an anti brane in the formalism. A closed string exchange between a
pair of $(D{\bar D})_4$-brane, underlying a closed/open string duality, is
argued to describe the Einstein vacuum in a low energy limit. We obtain
topological de Sitter and Schwarzschild brane universe in six dimensions. The
brane/anti-brane geometries are analyzed to explore some of their
characteristic and thermal behaviours in presence of the quantum effects. They
reveal an underlying nine dimensional type IIA and IIB superstring theories on
$S^1$.
