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We investigate an effective torsion curvature in a second order formalismunderlying a two form world-volume dynamics in a $D_5$-brane. In particular, weconsider the two form in presence of a background (open string) metric in a$U(1)$ gauge theory. Interestingly the formalism may be viewed via anon-coincident pair of $(D{\bar D})_5$-brane with a global NS two form on ananti brane and a local two form on a brane. The energy-momentum tensor iscomputed in the six dimensional CFT. It is shown to source a metric fluctuationon a vacuum created pair of $(D{\bar D})_4$-brane at a cosmological horizon bythe two form quanta in the gauge theory. The emergent gravity scenario is shownto 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 apair of $(D{\bar D})_4$-brane, underlying a closed/open string duality, isargued to describe the Einstein vacuum in a low energy limit. We obtaintopological de Sitter and Schwarzschild brane universe in six dimensions. Thebrane/anti-brane geometries are analyzed to explore some of theircharacteristic and thermal behaviours in presence of the quantum effects. Theyreveal an underlying nine dimensional type IIA and IIB superstring theories on$S^1$.

Quantum effects in topological and Schwarzschild de Sitter brane: Aspects of torsion on ${\mathbf{(D{\bar D})_4}}$-brane universe