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The U(1) gauge dynamics on a D4-brane is revisited, with a two form, toconstruct an effective curvature theory in a second order formalism. We exploitthe local degrees in a two form, and modify its dynamics in a gauge invariantway, to incorporate a non-perturbative metric fluctuation in an effectiveD4-brane. Interestingly, the near horizon D4-brane is shown to describe anasymptotic Anti de Sitter (AdS) in a semi-classical regime. Using Weylscaling(s), we obtain the emergent rotating geometries leading to primordial deSitter (dS) and AdS vacua in a quantum regime. Under a discrete transformation,we re-arrange the mixed dS patches to describe a Schwazschild-like dS (SdS) anda topological-like dS (TdS) black holes. We analyze SdS vacuum for Hawkingradiations to arrive at Nariai geometry, where a discrete torsion forms acondensate. We perform thermal analysis to identify Nariai vacuum with a TdS.Investigation reveals an AdS patch within a thermal dS brane, which may providea clue to unfold dS/CFT. In addition, the role of dark energy, sourced by adiscrete torsion, in the dS vacua is investigated using Painleve geometries. Itis argued that a D-instanton pair is created by a discrete torsion, with a BigBang/Crunch, at the past horizon in a pure dS. Nucleation, of brane/anti-branepair(s), is qualitatively analyzed to construct an effective space-time on aD4-brane and its anti brane. Analysis re-assures the significant role played bya non-zero mode, of NS-NS two form, to generalize the notion of branes within abrane.

Discrete torsion, de Sitter tunneling vacua and AdS brane: U(1) gauge theory on D4-brane and an effective curvature