Geometric torsion, 4-form, Riemann duals and Quintessence

We revisit an emergent gravity scenario in $(4+1)$ dimensions underlying a propagating geometric torsion ${\cal H}_3$ with a renewed interest. We show that a pair-symmetric $4$th order curvature tensor is sourced by a two-form Neveu-Schwarz (NS) in a $U(1)$ gauge theoretic formulation. Interestingly the new space-time curvature governs a torsion free geometry sourced by a two-form NS field and shares the properties of the Riemann tensor. On the other hand, a completely anti-symmetric $4$th order tensor in the formulation is shown to incorporate a dynamical geometric torsion correction and is argued to be identified with a non-perturbative correction. The four-form turns out to be $U(1)$ gauge invariant underlying an onshell NS form. We show that an emergent gravity theory may elegantly be described with an axionic scalar presumably signifying a quintessence coupling to the Riemann type geometries. The curvatures are appropriately worked out to obtain a $d$$=$$12$ emergent {\it form} theory. Investigation reveals that a pair of $(M{\bar M})_{10}$-brane is created across an event horizon. We show that an emergent $M$ theory in a decoupling limit identifies with the bosonic sector of $N$$=$$1$ Supergravity in $d$$=$$11$.