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Towards Nonperturbation Theory of Emergent Gravity

We investigate an emergent gravity in $(4+1)$ dimensions underlying a geometric torsion ${\cal H}_3$ in $1.5$ order formulation. We show that an emergent pair-symmetric $4$th order curvature tensor underlying a NS field theory governs a torsion free geometry and is identified as the Riemann type tensor. Interestingly a pair anti-symmetric $4$th order tensor is shown to incorporate a dynamical correction underlying a ${\cal H}_3$ gauge potential and is identified with a non-perturbative correction. The non-perturbative term in the emergent action is shown to be described by a $U(1)$ gauge invariant ${\cal F}_4$ in a second order theory underlying an onshell ${\cal H}_3$ in a first order. A complete emergent theory is elegantly described with an axion, and hence a quintessence, coupling to the Riemann type geometries. The curvatures are appropriately worked out to obtain a $12D$ emergent $F$ theory. Further investigation reveals that a pair of $(M{\bar M})_{10}$-brane are created across an event horizon. We obtain an emergent $M$ theory which in a decoupling limit identifies with the bosonic sector of $N=1$ SUGRA in $11D$.