Non-perturbative quantum effects in stringy degenerate geometries:   Vacuum created pair of $(D{\bar D})_3$-brane by a two form
  We obtain axionic charged primordial black holes on a vacuum created gravitational pair of $(3{\bar 3})$-brane by the Kalb-Ramond field on a $D_4$-brane in presence of a background open string metric. The new geometries on an emergent pair of $(3{\bar 3})$-brane universe is shown to be influenced by the non-perturbative quantum effects underlying a geometric torsion in a second order formalism. The presence of small extra dimensions transverse to the pair in the formalism ensures dynamical scalar fields hidden to a $3$-brane or an anti 3-brane universe. We investigate the non-perturbative quantum vacua for their characteristic properties to explain the accelerated expansion of our universe. Interestingly the emergent brane universe is shown to possess its origin in a degenerate stringy de Sitter vacua at an early epoch. A non-perturbative $D_p$-brane world volume correction for $p < 3$ is worked out to explain some of the quantum effects underlying a quintessence axion in the string-brane setup. Our analysis reveals that a $D$-instanton can be a potential candidate to source the dark energy in our brane universe. 
K. Priyabrat Pandey, Abhishek K. Singh, Sunita Singh and Supriya Kar
hep-th gr-qc hep-ph