Klebanov-Strassler warp-throat conifold-background will be the basis for our explicit analysis of warped D-brane inflationary cosmology

For a visual treat of the mathematical ‘picture’, scroll to the bottom of this post. In part two of this series on M-theoretic world-brane cosmology, I showed that a Klebanov-Strassler geometry naturally arises by considering string theory compactification on where is the Einstein manifold in five dimensions, with the interaction-Lagrangian of the massless Klebanov-Strassler field and the brane fields fermions is

then I showed that after integrating over the extra dimensional part, the effective 4-D Lagrangian reduces to

with the fundamental Planck scale and the 4-D Planck scale related as

Moreover, I demonstrated that the moduli spaces of compact Calabi-Yau spaces naturally contain conifold singularities and that the local description of these singularities is a conifold, a noncompact Calabi-Yau three-fold whose geometry is given by a cone, and that the orbifolded conifold equation

allows us to consider the orbifolded conifold as a fibration over the plane and is a chiral theory with the gauge group

with

the deep part being that this AdS background is an explicit realization of the Randall-Sundrum scenario in string theory

that I discussed here and here. And so in line with the AdS/CFT duality, the geometry

has a dual gauge theory interpretation

namely, an gauge theory coupled to bifundamental chiral superfields, and adding D5-branes wrapped over the inside , the gauge group becomes

giving a cascading gauge theory. The three-form flux induced by the wrapped D5-branes – fractional D3-branes – satisfies

and the Klebanov-Strassler warp-throat factor is

with

thus allowing explicit analysis of warped D-brane inflationary cosmology. Since **M-theory** remains the only promising paradigm for a corrective **UV-completion of the Standard Model that also unifies gauge and gravitational interactions in a consistent quantum field theory**, it is natural to analyze the theory for an explicit realization of inflationary cosmology. In small steps, in this post, let me do some warped-throat-analysis and set the stage for the next post on Klebanov-Strassler throat-analysis.

Now, the low-energy limit of type IIB superstring theory is type IIB supergravity, whose action is

by which is meant ‘fermion-terms’, with the 10-d reduced Planck mass, and the 10-d Einstein frame metric with Ricci scalar , and is the axio-dilaton, formed from the Ramond-Ramond axion , with the dilaton defined by

and the RR 4-form potential, whose field strength is . The fields and are constructed from the RR and Neveu-Schwarz 2-form potentials and with their respective field strengths and via

and

and since one can include additional localized sources of flux and energy density, such as D-branes and orientifold-planes, in the background, then

can be supplemented by a piece from these sources, containing the tensions and couplings to the p-form fields. Hence, generally, for a compactification background which preserves 4-d Poincaré invariance, the metric can be parametrized as

with is the 4-d Minkowski metric and are coordinates on a compact 6-d internal space and is the warp factor function. Note that can only have Picard-saddle-legs in the compact directions, and the self-dual must take the form

for scalar function of the internal coordinates. Since antibranes break supersymmetry, and orientifold planes break 4-d N = 2 SUSY, which is preserved by a pure Calabi-Yau flux-compactification, to N = 1, throughout this post-series, I will tacitly assume a Calabi-Yau orientifold, which is N = 1 …

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