5-D AdS/CFT Warped-Throat Calabi-Yau N-fold Analysis and Klebanov-Strassler Theory

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 Ad{S_5} \times {X_5} where {X_5} is the Einstein manifold in five dimensions, with the interaction-Lagrangian of the massless Klebanov-Strassler field and the brane fields fermions is

    \[\begin{array}{c}{{\not L}^{KS}}_{\psi \bar \psi {H^0}}\frac{1}{{{M^{3/2}}}}\bar \psi \left[ {i{\gamma ^\mu }} \right.{\sigma ^{\mu \nu }}H_{\mu \nu \lambda }^0\left( {{x^\mu }} \right)\\\left. {\frac{{{\chi ^0}(r)}}{{\sqrt {\tau c} }}} \right]\psi \end{array}\]

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

    \[\begin{array}{c}\not L_{\psi \bar \psi {H^0}}^{KS} = i\bar \psi {\gamma ^\mu }{\sigma ^{\mu \nu }}\left[ {\frac{{{e^{ - 4\pi K/{3_{{g_s}}}M}}}}{{{M_{pl}}}}} \right. \cdot \\\left. {\left( {\frac{{{r_{\max }}}}{{{r_0}}}} \right)} \right]H_{\mu \nu \lambda }^0\psi \end{array}\]

with the fundamental Planck scale M and the 4-D Planck scale {M_{pl}} related as

    \[{M_{pl}} = \frac{{{M^{3/2}}}}{{\sqrt {2R} }}{r_{\max }}{\left( {1 - \frac{{r_0^2}}{{r_{\max }^2}}} \right)^{1/2}}\]

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

Klebanov-Strassler Warped Spacetime Geometry and M-Theoretic Cosmology

Loosely put: Klebanov-Strassler spacetime geometry is the warped product of 4-D Minkowski spacetime with 6-D Calabi-Yau orientifold

In part one, I showed, in the context of 4-D low-energy effective description of the KKLT string flux compactifications proposal, that in the limit of N = 1 supergravity, where the moduli potential {V_F} is characterized by a superpotential W and a Kähler potential {\rm K}

    \[{V_F} = {e^{{\rm K}/M_{pl}^2}}\left[ {{{\rm K}^{i\overline j }}{D_i}W\overline {{D_j}W} - \frac{1}{{M_{pl}^2}}{{\left| W \right|}^2}} \right]\]

where W is defined by

    \[\left\{ {\begin{array}{*{20}{c}}{{D_i}W \equiv {{\not \partial }_i}W + \frac{1}{{M_{pl}^2}}\left( {{{\not \partial }_i}{\rm K}} \right)W}\\{{{\rm K}_{i\overline j }} \equiv {{\not \partial }_i}{{\not \partial }_{\overline j }}{\rm K}}\end{array}} \right.\]

yields a standard Calabi-Yau compactification containing 3-form flux {G_3} \equiv {F_3} - \tau {H_3} that contributes to the superpotential via the Gukov-Vafa-Witten 4-fold term

    \[W_{{\rm{Flux}}}^{GV} = \int {{G_3}} \wedge \Omega \]

with \Omega the holomorphic 3-form on the Calabi-Yau three-fold and

    \[\tau \equiv {C_0} + i{e^{ - \Phi }}\]