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

    \[\left\{ {\begin{array}{*{20}{c}}{C_{kl}^\wp :xy = {z^l}}\\{uv = {z^k}}\end{array}} \right.\]

allows us to consider the orbifolded conifold as a {C^ * } \times {C^ * } fibration over the z plane and is a chiral theory with the gauge group

    \[\prod\limits_{i,j}^2 {SU{{(M)}_{i,j}}} \times \prod\limits_{i,j}^2 {SU{{(M)'}_{i,j}}} \]

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 Ad{S_5} \times {T^{1,1}} geometry

has a dual gauge theory interpretation

namely, an SU(N) \times SU(N) gauge theory coupled to bifundamental chiral superfields, and adding M D5-branes wrapped over the {S^2} inside {T^{1,1}}, the gauge group becomes

    \[SU\left( {N + M} \right){\rm{ }} \times {\rm{ }}SU\left( N \right)\]

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

    \[\frac{1}{{{{\left( {2\pi } \right)}^2}\alpha '}}\int_{{S^3}} {{F_3}} = M\]

and the Klebanov-Strassler warp-throat factor is

    \[\begin{array}{c}h(r) = \frac{{27\pi {{\left( {\alpha '} \right)}^2}}}{{4{r^2}}}\left[ {{g_s}} \right.N + \frac{2}{{2\pi }}\\{\left( {{g_s}M} \right)^2}{\rm{In}}\left( {\frac{r}{{{r_0}}}} \right) + \frac{3}{{8\pi }}\left. {{{\left( {{g_s}M} \right)}^2}} \right]\end{array}\]

with

    \[{r_0} \sim {\varepsilon ^{2/3}}{e^{2\pi N/\left( {3{g_s}M} \right)}}\]

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 Ad{S_5} \times {S^5} 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

    \[\begin{array}{c}S = \frac{{M_{10}^8}}{2}\int {{d^{10}}} x\sqrt { - g} \left( {R - \frac{{{{\left| {{{\not \partial }_\tau }} \right|}^2}}}{{2{{\left( {{\rm{Im}}\tau } \right)}^2}}}} \right.\\ - \frac{{{{\left| {{G_3}} \right|}^2}}}{{12{\rm{Im}}\tau }} - \left. {\widetilde F_5^2} \right) + \\\frac{{M_{10}^8}}{{8i}}\int {\frac{{{C_4} \wedge {G_3} \wedge {{\bar G}_3}}}{{{\rm{Im}}\tau }}} + fermions\end{array}\]

by which is meant ‘fermion-terms’, with {M_{10}} the 10-d reduced Planck mass, and g the 10-d Einstein frame metric with Ricci scalar R, and \tau is the axio-dilaton, formed from the Ramond-Ramond axion {C_0}, with the dilaton \phi defined by

    \[\tau = {C_0} + i{e^{ - \phi }}\]

and {C_4} the RR 4-form potential, whose field strength is {F_5}. The fields {G_3} and {\widetilde F_5} are constructed from the RR and Neveu-Schwarz 2-form potentials {C_2} and {B_2} with their respective field strengths {F_3} and {H_3} via

    \[{G_3} = {F_3} - \tau {H_3}\]

and

    \[\begin{array}{c}{\widetilde F_5} = {F_5} - \frac{1}{2}{C_2} \wedge {H_3} + \\\frac{1}{2}{B_2} \wedge {F_3}\end{array}\]

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

    \[\begin{array}{c}{\widetilde F_5} = {F_5} - \frac{1}{2}{C_2} \wedge {H_3} + \\\frac{1}{2}{B_2} \wedge {F_3}\end{array}\]

can be supplemented by a piece {S_{{\rm{Loc}}}} 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

    \[\begin{array}{c}d{s^2} = {e^{2A(y)}}{\eta _{\mu \nu }}d{x^\mu }d{x^\nu } + \\{e^{ - 2A(y)}}{\widetilde g_{mn}}d{y^m}d{y^n}\end{array}\]

with {\eta _{\mu \nu }} is the 4-d Minkowski metric and {y^m} are coordinates on a compact 6-d internal space M and A(y) is the warp factor function. Note that {G_3} can only have Picard-saddle-legs in the compact directions, and the self-dual {\widetilde F_5} must take the form

    \[{\widetilde F_5} = \left( {1 + * } \right)\left( {d\alpha (x) \wedge d{x^0} \wedge d{x^2} \wedge d{x^2} \wedge d{x^3}} \right)\]

for scalar function \alpha (x) 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 …