Randall-Sundrum Cosmology and Dp-Brane Dynamics

Let me work in the large-warped Randall-Sundrum D-brane scenario and consider how an {S^1}/{Z_2} orbifolding leads to a non-singular cosmic bounce on the brane thus alleviating the rip of the big-bang singularity. The Randall-Sundrum 5-D braneworld action is:

    \[\begin{array}{*{20}{c}}{S = \int {{d^5}} x\sqrt { - {g_{\left( 5 \right)}}} \left[ {{M^3}{R_{\left( 5 \right)}} - \Lambda } \right]}\\{ + \int {{d^4}x} \sqrt { - {g_{\left( 4 \right)}}} {\rm{ \tilde L}}_{Brane}^{{\rm{large}}}}\end{array}\]

M the 5-D Planck mass, {M^3} = 1/\left( {16\pi {G_5}} \right), and \Lambda the cosmological constant in the bulk, yielding the metric on the brane:

    \[\begin{array}{c}d{s^2} = {e^{ - 2\left| y \right|/L}}\left[ {{\eta _{\mu \nu }} + {h_{\mu \nu }}\left( {x,y} \right)} \right] \cdot \\d{x^\mu }d{x^\nu } + d{y^2}\end{array}\]

with L being the radius of AdS, defined by:

    \[R_{MNPQ}^{\left( 5 \right)} = - \frac{1}{{{L^2}}}\left( {g_{MP}^{\left( 5 \right)}g_{NQ}^{\left( 5 \right)} - g_{MQ}^{\left( 5 \right)}g_{NP}^{\left( 5 \right)}} \right)\]

thus allowing us to derive the CFT-brane relation:

    \[\begin{array}{l}{h_{\mu \nu }}\left( p \right) = - \frac{2}{{L{M^3}}}{\Delta _3}\left[ {{T_{\mu \nu }}\left( p \right) - \frac{1}{2}{\eta _{\mu \nu }}{T^\alpha }_\alpha \left( p \right)} \right]\\ - \frac{1}{{{M^3}}}{\Delta _{KK}}\left( p \right)\left[ {{T_{\mu \nu }}\left( p \right) - \frac{1}{3}{\eta _{\mu \nu }}{T^\alpha }_\alpha \left( p \right)} \right]\end{array}\]

where the dynamics of N-parallel topologically intersecting Dp-branes with gauge group U\left( N \right) is:

    \[\begin{array}{l}S = - \frac{{{T_p}{g_s}{{\left( {2\pi \alpha '} \right)}^2}}}{4}\int {{{\rm{d}}^{p + 1}}} \xi {\rm{tr}}\left( {{F_{ab}}{F^{ab}} + } \right.\\2{D_a}{\Phi ^m}{D^a}{\Phi _m} + \sum\limits_{m \ne n} {{{\left[ {{\Phi ^m},{\Phi ^n}} \right]}^2} + \left. {{\rm{fermions}}} \right)} \end{array}\]

with the Yang-Mills potential

String Field Theory, Gauge Theory and the Landau-Stueckelberg action

Continuing from my last post where I discussed the triangular interplay between string-string duality, string field theory, and the action of Dp/M5-branes, here I shall discuss Stueckelberg string fields and derive the BRST invariance of the Landau-Stueckelberg action. Recalling that the action of M-theory in the Witten gauge is:

    \[\begin{array}{l}{S_M} = \frac{1}{{{k^9}}}\int\limits_{{\rm{world - volumes}}} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {T_p}^{10}d\Omega {\left( {{\phi _{INST}}} \right)^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right) + \\\sum\limits_{Dp} {D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\end{array}\]

with k the kappa symmetry term, {g_{mn}} the metric on {M^{11}}, and {x^m} the corresponding coordinates with {B_{mnp}} an antisymmetric 3-tensor. Hence, the worldvolume {M^3} is:

    \[R \times {S^1} \times {S^1}/{Z_2}\]

and the worldsheet action:

    \[{S_{het}} = {S_{st}} + {S_{KK}} + {S_{\bmod }}\]

being the sum of three terms:

    \[{S_{st}} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right){\partial _i}{x^m}{\partial _j}{x^n}\]

    \[{S_{KK}}\int {{d^2}} \sigma {\varepsilon ^{ij}}{\partial _i}{x^I}{\partial _j}{x^m}A_m^I\]

    \[{S_{\,\bmod \,}} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){\partial _i}{x^J}{\partial _j}{x^I}\]

and the index I = 1, … , 22 labels 22 gauge fields: 16 coming from the internal dimensions of the heterotic string, and the other 6 gauge fields are the KK modes of the metric and antisymmetric tensor. The action {S_{\bmod }} has a massless spectrum given by moduli fields corresponding to deformations of the Narain lattice and thus take values in the group manifold:

    \[\frac{{SO\left( {19,3} \right)}}{{SO\left( {19} \right) \times SO\left( 3 \right)}}\]