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

T-Branes and F-Theory: α’- Corrections and the D-Term-Equations

I last introduced T-branes, which are non-Abelian deformation of intersecting D-brane systems in the corresponding compactification manifold. Then I showed that we have a Kähler-equivalence of the derivatives in the pull-back with the gauge-covariant ones, which gave us:

    \[{W^0} = \int_{\tilde S} {S{\rm{Tr}}} \left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}} \right] \wedge {e^{\lambda F}}} \right\}\]

    \[{D^K} = \int_{\tilde S} {S\left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}{\mathop{\rm Im}\nolimits} {e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}} \right\}} \]

with \iota \Phi the inclusion of the complex Higgs field \Phi, and S representing the symmetrization over gauge indices.

Locally, the Higgs field is given by:

    \[\Phi \equiv \phi \frac{\partial }{{\bar \partial z}} + \bar \phi \frac{{\bar \partial }}{{\partial \bar z}}\]

where \phi is a matrix in the complexified adjoint representation of G and \bar \phi its Hermitian conjugate. Thus, I could derive:

    \[\gamma \equiv z{\rm{d}}x \wedge {\rm{d}}y\]


    \[\iota \Phi \gamma = 0\]

a Kähler coordiante expansion of \gamma and gives us, after inserting it in:

    \[{W^0} = \int_{\tilde S} {S{\rm{Tr}}} \left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}} \right] \wedge {e^{\lambda F}}} \right\}\]

the following:

    \[\begin{array}{l}{W^0} = {\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\phi dx \wedge dy \wedge F} \right\} = \\{\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\iota \Phi \Omega \wedge F} \right\}\end{array}\]

which is the exact 7-brane superpotential for F-theory and the integrand is independent of \lambda, entailing that the F-term conditions are purely topological and in no need for \alpha '-corrections