Randall-Sundrum Braneworld Scenario, Klebanov-Strassler Geometry and the Standard Model

In this post, the mathematics applies to both, Randall-Sundrum-1and-2 models, hence I will not distinguish between them here. One of the most powerful aspects of M-theory’s braneworld scenarios is that the bosonic and fermionic fields of the Standard Model of physics can be interpreted as low-lying Kaluza-Klein excitations of Randall-Sundrum bulk fields, after extra dimensional modulus stabilization, and recalling that Randall-Sundrum bulk/brane interactions yield a very deep solution to the EW hierarchy problem. Start with the theory defined by the following action:

    \[S = \frac{1}{2}\int {{d^4}} x\int\limits_{ - \pi }^\pi {d\phi \sqrt G } \left( {{G^{AB}}{\partial _A}\Phi {\partial _B}\Phi - {m^2}{\phi ^2}} \right)\]

with the bulk field given by:

    \[\Phi \left( {x,\phi } \right) = \sum\limits_n {\frac{{{\Upsilon _n}\left( \phi \right)}}{{\sqrt {{\tau _c}} }}} \]

where generally, the bulk action, with worldsheet-uplift, is given by:

    \[\begin{array}{l}{S_B} = - \frac{1}{{2{k^2}}}\int {{d^D}} X\left\{ {\frac{1}{2}} \right.\left[ {{\eta ^{\mu \rho }}} \right.{\eta ^{\nu \sigma }} + {\eta ^{\mu \sigma }}{\eta ^{\nu \rho }}\\ - \frac{2}{{D - 2}}{\eta ^{\mu \nu }}\left. {{\eta ^{\rho \sigma }}} \right]{h_{\mu \nu }}{\partial ^2}{h_{\rho \sigma }}\left. { + \frac{4}{{D - 2}}\bar \Phi {\partial ^2}\bar \Phi } \right\}\end{array}\]

and {\Upsilon _n}\left( \phi \right) satisfying:

    \[\int\limits_{ - \pi }^\pi {d\phi {e^{ - \sigma \left( \phi \right)}}} {\Upsilon _n}\left( \phi \right){\Upsilon _m}\left( \phi \right) = {\delta _{nm}}\]

with a Dirac-Born-Infeld brane interaction term:

    \[{S_{BI}} = - {\tau _\rho }\int {{d^{p + 1}}} \xi \left( {\left( {\frac{{2p - D + 4}}{{D - 2}}} \right)\bar \Phi - \frac{1}{2}{h_{aa}}} \right)\]

which, after integration by parts and upon substituting {e^{ - \sigma \left( \phi \right)}} in our action, we get the Horava-Witten action variant:

    \[\begin{array}{c}S = \frac{1}{2}\int {{d^4}} x\int\limits_{ - \pi }^\pi {{r_c}d\phi } \left( {{e^{ - 2\sigma \left( \phi \right)}}} \right.\\{\eta _{\mu \nu }}{\partial _\mu }\Phi {\partial _\nu }\Phi + \frac{1}{{r_c^2}}\left( {{e^{ - 4\sigma \left( \phi \right)}}\partial \Phi } \right)\\ - {m^2}{e^{ - 4\sigma \left( \phi \right)}}\left. {{\Phi ^2}} \right)\end{array}\]

Now, the bulk fields manifest themselves to 4-D ‘observers’ as infinite towers of scalars {\psi _n}\left( x \right) with masses {m_n}. After change of variables to:

    \[\left\{ {\begin{array}{*{20}{c}}{{z_n} = {m_n}{e^{\sigma \left( \phi \right)}}/k}\\{{f_n} = {e^{ - 2\sigma \left( \phi \right)}}{\Upsilon _n}}\end{array}} \right.\]

our actions reduce to two interaction terms:

    \[S_{{\mathop{\rm int}} }^G = \int {{d^4}} x\int\limits_{ - \pi }^\pi {d\phi \sqrt G } \frac{\lambda }{{{M^{5m - 5}}}}{\left( {{G^{AB}}{\partial _A}\Phi {\partial _B}\Phi } \right)^m}\]

and:

    \[\begin{array}{l}S_{{\mathop{\rm int}} }^\Upsilon = \int {{d^4}} x\int\limits_{ - \pi }^\pi {{r_c}} d\phi {e^{ - 4\sigma \left( \phi \right)}}\frac{\lambda }{{{M^{5m - 5}}}} \cdot \\\psi _n^{2m}\left( {\frac{{{{\left( {{\partial _\phi }{\Upsilon _n}} \right)}^2}}}{{r_c^3}}} \right)\end{array}\]

where we have

T-Branes, the Chern-Simons Action and the Kähler Pull-Back

T-branes are supersymmetric intersecting brane configurations such that the non-Abelian Higgs field \Phi that describes D-brane deformations is not diagonalisable and satisfies nilpotency conditions where the worldvolume flux has non-commuting expectation values and their worldvolume adjoint Higgs field is given a VEV that cannot be captured by its characteristic polynomial, and thus derive their importance from the fact that heterotic string compactifications are dual to T-branes in F-theory. Let’s probe their dynamics. Starting with the D-term potential:

    \[\begin{array}{c}{{\hat V}_D} = \frac{1}{{2{\mathop{\rm Re}\nolimits} \left( {{f_h}} \right)}}{\left( {\sum\limits_j {{q_{{\phi _j}}}{\phi _j}\frac{{\partial K}}{{\partial {\phi _j}}} + M_P^2\sum\limits_j {{q_{hj}}} } } \right)^2}\\ = \frac{\pi }{{{\mathop{\rm Re}\nolimits} \left( {{T_h}} \right)}}{\left( {\sum\limits_j {{q_{{\phi _j}}}\frac{{{{\left| {{\phi _j}} \right|}^2}}}{s} - {\xi _h}} } \right)^2}\end{array}\]

with the U\left( 1 \right)-charge:

    \[{q_{hj}} = \frac{1}{{l_s^4}}\int_{{D_h}} {{{\hat D}_j}} \wedge {F^G}\]

and {F^G} the gauge flux that yields the Fayet-Iliopoulos term:

    \[\begin{array}{l}\frac{{{\xi _h}}}{{M_P^2}} = \frac{{{e^{ - \phi /2}}}}{{4\pi \mathcal{V}}}\frac{1}{{l_s^4}}\int_{{D_h}} {J \wedge {F^G}} = \frac{1}{{2\pi }}\sum\limits_j {\frac{{{q_{hj}}}}{\mathcal{V}}} \\ = - \sum\limits_j {{q_{hj}}} \frac{{\partial K}}{{\partial {T_j}}}\end{array}\]

where the D-brane partition function for closed strings is given by:

    \[P_{{\rm{int}}}^{Dp} \equiv Z = \sum\limits_{\gamma = 0}^\infty {\underbrace {\int {{D^K}\gamma {{D'}^K}X{e^{S_{cld}^s}}} }_{{\rm{Topologies}}}} \]

with a non-Abelian D-term:

    \[D_{\hat A}^K = \int_{\tilde S} {\rm{P}} \left[ {{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}} \wedge \sqrt {{{\rm{A}}^\prime }(\Gamma )/{{\overline {\rm{A}} }^\prime }({\rm{N}})} {\rm{ }}\]