M-Theory/Type-IIB Duality, Brane-Dynamics and EFT

Toroidal compactifications of 11-D supergravity naturally induce {{{\mathrm E}}_{{n(n)}}}{{\left( \mathbb{R} \right)}_{{n=6,7,8}}} exceptional symmetries in \text{D}=11-n that can be realized as U-duality symmetries of M-theory upon Z-discretization and without Betti-truncations. Hence, exceptional field theory based on the modular group S{{L}_{2}}\left( \mathbb{R} \right) uses a dimensionally extended spacetime to 12-D that fully covariantizes supergravity under the U-duality symmetry groups of M-theory. By mirror symmetry, there ought to be a deep internal symmetry induced between M-theory and F-theory upon KK-reduction to Type-IIB SUGRA. In the {{{\mathrm E}}_{{n(n)}}}/USp(n+2) formalism taking the 6-8 limit, the content of the theory is given by the action:

\displaystyle {{S}_{{EFT}}}=\int{{{{d}^{5}}}}x{{d}^{{27}}}e\left( {g_{R}^{g}+\tilde{F}_{V}^{g}\left( {{{\mathcal{L}}_{{top}}}} \right)} \right)

with:

\displaystyle g_{R}^{g}\equiv \hat{R}+\frac{1}{{24}}{{g}^{{\mu \nu }}}{{D}_{\mu }}{{{\hat{M}}}^{{MN}}}{{D}_{\nu }}{{{\hat{M}}}_{{MN}}}

and:

\displaystyle \tilde{F}_{V}^{g}\left( {{{\mathcal{L}}_{{top}}}} \right)\equiv -\frac{1}{4}{{{\hat{M}}}_{{MN}}}{{{\hat{F}}}^{{\mu \nu M}}}{{{\hat{F}}}_{{\mu \nu }}}^{N}+{{e}^{{-1}}}{{\mathcal{L}}_{{top}}}-V\left( {{{{\hat{M}}}_{{MN}}},{{g}_{{\mu \nu }}}} \right)

where the Chern-Simons-topological Lagrangian has covariant variational form:

\displaystyle \delta {{\mathcal{L}}_{{top}}}=\kappa {{\varepsilon }^{{\mu \nu \rho \sigma \tau }}}\left( {A_{{\tilde{F}}}^{\delta }+\mathcal{H}_{\Delta }^{{{{\partial }_{N}}}}} \right)

with:

\displaystyle A_{{\tilde{F}}}^{\delta }\equiv \frac{3}{4}{{d}_{{MNK}}}{{{\tilde{F}}}_{{\mu \nu }}}^{M}{{{\tilde{F}}}_{{\rho \sigma }}}^{N}\delta {{A}_{\tau }}^{K}

\displaystyle \mathcal{H}_{\Delta }^{{{{\partial }_{N}}}}\equiv 5{{d}^{{MNK}}}{{\partial }_{N}}{{\mathcal{H}}_{{\mu \nu \rho M}}}\Delta {{B}_{{\sigma \tau K}}}

and the Yang-Mills field equation for the covariant field strength form {{\tilde{F}}_{{\mu \nu }}}^{M} is:

\displaystyle {{d}^{{PML}}}{{\partial }_{L}}\left( {e{{{\hat{M}}}_{{MN}}}{{{\tilde{F}}}^{{\mu \nu N}}}+\kappa {{\varepsilon }^{{\mu \nu \rho \sigma \tau }}}{{\mathcal{H}}_{{\rho \sigma \tau M}}}} \right)=0

Thus, we can derive the Chern-Simons-type topological action:

\displaystyle {{S}_{{top}}}=\kappa \int{{{{d}^{{27}}}}}Y\int_{{{{{\hat{M}}}_{6}}}}{{\left( {dF-\partial \mathcal{H}} \right)}}

with:

\displaystyle dF\equiv {{d}_{{MNK}}}{{{\tilde{F}}}^{M}}\wedge {{{\tilde{F}}}^{N}}\wedge {{{\tilde{F}}}^{K}}

and:

\displaystyle \partial \mathcal{H}\equiv -40{{d}^{{MNK}}}{{\mathcal{H}}_{{\hat{M}}}}\wedge {{\partial }_{N}}{{\mathcal{H}}_{K}}

and the covariant curvature form {{\tilde{F}}^{M}} and holomorphic curvature form {{\mathcal{H}}_{M}} are, respectively:

\displaystyle {{\tilde{F}}^{M}}\equiv \frac{1}{2}{{\tilde{F}}_{{\mu \nu }}}^{M}d{{x}^{\mu }}\wedge d{{x}^{\nu }}

and:

\displaystyle {{\mathcal{H}}_{M}}\equiv \frac{1}{{3!}}{{\mathcal{H}}_{{\mu \nu \rho M}}}d{{x}^{\mu }}\wedge d{{x}^{\nu }}\wedge d{{x}^{\rho }}

where the Ramond-Ramond gauge-coupling sector is given by the action:

\displaystyle \mathcal{L}_{G}^{{Loc}}=\sum\limits_{{b=1}}^{{N-1}}{{\frac{1}{{2g_{b}^{2}}}}}{{\int{\text{d}}}^{2}}\theta {{W}^{\alpha }}{{W}_{\alpha }}{{\delta }^{2}}\left( {\left( {1-{{e}^{{ib\phi }}}} \right)z} \right)

and the Ramond-Ramond term being:

\displaystyle {{S}_{{CS}}}=\frac{{{{T}_{p}}}}{2}\int\limits_{{{{\Sigma }_{{p+1}}}}}{{C\wedge \text{Tr}}}\left( {{{e}^{{F/2\pi }}}} \right)

thus giving us the Type-IIB Calabi-Yau three-fold superpotential:

\displaystyle {{V}_{W}}=\int\limits_{X}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}+\sum\limits_{{i=1}}^{{{{h}^{{1,1}}}}}{{{{A}_{i}}}}\left( {\left( {{{e}^{{-\phi }}}+i{{C}_{0}}} \right),U} \right){{e}^{{-a\left( {{{e}^{{-\phi }}}{{\tau }_{i}}+i{{\rho }_{i}}} \right)}}}

Before we can see the duality relations between M-theory and F-theory elliptic fibrational Standard-Model constructions, note that the topologically mixed Yang-Mills action:

\displaystyle {{\mathcal{L}}_{{TYM}}}\equiv -\frac{1}{4}e{{\tilde{F}}_{{\mu \nu }}}^{M}{{\tilde{F}}^{{\mu \nu N}}}{{\hat{M}}_{{MN}}}+\kappa {{\mathcal{L}}_{{CS}}}

where the corresponding Chern-Simons action is:

\displaystyle {{S}_{{CS}}}=\frac{{{{T}_{p}}}}{2}\int\limits_{{{{\Sigma }_{{p+1}}}}}{{C\wedge \text{ch}}}\left( {\tilde{F}} \right)\wedge \sqrt{{\frac{{\hat{A}\left( {{{R}_{T}}} \right)}}{{\hat{A}\left( {{{R}_{N}}} \right)}}}}

with the Ramond-Ramond coupling-term:

\displaystyle {{S}_{{CS}}}=\frac{{{{T}_{p}}}}{2}\int\limits_{{{{\Sigma }_{{p+1}}}}}{{C\wedge \text{Tr}}}\left( {{{e}^{{\tilde{F}/\pi }}}} \right)

has variational action:

\displaystyle \begin{array}{c}\delta {{\mathcal{L}}_{{TYM}}}=\left( {\Theta _{F}^{\kappa }-\Xi _{D}^{M}} \right)\delta {{A}_{\mu }}^{M}+\\5{{d}^{{MKN}}}{{\partial }_{K}}\left( {\tilde{\Theta }_{F}^{\kappa }+\mathcal{H}} \right)\Delta {{B}_{{\mu \nu N}}}+\vartheta \left( {\delta {{g}_{{\mu \nu }}}} \right)+\vartheta \left( {\delta {{{\hat{M}}}_{{MN}}}} \right)\end{array}

with:

\displaystyle \Theta _{F}^{\kappa }\equiv \kappa {{\varepsilon }^{{\mu \nu \rho \sigma \tau }}}{{d}_{{MNK}}}{{{\tilde{F}}}_{{\nu \rho }}}^{K}{{{\tilde{F}}}_{{\sigma \tau }}}^{N}

\displaystyle \Xi _{D}^{M}\equiv -{{D}_{\nu }}\left( {e{{{\hat{M}}}_{{MN}}}{{{\tilde{F}}}^{{\mu \nu N}}}} \right)

\displaystyle \tilde{\Theta }_{F}^{\kappa }\equiv e{{{\tilde{F}}}^{{\mu \nu N}}}{{{\hat{M}}}_{{MN}}}

\displaystyle \mathcal{H}\equiv \frac{{4\kappa }}{3}{{\varepsilon }^{{\mu \nu \rho \sigma \tau }}}{{\mathcal{H}}_{{\rho \sigma \tau M}}}

Now, since 11-D SUGRA on a torus is equivalent to Type-IIB string-theory on a circle, the action of the modular group on the Type-IIB axio-dilaton…