F-theory as the Modular Equivariant 12-D Lift of Type-IIB String Theory

Orbifold compactifications of 11-D SUGRA induce ${{{\mathrm E}}_{{n(n)}}}{{\left( \mathbb{R} \right)}_{{n=6,7,8}}}$ exceptional symmetries in $\text{D}=11-n$ that are realized as U-duality symmetries of M-theory upon Z-discretization and without Betti-truncations. Therefore, exceptional field theory based on the modular group $S{{L}_{2}}\left( \mathbb{R*} \right)$ utilizes a dimensionally extended spacetime to 12-D fully covariantizing SUGRA under the U-duality symmetry groups of M-theory. Homological mirror symmetry hence induces an internal symmetry between M-theory and F-theory upon KK-reduction to Type-IIB SUGRA. In the ${{{\mathrm E}}_{{n(n)}}}/USp(n+2)$ Witten limit, the action is given as such:

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

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 ${{\tilde{F}}_{{\mu \nu }}}^{M}$ is given by:

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

Hence, 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:

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

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:

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

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

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

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

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 allows us to take the zero limit of:

$\displaystyle \delta {{S}_{{EFT({{E}_{{n(n)}}})}}}/\delta {{\mathcal{L}}_{{TYM}}}$

and by homological mirror symmetry, we get a Type-IIA dimensional uplift to M-theory, given that in the Einstein frame, the Type-IIB bosonic SUGRA action is:

$\displaystyle S_{{bos}}^{{{{\text{T}}_{{IIB}}}}}=\frac{1}{{2{{\kappa }^{2}}}}\int{{{{d}^{{10}}}}}x\sqrt{{-{{g}^{E}}}}\left[ {\partial _{{R,\tau }}^{M}+{{\Xi }_{\tau }}+{{\Pi }_{F}}} \right]+C_{G}^{{\tilde{G}}}$

with:

$\displaystyle \partial _{{R,\tau }}^{M}\doteq R-\frac{1}{2}\frac{{{{\partial }_{M}}\tau \,{{\partial }^{M}}\bar{\tau }}}{{{{{\left( {\text{Im}\tau } \right)}}^{2}}}}$

$\displaystyle {{\Xi }_{\tau }}\doteq \frac{1}{2}\frac{{{{{\left[ {{{G}_{{\left( 3 \right)}}}} \right]}}^{2}}}}{{\text{Im}\tau }}$

$\displaystyle {{\Pi }_{F}}\doteq \frac{1}{{8i{{\kappa }^{2}}}}\int{{{{C}_{{\left( 4 \right)}}}\wedge {{G}_{{\left( 3 \right)}}}\wedge {{{\hat{G}}}_{{\left( 3 \right)}}}}}$

One can appreciate then that the essence of Exceptional Field Theory is that it characterizes a deeper double duality relating M-theory/Type-IIA and F-theory/Type-IIB. The key is the role of U-duality in the modular holomorphic action on the Neveu-Schwarz sector of Type-IIB string-theory. The generalized diffeomorphisms generated by a vector ${{\Lambda }^{M}}$ act fully locally on $S{{L}_{2}}\times {{\mathbb{R}}^{+}}$ yielding the Lie derivative $\mathcal{L}_{\Lambda }^{D}$ that differs from the classic Lie derivative ${{L}_{\Lambda }}$ by a Calabi-Yau induced $Y$ -tensor and is defined by the following transformation rules:

$\displaystyle \begin{array}{l}\mathcal{L}_{\Lambda }^{D}{{V}^{\alpha }}={{\Lambda }^{M}}{{\partial }_{M}}{{V}^{\alpha }}-{{V}^{\beta }}{{\partial }_{\beta }}{{\Lambda }^{\alpha }}\\-\frac{1}{7}{{V}^{\alpha }}{{\partial }_{\beta }}{{\Lambda }^{\beta }}+\frac{6}{7}{{V}^{\alpha }}{{\partial }_{s}}{{\Lambda }^{s}}\end{array}$

$\displaystyle \mathcal{L}_{\Lambda }^{D}{{V}^{s}}={{\Lambda }^{M}}{{\partial }_{M}}{{V}^{s}}+\frac{6}{7}{{V}^{s}}{{\partial }_{\beta }}{{\Lambda }^{\beta }}-\frac{8}{7}{{V}^{s}}{{\partial }_{s}}{{\Lambda }^{s}}$

The corresponding diffeomorphism algebra has an exceptional field bracket:

$\displaystyle {{\left[ {U,V} \right]}_{E}}=\frac{1}{2}\left( {\mathcal{L}_{{\Lambda ,U}}^{D}V-\mathcal{L}_{{\Lambda ,V}}^{D}U} \right)$

with closure condition:

$\displaystyle \mathcal{L}_{{\Lambda ,U}}^{D}\mathcal{L}_{{\Lambda ,V}}^{D}-\mathcal{L}_{{\Lambda ,V}}^{D}\mathcal{L}_{{\Lambda ,U}}^{D}=\mathcal{L}_{{\Lambda ,{{{\left[ {U,V} \right]}}_{E}}}}^{D}$

The action of the diffeomorphism-symmetries are parametrized by vector bundles over the metaplectic space and take the form:

$\displaystyle \begin{array}{l}{{\delta }_{\xi }}{{V}^{\mu }}\equiv {{L}_{\xi }}{{V}^{\mu }}={{\xi }^{\nu }}{{D}_{\nu }}{{V}^{\mu }}\\-{{V}^{\nu }}{{D}_{\nu }}{{\xi }^{\mu }}+{{{\hat{\lambda }}}_{V}}{{D}_{\nu }}{{\xi }^{\nu }}{{V}^{\mu }}\end{array}$

with:

$\displaystyle {{D}_{\mu }}={{\partial }_{\mu }}-{{\delta }_{{{{A}_{\mu }}}}}$

where the gauge vector transforms as:

$\displaystyle {{\delta }_{\Lambda }}{{A}_{\mu }}^{M}={{D}_{\mu }}{{\Lambda }^{M}}$

The corresponding generalized exceptional scalar metric $\wp$ satisfies the following property:

$\displaystyle {{\delta }_{\xi }}{{\wp }_{{MN}}}={{\xi }^{\mu }}{{D}_{\mu }}{{\wp }_{{MN}}}$

which decomposes in light of the orbifold blow-up:

$\displaystyle S{{L}_{2}}\times \mathbb{R*}/SO\left( 2 \right)$

as:

$\displaystyle {{\wp }_{{MN}}}={{\wp }_{{\alpha \beta }}}\otimes {{\wp }_{{ss}}}$

Hence, we can define the exceptional metric:

$\displaystyle {{\Omega }_{{\alpha \beta }}}\equiv {{\left( {{{\wp }_{{ss}}}} \right)}^{{3/4}}}/{{\wp }_{{\alpha \beta }}}$

Since the full Type-IIB Calabi-Yau superpotential is given by:

$\displaystyle W=\int_{Y}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}+\sum\limits_{{i=1}}^{{{{h}^{{1,3}}}}}{{{{A}_{i}}}}\left( {S,U} \right){{e}^{{-{{a}_{i}}{{T}_{i}}}}}$

with Kähler Type-IIB orientifold moduli:

$\displaystyle {{T}_{i}}={{e}^{{-\phi }}}{{\tau }_{1}}+i{{\rho }_{i}}$

and:

$\displaystyle S={{e}^{{-\phi }}}+i{{C}_{0}}$

and where the volume of the divisor, ${{\tau }_{1}}$ , is:

$\displaystyle {{\tau }_{1}}=\frac{1}{2}\int_{D}{{J\wedge J}}$

with:

$\displaystyle {{\rho }_{i}}=\int_{{{{D}_{i}}}}{{{{C}_{4}}}}$

we thus have the ingredients to write the modular exceptional field theory action as:

$\displaystyle S=\int{{{{d}^{9}}}}x{{d}^{3}}Y\sqrt{g}\left( {\hat{R}+{{\mathcal{L}}_{{skin}}}+{{\mathcal{L}}_{{gkin}}}+\frac{1}{{\sqrt{g}}}{{\mathcal{L}}_{{top}}}+V} \right)$

with the exceptional Ricci scalar:

$\displaystyle \begin{array}{l}\hat{R}=\frac{1}{4}{{g}^{{\mu \nu }}}{{D}_{\mu }}{{g}_{{\rho \sigma }}}{{D}_{\nu }}{{g}^{{\rho \sigma }}}\frac{1}{2}{{g}^{{\mu \nu }}}{{D}_{\mu }}{{g}^{{\rho \sigma }}}{{D}_{\rho }}{{g}_{{\nu \sigma }}}\\+\frac{1}{4}{{g}^{{\mu \nu }}}{{D}_{\mu }}\text{In}g{{D}_{\nu }}\text{In}g+\frac{1}{2}{{D}_{\mu }}\text{In}g{{D}_{\nu }}{{g}^{{\mu \nu }}}\end{array}$

the kinetic part:

$\displaystyle \begin{array}{l}{{\mathcal{L}}_{{skin}}}=\frac{7}{{32}}{{g}^{{\mu \nu }}}{{D}_{\mu }}\text{In}{{\wp }_{{ss}}}{{D}_{\nu }}\text{In}{{\wp }_{{ss}}}\\+\frac{1}{4}{{g}^{{\mu \nu }}}{{D}_{\mu }}{{\Omega }_{{\alpha \beta }}}{{D}_{\nu }}{{\Omega }^{{\alpha \beta }}}\end{array}$

and the gauge term:

$\displaystyle \begin{array}{l}{{\mathcal{L}}_{{gkin}}}=-\frac{1}{{2\cdot 2!}}{{\wp }_{{MN}}}{{\Gamma }_{{\mu \upsilon }}}^{M}{{\Gamma }^{{\mu MN}}}-\frac{1}{{2\cdot 3!}}{{\wp }_{{\alpha \beta }}}{{\wp }_{{ss}}}{{\Omega }_{{\mu \nu \rho }}}^{{\alpha s}}{{\Omega }^{{\mu \nu \rho \beta s}}}\\-\frac{1}{{2\cdot 2!4!}}{{\wp }_{{ss}}}{{\wp }_{{\alpha \gamma }}}{{\wp }_{{\beta \delta }}}{{J}_{{\mu \nu \rho \sigma }}}^{{\left[ {\alpha \beta } \right]s}}{{J}^{{\mu \nu \rho \sigma }}}^{{\left[ {\gamma \delta } \right]s}}\end{array}$

and the 10+3-D Chern-Simons topological term:

$\displaystyle \begin{array}{l}{{\mathcal{L}}_{{top}}}=\kappa \int{{{{d}^{{10}}}}}x{{d}^{3}}Y{{\varepsilon }^{{{{\mu }_{1}}...{{\mu }_{{10}}}}}}\frac{1}{4}{{\varepsilon }_{{\alpha \beta }}}{{\varepsilon }_{{\gamma \delta }}}\left[ {\frac{1}{5}} \right.{{\partial }_{s}}{{\Omega }_{{{{\mu }_{1}}...{{\mu }_{5}}}}}^{{\alpha \beta ss}}\\{{\Omega }_{{{{\mu }_{6}}...{{\mu }_{{10}}}}}}^{{\gamma \delta ss}}-\frac{5}{2}{{\Gamma }_{{{{\mu }_{1}}{{\mu }_{2}}}}}^{s}{{J}_{{{{\mu }_{3}}...{{\mu }_{6}}}}}^{{\alpha \beta s}}{{J}_{{{{\mu }_{7}}...{{\mu }_{{10}}}}}}^{{\gamma \delta }}\\+\frac{{10}}{3}2{{\Omega }_{{{{\mu }_{1}}...{{\mu }_{3}}}}}^{{\alpha s}}{{\Omega }_{{{{\mu }_{4}}...{{\mu }_{6}}}}}^{{\beta s}}\left. {{{J}_{{{{\mu }_{7}}...{{\mu }_{{10}}}}}}^{{\gamma \delta }}} \right]\end{array}$

where the potential has the form:

$\displaystyle \begin{array}{l}V=\frac{1}{4}{{\wp }^{{ss}}}\left( {{{\partial }_{s}}{{\Omega }^{{\alpha \beta }}}{{\partial }_{s}}{{\Omega }_{{\alpha \beta }}}+{{\partial }_{s}}{{g}^{{\mu \nu }}}{{\partial }_{s}}{{g}_{{\mu \nu }}}+{{\partial }_{s}}\text{In}g{{\partial }_{s}}\text{In}g} \right)\\+\frac{9}{{32}}{{\wp }^{{ss}}}{{\partial }_{s}}\text{In}{{\wp }_{{ss}}}{{\partial }_{s}}\text{In}{{\wp }_{{ss}}}-\frac{1}{2}{{\wp }^{{ss}}}{{\partial }_{s}}\text{In}{{\wp }_{{ss}}}\text{In}g\\+\wp _{{ss}}^{{3/4}}\left[ {\frac{1}{4}} \right.{{\Omega }^{{\alpha \beta }}}{{\partial }_{\alpha }}{{\Omega }^{{\gamma \delta }}}{{\partial }_{\beta }}{{\Omega }_{{\gamma \delta }}}+\frac{1}{2}{{\Omega }^{{\alpha \beta }}}{{\partial }_{\alpha }}{{\Omega }^{{\gamma \delta }}}{{\partial }_{\gamma }}{{\Omega }_{{\delta \beta }}}\\+{{\partial }_{\alpha }}{{\Omega }^{{\alpha \beta }}}{{\partial }_{\beta }}\text{In}\left( {{{g}^{{1/2}}}\wp _{{ss}}^{{3/4}}} \right)+\\\frac{1}{4}{{\Omega }^{{\alpha \beta }}}\left( {{{\partial }_{\alpha }}} \right.{{g}^{{\mu \nu }}}{{\partial }_{\beta }}{{g}_{{\mu \nu }}}+{{\partial }_{\alpha }}\text{Ing}{{\partial }_{\beta }}\text{In}g+\frac{1}{4}{{\partial }_{\alpha }}\text{In}{{\wp }_{{ss}}}{{\partial }_{\beta }}\text{In}{{\wp }_{{ss}}}\\+\left. {\frac{1}{2}\left. {{{\partial }_{\alpha }}\text{In}g{{\partial }_{\beta }}\text{In}{{\wp }_{{ss}}}} \right)} \right]\end{array}$

Essentially, we have deduced a theory that is dynamically equivalent to 11-D SUGRA and Type-IIB under the covariantized U-duality group-action. However, the gauged kinetic terms ${{\Omega }_{{\mu \nu \rho \sigma \kappa }}}$ corresponding to the gauge form ${{D}_{{\mu \nu \rho \sigma \kappa }}}$ appear only topologically in:

$\displaystyle \begin{array}{*{20}{l}} {{{\mathcal{L}}_{{top}}}=\kappa \int{{{{d}^{{10}}}}}x{{d}^{3}}Y{{\varepsilon }^{{{{\mu }_{1}}...{{\mu }_{{10}}}}}}\frac{1}{4}{{\varepsilon }_{{\alpha \beta }}}{{\varepsilon }_{{\gamma \delta }}}\left[ {\frac{1}{5}} \right.{{\partial }_{s}}{{\Omega }_{{{{\mu }_{1}}...{{\mu }_{5}}}}}^{{\alpha \beta ss}}} \\ {{{\Omega }_{{{{\mu }_{6}}...{{\mu }_{{10}}}}}}^{{\gamma \delta ss}}-\frac{5}{2}{{\Gamma }_{{{{\mu }_{1}}{{\mu }_{2}}}}}^{s}{{J}_{{{{\mu }_{3}}...{{\mu }_{6}}}}}^{{\alpha \beta s}}{{J}_{{{{\mu }_{7}}...{{\mu }_{{10}}}}}}^{{\gamma \delta }}} \\ {+\frac{{10}}{3}2{{\Omega }_{{{{\mu }_{1}}...{{\mu }_{3}}}}}^{{\alpha s}}{{\Omega }_{{{{\mu }_{4}}...{{\mu }_{6}}}}}^{{\beta s}}\left. {{{J}_{{{{\mu }_{7}}...{{\mu }_{{10}}}}}}^{{\gamma \delta }}} \right]} \end{array}$

Hence, the EoM for the field $J$ are given as such:

$\displaystyle {{\partial }_{s}}\left( {\frac{\kappa }{2}{{\varepsilon }^{{{{\mu }_{1}}...{{\mu }_{9}}}}}{{\varepsilon }_{{\alpha \beta }}}{{\varepsilon }_{{\gamma \delta }}}{{\Omega }_{{{{\mu }_{5}}...{{\mu }_{9}}}}}^{{\gamma \delta ss}}-e\frac{1}{{48}}{{\wp }_{{ss}}}{{\wp }_{{\alpha \lambda }}}{{\wp }_{{\beta \delta }}}{{J}^{{{{\mu }_{1}}...{{\mu }_{4}}\gamma \delta s}}}} \right)=0$

By U-duality, homological mirror symmetry then entails an internal symmetry between M-theory and F-theory upon dimensional-reduction to Type-IIB SUGRA, which in the ${{{\mathrm E}}_{{n(n)}}}/USp(n+2)$ formalism, taking the Klebanov-Witten limit, is defined by the action:

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

Let us delve deeper into this M/F-U-duality, noting that it is a duality that is rich in F-theory phenomenology, and by the Type-IIB duality, such phenomenology is inherited by 11-D SUGRA under U-duality. First note that $S{{L}_{2}}\times \mathbb{R*}$ -EFT is equivalent to both, 11-D SUGRA and Type-IIB SUGRA. The field content for EFT consists of:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{g}_{{\mu \nu }}}} \\ {{{\wp }_{{MN}}}} \\ {{{A}_{\mu }}^{M}} \\ {{{B}_{{\mu \nu }}}^{{\alpha ,s}}} \\ {{{C}_{{\mu \nu \rho }}}^{{\alpha \beta ,s}}} \\ {{{D}_{{\mu \nu \rho \sigma }}}^{{\alpha \beta ,ss}}} \end{array}} \right.$

whereas the field content for M-theory consists of:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{G}_{{\mu \nu }}}} \\ {{{\gamma }_{{\alpha \beta }}}} \\ {{{A}_{\mu }}^{\alpha }} \\ {{{{\hat{C}}}_{{\hat{\mu }\hat{\nu }\hat{\rho }}}}} \end{array}} \right.$

while that of Type-IIB is given by:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{G}_{{\mu \nu }}}} \\ \phi \\ {{{A}_{\mu }}^{s}} \\ \varphi \\ {{{C}_{0}}} \\ {{{{\hat{C}}}_{{\hat{\mu }\hat{\nu }}}}^{\alpha }} \\ {{{{\hat{C}}}_{{\hat{\mu }\hat{\nu }\hat{\rho }\hat{\lambda }}}}} \end{array}} \right.$

The Kaluza-Klein and gauge fields for M-theory are, respectively:

$\displaystyle {{\gamma }^{{1/7}}}\left( {{{G}_{{\mu \nu }}}-{{\gamma }_{{\alpha \beta }}}{{A}_{\mu }}^{\alpha }{{A}_{\nu }}^{\beta }} \right)$

$\displaystyle {{\gamma }^{{-1/7}}}{{\gamma }_{{\alpha \beta }}}={{\Omega }_{{\alpha \beta }}}$

$\displaystyle {{\gamma }^{{-6/7}}}={{\wp }_{{ss}}}$

and:

$\displaystyle {{A}_{\mu }}^{s}=\frac{1}{2}{{\varepsilon }^{{\alpha \beta }}}{{{\hat{C}}}_{{\mu \alpha \beta }}}$

$\displaystyle {{B}_{{\mu \nu }}}^{{\alpha ,s}}={{\varepsilon }^{{\alpha \beta }}}{{{\hat{C}}}_{{\mu \nu \beta }}}+\frac{1}{2}{{\varepsilon }^{{\beta \gamma }}}{{A}_{{\left[ \mu \right.}}}^{\alpha }{{{\hat{C}}}_{{\left. \nu \right]\beta \gamma }}}$

$\displaystyle {{C}_{{\mu \nu \rho }}}^{{\alpha \beta ,s}}={{\varepsilon }^{{\alpha \beta }}}\left( {{{{\hat{C}}}_{{\mu \nu \beta }}}-3{{A}_{{\left[ \mu \right.}}}^{\gamma }{{{\hat{C}}}_{{\left. {\nu \rho } \right]\gamma }}}+2{{A}_{{\left[ \mu \right.}}}^{\gamma }{{A}_{{\left. \nu \right]}}}^{\delta }{{{\hat{C}}}_{{\rho \gamma \delta }}}} \right)$

For the split-Type-IIB theory, the Kaluza-Klein and gauge fields are respectively:

$\displaystyle {{g}_{{\mu \nu }}}={{\phi }^{{1/7}}}\left( {{{G}_{{\mu \nu }}}-\phi {{A}_{\mu }}^{s}{{A}_{\nu }}^{s}} \right)$

$\displaystyle {{\wp }_{{\alpha \beta }}}={{\phi }^{{-6/7}}}{{\Omega }_{{\alpha \beta }}}$

$\displaystyle {{\wp }_{{ss}}}={{\phi }^{{8/7}}}$

where we have:

$\displaystyle {{{A}_{\mu }}^{s}={{\phi }^{{-1}}}{{G}_{{\mu s}}}}$

and we parametrize $\Omega$ in terms of the axio-dilaton $\tau ={{C}_{0}}+i{{e}^{{-\varphi }}}$ as such:

$\displaystyle {{\Omega }_{{\alpha \beta }}}=\frac{1}{{{{\tau }_{2}}}}\left( {\begin{array}{*{20}{c}} 1 & {{{\tau }_{1}}} \\ {{{\tau }_{1}}} & {{{{\left| \tau \right|}}^{2}}} \end{array}} \right)={{e}^{\varphi }}\left( {\begin{array}{*{20}{c}} 1 & {{{C}_{0}}} \\ {{{C}_{0}}} & {C_{0}^{2}+{{e}^{{-2\varphi }}}} \end{array}} \right)$

The $S{{L}_{2}}\times \mathbb{R*}$ -EFT/11-D/Type-IIB duality can now be formulated in terms of the gauge field equations as such:

$\displaystyle {{A}_{\mu }}^{\alpha }={{{\hat{C}}}_{{\mu s}}}^{\alpha }$

$\displaystyle {{B}_{{\mu \nu }}}^{{\alpha ,s}}={{{\hat{C}}}_{{\mu \nu }}}^{\alpha }+{{A}_{{\left[ \mu \right.}}}^{\alpha }{{{\hat{C}}}_{{\left. \nu \right]s}}}^{\alpha }$

$\displaystyle \begin{array}{l}{{C}_{{\mu \nu \rho }}}^{{\alpha \beta ,s}}={{\varepsilon }^{{\alpha \beta }}}{{{\hat{C}}}_{{\mu \nu \rho s}}}+3{{{\hat{C}}}_{{\left[ {\left. \mu \right|\left. s \right|} \right.}}}^{{\left[ \alpha \right.}}{{C}_{{\left. {\nu \rho } \right]}}}^{{\left. \beta \right]}}\\-2{{{\hat{C}}}_{{\left[ {\left. \mu \right|s} \right.}}}^{\alpha }{{{\hat{C}}}_{{\left. \nu \right|\left. s \right|}}}^{\beta }{{A}_{{\left. \rho \right]}}}^{s}\end{array}$

$\displaystyle \begin{array}{l}{{D}_{{\mu \nu \rho \sigma }}}^{{\alpha \beta ,ss}}={{\varepsilon }^{{\alpha \beta }}}\left( {{{{\hat{C}}}_{{\mu \nu \rho \sigma }}}+4{{A}_{{\left[ \mu \right.}}}^{s}{{{\hat{C}}}_{{\left. {\nu \rho \sigma } \right]s}}}} \right)\\+6{{{\hat{C}}}_{{\left[ {\mu \nu } \right.}}}^{{\left[ \alpha \right.}}{{{\hat{C}}}_{{\rho \left| s \right|}}}^{{\left. \beta \right]}}{{A}_{{\left. \sigma \right]}}}^{s}\end{array}$

Hence we can now define F-theory as the modular equivariant 12-D lift of Type-IIB with varying axio-dilaton and 7-brane geometric backreaction where the $S{{L}_{2}}\left( {\mathrm X} \right)$ duality U-action yields a monodromy-group representation induced by elliptic fibrations that admit a duality with M-theory via KK-reduction. The $S{{L}_{2}}\times \mathbb{R*}$ -EFT diffeomorphism group action has two Sen singularity solutions, one corresponding to F-theory, one to 11-D SUGRA. Hence, such a 12-D field theory dimensionally reduces in the Witten-Vafa limit, to 11-D SUGRA and 10-D Type-IIB.

Since the D7-brane backreaction is central to F-theory, let’s study it under the $S{{L}_{2}}\times \mathbb{R*}$ -EFT/11-D/Type-IIB duality. Solutions to the D7-brane worldvolume action in Type-IIB SUGRA possess non-trivial axio-dilaton and metric. Let a D7-brane extend along the six-dimensions of the internal manifold in the $\overrightarrow{{{{x}_{6}}}}$ and ${{{y}^{s}}}$ directions, and we represent the time direction transverse to the brane in polar coordinates $\left( {\tau ,\theta } \right)$ . Thus, the harmonic functional of the D7-brane is $H\approx h\text{In}\left[ {{{r}_{0}}/r} \right]$ and the solutions to our system are given by:

$\displaystyle \text{d}s_{{\left( 9 \right)}}^{2}=-\text{d}{{t}^{2}}+\text{d}\overrightarrow{{x_{{\left( 6 \right)}}^{2}}}+H\left( {\text{d}{{r}^{2}}+{{r}^{2}}\text{d}{{\theta }^{2}}} \right)$

$\displaystyle \text{d}s_{{\left( 3 \right)}}^{2}={{H}^{{-1}}}\left[ {{{{\left( {\text{d}{{y}^{1}}} \right)}}^{2}}+2h\theta \text{d}{{y}^{1}}\text{d}{{y}^{2}}+K{{{\left( {\text{d}{{y}^{2}}} \right)}}^{2}}} \right]+{{\left( {\text{d}{{y}^{s}}} \right)}^{2}}$

$\displaystyle {{A}_{\mu }}^{M}=0$

$\displaystyle K={{H}^{2}}+{{h}^{2}}{{\theta }^{2}}$

Holomorphically circling the transverse dimension yields, by the monodromy group action on $\wp$ , $\Omega$ :

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\wp \to {{\Omega }^{T}}\wp \Omega } \\ {\Omega =\left( {\begin{array}{*{20}{c}} 1 & {2\pi h} \\ 0 & 1 \end{array}} \right)} \end{array}} \right.$

The D7-brane solution is derived via Type-IIB dimensional reduction under the elliptic fibrational structure underlying the monodromy action on the CY fourfold geometry:

$\displaystyle \text{d}s_{{\left( {10} \right)}}^{2}=-\text{d}{{t}^{2}}+\text{d}\overrightarrow{{x_{{\left( 6 \right)}}^{2}}}+H\left( {\text{d}{{r}^{2}}+{{r}^{2}}\text{d}{{\theta }^{2}}} \right)+{{\left( {\text{d}{{y}^{s}}} \right)}^{2}}$

$\displaystyle {{C}_{0}}=h\theta$

$\displaystyle {{e}^{{2\varphi }}}={{H}^{{-2}}}$

On the M-section, we have a monopole-smearing solution:

$\displaystyle \begin{array}{l}\text{d}s_{{\left( {11} \right)}}^{2}=-\text{d}{{t}^{2}}+\text{d}\overrightarrow{{x_{{\left( 6 \right)}}^{2}}}+H\left( {\text{d}{{r}^{2}}+{{r}^{2}}\text{d}{{\theta }^{2}}+{{{\left( {\text{d}{{y}^{1}}} \right)}}^{2}}} \right)\\+{{H}^{{-1}}}{{\left[ {\text{d}{{y}^{2}}+h\theta \text{d}{{y}^{1}}} \right]}^{2}}\end{array}$

Hence, by the M/Type-IIB duality (via F-theory), the smearing relates the first Chern class of the Witten-Vafa fibration to the co-dimension two monodromy hypercharges. By modularity, the D7-brane hyperdoublet and the D3-brane multiplet are smeared monopoles that p-q-7-branes in Type-IIB with p-cycles and q-cycles holomorphically wrap. The metric of a p-q-Type-IIB theory is given as such:

$\displaystyle \begin{array}{c}\text{d}s_{{\left( 3 \right)}}^{2}=\frac{{{{H}^{{-1}}}}}{{{{p}^{2}}+{{q}^{2}}}}\left\{ {\left[ {{{p}^{2}}{{H}^{2}}+{{{\left( {ph\theta -q} \right)}}^{2}}} \right]{{{\left( {\text{d}{{y}^{1}}} \right)}}^{2}}} \right.\\+\left[ {{{{\left( {p+qh\theta } \right)}}^{2}}+{{q}^{2}}{{H}^{2}}} \right]{{\left( {\text{d}{{y}^{2}}} \right)}^{2}}\\-2\left[ {\left( {{{p}^{2}}-{{q}^{2}}} \right)h\theta +pq\left( {K-1} \right)} \right]\left. {\text{d}{{y}^{1}}\text{d}{{y}^{2}}} \right\}+{{\left( {{{\text{d}}^{s}}} \right)}^{2}}\end{array}$

Thus, the exceptionality property determines two holomorphic functions $\tau \left( z \right)$ and $f\left( z \right)$ where the modulus takes the form:

$\displaystyle \tau ={{j}^{{-1}}}\left( {\frac{{{{P}_{{oly}}}\left( z \right)}}{{{{Q}_{{oly}}}\left( z \right)}}} \right)$

with elliptic invariant Jacobian. The polynomial roots give rise to Type-IIB singularities localizing the 7-branes. Hence we get:

$\displaystyle \text{d}s_{{\left( {10} \right)}}^{2}=-\text{d}{{t}^{2}}+\text{d}\overrightarrow{{x_{{\left( 6 \right)}}^{2}}}+\text{d}y_{s}^{2}+{{\tau }_{2}}{{\left| f \right|}^{2}}\text{d}z\text{d}\bar{z}$

and for the M-section, we have:

$\displaystyle \begin{array}{l}\text{d}s_{{\left( {11} \right)}}^{2}=-\text{d}{{t}^{2}}+\text{d}\overrightarrow{{x_{{\left( 6 \right)}}^{2}}}+{{\tau }_{2}}{{\left| f \right|}^{2}}\text{d}z\text{d}\bar{z}+{{\tau }_{2}}{{\left( {\text{d}{{y}^{1}}} \right)}^{2}}\\+\frac{1}{{{{\tau }_{2}}}}\left( {\text{d}{{y}^{2}}+{{\tau }_{1}}\text{d}{{y}^{1}}} \right)2\end{array}$

Both maintain the elliptic fibrational base singularities under which the action of the M/Type-IIB duality symmetry gives rise to generalized Klebanov-Witten quiver gauge theories that yield the Yang-Mills gauge theories of the Standard Model. Exceptionality hence eliminates the need to go to the full-blown 12-D F-theory and thus eliminates both Betti-truncations and the KK-conical blow-up sector that otherwise would have to be Higgsed away upon dimensional reduction to 11-D SUGRA. This is phenomenologically central in deriving the Standard Model of physics from F-theory, and hence M-theory in light of the M/F-theory duality.

F-theory as the Modular Equivariant 12-D Lift of Type-IIB String Theory

George Shiber

Exceptional Field Theory and M-Theory on Kovalev TCS G2-Manifolds

George Shiber

Exceptional Field Theory and M-Theory on Kovalev TCS G2-Manifolds

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