M-Theory on Kovalev TCS G2-Manifolds, Yukawa Couplings, and Chirality

Eleven-dimensional supergravity on Calabi-Yau orbifolds naturally induces ${{{\mathrm E}}_{{n(n)}}}{{\left( \mathbb{R} \right)}_{{n=6,7,8}}}$ exceptional symmetries in $\text{D}=11-n$ that can be realized as U-duality symmetries of M-theory upon $\mathbb{Z}$ -discretization and without Betti-truncations. Hence, exceptional field theory based on the modular group $S{{L}_{2}}\left( \mathbb{Z} \right)$ uses a dimensionally extended spacetime to 12-D that fully covariantizes supergravity under the U-duality symmetry group of M-theory. Taking full advantage of the M/F-theory duality, and Kovalev’s twisted connected sum constructed ${{G}_{2}}$ manifolds via gluing pairs of asymptotically cylindrical Calabi–Yau threefolds, M-theory is hence up to isomorphism the unique UV completion of the Standard Model of physics coupled to gravity. Moreover, by mirror symmetry, there exists an internal symmetry induced between M-theory and F-theory upon KK-reduction to Type-IIB SUGRA. Here I shall discuss singular transitions to chirality and Yukawa couplings in M-theory on Calabi-Yau manifolds that arise in the Kovalev TCS construction involving an action with half-maximal gauged supergravity:

$\displaystyle {S=\int{{{{d}^{d}}}}X\sqrt{{-\tilde{G}}}{{e}^{{-2\varphi }}}\left[ {R+4{{D}_{\mu }}} \right.\varphi {{D}^{\mu }}\varphi -}$

$\displaystyle {\frac{1}{{12}}{{H}_{{\mu \nu \rho }}}{{H}^{{\mu \nu \rho }}}-{{{\tilde{F}}}_{{\mu \nu }}}^{M}{{{\tilde{F}}}^{{\mu \nu }}}^{N}{{{\tilde{M}}}_{{MN}}}+}$

$\displaystyle {\frac{1}{8}{{D}_{\mu }}{{{\tilde{M}}}_{{MN}}}{{D}^{\mu }}{{{\tilde{M}}}^{{MN}}}-\left. V \right]}$

where we have:

$\displaystyle {{H}_{{\mu \nu \rho }}}=3\left( {{{\partial }_{{\left[ \mu \right.}}}{{{\tilde{B}}}_{{\left. {\nu \rho } \right]}}}-\tilde{A}_{{\left[ \mu \right.}}^{M}{{\partial }_{\nu }}{{{\tilde{A}}}_{{\left. \rho \right]M}}}-\frac{1}{3}{{f}_{{MNP}}}\tilde{A}_{{\left[ \mu \right.}}^{M}\tilde{A}_{\nu }^{N}\tilde{A}_{{\left. \rho \right]}}^{P}} \right)$

$\displaystyle \tilde{F}_{{\mu \nu }}^{M}=2{{\partial }_{{\left[ \mu \right.}}}\tilde{A}_{{\left. \nu \right]}}^{M}+{{f}^{M}}_{{NP}}\tilde{A}_{\mu }^{N}\tilde{A}_{\nu }^{P}$

$\displaystyle {{D}_{\mu }}{{{\tilde{M}}}_{{MN}}}={{\partial }_{\mu }}{{{\tilde{M}}}_{{MN}}}+{{f}_{{MP}}}^{Q}\tilde{A}_{\mu }^{P}{{{\tilde{M}}}_{{QN}}}+{{f}_{{NP}}}^{Q}\tilde{A}_{\mu }^{P}{{{\tilde{M}}}_{{MQ}}}$

with $V$ the scalar potential. In the ${{{\mathrm E}}_{{n(n)}}}/USp(n+2)$ formalism taking the Klebanov-Witten limit, the theory is given by the action:

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

where we have:

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

and:

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

and 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 elliptically fibered Calabi-Yau 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 the following 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 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:

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

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

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

Central in EFT is the role of U-duality in the modular holomorphic action on the Neveu-Schwarz sector of Type-IIB. The generalized diffeomorphisms, generated by a vector ${{\Lambda }^{M}}$ , act 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 implicitly defined by the transformation rules for a generalized vector:

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

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

The associated 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 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 $\wp$ hence has 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 such:

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

thus allowing us to define the crucial 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:

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

where the Kähler Type-IIB orientifold moduli is:

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

with:

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

and 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 now hence 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:

$\displaystyle {\hat{R}=\frac{1}{4}{{g}^{{\mu \nu }}}{{D}_{\mu }}{{g}_{{\rho \sigma }}}{{D}_{\nu }}{{g}^{{\rho \sigma }}}\frac{1}{2}{{g}^{{\mu \nu }}}{{D}_{\mu }}{{g}^{{\rho \sigma }}}{{D}_{\rho }}{{g}_{{\nu \sigma }}}}$

$\displaystyle {+\frac{1}{4}{{g}^{{\mu \nu }}}{{D}_{\mu }}\text{In}g{{D}_{\nu }}\text{In}g+\frac{1}{2}{{D}_{\mu }}\text{In}g{{D}_{\nu }}{{g}^{{\mu \nu }}}}$

the kinetic part:

$\displaystyle {{{\mathcal{L}}_{{skin}}}=\frac{7}{{32}}{{g}^{{\mu \nu }}}{{D}_{\mu }}\text{In}{{\wp }_{{ss}}}{{D}_{\nu }}\text{In}{{\wp }_{{ss}}}}$

$\displaystyle {+\frac{1}{4}{{g}^{{\mu \nu }}}{{D}_{\mu }}{{\Omega }_{{\alpha \beta }}}{{D}_{\nu }}{{\Omega }^{{\alpha \beta }}}}$

and the gauge term:

$\displaystyle {{{\mathcal{L}}_{{g/kin}}}=-\frac{1}{{2\cdot 2!}}{{\wp }_{{MN}}}{{\Gamma }_{{\mu \upsilon }}}^{M}{{\Gamma }^{{\mu MN}}}-\frac{1}{{2\cdot 3!}}{{\wp }_{{\alpha \beta }}}{{\wp }_{{ss}}}{{\Omega }_{{\mu \nu \rho }}}^{{\alpha s}}{{\Omega }^{{\mu \nu \rho \beta s}}}}$

$\displaystyle {-\frac{1}{{2\cdot 2!4!}}{{\wp }_{{ss}}}{{\wp }_{{\alpha \gamma }}}{{\wp }_{{\beta \delta }}}{{J}_{{\mu \nu \rho \sigma }}}^{{\left[ {\alpha \beta } \right]s}}{{J}^{{\mu \nu \rho \sigma }}}^{{\left[ {\gamma \delta } \right]s}}}$

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

$\displaystyle {{{\mathcal{L}}_{{top}}}=\kappa \int{{{{d}^{{10}}}}}x{{d}^{3}}Y{{\varepsilon }^{{{{\mu }_{1}}...{{\mu }_{{10}}}}}}\frac{1}{4}{{\varepsilon }_{{\alpha \beta }}}{{\varepsilon }_{{\gamma \delta }}}\left[ {\frac{1}{5}} \right.{{\partial }_{s}}{{\Omega }_{{{{\mu }_{1}}...{{\mu }_{5}}}}}^{{\alpha \beta ss}}}$

$\displaystyle {{{\Omega }_{{{{\mu }_{6}}...{{\mu }_{{10}}}}}}^{{\gamma \delta ss}}-\frac{5}{2}{{\Gamma }_{{{{\mu }_{1}}{{\mu }_{2}}}}}^{s}{{J}_{{{{\mu }_{3}}...{{\mu }_{6}}}}}^{{\alpha \beta s}}{{J}_{{{{\mu }_{7}}...{{\mu }_{{10}}}}}}^{{\gamma \delta }}}$

$\displaystyle {+\frac{{10}}{3}2{{\Omega }_{{{{\mu }_{1}}...{{\mu }_{3}}}}}^{{\alpha s}}{{\Omega }_{{{{\mu }_{4}}...{{\mu }_{6}}}}}^{{\beta s}}\left. {{{J}_{{{{\mu }_{7}}...{{\mu }_{{10}}}}}}^{{\gamma \delta }}} \right]}$

where the potential has the form:

$\displaystyle {V=\frac{1}{4}{{\wp }^{{ss}}}\left( {{{\partial }_{s}}{{\Omega }^{{\alpha \beta }}}{{\partial }_{s}}{{\Omega }_{{\alpha \beta }}}+{{\partial }_{s}}{{g}^{{\mu \nu }}}{{\partial }_{s}}{{g}_{{\mu \nu }}}+{{\partial }_{s}}\text{In}g{{\partial }_{s}}\text{In}g} \right)}$

$\displaystyle {+\frac{9}{{32}}{{\wp }^{{ss}}}{{\partial }_{s}}\text{In}{{\wp }_{{ss}}}{{\partial }_{s}}\text{In}{{\wp }_{{ss}}}-\frac{1}{2}{{\wp }^{{ss}}}{{\partial }_{s}}\text{In}{{\wp }_{{ss}}}\text{In}g}$

$\displaystyle {+\wp _{{ss}}^{{3/4}}\left[ {\frac{1}{4}} \right.{{\Omega }^{{\alpha \beta }}}{{\partial }_{\alpha }}{{\Omega }^{{\gamma \delta }}}{{\partial }_{\beta }}{{\Omega }_{{\gamma \delta }}}+\frac{1}{2}{{\Omega }^{{\alpha \beta }}}{{\partial }_{\alpha }}{{\Omega }^{{\gamma \delta }}}{{\partial }_{\gamma }}{{\Omega }_{{\delta \beta }}}}$

$\displaystyle {+{{\partial }_{\alpha }}{{\Omega }^{{\alpha \beta }}}{{\partial }_{\beta }}\text{In}\left( {{{g}^{{1/2}}}\wp _{{ss}}^{{3/4}}} \right)+}$

$\displaystyle {\frac{1}{4}{{\Omega }^{{\alpha \beta }}}\left( {{{\partial }_{\alpha }}} \right.{{g}^{{\mu \nu }}}{{\partial }_{\beta }}{{g}_{{\mu \nu }}}+{{\partial }_{\alpha }}\text{Ing}{{\partial }_{\beta }}\text{In}g+\frac{1}{4}{{\partial }_{\alpha }}\text{In}{{\wp }_{{ss}}}{{\partial }_{\beta }}\text{In}{{\wp }_{{ss}}}}$

$\displaystyle {+\left. {\frac{1}{2}\left. {{{\partial }_{\alpha }}\text{In}g{{\partial }_{\beta }}\text{In}{{\wp }_{{ss}}}} \right)} \right]}$

This is a theory 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 ${{D}_{{\mu \nu \rho \sigma \kappa }}}$ appears only topologically in:

Hence, the EoM for the field $J$ is given by:

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

Since exceptional field theory based on the modular group $S{{L}_{2}}\left( \mathbb{Z} \right)$ uses a dimensionally extended spacetime to 12-D that fully covariantizes supergravity under the U-duality symmetry groups of M-theory, homological mirror symmetry entails the existence of an internal symmetry induced 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:

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

We can now derive the dimensional reduction of M-theory on ACCY threefolds that are building blocks of twisted connected sum G2 manifolds. We begin with the two-derivative action for 11d SUGRA, unique up to isomorphism, with a pure bosonic action given as such:

$\displaystyle {{\tilde{S}}^{{\left( {11} \right)}}}=\int_{{\tilde{M}}}{{\frac{1}{2}}}\hat{R}\hat{*}1-\frac{1}{4}{{\hat{F}}_{4}}\wedge \hat{*}{{\hat{F}}_{4}}-\frac{1}{{12}}{{\hat{C}}_{3}}\wedge {{\hat{F}}_{4}}\wedge {{\hat{F}}_{4}}$

admitting a quantum gauge-gravitational correction:

$\displaystyle \tilde{S}_{{{{{\hat{R}}}^{4}}c}}^{{\left( {11} \right)}}=\mathfrak{g}{{\mathfrak{l}}^{{1/96}}}_{{KK}}\int_{{{{{\tilde{M}}}_{{11}}}}}{{{{{\hat{C}}}_{3}}}}\wedge \left[ {\text{tr}{{{\hat{R}}}^{4}}-\frac{1}{4}{{{\left( {\text{tr}{{{\hat{R}}}^{2}}} \right)}}^{2}}} \right]$

where we have:

$\displaystyle {{{\hat{F}}}_{4}}=d{{{\hat{C}}}_{3}}$

We then expand the 11D fields in a dimensional reduction on a Calabi-Yau threefold ${{Y}_{3}}$ on a basis of zero-modes of the Dirac-Dolbeault differential operator on the ACCY internal threefold with metric:

$\displaystyle ds_{{11}}^{2}={{{\tilde{g}}}_{{\mu \nu }}}\left( x \right)d{{x}^{\mu }}d{{y}^{\nu }}+2{{g}_{{\bar{i}j}}}\left( y \right)d{{{\bar{y}}}^{{\bar{i}}}}d{{x}^{j}}$

whose external part describes a maximally symmetric spacetime. Note that the fluctuations of the internal metric ${{g}_{{\bar{i}j}}}$ are zero-modes of the Lichnerowicz operator satisfying the relations:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\delta {{g}_{{\bar{i}j}}}=-i{{{\left( {{{\omega }_{\Lambda }}} \right)}}_{{i\bar{j}}}}\delta {{v}^{\Lambda }}} \\ {\delta {{g}_{{ij}}}={{{\left( {{{{\bar{b}}}_{{\bar{\kappa }}}}} \right)}}_{{ij}}}\delta {{{\bar{z}}}^{{\bar{\kappa }}}}} \\ {{{{\left( {{{{\bar{b}}}_{{\bar{\kappa }}}}} \right)}}_{{ij}}}=\frac{i}{{{{{\left\| \Omega \right\|}}^{2}}}}{{{\left( {{{{\bar{\chi }}}_{{\bar{\kappa }}}}} \right)}}_{{i\bar{k}\bar{l}}}}{{\Omega }^{{\bar{k}\bar{l}}}}_{j}} \\ {\Omega \doteq \ holomorphic\ \left( {3,0} \right)\,form} \end{array}} \right.$

parameterized by the complex structure moduli ${{z}^{\kappa }}$ and the Kähler moduli ${{v}^{\Lambda }}$ defined implicitly by the Kähler expansion:

$\displaystyle J={{v}^{\Lambda }}{{\omega }_{\Lambda }}$

with the following cohomological Laplacian Calabi-Yau expansion:

$\displaystyle {{{\hat{C}}}_{3}}={{\xi }^{K}}{{\alpha }_{K}}-{{{{\tilde{\xi }}'}}_{K}}{{\beta }^{K}}+{{A}^{\Lambda }}\wedge {{\omega }_{\Lambda }}+{{C}_{3}}$

and where all our fields are naturally embeddable in 5d N=2 SUSY multiplets. To see that, recall that the following identity holds:

$\displaystyle n_{V}^{{\left( 5 \right)}}={{h}^{{1,1}}}\left( {Y_{3}^{{CY}}} \right)-1$

since the total volume of our Calabi-Yau threefold is given as such:

$\displaystyle \mathcal{V}=\frac{1}{{3!}}\int_{{Y_{3}^{{CY}}}}{{J\wedge J\wedge J}}=\frac{1}{{3!}}{{\mathcal{V}}_{{\Lambda \Sigma \Theta }}}{{v}^{\Lambda }}{{v}^{\Sigma }}{{v}^{\Theta }}$

with ${{\mathcal{V}}_{{\Lambda \Sigma \Theta }}}$ being the threefold Cartan-Weyl intersection numbers. Furthermore, we recall the decomposition of the third CY-cohomology into complex cohomologies:

$\displaystyle \begin{array}{l}{{H}^{3}}\left( {Y_{3}^{{CY}}} \right)=\left[ {{{H}^{{1,2}}}\left( {Y_{3}^{{CY}}} \right)\oplus {{H}^{{2,1}}}\left( {Y_{3}^{{CY}}} \right)} \right]\\\oplus \left[ {{{H}^{{0,3}}}\left( {Y_{3}^{{CY}}} \right)\oplus {{H}^{{3,0}}}\left( {Y_{3}^{{CY}}} \right)} \right]\end{array}$

Now since we have:

$\displaystyle \mathcal{V}=\frac{1}{{3!}}\int_{{Y_{3}^{{CY}}}}{{J\wedge J\wedge J}}=\frac{1}{{3!}}{{\mathcal{V}}_{{\Lambda \Sigma \Theta }}}{{v}^{\Lambda }}{{v}^{\Sigma }}{{v}^{\Theta }}$

the scalar fields satisfy the following relation:

$\displaystyle {{L}^{\Lambda }}={{\mathcal{V}}^{{-1/3}}}{{v}^{\Lambda }}$

leading to the natural interpretation of ${{L}^{\Lambda }}$ as 5-D holomorphic coordinates inducing a potential of the form:

$\displaystyle \tilde{N}=\frac{1}{{3!}}{{\mathcal{V}}_{{\Lambda \Sigma \Theta }}}{{L}^{\Lambda }}{{L}^{\Sigma }}{{L}^{\Theta }}$

yielding the dimensionally reduced bosonic action:

$\displaystyle S_{M}^{{\left( 5 \right)}}=\int_{{{{{\tilde{M}}}_{5}}}}{{\frac{1}{2}}}R*1-\frac{1}{2}{{G}_{{\Lambda \Sigma }}}d{{L}^{\Lambda }}\wedge *d{{L}^{\Sigma }}-{{h}_{{uv}}}d{{q}^{u}}\wedge *d{{q}^{v}}$

$\displaystyle -\frac{1}{2}{{G}_{{\Lambda \Sigma }}}{{F}^{\Lambda }}\wedge *{{F}^{\Sigma }}-\frac{1}{{12}}{{\mathcal{V}}_{{\Lambda \Sigma \Theta }}}{{A}^{\Lambda }}\wedge {{F}^{\Sigma }}\wedge {{F}^{\Theta }}$

with the following being a logical consequence:

$\displaystyle {{\bar{G}}_{{\Lambda \Sigma }}}={{\left[ {-\frac{1}{2}{{\partial }_{{{{L}^{\Lambda }}}}}{{\partial }_{{{{L}^{\Sigma }}}}}\log \tilde{N}} \right]}_{{N=1}}}$

In light of the threefold Cartan-Weyl intersection numbers ${{\mathcal{V}}_{{\Lambda \Sigma \Theta }}}$ , the elliptically fibered Calabi-Yau geometry satisfies the following relation:

$\displaystyle \tilde{N}=\frac{1}{2}{{\eta }_{{\alpha \beta }}}R{{L}^{\alpha }}{{L}^{\beta }}+\frac{1}{2}{{\eta }_{{\alpha \beta }}}{{K}^{\alpha }}{{R}^{2}}{{L}^{\beta }}+\frac{1}{6}{{\eta }_{{\alpha \beta }}}{{K}^{\alpha }}{{K}^{\beta }}{{R}^{3}}$

$\displaystyle -\frac{1}{2}{{\eta }_{{\alpha \beta }}}{{C}^{\alpha }}{{C}_{{ij}}}{{L}^{\beta }}{{\xi }^{i}}{{\xi }^{j}}+\frac{1}{6}{{\mathcal{V}}_{{ijk}}}{{\xi }^{i}}{{\xi }^{j}}{{\xi }^{k}}$

To allow the 5D/6D lift, one defines T-shifted fields:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{{\hat{L}}}^{{\dagger \alpha }}}={{L}^{\alpha }}+\frac{1}{2}{{K}^{\alpha }}R} \\ {{{{\hat{A}}}^{{\dagger \alpha }}}={{A}^{\alpha }}+\frac{1}{2}{{K}^{\alpha }}{{A}^{0}}} \end{array}} \right.$

as required by supersymmetry, which yield the following correction to our cubic potential:

$\displaystyle {{{\tilde{N}}}^{M}}=\frac{1}{2}{{\eta }_{{\alpha \beta }}}R{{{\hat{L}}}^{{\dagger \alpha }}}{{{\hat{L}}}^{{\dagger \beta }}}+\frac{1}{{24}}{{\eta }_{{\alpha \beta }}}{{K}^{\alpha }}{{K}^{\beta }}{{R}^{3}}-\frac{1}{2}{{\eta }_{{\alpha \beta }}}{{C}^{\alpha }}{{C}_{{ij}}}{{{\hat{L}}}^{{\dagger \beta }}}{{\xi }^{i}}{{\xi }^{j}}$

$\displaystyle +\frac{1}{4}{{\eta }_{{\alpha \beta }}}{{C}^{\alpha }}{{C}_{{ij}}}{{K}^{\beta }}R{{\xi }^{i}}{{\xi }^{j}}+\frac{1}{6}{{\mathcal{V}}_{{ijk}}}{{\xi }^{i}}{{\xi }^{j}}{{\xi }^{k}}$

Hence, our Chern-Simons term reduces to:

$\displaystyle S_{{SC}}^{{\left( 5 \right)M}}=\int_{{{{{\tilde{M}}}_{5}}}}{{-\frac{1}{4}}}{{\eta }_{{\alpha \beta }}}{{A}^{0}}\wedge {{{{\hat{F}}'}}^{\alpha }}\wedge {{{{\hat{F}}'}}^{\beta }}+\frac{1}{4}{{\eta }_{{\alpha \beta }}}{{C}^{\alpha }}{{C}_{{ij}}}{{{\tilde{A}}}^{\alpha }}\wedge {{F}^{i}}\wedge {{F}^{j}}$

$\displaystyle -\frac{1}{{48}}{{\eta }_{{\alpha \beta }}}{{K}^{\alpha }}{{K}^{\beta }}{{A}^{0}}\wedge {{F}^{0}}\wedge {{F}^{0}}-\frac{1}{8}{{\eta }_{{\alpha \beta }}}{{C}^{\alpha }}{{C}_{{ij}}}{{K}^{\beta }}{{A}^{0}}\wedge {{F}^{i}}\wedge {{F}^{j}}$

$\displaystyle -\frac{1}{{12}}{{\eta }_{{\alpha \beta }}}{{\mathcal{V}}_{{ijk}}}{{A}^{i}}\wedge {{F}^{j}}\wedge {{F}^{k}}$

with:

$\displaystyle {{{{\hat{F}}'}}^{\alpha }}=d{{{\tilde{A}}}^{\alpha }}$

Hence, for M-theory on G2 TCS Kovalev manifolds with ACCY building blocks, the 4D action takes the following form:

$\displaystyle S_{{bos}}^{{4D}}=\frac{1}{{2\kappa _{4}^{2}}}\int{{\left[ {{{*}_{4}}} \right.}}{{R}_{S}}+\frac{{{{\kappa }_{{IJk}}}}}{{2{{\mathcal{V}}_{{{{Y}_{0}}}}}}}\left( {{{S}^{k}}{{F}^{I}}\wedge {{*}_{4}}{{F}^{J}}-{{P}^{k}}{{F}^{I}}\wedge F} \right)$

$\displaystyle -\frac{7}{{2{{\mathcal{V}}_{{{{Y}_{0}}}}}}}\int_{Y}{{\rho _{i}^{{\left( 3 \right)}}}}\wedge {{*}_{{{{g}_{\varphi }}}}}\rho _{j}^{{\left( 3 \right)}}\left( {d{{P}^{i}}\wedge {{*}_{4}}d{{P}^{j}}-d{{S}^{i}}\wedge {{*}_{4}}d{{S}^{j}}} \right)$

We are now in a position to discuss Yukawa couplings and chirality. Noting that in M-theory, interactions between localized matter fields derive from M2-instantons wrapped on calibrated 3-spheres of local ALE-fibrations. Taking the G2 Higgs bundle spectral background on a 3-cycle $M3$ :

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\langle \phi \rangle =diag\left( {{{\lambda }_{1}},...,{{\lambda }_{n}}} \right)=\sum\limits_{{i=1}}^{n}{{{{\text{t}}^{i}}d{{f}_{i}}}}} \\ {\Delta {{f}_{i}}={{\rho }_{i}}} \\ {\int_{{M3}}{{{{\rho }_{i}}=0}}} \end{array}} \right.$

we expand the topologically twisted 7d action:

$\displaystyle {{S}_{{7d}}}=\frac{1}{{g_{{7d}}^{2}}}\int_{{{{\mathbb{R}}^{{1,3}}}}}{{\int_{{M3}}{{{{d}^{7}}}}}}x\left[ {-\frac{1}{4}} \right.{{F}_{{\mu \nu }}}{{F}^{{\mu \nu }}}-{{D}_{\mu }}{{\bar{\varphi }}_{k}}{{D}^{\mu }}{{\varphi }^{k}}+$

$\displaystyle i{{D}_{\mu }}\chi {{\sigma }^{\mu }}\bar{\chi }-i{{D}_{\mu }}{{\psi }_{k}}{{\sigma }^{\mu }}{{{\bar{\psi }}}^{k}}+\frac{1}{2}{{D}^{2}}+{{H}^{k}}{{{\bar{H}}}_{k}}-D{{I}_{{\varphi ,\bar{\varphi }}}}$

$\displaystyle +\frac{i}{{\sqrt{2}}}{{\left( {{{F}_{{\bar{\varphi }}}}} \right)}_{{ij}}}{{\epsilon }^{{ijk}}}{{\bar{H}}_{k}}-\frac{i}{{\sqrt{2}}}{{\left( {{{F}_{\varphi }}} \right)}_{{ij}}}{{\epsilon }^{{ijk}}}{{H}_{k}}-\frac{i}{{\sqrt{2}}}{{\epsilon }^{{ijk}}}{{\psi }_{i}}{{\hat{D}}_{j}}{{\psi }_{k}}$

$\displaystyle \left. {+\frac{i}{{\sqrt{2}}}{{\epsilon }^{{ijk}}}{{{\bar{\psi }}}_{i}}{{{{\hat{D}}'}}_{j}}{{{\bar{\psi }}}_{k}}+\sqrt{2}i\chi {{{{\hat{D}}'}}_{i}}{{\psi }^{i}}-\sqrt{2}i\bar{\chi }{{{\hat{D}}}_{i}}{{{\bar{\psi }}}^{i}}} \right]$

Gauge symmetry restricts matter interactions to:

$\displaystyle {{\text{R}}_{Q}}\otimes {{\text{\bar{R}}}_{{-Q}}}\otimes \left( {\text{Ad}G\oplus \text{Ad}U{{{(1)}}^{n}}} \right)$

determined by:

$\displaystyle {{Z}^{{\left( {ab,k} \right)}}}=\int_{{M3}}{{{{\varphi }^{{\left( a \right)}}}}}\wedge {{\varphi }^{{\left( b \right)}}}\wedge {{h}^{{\left( k \right)}}}$

where $\left( {{{\varphi }^{a}},{{\varphi }^{b}}} \right)$ are 1-forms that describe the bosonic ground states along the 3-cycle, and ${{h}^{{\left( k \right)}}}_{{,\,\,k=1,...,{{b}^{1}}\left( {{{M}_{3}}} \right)}}$ are harmonic basis. Dimensional reduction to 4D yields at all critical Morse-Witten indices a chiral multiplet $\Phi _{\varphi }^{a}$ in ${{\text{R}}_{Q}}$ . Moreover, there are ${{b}^{1}}\left( {{{M}_{3}}} \right)$ chiral multiplets in $\text{Ad}G$ and $\text{Ad}U{{(1)}^{n}}$ derived from bulk-fields $\left( {{{{{\Phi }'}}_{k}},{{{\tilde{\Phi }}}_{k}}} \right)$ harmonic expansion yielding the 4D interaction term:

$\displaystyle {{\mathcal{L}}_{{4d,\operatorname{int}}}}=\text{Tr}\left[ {\frac{i}{{2\sqrt{2}}}\sum\limits_{{ab,k}}{{{{Z}^{{\left( {ab,k} \right)}}}}}} \right.\left( {{{\Phi }_{a}}{{{{\Phi }'}}_{k}}{{\Phi }_{b}}} \right)\left| {_{{\theta \theta }}} \right.-$

$\displaystyle \left. {\frac{i}{{2\sqrt{2}}}\sum\limits_{{ab,k}}{{{{Z}^{{\left( {ab,k} \right)}}}}}\left( {{{\Phi }_{a}}{{{\tilde{\Phi }}}_{k}}{{\Phi }_{b}}} \right)\left| {_{{\theta \theta }}} \right.+\,\text{h}\text{.c}\text{.}} \right]$

Yukawa couplings require a rank n = 2 Higgs bundle or higher. The Morse-Witten functions:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{f}_{1}}} \\ {{{f}_{2}}} \\ {{{f}_{Q}}={{q}_{1}}{{f}_{1}}+{{q}_{2}}{{f}_{2}}} \end{array}} \right.$

effectively yield a coupling by expanding the action:

in zero modes:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {Y_{{pqr}}^{{abc}}=\int_{{{{M}_{3}}}}{{{{\psi }^{{\left( {a,p} \right)}}}\wedge {{\varphi }^{{\left( {b,q} \right)}}}\wedge {{\psi }^{{\left( {c,r} \right)}}}}}} \\ {{{Q}_{p}}+{{Q}_{q}}+{{Q}_{r}}=0} \end{array}} \right.$

Hence, Yukawa couplings arise from M2-instantons wrapping associative three-cycle that describe $SU(3)$ ALE-fibrations over the base ${{M}_{3}}$ . M2 branes wrapped on such cycles yield the Yukawa couplings between modes hyper-localized at the critical points on the $SU(3)$ ALE-fibers. The interactions are determined by the Higgs-bundle Hitchins functions ${{f}_{i}}$ . Three-spheres supporting the Yukawa couplings are G2-manifold supersymmetric stable homology spheres determined by a superpotential class of the form:

$\displaystyle \Omega W\sim {{\tilde{n}}_{\gamma }}\exp \left( {i\int_{{S_{\gamma }^{3}}}{{\left( {C+i\Phi } \right)}}} \right)$

Hence, one can identify the Higgs field ${{\phi }_{i}}$ and the gauge field ${{W}_{i}}$ with the SUGRA 3-form $C$ and the G2-associative 3-form $\Phi$ to derive:

$\displaystyle \int_{{S_{\gamma }^{3}}}{{\left( {C+i\Phi } \right)}}=\sum\limits_{{j=1}}^{3}{{\int_{{\gamma \left( {{{f}_{j}}} \right)}}{{\int_{{{{\alpha }_{j}}}}{{\left( {C+i\Phi } \right)}}}}}}$

$\displaystyle =\sum\limits_{{j=1}}^{3}{{\int_{{\gamma \left( {{{f}_{j}}} \right)}}{{\left( {{{W}_{j}}+i{{\phi }_{j}}} \right)}}}}=i\sum\limits_{{j=1}}^{3}{{\int_{{\gamma \left( {{{f}_{j}}} \right)}}{{td{{f}_{{{{Q}_{j}}}}}}}}}$

Now since the following homological relation:

$\displaystyle \sum\nolimits_{i}^{3}{{f_{i}^{\alpha }}}$

holds, the superpotential assumes the following form:

$\displaystyle \Omega W={{\tilde{n}}_{\gamma }}\exp \left( {-\sum\limits_{{i=1}}^{m}{{t{{f}_{{{{Q}_{i}}}}}\left( {{{p}_{i}}} \right)}}} \right)$

thus satisfying the SM criteria on chirality and Yukawa couplings and their associated interaction strengths.

‘Tops as Building Blocks for G2 Manifolds’ is an excellent read on M-theory on manifolds of G2 holonomy that serves to support the main claims of this post. Below is a visualization of a Kovalev TCS construction:

M-Theory on Kovalev TCS G2-Manifolds, Yukawa Couplings, and Chirality

George Shiber

M-Theory, Exceptional Field Theory, U-Duality, and F-Theory

The Wess–Zumino–Witten–Novikov Model, Taub-NUT Spaces, and M-Theory

George Shiber

M-Theory, Exceptional Field Theory, U-Duality, and F-Theory

The Wess–Zumino–Witten–Novikov Model, Taub-NUT Spaces, and M-Theory

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