M-Theory, Peccei-Quinn Analysis, Calabi-Yau fourfolds and Kähler ‘Kaluza-Klein’ Reduction

I know that two and two make four – and should be glad to prove it too if I could – though I must say if by any sort of process I could convert 2 and 2 into five it would give me much greater pleasure ~ George Gordon Byron

Great news for me personally – after nearly 20 years of researching M-Theory: see bottom link on M-Theory and my last formula in this post involving supersymmetry. I ended my last post by showing how Peccei-Quinn invariance leads to the compactification smoothness required for M-theory to isomorphically embed an ‘Einsteinian-Minkowski‘ 4-D space-time in a Calabi-Yau fourfold in a way necessitated by quantum gravity. Let me address here the Kaluza-Klein reduction of string theory on Calabi-Yau fourfolds. Keep your eyes on

    \[\begin{array}{l}\not L_{IIA}^{(2)} = {e^{ - 2\Phi _{IIA}^{(2)}}}\\\left( { - \frac{1}{2}{{\not \partial }_\mu }{{\not \partial }^\mu }\sigma - \left( {{{\not \partial }_{{\Phi ^\Sigma }}}{{\overline {\not \partial } }_{{{\overline \Phi }_\Lambda }}}K_1^{IIA}} \right){{\not \partial }_\mu }{\Phi ^\Sigma }{{\not \partial }^\mu }{{\overline \Phi }^\Lambda } + \left( {{{\not \partial }_{{t^A}}}{{\overline {\not \partial } }_{\overline \Phi \Lambda }}K_2^{IIA}} \right){{\not \partial }_{\mu {\Phi ^\Sigma }}}} \right)\\{{\not \partial }^\mu }{\overline \Phi ^{\overline \Lambda }} - \varepsilon \left( {\left( {{{\overline {\not \partial } }_{{{\overline \Phi }^{\overline \Lambda }}}}{{\not \partial }_{{t^A}}}K_2^{IIA}} \right){{\not \partial }_\mu }{{\overline \Phi }^{\overline \Lambda }}{{\not \partial }_\nu }{t^A} + \left( {{{\not \partial }_{{\Phi ^\Lambda }}}{{\overline {\not \partial } }_{{{\overline t }^{\overline A }}}}K_2^{IIA}} \right){{\not \partial }_\mu }{\Phi ^\Lambda }{{\overline t }^{\overline A }}} \right)\end{array}\]

and

    \[\begin{array}{l}K_{IIA}^{(2)} = \, - {\rm{In}}\left( {\int_{{Y_4}} {\Omega \wedge \overline \Omega } } \right)\\ - {\rm{In}}\not V{{\not D}^{SuSy}}{{\not L}^{(2)}}{e^{\frac{{\not L_{IIA}^{(2)}}}{{\sqrt { - {g^{(2)}}} }}}}{{\not D}^{SuSy}}{e^{\kappa _{10}^2S_{IIA}^{(10)}}}\end{array}\]

to be explained below, throughout, to really appreciate the ‘M’agic of M-theory.

My starting point is the low-energy effective D=10 action in the string-frame

    \[\begin{array}{l}\kappa _{10}^2S_{IIA}^{(10)} = \int {{d^{10}}} x\sqrt { - {g^{(10)}}} {e^{ - \Phi _{IIA}^{(10)}}}\\ \cdot \left( {\frac{1}{2}{R^{(10)}} + 2{{\not \partial }_M}\Phi _{IIA}^{(10)}{{\not \partial }^M}\Phi _{IIA}^{(10)} - \frac{1}{{4!}}{{\left| {{H_3}} \right|}^2}} \right)\\ - \frac{1}{4}\int {{d^{10}}} x\sqrt { - g_{(10)}^{11}} \left( {{{\left| {{F_2}} \right|}^2} + {{\left| {{{\widetilde F}_4}} \right|}^2}} \right)\\ - \frac{1}{4}\int {{B_2}} \wedge {F_4} \wedge {F_4}\end{array}\]

with 

    \[{\Phi _{IIA}^{(10)}}\]

the D=10 dilaton{H_3} = d{B_2} being the field strength of the anti-symmetric tensor {B_2} and {F_2} = d{A_1} the field strength of the RR vector {A_1} and {F_4} = d{A_3} the field strength of the RR/3 form {A_3} and by convention:

    \[{\widetilde F_4} = {F_4} - {A_1} \wedge {H_3}\]

Also, and key, the term proportional to

    \[\int {{B_2}} \wedge {X_8}\]

which is analytically related to fourfold-dimensional reduction to the higher derivative term of M-theory

    \[\delta S_1^{(11)} = \, - {T_2}\int {{A_3}} \wedge {X_8}\]

with

    \[{X_8} = \frac{1}{{{{(2\pi )}^4}}}\left( { - \frac{1}{{768}}{{\left( {{\rm{Tr}}{R^2}} \right)}^2} + \frac{1}{{192}}{\rm{Tr}}{R^4}} \right)\]

imposes a consistency condition on compactification, and the absence of a {B_2} tadpole requires hence

    \[\frac{\chi }{{24}} = n + \frac{1}{{8{\pi ^2}}}\int_{{Y_4}} {{F_4}} \wedge {F_4}\]

to be solvable on manifolds isomorphic to {Y_4}, with n the number of string-fittings. With no loss of generality, let me focus on the conditions

    \[\left\{ {\begin{array}{*{20}{c}}{\chi = n = {F_4} = 0}\\{{K_{10}} \equiv 1}\end{array}} \right.\]

Now, realize, the spectrum of the 2-D theory is determined by super-deformations of the Calabi-Yau metric. So, for the D=10 metric, my ansatz

    \[g_{MN}^{(10)}(x,y) = \left( {\begin{array}{*{20}{c}}{g_{\mu \nu }^{(2)}}&0\\0&{g_{ab}^{(8)}(x,y)}\end{array}} \right)\]

and since the vectors contain no physical degree of freedom in D=2, then, in light of the lack of 1-forms on {Y_4}{A_1} does not contribute any D=2 massless modes: so the anti-symmetric tensors {B_2}  expand in terms of \left( {1,1} \right) forms

    \[\left\{ {\begin{array}{*{20}{c}}{{e_A}}\\{{B_{i\overline j }} = \sum\limits_{A = 1}^{{h^{1,1}}} {{a^A}(x){e_{{A_{i\overline j }}}}} }\end{array}} \right.\]

which leads to {h^{1,1}} real scalar field aA while {A_3} contributes {h^{1,2}} complex scalars {N^I}I = 1,...,{h^{1,2}}

The \left( {1,1} \right) moduli reside in the twisted chiral multiplets where all others are not. Therefore, by twistor-algebra, the dimensional reduction of

    \[\begin{array}{l}\kappa _{10}^2S_{IIA}^{(10)} = \int {{d^{10}}} x\sqrt { - {g^{(10)}}} {e^{ - \Phi _{IIA}^{(10)}}}\\ \cdot \left( {\frac{1}{2}{R^{(10)}} + 2{{\not \partial }_M}\Phi _{IIA}^{(10)}{{\not \partial }^M}\Phi _{IIA}^{(10)} - \frac{1}{{4!}}{{\left| {{H_3}} \right|}^2}} \right)\\ - \frac{1}{4}\int {{d^{10}}} x\sqrt { - g_{(10)}^{11}} \left( {{{\left| {{F_2}} \right|}^2} + {{\left| {{{\widetilde F}_4}} \right|}^2}} \right)\\ - \frac{1}{4}\int {{B_2}} \wedge {F_4} \wedge {F_4}\end{array}\]

gives us, by Teichmüller-integration

    \[\begin{array}{l}\begin{array}{*{20}{l}}{\frac{{\not L_{IIA}^{(2)}}}{{\sqrt { - {g^{(2)}}} }} = {e^{ - 2\Phi _{IIA}^{(2)}}} \cdot }\\{\left( {\frac{1}{2}{R^{(2)}} + 2{{\not \partial }_\mu }\Phi _{IIA}^{(2)}{{\not \partial }^\mu }\Phi _{IIA}^{(2)} - {G_{A\bar B}}{{\not \partial }_\mu }{t^A}{{\not \partial }_\mu }{{\bar t}^{\bar B}} - {G_{\alpha \bar \beta }}{{\not \partial }_\mu }{{\not Z}^\alpha }{{\not \partial }^\mu }{{\bar Z}^{\bar \beta }}} \right)}\end{array}\\ - \left( {{G_{I\overline J }}{{\not D}_\mu }{N^I}{{\not D}^\mu }{{\overline N }^{\overline J }} + \frac{1}{4}{d_{AI\overline J }}{\varepsilon ^{\mu \nu }}{{\not \partial }_\mu }{a^A}\left[ {{N^I}{{\not D}_\nu }{{\overline N }^{\overline J }}{{\not D}_\nu }{N^{\overline I }}} \right]} \right)\end{array}\]

with

    \[{G_{A\overline B }}{\not \partial _\mu }{t^A}{\not \partial _\mu }{\overline t ^{\overline B }} - {G_{\alpha \overline \beta }}{\not \partial _\mu }{\not Z^\alpha }{\not \partial ^\mu }{\overline Z ^{\overline \beta }}\]

being crucial for smoothness and

    \[{\frac{1}{4}{d_{AI\overline J }}{\varepsilon ^{\mu \nu }}{{\not \partial }_\mu }{a^A}}\]

for {Y_4} integrability.

In such a derivation, the key are the 2 identities

    \[\left\{ {\begin{array}{*{20}{c}}{{e^{ - 2\Phi _{IIA}^{(2)}}} \equiv {e^{ - 2\Phi _{IIA}^{(10)}}}\not V}\\{{t^A} = \frac{1}{{\sqrt 2 }}\left( {{M^A} + i{a^A}} \right)}\end{array}} \right.\]

Now, both can be separated via a Kähler potential. Hence, define

    \[K_1^{IIA} = \, - {\rm{In}}\left[ {\int_{{Y_4}} {\Omega \wedge \overline \Omega } } \right] + {\rm{In}}\not V\]

and

    \[\begin{array}{l}K_2^{IIA} = \, - \frac{1}{{\sqrt 2 }}\left( {{t^A} + {{\overline t }^A}} \right) \cdot \\\left( {\frac{i}{2}{d_{AM\overline L }}\widehat G_{\overline J M}^{ - 1}\widehat G_{\overline L I}^{ - 1}{{\overline {\widehat N} }^{\overline J }} - {\omega _{AIK}}{{\widehat N}^I}{{\widehat N}^K} - {{\overline \omega }_{A\overline J L}}{{\overline {\widehat N} }^{\overline J }}{{\overline {\widehat N} }^{\overline L }}} \right)\end{array}\]

with K_1^{IIA} and K_2^{IIA} and letting the fields \left( {{{\not Z}^\alpha },{{\widehat N}^I}} \right) denote {\Phi ^\Sigma }

one can see that

    \[\begin{array}{l}\begin{array}{*{20}{l}}{\frac{{\not L_{IIA}^{(2)}}}{{\sqrt { - {g^{(2)}}} }} = {e^{ - 2\Phi _{IIA}^{(2)}}} \cdot }\\{\left( {\frac{1}{2}{R^{(2)}} + 2{{\not \partial }_\mu }\Phi _{IIA}^{(2)}{{\not \partial }^\mu }\Phi _{IIA}^{(2)} - {G_{A\bar B}}{{\not \partial }_\mu }{t^A}{{\not \partial }_\mu }{{\bar t}^{\bar B}} - {G_{\alpha \bar \beta }}{{\not \partial }_\mu }{{\not Z}^\alpha }{{\not \partial }^\mu }{{\bar Z}^{\bar \beta }}} \right)}\end{array}\\ - \left( {{G_{I\overline J }}{{\not D}_\mu }{N^I}{{\not D}^\mu }{{\overline N }^{\overline J }} + \frac{1}{4}{d_{AI\overline J }}{\varepsilon ^{\mu \nu }}{{\not \partial }_\mu }{a^A}\left[ {{N^I}{{\not D}_\nu }{{\overline N }^{\overline J }}{{\not D}_\nu }{N^{\overline I }}} \right]} \right)\end{array}\]

becomes

    \[\begin{array}{l}\not L_{IIA}^{(2)} = {e^{ - 2\Phi _{IIA}^{(2)}}}\\\left( { - \frac{1}{2}{{\not \partial }_\mu }{{\not \partial }^\mu }\sigma - \left( {{{\not \partial }_{{\Phi ^\Sigma }}}{{\overline {\not \partial } }_{{{\overline \Phi }_\Lambda }}}K_1^{IIA}} \right){{\not \partial }_\mu }{\Phi ^\Sigma }{{\not \partial }^\mu }{{\overline \Phi }^\Lambda } + \left( {{{\not \partial }_{{t^A}}}{{\overline {\not \partial } }_{\overline \Phi \Lambda }}K_2^{IIA}} \right){{\not \partial }_{\mu {\Phi ^\Sigma }}}} \right)\\{{\not \partial }^\mu }{\overline \Phi ^{\overline \Lambda }} - \varepsilon \left( {\left( {{{\overline {\not \partial } }_{{{\overline \Phi }^{\overline \Lambda }}}}{{\not \partial }_{{t^A}}}K_2^{IIA}} \right){{\not \partial }_\mu }{{\overline \Phi }^{\overline \Lambda }}{{\not \partial }_\nu }{t^A} + \left( {{{\not \partial }_{{\Phi ^\Lambda }}}{{\overline {\not \partial } }_{{{\overline t }^{\overline A }}}}K_2^{IIA}} \right){{\not \partial }_\mu }{\Phi ^\Lambda }{{\overline t }^{\overline A }}} \right)\end{array}\]

with the conformal gauge being

    \[g_{\mu \nu }^{(2)} = {e^\sigma }{e^{2\Phi _{IIA}^{(2)}}}{\eta _{\mu \nu }}\]

and this results, after noting that {h^{2,1}}\left( {{Y_4}} \right) = 0, in the Minkowski-Sylvester space-time Lagrange interpolation continuity condition required of M-theory and only M-theory can thus meet if gravity is to be globally quantized at both scales: the cosmological and the Planck ones, with

    \[\begin{array}{l}\not L_{IIA}^{(2)} = \sqrt { - {g^{(2)}}} {e^{ - 2\Phi _{IIA}^{(2)}}}\\\left( {\frac{1}{2}{R^{(2)}} + 2{{\not \partial }_\mu }\Phi _{IIA}^{(2)}{{\not \partial }^\mu }\Phi _{IIA}^{(2)} - {G_{A\overline B }}{{\not \partial }_\mu }{t^A}{{\not \partial }^\mu }{{\overline t }^{\overline B }} - {G_{\overline \alpha \beta }}{{\not \partial }_\mu }{{\overline {\not Z} }^{\overline \alpha }}{{\not Z}^\beta }} \right)\end{array}\]

and since the moduli space factorizes into chiral and twisted chiral multiplets, which is to say, it is Kähler, we get a fourfolding of M-theory in a way that leads to a finite theory of quantum gravity at the cosmological and Planck scales, as evidenced by

    \[\begin{array}{l}K_{IIA}^{(2)} = \, - {\rm{In}}\left( {\int_{{Y_4}} {\Omega \wedge \overline \Omega } } \right)\\ - {\rm{In}}\not V{{\not D}^{SuSy}}{{\not L}^{(2)}}{e^{\frac{{\not L_{IIA}^{(2)}}}{{\sqrt { - {g^{(2)}}} }}}}{{\not D}^{SuSy}}{e^{\kappa _{10}^2S_{IIA}^{(10)}}}\end{array}\]

And here is Physicists Sebastian A.R. Ellis, Gordon L. Kane, and Bob Zheng’s paradigm-shifting revolutionary paper on supersymmetry and hence by U-Duality, via string-theory, M-theory:

LHC and Predictions from Constrained Compactified M-Theory

By U-duality, and {Y_4} fibration, one can generalize the above reduction and compactification arguments. What more can one ask of a theory.

A mathematician who is not also something of a poet will never be a complete mathematician ~ Karl