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String-Theory and Calabi-Yau Fourfolding of M-Theory

By Posted on In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory, Supersymmetry ,
M-Theory

In this post, I will carry a Calabi-Yau fourfold compactification of M-theory in a topologically smooth way. Since M-theory is the only quantum theory of gravity that provably has a finite renormalization group and is the only complete self-consistent GUT, such 4-D compactifications are essential in order to have a correspondence with 4-D spacetime. Recall I derived, via Clifford algebraic symmetry, the total action:

    \[\begin{array}{l}{S^{Total}} = \frac{1}{{2\pi {\alpha ^\dagger }12}}\int\limits_{{\rm{world - volumes}}} {{d^{26}}} x\,d\,\Omega {\left( {{\phi _{INST}}} \right)^2}\sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} \,{e^{ - {c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}} \cdot \\\left( {{R_{icci}} - 4{{\left( {{{\not D}^{SuSy}}\left( {{\phi _{INST}}} \right)} \right)}^2}} \right) + \frac{1}{{12}}H_{3,\mu \nu \lambda }^bH_3^{b,\mu \nu \lambda }/A_\mu ^H + \sum\limits_{D - p - branes} {S_{Dp}^{WV}} \end{array}\]

which is deep since Clifford algebras are a quantization of exterior algebras, and applying to the ‘Einstein-Minkowski’ tangent bundle, we get via Gaussian matrix elimination, an expansion of {\not D^{SuSy}} via Green-functions, yielding M-Theory’s action:

    \[{S_M} = \frac{1}{{{k^9}}}\int\limits_{{\rm{world - volumes}}} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {T_p}^{10}{\mkern 1mu} d{\mkern 1mu} \Omega {\left( {{\phi _{INST}}} \right)^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right) + \sum\limits_{Dp} {\not D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {\not D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\]

with k the kappa symmetry gravitonic term and the supergravitational Hamiltonian term being:

    \[\sum\limits_{Dp} {\not D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {\not D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\]

Let {Y_4} be a smooth Calabi-Yau fourfold and start with the bosonic 11-D SuGra sector

Start with the total M-theory action

Beginning with dimensional reduction of 11-D supergravity on {Y_4}, the bosonic part is given by:

    \[{S^{\left( {11} \right)}} = \frac{1}{2}\int {\left( {\hat R\hat * 1 - \frac{1}{2}\hat G \wedge \hat * \,G - \frac{1}{6}\hat C \wedge \hat G \wedge \hat G} \right)} \]

with \hat R the 11-D Ricci scalar and \hat G = d\hat C the 4-form field strength for the 4-form \hat C. Let {g_{m\bar n}} be a metric on the Calabi-Yau 4-fold {Y_4} and pick the following backgrounds:

    \[\left\langle {{\rm{d}}{{\hat s}^2}} \right\rangle = {\eta _{\mu \nu }}{\rm{d}}{x^\mu }{\rm{d}}{x^\nu } + 2{g_{m\bar n}}{\rm{d}}{y^m}{\rm{d}}{y^{\bar n}}\]

and

    \[\left\langle {{\rm{d}}\hat C} \right\rangle = 0\]

which ensure that the effective theory is a three-dimensional N = 2 supergravity one. Massless modes that arise from fluctuations of the metric are encoded by a Kähler form J expanded as

    \[\left\{ {\begin{array}{*{20}{c}}{J = {v^\Sigma }{\omega _\Sigma }}\\{\Sigma = 1,...,{h^{1,1}}\left( {{Y_4}} \right)}\end{array}} \right.\]

with {\omega _\Sigma } the basis of harmonic two-forms and {v^\Sigma } are 3-D scalar fields parametrizing the Kähler structure-deformations of {Y_4}.

Now, the massless modes that come from Heisenberg-fluctuations of M-theory’s three-form \hat C are

    \[\hat C = {A^\Sigma } \wedge {\omega _\Sigma } + {N_{\tilde A}}{\Psi ^{\tilde A}} + {\bar N_{\tilde A}}{\bar \Psi ^{\tilde A}}\]

with

    \[\tilde A = 1,...,{h^{2,1}}\left( {{Y_4}} \right)\]

and {\Psi ^{\tilde A}} a basis of harmonic (1, 2)-forms, with parametrization

    \[{\Psi ^{\tilde A}} = \frac{1}{2}{\mathop{\rm Re}\nolimits} {f^{\tilde A\tilde B}}\left( {{\alpha _{\tilde B}} - i{{\bar f}_{\tilde A\tilde B}}\beta {\,^{C'}}} \right)\]

with \left( {{\alpha _{\tilde A}},{\beta ^{\tilde B}}} \right) an integral-harmonic three-forms basis and {f^{\tilde A\tilde B}} holomorphic in complex structure.

Thus, we have 

    \[\begin{array}{l}\hat G = {\rm{d}}{A^\Sigma } \wedge {\omega _\Sigma } + D{N_{\tilde A}} \wedge {\Psi ^{\tilde A}} + \\D{{\bar N}_{\tilde A}} \wedge {{\bar \Psi }^{\tilde A}}\end{array}\]

with the following holding:

    \[D{N_{\tilde A}}{\rm{ = }}\,{\rm{d}}{N_{\tilde A}} - {\mathop{\rm Re}\nolimits} {N_{\tilde B}}{\mathop{\rm Re}\nolimits} {f^{\tilde A\tilde B}}{\not \partial _K}{f_{C'\tilde A}}\,{\rm{d}}{z^K}\]

and

    \[D{\bar N_{\tilde A}} = {\overline {DN} _{\tilde A}}\]

Now, substituting the ansatz

    \[\left\{ {\begin{array}{*{20}{c}}{J = {v^\Sigma }{\omega _\Sigma }}\\{\Sigma = 1,...,{h^{1,1}}\left( {{Y_4}} \right)}\end{array}} \right.\]

and

    \[\hat C = {A^\Sigma } \wedge {\omega _\Sigma } + {N_{\tilde A}}{\Psi ^{\tilde A}} + {\bar N_{\tilde A}}{\bar \Psi ^{\tilde A}}\]

into:

    \[{S^{\left( {11} \right)}} = \frac{1}{2}\int {\left( {\hat R\hat * 1 - \frac{1}{2}\hat G \wedge \hat * \,G - \frac{1}{6}\hat C \wedge \hat G \wedge \hat G} \right)} \]

After a Weyl rescaling, thus bringing the effective action into Einstein-Sasaki frame, the 3-D effective theory is hence given by

 

eqa

 

The central mathematical entities in this expression, since I am working in M-theory, is the rescaled Kähler moduli

    \[\left\{ {\begin{array}{*{20}{c}}{{{L'}^{\,\Sigma }} = \frac{{{v^\Sigma }}}{{\tilde V}}}\\{\tilde V = \frac{1}{{4!}}\int_{{Y_4}} {{J^4}} }\end{array}} \right.\]

with

    \[V = \frac{1}{{4!}}{K_{\Sigma \Lambda \Gamma \Delta }}{L'^{\,\Sigma }}{L'^{\,\Lambda }}{L'^{\,\Gamma }}{L'^{\,\Delta }}\]

and the following identity holding:

    \[{K_{\Sigma \Lambda \Gamma \Delta }} = \int_{{Y_4}} {{\omega _\Sigma }} \wedge {\omega _\Lambda } \wedge {\omega _\Gamma } \wedge {\omega _\Delta }\]

and expresses the intersection number of two-forms. Since the kinetic term for complex structure moduli {z^K} depends on the Sasaki-Kähler metric:

    \[{G_{K\bar L}} = - \frac{{\int_{{Y_4}} {\chi K \wedge \chi \bar L} }}{{\int_{{Y_4}} {\Omega \wedge \Omega } }} = - {\not \partial _z}k{\not \partial _{{{\bar z}^{\bar L}}}}\log \left( {\int_{{Y_4}} {\Omega \wedge \bar \Omega } } \right)\]

with \chi L a basis of harmonic (3, 1)-forms and L = 1,...,{h^{1,3}}\left( {{Y_4}} \right).

We can derive:

    \[\begin{array}{l}{G_{\Sigma \Lambda }} = \frac{{\bar V}}{4}\int_{{Y_4}} {{\omega _\Sigma }} \wedge * {\omega _\Lambda } = - \frac{1}{{8V}} \cdot \\\left( {{K_{\Sigma \Lambda }} - \frac{1}{{18V}}{K_\Sigma }{K_\Lambda }} \right)\\ = - \frac{1}{4}{{\not \partial }_{{{L'}^{\,\Sigma }}\,}}{{\not \partial }_{{{L'}^{\,\Lambda }}}}\log V\end{array}\]

with the definitions:

    \[\left\{ {\begin{array}{*{20}{c}}{{K_\Sigma } = {K_{_{\Sigma \Lambda \Gamma \Delta }}}{{L'}^{\,\Lambda }}{{L'}^{\,\Gamma }}L'{\,^\Delta }}\\{{K_{_{\Sigma \Lambda }}} = {K_{_{\Sigma \Lambda \Gamma \Delta }}}{{L'}^{\,\Gamma }}L'{\,^\Delta }}\end{array}} \right.\]

and I explicitly used

    \[\begin{array}{l} * {\omega _\Sigma } = - \frac{1}{2}J \wedge J \wedge {\omega _\Sigma } + \\\frac{{{{\tilde V}^2}}}{{36}}{K_\Sigma }J \wedge J \wedge J\end{array}\]

and the couplings:

    \[\int_{{Y_4}} {{\Psi ^{\tilde A}}} \wedge * {\bar \Psi ^{\bar B}} = {L'^{\,\Sigma }}d_\Sigma ^{\tilde A\tilde B}\]

and

    \[d_\Sigma ^{\tilde A\tilde B} = i\int_{{Y_4}} {{\omega _\Sigma }} \wedge {\Psi ^{\tilde A}} \wedge {\bar \Psi ^{\tilde B}}\]

and made a green-functional use of

    \[ * {\Psi ^{\tilde A}} = iJ \wedge {\Psi ^{\tilde A}}\]

So we get then the following identities:

    \[d_\Sigma ^{\tilde A\tilde B} = - \frac{1}{2}{\mathop{\rm Re}\nolimits} {f^{\tilde A\tilde B}}{Q_{\Sigma {{C'}^{\tilde A}}}}\]

and

    \[Q_{\Sigma C'}^{\tilde A} = {M_{\Sigma C'}}^{\tilde A} + i{f_{C'\tilde B}}^{\tilde B\tilde A}\]

with complex and Kähler structure independent intersection numbers:

    \[{M_{\Sigma \tilde A}}^{\tilde B} = \int_{{Y_4}} {{\omega _\Sigma }} \wedge {\alpha _{\tilde A}} \wedge {\beta ^{\tilde B}}\]

and

    \[{M_\Sigma }^{\tilde A\tilde B} = \int_{{Y_4}} {{\omega _\Sigma } \wedge {\beta ^{\tilde A}}} \wedge {\beta ^{\tilde B}}\]

by solving

    \[\int_{{Y_4}} {{\omega _\Sigma }} \wedge {\Psi ^{\tilde A}} \wedge {\Psi ^{\tilde B}} = 0\]

and via change-of-variables, substituting in

    \[{S_M} = \frac{1}{{{k^9}}}\int\limits_{{\rm{world - volumes}}} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {T_p}^{10}{\mkern 1mu} d{\mkern 1mu} \Omega {\left( {{\phi _{INST}}} \right)^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right) + \sum\limits_{Dp} {\not D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {\not D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\]

one gets a solution to {S_M} that dimensionally reduces to a Calabi-Yau hypersurface in weighted projective four-dimensional space, which is isomorphic to compactification on {Y_4}

Untitled1

 

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String Theory on AdS and Super-Yang–Mills Conformal Field Theory: the Witten Map
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String-Theory and the String/Non-Local-'Particles' Duality
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