M-Theory, Compactification, and Calabi–Yau FourFolds

By George Shiber Wednesday, November 4, 2015Saturday, August 17, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Physics Calabi-Yau Fourfold, compactification

Mathematics is less related to science than it is to philosophy ~ George Shiber

The philosophically essential point when it comes to M-theory is that it is uniquely constrained in its unifying description of all the forces of nature as well as group-theoretically accounting, in a monistic way, for all the particle spectrum and metaplectically accounts for all the particles’ nomological properties to all arbitrary explanatory and causal orders and analytically entails the action(s) of quantum gravity, and thus includes in its Picard-group, the graviton, and completely subsumes the Standard Model of physics. A philosopher will, however, want to ask: can M-theory be evidentially compatible with 4-dimensional phenomenology. The answer is yes, and thus the importance of a fourfold ${Y_4}$ Calabi-Yau M-theoretic compactification. A bonus is that one obtains a D = 3 effective Lagrangian without having to contextually embed the theory in 11-dimensional super-gravity theory.

The starting context is 11-dimensional SuGra action:

$\begin{array}{c}{S^{(11)}} = \frac{1}{{2\kappa _{11}^2}}\int {{d^{11}}} x\sqrt { - {g^{(11)}}} \cdot \\\left( {{R^{(11)}}\frac{1}{2}{{\left| {{F_4}} \right|}^2}} \right) - \frac{1}{{12\kappa _{11}^2}} \cdot \\\int {{A_3}} \wedge {F_4} \wedge {F_4}\end{array}$

with ${A_3}$ a 3-form, ${F_4}$ its field strength and ${g^{(11)}}$ the Gaussian determinant of the 11-D metric that allows us to associate it to the sigma-model 6-D 5-brane anomaly of the world-volume:

$\delta S_1^{11} = \, - {T_2}\int {{A_3}} \wedge {X_8}$

with:

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

where:

${T_2} \equiv {\left( {2\pi } \right)^{2/3}}/{\left( {2\kappa _{11}^2} \right)^{ - 1/3}}$

is the super-membrane world-volume tension and $R$ is the 2-form curvature such that ${R^n}$ has symplectically both, matrix and wedge product properties. Note that ${X_8}$ is centrally key: it uniquely constraints Calabi-Yau fourfold reductions of:

thus inducing a tadpolic potential term for ${A_3}$ . One immediately realizes that this entails that the corresponding vacuum is totally degenerate. To go any further, realize that the Weyl-Polyakov anomaly-coefficient is determined by the Euler number $\chi$ of ${Y_4}$ :

$\int_{{Y_4}} {{X_8}} = \, - \frac{\chi }{{24}} = \frac{1}{4}\left( {8 + {h^{1,1}} + {h^{1,3}} + {h^{1,2}}} \right)$

and hence, setting $\chi = 0$ , one avoids the vacuum-degeneracy problem, giving us:

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

For the remainder of this post, let me set $n = 0$ and ${\kappa _{11}} \equiv 1$ and I will analyse a lowest possible order Kaluza–Klein reduction of the 11-dimensional SuGra Calabi-Yau fourfold. Now, taking the 11-dimensional metric the ansatz:

$g_{MN}^{(11)}(x,y) = \left( {\begin{array}{*{20}{c}}{g_{\mu \nu }^3(x)}&0\\0&{g_{ab}^{(8)}}\end{array}} \right)$

with ${x^\mu }$ , $\left( {\mu = 0,1,2} \right)$ , denoting the 3-D Minkowski space coordinates, and ${y^a}$ , $\left( {a = 3,...,10} \right)$ the internal Calabi-Yau coordinates. Therefore, and crucial for compactification, is that:

$g_{ab}^{(8)} = {\widehat g_{ab}}^{(8)}\left( {\left\langle M \right\rangle } \right) + \delta {g_{ab}}^{(8)}\left( {\delta M(x)} \right)$

splits metaplectically into Einstein-background solution as a function of the vacuum expectation values of the metric moduli and the quantum micro-fluctuations induced by the moduli variations and determined by the zero-modes of the Lichnerowicz operator for $\widehat g$ . Let ${\xi ^j}$ $\left( {j = 1,...,4} \right)$ be complex coordinates for ${Y_4}$ that uniquely characterize and define:

${\xi ^j} = \frac{1}{{\sqrt 2 }}\left( {{y^{2j + 1}} + i{y^{2i + 2}}} \right)$

and for the deformation of the Kähler form, we have:

$i\delta {g_{i\overline j }} = \sum\limits_{A = 1}^{{h^{1,1}}} {\delta {M^A}} (x){e_{{A_{i\overline j }}}}$

with ${e_A}$ a basis of ${H^{1,1}}\left( {{Y_4}} \right)$ and ${M^A}(x)$ the corresponding moduli. For complex structure deformations, one gets:

${\delta _{{g_{i\overline j }}}}\sum\limits_{\alpha = 1}^{{h^{3,1}}} {\delta {{\not Z}^\alpha }} (x){b_{{\alpha _{i\overline j }}}}$

with ${\not Z^\alpha }(x)$ being complex moduli and ${b_{{\alpha _{i\overline j }}}} \sim {\Phi _\alpha }$ , with ${\Phi _\alpha }$ the basis of ${H^{3.1}}\left( {{Y_4}} \right)$ determined by a Hilbert-contraction corresponding to the anti-holomorphic 4-form $\overline \Omega$ on ${Y_4}$ defined implicitly by:

${b_{{\alpha _{\overline {i\overline j } }}}} = \, - \frac{1}{{3{{\left| {\overline \Omega } \right|}^2}}}\overline \Omega _{\overline I }^{klm}{\Phi _{\alpha ,klm\overline j }}$

and:

${\left| {\overline \Omega } \right|^2} \equiv \frac{1}{{4!}}{\Omega _{ijkl}}\,{\overline \Omega ^{ijkl}}$

Hence, we finally, for this post, get the 3-form ${A_3}$ in expansion form via $\left( {1,1} \right)$ form ${e_A}$ and the $\left( {2,1} \right)$ form ${\Psi _{{I_{ij\overline k }}}}$ , yielding:

$\left\{ {\begin{array}{*{20}{c}}{{A_{{\mu _{i\overline j }}}} = \sum\limits_{A = 1}^h {A_\mu ^A(x){e_{{A_{i\overline j }}}}} }\\{{A_{ij\overline k }} = \sum\limits_{I = 1}^{{h^{2,1}}} {{N^I}(x){\Psi _{{I_{ij\overline k }}}}} }\\{{A_{\overline {i\overline {jk} } }} = \sum\limits_{I = 1}^{{h^{2,1}}} {{{\overline N }^{{J_\alpha }}}(x){{\overline \Psi }_{\overline J ij\overline k }}} }\end{array}} \right.$

Hence, until the next post, the lesson is that ${Y_4}$ compactification yields ${h^{1,1}}$ Euclidean Supersymmetry vector multiplets $\left( {A_\mu ^A,{M^A}} \right)$ and ${h^{2,1}} + {h^{3,1}}$ chiral multiplets $\left( {{N^I}} \right)$ , $\left( {{{\not Z}^\alpha }} \right)$ .