Calabi-Yau FourFolds, Kähler Analysis, and Compactification

By George Shiber Tuesday, November 10, 2015Friday, October 26, 2018 In Grand Unification Physics, M Theory, Mathematical Physics, Physics, Super String Theory, Supersymmetry Branes, Calabi-Yau Fourfold

I accept no principles of physics which are not also accepted in mathematics ~ René Descartes

Continuing with my M-theoretic Calabi-Yau fourfold compactification series, let me note that a C-‘fourfolding’ of M-Theory, as I will show ultimately, is equivalent to a proof of its testability and predictive power. Before I delve into the mathematics, there is a great quote at the bottom of this post of a extract from a forthcoming publication by Professor Stephen Law: ‘Is Philosophy a Grand Waste of Time?‘, which I highly recommend you read. Let me pick up where I left off at the end: On the space of $\left( {1,1} \right)$ forms, let me define the central metric:

$\begin{array}{c}{G_{AB}} = \frac{1}{{2\not V}}\int_{{Y_4}} {{e_A}} \wedge * \,{e_B}\\ = \, - \frac{1}{{2\not V}}\int_{{Y_4}} {{d^8}\xi \sqrt g } \,{e_{{A_{i\overline j }}}}\,{e_{BL\overline m }}\\ \cdot {g^{i\overline m }}{g^{k\overline m }}\end{array}$

with $\not V$ being the volume of ${Y_4}$ and given as:

$\begin{array}{c}\not V = \int_{{Y_4}} {{d^8}} \xi \sqrt g = \\\frac{1}{{4!}}\int_{{Y_4}} {J \wedge J \wedge J \wedge J = } \\\frac{1}{{4!}}{d_{ABCD}}{M^A}{M^B}{M^C}{M^D}\end{array}$

where $J$ is the super-Kähler Calabi-Yau fourfold form:

$J = i{g_{i\overline j }}\,d{\xi ^i} \wedge d{\xi ^{\overline j }} = M_{{e_A}}^A$

Now it follows that:

${d_{ABCD}} = \int_{{Y_4}} {{e_A}} \wedge {e_B} \wedge {e_C} \wedge {e_D}$

are the Euler-intersection numbers of ${Y_4}$ on the space of $\left( {3,1} \right)$ forms, and one defines the super-metric:

$\begin{array}{c}{G_{\alpha \overline \beta }} \equiv \, - \frac{{{{\int_{{Y_4}} {{\Phi _\alpha } \wedge \overline \Phi } }_{\overline \beta }}}}{{\int_{{Y_4}} {\Omega \wedge \Omega } }} = \\\frac{1}{{4\not V}}\int_{{Y_4}} {{d^8}} \xi \sqrt g \,{b_{\overline {i\overline j } }}\,{b_{\beta \overline {i\overline j } }}\,{g^{\overline i \overline j }}{g^{^k\overline m }} \cdot \\{{\not \partial }_\alpha }{{\not \partial }_\beta }{K_{3,1}}\end{array}$

that analytically defines a Kähler metric potential:

${K_{3,1}} = {\rm{In}}\left[ {\int_{{Y_4}} {\Omega \wedge \Omega } } \right]$

and, on the space of $\left( {2,1} \right)$ forms, one has the metric ${G_{I\overline J }}$ and Euler-intersection numbers ${d_{AI\overline J }}$ :

$\begin{array}{c}{G_{I\overline J }} \equiv \,\frac{1}{2}\int_{{Y_4}} {{\Psi _I}} \wedge \overline * \,{\Psi _J} = \\\frac{1}{4}\int_{{Y_4}} {{d^8}} \xi \sqrt g \,{\Psi _{Iij\overline k }}{\overline \Psi _{\overline J l\overline m \overline n }} \cdot \\{g^{i\overline m }}{g^{i\overline n }}{g^{I\overline K }}\end{array}$

and:

$\begin{array}{c}{d_{AI\overline J }} \equiv \int_{{Y_4}} {{e_A}} \wedge {\Psi _I} \wedge {\Psi _{\overline J }} = \\\frac{1}{4}\int_{{Y_4}} {{d^8}} \xi \sqrt g \,{\varepsilon ^{ikls}}{\varepsilon ^{\overline i \overline m \overline n \overline r }}{\varepsilon _{{A_{i\overline J }}}} \cdot \\{\Psi _{Ikl\overline m }}{\overline \Psi _{\overline J snr}}\end{array}$

which, and this is key, related as:

${G_{I\overline J }} = \frac{i}{2}{d_{AI\overline J }}{M^A}$

or equivalently:

${d_{AI\overline J }} = 2i\frac{{\not \partial {G_{I\overline J }}}}{{\not \partial {M^A}}}$

Now, define a useful metric:

${\widehat G_{I\overline J }} = \frac{i}{2}{c^A}{d_{AI\overline J }}$

with ${c^A}$ being the constant real vectors with no vanishing Hilbert-entries. Note now that ${G_{AB}}$ and ${d_{ABCD}}$ are independent of the complex structure, however, ${G_{I\overline J }}$ and ${d_{AI\overline J }}$ are a function of ${Z^\alpha }$ and ${\overline Z ^{\overline \alpha }}$ . The ${\Psi ^I}$ basis of $\left( {2,1} \right)$ forms are local and depend holomorphically on the complex structure; hence:

$\left\{ {\begin{array}{*{20}{c}}{{{\overline {\not \partial } }_{{{\overline Z }^\alpha }}}{\Psi _I} = 0}\\{{{\not \partial }_{{Z^{\overline \alpha }}}} = {\Psi _I} = 0}\end{array}} \right.$

with:

$\begin{array}{c}{{\not \partial }_{{Z^{\overline \alpha }}}}{\Psi _I} = \sigma _{\alpha I}^K\left( {Z,\overline Z } \right){\Psi _K} + \\{\tau _{\alpha I}}^{\overline L }\left( {Z,\overline Z } \right){\overline \Psi _{\overline L }}\end{array}$

By differentiation in the above with respect to ${\overline Z ^{\overline \alpha }}$ gives us the differential constraints for $\sigma$ and $\tau$ :

$\left\{ {\begin{array}{*{20}{c}}{{{\overline {\not \partial } }_{{{\overline Z }^{\overline \beta }}}}\sigma _{\alpha I}^{\overline L } = \, - \tau _{\alpha I}^{\overline L }\overline \tau _{\overline \beta \overline L }^K}\\{{{\overline {\not \partial } }_{{{\overline Z }^{\overline \beta }}}}\tau _{\alpha I}^{\overline K } = \, - \tau _{\alpha I}^{\overline L }\overline \sigma _{\overline \beta \overline L }^{\overline K }}\end{array}} \right.$

From:

and

${d_{AI\overline J }} = 2i\frac{{\not \partial {G_{I\overline J }}}}{{\not \partial {M^A}}}$

it follows that the complex structure functional dependence of ${G_{I\overline J }}$ , ${\widehat G_{I\overline J }}$ and ${d_{AI\overline J }}$ is completely constrained by the differential equations:

$\left\{ {\begin{array}{*{20}{c}}{{{\not \partial }_{{Z^\alpha }}}{G_{I\overline J }} = {\sigma _{\alpha I}}^K{G_{K\overline J }}}\\{{{\not \partial }_{{Z^\alpha }}}{{\widehat G}_{I\overline J }} = {\sigma _{\alpha I}}^K{{\widehat G}_{K\overline J }}}\\{{{\not \partial }_{{Z^\alpha }}}{d_{AI\overline J }} = {\sigma _{\alpha I}}^K{d_{AK\overline J }}}\end{array}} \right.$

To be able to move on with the compactification, one must note that the vectors $A_\mu ^A$ are dualized to scalar fields denoted by ${P^A}$ and after dualization the vector super-multiplet becomes a chiral multiplet with scalars $\left( {{M^A},{P^A}} \right)$ , and altogether are:

${h^{1,1}} + {h^{1,2}} + {h^{1,3}}$

Supersymmetry demands now that the Lagrangian be the form:

${\not L^3} = \sqrt { - {g^{(3)}}} \left( {\frac{1}{2}{R^{(3)}} - {G_{\overline \Lambda \sum }}{{\not \partial }_\mu }{{\overline {\not Z} }^{\overline \Lambda }}{{\not \partial }^\mu }{{\not Z}^\sum }} \right)$

with

${{{\not \partial }_\mu }{{\overline {\not Z} }^{\overline \Lambda }}{{\not \partial }^\mu }{{\not Z}^\sum }}$

being crucial and $\Lambda$ , $\sum$ $= 1,...,{h^{1,1}} + {h^{1,2}} + {h^{1,3}}$ and ${G_{\overline \Lambda \sum }}$ is a super-Kähler metric:

${G_{\overline \Lambda \sum }} \equiv {\overline {\not \partial } _{\overline \Lambda }}\,{\not \partial _\sum }K_M^{(3)}$

Realize though that the scalar fields that appear in:

$\left\{ {\begin{array}{*{20}{c}}{i\delta {g_{i\overline j }} = \sum\limits_{A = 1}^{{h^{1,1}}} {\delta {M^A}} (x)\,{e_{{A_{i\overline j }}}}}\\{\delta {g_{\overline i \overline j }} = \sum\limits_{\alpha = 1}^{{h^{3,1}}} {\delta {{\not Z}^\alpha }(x){b_{\alpha \overline i \overline j }}} }\end{array}} \right.$

$\left\{ {\begin{array}{*{20}{c}}{{b_{\alpha \overline i \overline j }} = \, - \frac{1}{{3{{\left| \Omega \right|}^2}}}\overline \Omega _{\overline I }^{klm}{\Phi _{\alpha klm\overline j }}}\\{{{\left| \Omega \right|}^2} \equiv \frac{1}{{4!}}{\Omega _{ijkl}}\,{{\overline \Omega }^{ijkl}}}\end{array}} \right.$

in the expansion of the harmonic forms on ${Y_4}$ are not Kähler admissible coordinates. We need a class of field redefinition to be performed in order to get the 3-D Lagrangian into the form:

${\not L^{(3)}} = \sqrt { - {g^{(3)}}} \left( {\frac{1}{2}{R^{(3)}} - {G_{\overline \Lambda \sum }}{{\not \partial }_\mu }{{\overline {\not Z} }^{\overline \Lambda }}{{\not \partial }^\mu }{{\not Z}^\sum }} \right)$

with:

${{{\not \partial }_\mu }{{\overline {\not Z} }^{\overline \Lambda }}{{\not \partial }^\mu }{{\not Z}^\sum }}$

being key. So, we can now get, along our journey to fourfold compactification, the Kähler coordinates ${T^A}$ and ${\widehat N^I}$ as:

${T^A} = \frac{1}{{\sqrt 8 }}\left( {i{P^A} + \not VG_B^A{M^B} - \frac{i}{4}d_{M\overline L }^A\widehat G_{\overline J }^{ - 1}{M^A}{{\widehat G}^{ - 1,\overline L }} + \omega _{IK}^A{{\widehat N}^I}{{\widehat N}^K}} \right)$

where:

${\frac{i}{4}d_{M\overline L }^A\widehat G_{\overline J }^{ - 1}{M^A}{{\widehat G}^{ - 1,\overline L }}}$

and:

${\omega _{IK}^A{{\widehat N}^I}{{\widehat N}^K}}$

being central to uniqueness and existence, and ${\widehat N^I}$ is given by:

${\widehat N^I} = \widehat G_{\overline J }^I\left( {{{\not Z}^\alpha },{{\overline Z }^{\overline \alpha }}} \right){\overline N ^{\overline J }}$

where $\omega _{IK}^A$ are functions of

$\left\{ {\begin{array}{*{20}{c}}{{{\not Z}^\alpha }}\\{{{\overline {\not Z} }^\alpha }}\end{array}} \right.$

and obey:

${\overline {\not \partial } _{{{\overline {\not Z} }^{\overline \alpha }}}}{\omega _{AIK}} = \frac{i}{4}\widehat G_I^{ - 1,\overline L }\widehat G_K^{ - 1,\overline J }{A_{AM\overline L }}{\overline \tau _{\overline J \overline L }}^M$

So, until the next post in this series, we can conclude that in terms of ${T^A},{\widehat N^I}{\rm{,}}\,{{\rm{\not Z}}^\alpha }$ , the ${Y_4}$ Kähler metric with Kähler potential:

$K_M^{(3)} = {K_{3,1}} - \left[ {{\Xi ^A}\not V\,G_{AB}^{ - 1,}\,{\Xi ^B}} \right]$

with :

${\Xi ^A} \equiv \left( {{T^A}{{\overline T }^A} + \frac{i}{{2\sqrt 8 }}d_{M\overline L }^A\widehat G_{\overline J }^{ - 1,M}\widehat G_I^{ - 1,\overline L } - \frac{1}{{\sqrt 8 }}\omega _{IK}^A\widehat N{{\widehat N}^K} - \frac{1}{{\sqrt 8 }}\overline \omega _{\overline J \overline L }^A\widehat {\overline N }} \right)$

${ - \frac{1}{{\sqrt 8 }}\omega _{IK}^A\widehat N{{\widehat N}^K} - \frac{1}{{\sqrt 8 }}\overline \omega _{\overline J \overline L }^A\widehat {\overline N }}$