Kähler Analysis, Peccei-Quinn-Symmetries, and Compactification

By George Shiber Monday, November 16, 2015Saturday, August 17, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Physics, Super String Theory calabi yau manifold, compactification

If a man’s wit be wandering, let him study mathematics ~ Francis Bacon

In my last two posts, I launched a series where M-Theory can be Calabi-Yau fourfold-compactified, let me make a connection with PQ-symmetries (for ‘Peccei-Quinn’) in this post. Why the number 3 keeps appearing in those two posts? As will be shown in upcoming posts, in order to make phenomenological quasi-morphic contact with ‘physical’ 3-dimensional space. Crucial is the Kähler potential

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

where

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

Now, in purely geometric terms, the Kähler argument of the logarithm in the cube of the Calabi-Yau 4-fold volume measured by the Weil-Petersson metric of M-theory is

$K_M^{(3)} = {K_{3,1}} - 3\,{\rm{In}}\not V'$

Now, by using the re-scaled Kähler form: $\overline J = \widetilde {M\,}{e^{A,A}}$ in

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

and

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

one can derive

$\left\{ {\begin{array}{*{20}{c}}{\widetilde {\not V} = {{\not V'}^3}}\\{{{\widetilde G}_{AB}} = {{\not V}^{ - 1}}{G_{AB}}}\end{array}} \right.$

with $\widetilde {\not V}$ and ${\widetilde G_{AB}}$ equi-functionally dependent on ${\widetilde M^A}$ . So, we get

$\begin{array}{c}K_M^{(3)} = {K_{3,1}} - {\rm{In}}\left[ {{\Xi ^A}\,\widetilde G_{AB}^{ - 1}\,{\Xi ^B}} \right]\\ = {K_{3,1}} - {\rm{In}}\not V\end{array}$

even though the sum of the two terms in the moduli space cannot factorize! Note, $K_M^{(3)}$ , expressed via the super-Kähler coordinates

${\rm{In}}\left[ {{\Xi ^A}\,\widetilde G_{AB}^{ - 1}\,{\Xi ^B}} \right]$

is independent of ${T^A}$ , ${\widehat N^I}$ , and ${\not Z^\alpha }$ . Thus, the M-theoretic metric is not a block-diagonal. However, from

one notes that for ${\widetilde N^I} = 0$ , the Weil-Petersson moduli space indeed does factorize locally and the Kähler potential becomes

$\left\{ {\begin{array}{*{20}{c}}{K_M^{(3)} = {K_{3,1}} + {K_{1,1}}}\\{{K_{1,1}} = \, - {\rm{In}}\left[ {\left( {{T^A} + {{\overline T }^A}} \right)\widetilde G_{AB}^{ - 1}\left( {{T^B} + {{\overline T }^B}} \right)} \right]}\\{{K_{3,1}} = \, - {\rm{In}}\left[ {\int_{{Y_4}} {\Omega \wedge \overline \Omega } } \right]}\end{array}} \right.$

This is excellent as it implies that quantum gravitational effects can be completely causally explained by the topology of ${Y_4}$ !

Now let me get into the PQ-symmetries of M-theoretic vacuua in the super-large volume limit. Realize that, first, all the scalars ${P^A}$ that are dualized relative to $A_\mu ^A$ vectors inherit PQ-symmetry as guaranteed by gauge group-renormalizability. So

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

is metaplectically invariant under super-shifts ${P^A} \to {P^A} + {\widetilde \gamma ^A}$ , with ${\widetilde \gamma ^A}$ arbitrarily real constants. Also, ${N^I}$ naturally emerges from 3-form expansion of

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

hence inheriting 3-form gauge-invariance. That is:

$\begin{array}{c}{N^I}{\Psi _I} + {\overline N ^I}{\overline \Psi _I} \to {N^I}{\Psi _I} + {\overline N ^I}{\overline \Psi _I}\\ + \,{\rm{constants}}\end{array}$

Focusing on ${N^I}$ , we get

$\left\{ {\begin{array}{*{20}{c}}{{N^I} \to {N^I} + \overline \gamma _{\overline I }^I\left( {\not Z,\widetilde {\not Z}} \right)}\\{{N^I} \to {N^I} + \overline \gamma _I^{\overline I }\left( {\not Z,\widetilde {\not Z}} \right)}\end{array}} \right.$

with ${\gamma ^I}$ a function of the complex structural Kähler potential phase factor, leading to satisfying

$\left\{ {\begin{array}{*{20}{c}}{{{\not \partial }_{{Z^\alpha }}}{\gamma ^I} = \, - {\gamma ^I}\sigma _{{\alpha _1}}^J}\\{{{\overline {\not \partial } }_{{{\overline Z }^{\overline \alpha }}}}{\gamma ^J} = \, - {{\overline \gamma }^{\overline I }}{{\overline \tau }_{\overline \alpha }}^{{{\overline I }^{\overline J }}}}\end{array}} \right.$

and hence, we have a mathematically deep fact and philosophically crucial point, and that is

${\widehat N^I} \to {\widehat N^I} + \widetilde \gamma \left( {\not Z} \right)$

follows, with

${\widetilde \gamma ^I} = {\widehat G_{\overline J {{\overline \gamma }^I}}} = 0$

and

$\begin{array}{c}{{\not \partial }_{{{\not Z}^\alpha }}}{\widetilde \gamma ^I} = \sigma _\alpha ^I\kappa {\widetilde \gamma ^\kappa }\, - {\widehat {\overline \gamma }^L}\widehat G_L^{ - 1}{\tau _{\alpha \kappa }} \cdot \\\widetilde N{\widehat G^I}\overline N \end{array}$

Hence getting the compactification smoothness required for space-time continuum via a ${h^{1,1}} + 2{h^{2,1}}$ continuous PQ-symmetries in the large volume limit of M-theoretic fourfolding, hence allowing 4-dimensional phenomenological contact with ‘reality’!