SuSy, Dp-Branes, And Green-Schwarz Analysis: The Sparticle Vacua And The Big Bang

One may wonder, What came before the Big Bang? If space-time did not exist then, how could everything appear from nothing? . . . Explaining this initial singularity —where and when it all began—still remains the most intractable problem of modern cosmology, science and philosophy in general. ~ Andrei Linde: I might add – will always remain so, never to be ‘solved’!

An equation means nothing to me unless it expresses a thought of God. S. Ramanujan!

In my last post, I showed that the kappa symmetric algebraic equation:

    \[{\Gamma _\kappa }\left| {_{{\rm{Bo}}{{\rm{s}}^\varepsilon }}} \right. = \varepsilon \]

has a Sasaki-Einstein solution. It can be seen that any such solution generates the SUSY vacuum for the sparticle spectrum:

    \[{\not {\rm Z}_V} = {T_{M_L^{p + 1}}}dy \wedge dz = {T_{{M^{p + n}}}}\frac{1}{{2\pi ik}}\int_{M_L^{p + 1}} {d\,\Omega } {({\phi _{si}})^{p + n}} \wedge d\;\widetilde \Omega {({\phi _{si}})^{ - p + 1}}\]

with {\phi _{si}} being the string variable on the corresponding 2-D world-sheet, and M the super-Lagrangian manifold, and solving gives us a exact and complete description of the sparticle fields ‘living’ on:

    \[Ad{S_5} \times E_S^5\]

with E_S^5 being a 5-D Sasaki-Einstein manifold. Let me now, in this context, raise deep philosophical problems, for both, {\not {\rm Z}_V}: which is critical for explaining Higgs physics, and for the Big Bang Theory. Note first, that by e = m{c^2} and the Heisenberg uncertainty principle for time and energy, denoted here by {H_{E,t}}, the expression in {\not {\rm Z}_V}, namely:

    \[\int_{M_L^{p + n}} {d\,\Omega } {({\phi _{si}})^{p + n}}\]

can be diffeomorphically transformed into a derivative funtor of the string variable as a function of time: {\phi _{si}}(t), and thus, we get:

    \[\frac{d}{{d{\sigma _t}}}\int_{M_L^{p + n}} {d\,\Omega } ({\phi _{si}}^{p + n}(t))\]

Now, by {H_{E,t}}, given that the Dp+n brane tension {T_{Dp + n}}, by unitarity, always has positive-definite energy, ‘time‘ is hence in quantum superposition with respect to energy, thus

    \[\frac{d}{{d{\sigma _t}}}\int_{M_L^{p + n}} {d\,\Omega } ({\phi _{si}}^{p + n}(t))\]

will have no solution, entailing that the super-Lagrangian manifold M_L^{p + n} is degenerate topologically and can neither admit a Ricci tensor nor a Minkowskian metric: hence, {\not {\rm Z}_V} cannot be the Kähler vacua that generates the sparticles, and since the existence of the Higgs superpartner is a necessary condition for a complete explanation of the Gibbs-cosmological mass distribution, we have a catastrophe, since that implies the universe has no matter, and that really matters! Moreover, and more severe, is that ‘time‘ as it occurs in the string variable, converges to zero, in the initial Dp+n singularity integral, at Big Bang ‘time’:

    \[{\not \Delta ^D}\varsigma = \underbrace {\int_0^{\delta {f_K}} {\frac{{d{t^ \circ }}}{{a({t^ \circ })}}\not D{{\not {\rm Z}}_V}} }_{{\rm{Quantum Gravity}}} + \underbrace {\int_{\delta f}^{{f_K}} {\frac{{d{t^ \circ }}}{{a({t^ \circ })}}d\,\Omega {{({\phi _{si}})}^{e{\phi _{si}}({t^ \circ })}}d{\Phi _I}} }_{{\rm{Inflation}}}\]

where {f_K} is the Yukawa interactional entropic measure of curvature ‘near’ the singularity, and a(t) describes the inflationary expansion, with:

    \[d\,\Omega = d{\theta ^2} + \sin d\Phi _I^2\]

with {\Phi _I} the instanton scalar field and K the Guassian curvature. So, by {H_{E,t}}:

    \[\frac{d}{{d{\sigma _t}}}\underbrace {\int_0^{\delta {f_K}} {\frac{{d{t^ \circ }}}{{a({t^ \circ })}}\not D{{\not {\rm Z}}_V}} }_{{\rm{Quantum Gravity}}}d({\phi _{si}}({t^ \circ })\]

has four solutions: one describing the expansion of space in positive-time direction, one in negative, and one in both (the Robinson solution) and one in none. The last 2 solutions are clearly incoherent since that would imply not only that the Big Bang did not ‘occur’ by the right-hand-side of {\not \Delta ^D}\varsigma, but also by e = m{c^2} applied to {\not {\rm Z}_V}, space not only could not have ‘started’ to expand, but actually we get an anti-‘Big-Bang’: space started, at initial time = 0, to contract into the ‘past’! The only way to solve the above-mentioned problems in a way that is consistent with any theory of quantum gravity is solving the kappa symmteric p-brane Green-Schwarz action with chiral anomaly cancellation. Let us consider a p-brane in a coset G/H with G a super-Poincaré group, H the corresponding Lorentz proper subgroup. A parametrization can be expressed as:

    \[g(z) = {e^{i({x^a}{P_a} + \widetilde \theta Q)}}\]

and the vielbein on G/H can be derived via the Cartan form:

    \[g{(z)^{ - 1}}dg(z) = i{\prod ^a}{P_a} + id\widetilde \theta Q\,\not \partial \phi _{si}^{ - 1} = i\left( {d{x^a} - i\left( {\widetilde \theta {\Gamma ^a}d\theta \,\not \partial \phi _{si}^{ + 1}} \right)} \right){P_a} + id\widetilde \theta Q\]

which is invariant under SySy transformations:

    \[{\delta ^i}{x^a} = i\left( {\widetilde \varepsilon {\Gamma ^\alpha }\widetilde \theta \not D{{(p_i^{ \pm 1})}^{ - 1/2}}} \right)\gamma _{{a^ \circ }a}^i\sum\limits_{n = - \infty }^\infty {S_{GS}^{10}} \alpha _n^i\]

with the Green-Schwarz 10-D action:

    \[S_{GS}^{10} = \frac{{ - 1}}{{4{\alpha ^ * }\pi }}\int {d\sigma \,d\tau } \left\{ {\sqrt {{g_{\mu \nu }}} {g^{\alpha \beta }}{\prod _\alpha }{\prod ^\beta } + {{\not {\rm Z}}_i}{\varepsilon ^{\alpha \beta }}\not \partial {X^\mu }\left( {{{\widetilde \theta }^1}{\Gamma _{\mu \nu }}\widetilde {\not \partial \,}{\theta ^{ - 1}}{\Gamma _\mu }{{\not \partial }_\beta }{\theta ^2}} \right)} \right\}\]

and with \delta \theta = \varepsilon holding. The Vielbein now transforms as:

\delta {\prod ^a} = - 2\left( {\widetilde \varepsilon {\Gamma ^a}d\theta } \right)


\delta d\theta = d\varepsilon