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M-Theory, BraneWorld Cosmology, and Calabi-Yau Compactification

By Posted on In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Philosophy Of Science, Physics, Super String Theory, Supersymmetry, Time ,
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While we should not view brane-world Calabi-Yau-based cosmology as the be-all and end-all of quantum cosmological unification physics, it certainly qualifies as mathematical ‘evidence’ that M-theory is on the right track

Interest in M-theoretic cosmology has myriad motivational angles, and besides from being by far the best candidate for a unified quantum gravity theory, most importantly being the almost-total inadequacy and incompleteness of the Standard Model of physics as well as the irresolvable problems inflationary cosmology faces. In this series of posts, I will show how M-theory can handle those crises. Before I move on to string compactifications and the moduli problem, keep your eyes on the dilaton-field \Phi defined by

    \[\left\{ {\begin{array}{*{20}{c}}{\delta {A_a} = {\Lambda _a} - {\varepsilon ^c}{{\not F}_{ca}} + {\Lambda _k}{{\not \partial }_a}{\Phi ^k}}\\{\delta \,{\Phi ^i} = {\varepsilon ^i} - {\varepsilon ^c}{{\not \partial }_c}{\Phi ^i}}\end{array}} \right.\]

and the Dirac-Born-Infeld action

    \[{S_{DBI}} = \int_{\varphi \Phi \left( \Sigma \right)} {\sqrt {{\rm{det}}\left( {\varphi _\Phi ^ * {{\left( {g + B - F} \right)}_{ab}}} \right)} } d{x^0} \wedge ... \wedge d{x^p}\]

which, over the target space M, reduces to

    \[{S_{DBI}} = {\int_M {\widehat {\not L}} _{DBI}}{\delta ^{\left( {D - p - 1} \right)}}\left( {{x^i} - {\Phi ^i}({x^a})} \right)d{x^0} \wedge ... \wedge d{x^{D - 1}}\]

where

    \[{\delta ^{\left( {D - p - 1} \right)}}\left( {{x^i} - {\Phi ^i}({x^a})} \right)\]

is a Dirac delta function interpreted as a distribution along {x^i} directions. In this post, I will reveal a deep relation between Calabi-Yau compactification, AdS-metastability, anti-branes, and the cosmological constant problem. Now, note that

the Dirac-Born-Infeld action is not only invariant under the world-volume diffeomorphism but also under the full target space diffeomorphism and the B-field gauge transformation

with

    \[{\varphi _\Phi }:\underbrace \to _{pullback}M\]

and thus we have

    \[\begin{array}{l}\delta \left[ {{{\widehat {\not L}}_{DBI}}{\delta ^{\left( {D - p - 1} \right)}}\left( {{x^i} - {\Phi ^i}} \right)} \right] = \\ - {{\not \partial }_M}\left[ {{\varepsilon ^M}{{\widehat {\not L}}_{DBI}}{\delta ^{\left( {D - p - 1} \right)}}\left( {{x^i} - {\Phi ^i}} \right)} \right]\end{array}\]

where {L_{DBI}} is the DBI Lagrangian. The foundations of type IIB flux compactifications are warped extra dimensions, branes and fluxes where in the Witten-limit, contain D3, D5, and D7-branes as well as orientifold-planes and moreover a Dp-brane is the sum of a Dirac-Born-Infeld term and a Chern-Simons term

    \[{S_{Dp}} = {S_{DBI}} + {S_{CS}}\]

and in

the brane-world scenario, the Dirac-Born-Infeld action describes the world-volume of a brane relative to a string-frame, and takes the following form

    \[{S_{DBI}} = - {T_p}\int {{d^{p + 1}}} \xi {e^{ - \Phi }}\sqrt { - {\rm{det}}\left( {{G_{AB}}} \right)} \]

with {T_p} being the D-brane tension given in terms of the string coupling {g_s} and the string length is \sqrt {\alpha '} and {G_{AB}} the Siegel-Eisenstein-pullback of the metaplectic-metric onto the brane worldvolume. Now, a Chern-Simons-term describes a Maxwellian-coupling of a Dp-brane to the R-R (p + 1)-form {C_{p + 1}}

    \[\left\{ {\begin{array}{*{20}{c}}{{S_{CS}} = {\mu _p}\int {{C_{p + 1}}} }\\{\left| {{\mu _p}} \right| = {T_p}{e^{ - (p - 3)\Phi /4}}}\end{array}} \right.\]

Therefore, branes are sources for form-field fluxes

    \[{F_{p + 2}} = d{C_{p + 1}}\]

The complete path-summation flux of these fields through topologically non-trivial surfaces in the extra dimensions is quantized

    \[\left\{ {\begin{array}{*{20}{c}}{\frac{1}{{{{\left( {2\pi } \right)}^2}\alpha '}}\int_A {{F_3} = M} }\\{\frac{1}{{{{\left( {2\pi } \right)}^2}\alpha '}}\int_B {{H_3} = - K} }\end{array}} \right.\]

where A and B are 3-cycles of the compact manifold, {F_3} = d{C_2} and {H_3} = d{B_2} are 3-form fluxes and M, K are integers and the presence of branes and fluxes sources locally warped spacetime regions

    \[\begin{array}{c}d{s^2} = {h^{1/2}}(y)\underbrace {{g_{\mu \nu }}d{x^\mu }d{x^\nu }}_{4D} + \\{h^{1/2}}(y)\underbrace {{g_{ij}}d{y^i}d{y^j}}_{6D}\end{array}\]

are essential backgrounds for the action(s) of string cosmology.

Now I must set up the fundamentals of 4-D low-energy effective description of the KKLT string flux compactifications proposal. In the limit of N = 1 supergravity, where the moduli potential {V_F} is characterized by a superpotential W and a Kähler potential {\rm K}

    \[{V_F} = {e^{{\rm K}/M_{pl}^2}}\left[ {{{\rm K}^{i\overline j }}{D_i}W\overline {{D_j}W} - \frac{1}{{M_{pl}^2}}{{\left| W \right|}^2}} \right]\]

where

    \[\left\{ {\begin{array}{*{20}{c}}{{D_i}W \equiv {{\not \partial }_i}W + \frac{1}{{M_{pl}^2}}\left( {{{\not \partial }_i}{\rm K}} \right)W}\\{{{\rm K}_{i\overline j }} \equiv {{\not \partial }_i}{{\not \partial }_{\overline j }}{\rm K}}\end{array}} \right.\]

Standard Calabi-Yau compactification contains 3-form flux {G_3} \equiv {F_3} - \tau {H_3} that contributes to the superpotential via the Gukov-Vafa-Witten term

    \[W_{{\rm{Flux}}}^{GV} = \int {{G_3}} \wedge \Omega \]

with \Omega the holomorphic 3-form on the Calabi-Yau three-fold and

    \[\tau \equiv {C_0} + i{e^{ - \Phi }}\]

is the axionic-dilaton, and the Kähler potential for the complex structure moduli and the dilaton is

    \[\begin{array}{c}{\rm K} = - M_{pl}^2\,{\rm{In}}\left[ {\int {\Omega \wedge \overline \Omega } } \right] - \\M_{pl}^2\,{\rm{In}}\left[ {\tau + \overline \tau } \right]\end{array}\]

The KKLT-model gives a framework for stabilizing the overall size of the compact manifold by including non-perturbative quantum effects on Dp-branes or Euclidean Dn-instantons and are parameterized via the following superpotential

    \[{W_{\pi {\rm{p}}}} = A{e^{ - a\rho }}\]

with a a constant. Now,

    \[K = - 3M_{pl}^2\,{\rm{In}}\left[ {\rho + \bar \rho } \right]\]

the F-term potential in

    \[{V_F} = {e^{{\rm K}/M_{pl}^2}}\left[ {{{\rm K}^{i\overline j }}{D_i}W\overline {{D_j}W} - \frac{1}{{M_{pl}^2}}{{\left| W \right|}^2}} \right]\]

leads to a supersymmetric anti-de Sitter vacua

    \[{D_\rho }W = {D_\rho }\left( {W_{{\rm{Flux}}}^{GV} + {W_{\pi \rho }}} \right) = 0\]

with a stable Kähler modulus!

Now, the Calabi-Yau compactification is stabilized at large volume

{\rho _ * } \gg 1  if and only if

the flux superpotential is a small negative constant

    \[W_{{\rm{Flux}}}^{GV}\left( {\chi _\alpha ^ * ,{\tau ^ * }} \right) \equiv {W_0} \sim {10^{ - 4}}\]

With negative cosmological constant, in order for these solutions to describe ‘our’ universe

    \[{V_F} = - \frac{3}{{M_{pl}^2}}{\left| W \right|^2}{e^{{\rm K}/M_{pl}^2}}\]

the KKLT-model uplifts the AdS minima to positive energies by adding anti-D3-branes

which adds the Witten-term to the moduli potential

    \[{V_D} = \frac{D}{{{{\left( {\rho + \bar \rho } \right)}^2}}}\]

with D a constant that is a function of the D3-brane tension and the warping of the background, thus getting us a realistic cosmological description via the potential

    \[\begin{array}{c}V(\sigma ) = \frac{{aA}}{{2M_{pl}^2}}\frac{{{e^{ - a\sigma }}}}{{{\sigma ^2}}}\left( {\frac{1}{3}} \right.a\sigma A{e^{ - a\sigma }}\\{W_0} + \left. {A{e^{ - a\sigma }}} \right) + \frac{1}{4}\frac{D}{{{\sigma ^2}}}\end{array}\]

and the de Sitter minimum is metastable!

and thus the magnitude of the cosmological constant associated with the minimum depends on the choice of flux quanta

    \[\left\{ {\begin{array}{*{20}{c}}{\frac{1}{{{{\left( {2\pi } \right)}^2}\alpha '}}\int_A {{F_3} = M} }\\{\frac{1}{{{{\left( {2\pi } \right)}^2}\alpha '}}\int_B {{H_3} = - K} }\end{array}} \right.\]

and is ‘tunable’ and the discretuum of vacua in type IIB flux Calabi-Yau compactifications is used for an anthropic solution to the cosmological constant problem, and that is really something to behold. Next post, I will discuss Klebanov-Strassler geometry and M-theoretic braneworld cosmology.

Natural science does not simply describe and explain nature; it is part of the interplay between nature and ourselves ~ Werner Heisenberg

previously

The Randall-Sundrum Model, a Brane-Singularity Problem and AdS-Compactification

up next

Klebanov-Strassler Warped Spacetime Geometry and M-Theoretic Cosmology
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