A natural and intuitive way of deriving the standard inflationary ΛCDM cosmological model is by identifying the Type-IIB Kähler modulus with the ΛCDM-inflaton. The -brane interaction is of particular importance because the worldvolume theory on a stack of -branes probing a Calabi-Yau 3-fold is a 4-D  supersymmetric gauge theory and the importance of a  worldvolume theory derives, via duality-relations, from F-theory compactification by availing ourselves to the heterotic/F-theory duality in eight dimensions. Generally, for a -brane, , there naturally is a -D  SYM theory associated with its worldvolume , where the Dirac-Born-Infeld action is given by:

coupled to a Chern-Simons Wess-Zumino term:

the worldvolume pullback, and the corresponding p-orientifold action is:

with the string-coupling given by:

and  the pullbacks SUGRA fields to the -brane worldvolume. Thus, we can derive the action:

with:

which yields a 10-D SYM action:

with:

and the non-abelian field strength of the gauge field :

where the gauge covariant derivative is defined by:

giving us the -worldvolume SYM action:

with:

with the potential:

that will figure in the Kähler form that will characterize -brane cosmology. -branes are central not simply because of F-theory compactification due to  backreaction and Type-IIB axio-dilaton, but also because the Kähler moduli space induced by the -brane F-theoretic backreaction has a modulus that can naturally be identified with the ΛCDM-inflaton. Thus, inflationary cosmology can be derived, up to isomorphism, from 6D F-theory compactification using the Type-IIB string-theory/11D SUGRA duality upon dimensional reduction to 4D, where the Type-IIB SUGRA action in the string frame is given by:

with:

In light of flux moduli stabilization, this is a natural framework since the Kähler moduli potential apriori allows a Type-IIB-modulus/ΛCDM-inflaton identification. In such a SYM -brane scenario, we have an -flaton model in the Large Volume Scenario that evades the -problem, and periodicity of the modulus is achieved by defluxing the pullback of the -term on the superstring worldsheet. This can be readily seen since the total supersymmetric worldsheet action in conformal gauge:

with:

has modular-invariance. The skeleton of Type-IIB flux compactification on a Calabi-Yau threefold  is constructed from a superpotential of the form:

where:

is the Gukov-Vafa-Witten superpotential stabilization complex term, as well as the axio-dilaton field:

Given the presence of -brane instantons, are of Kähler moduli Type-IIB-orbifold class:

with  being the volume of the divisor  and  the 4-form Ramond-Ramond axion field corresponding to:

and:

where is the Kähler form:

and:

an integral-form basis and  the associated intersection coefficients. Hence, the Kähler potential is given by:

with  the Calabi-Yau volume, and in the Einstein frame, is given by:

The -term is given by:

and in the large volume scenario (LVS) is:

Thus, the LVS -term is given by:

with:

and the Fayet-Illopoulos terms being:

where are the -brane charge-vectors. The Kähler LVS potential is now derivable, and takes the form:

One can now choose any as the Kähler-modulus-inflaton, and inflation takes place in:

We can build our model now from the Kähler form and the superpotential. For a Witten-deformed -brane, the Kähler potential is thus derivable in the weak string-coupling limit of the F-theory Kähler potential arising from the elliptic fibration base Calabi-Yau four-fold:

Expanded in the weak string coupling limit gives us:

with  the periods of integrals of the holomorphic (3,0)-form over the symplectic basis of 3-cycles with intersection matrix  and is a function of symplectic and brane-moduli that figures in the flux-compactification moduli-stabilization but is independent of the axio-dilaton. Using mirror symmetry, the Kähler potential:

can be identified with the Kähler-moduli-potential of the mirror fourfold, which in the LVS-limit, involves the volume moduli of the elliptic fourfold, but not the corresponding axions due to shift-symmetry. Hence, the shift-invariant potential takes the form:

the self-intersecting matrix of the mirror four-fold divisors and the complex structure of the moduli of the three-form. We can now identify any  with the axio-dilaton and the remaining identified with the -brane position-moduli  and the three-fold complex moduli . Their interdependence can be expressed by the relation:

Hence, any can be identified with the deformation complex moduli term of a -brane, and Type-IIB N-flaton cosmology can reproduce ΛCDM by integrating out all the other moduli terms, and by substitution, the Kähler potential:

takes the following form:

The F-theory backreacted counterpart Kähler potential hence takes the form:

where  are the flux compactification quantum numbers and  is the period four-fold Calabi-Yau vector. In terms of the brane modulus , we can write the potential as:

We now include the Kähler moduli term to the potential:

and the superpotential:

Then, instanton correction terms from the four-fold blow-up cycles yield:

from which the F-term potential:

can be derived.  the complexified Kähler moduli that measure the 3-fold four-cycles when restricted to the reals, in string-length unit measure and as standard,  the Calabi-Yau 3-fold volume. Now since the Kähler metric is block-diagonal in the total space, no mixed derivatives in and occur in:

hence the term in the F-term potential dominates and stabilizes  in a supersymmetric minimum:

Thus, we have a classic moduli stabilization and applies to our Large Volume Scenario straightforwardly and yields the minimum:

that can be uplifted to a Minkowski minimum. Since inflation occurs in  and the terms:

are subleading in the inverse of  with respect to , it follows that the leading inflaton mass term is part of:

Since the LVS large-scale structure of the Kähler moduli space guarantees that loop-corrections are stably subleading with respect to moduli-stabilization -corrections, the Type-IIB axio-dilaton and brane moduli are integrated out at higher scales. Central is the -stabilization, since from:

we can derive:

with . At the ground level,

vanishes and we have:

and imposing the condition that the real part of:

at first-order in the and terms vanishes gives us:

We are now in position to derive the inflaton mass which arises from:

Moreover, in the minimum, we have:

and given that the Chern-Simons orientifold action gets the following Calabi-Yau curvature-correction:

we hence only need to expand it in the leading order with respect to the real part  and moduli stabilization enforces  which scales linearly. Hence we have:

and:

Thus, the SM renormalization parameter connects the inflaton to . Since the kinetic term for  is contained in:

then from:

and:

we can deduce:

In the ΛCDM Standard Model, quadratic inflation has a potential  with slow-roll parameters obeying the following relations:

with spectral index:

Phenomenologically, the ΛCDM Standard Model sets this term at  with e-fold field displacement  and . Moreover, the amplitude of the curvature perturbations (chapter 2/2.11) yields  and hence we have .

In Type-IIA/B string theories in the LVS, this remarkably yields:

In order not to destroy the moduli stabilization, we impose the harmless condition:

along the inflaton field flow. Now during the inflationary phase, we have:

and the imaginary part does not destroy moduli stability since:

holds, giving:

and since the LVS F-term is given by:

we have:

To handle the cubic and quartic terms in , we expand:

in  where at the minimum we have:

Solving:

with respect to  gives us:

and:

with respect to the kappa symmetric shift Type-IIB parameter, we get the Type-IIA dual instanton alpha-corrections to the Kähler potential thus yielding the correct Hilbert inflaton spectrum correction to the F-term:

Now, given:

the Kaluza-Klein inter-brane mode loop corrections give us the correct periodicity condition:

due to the Gauss–Codazzi equations for:

and are parametrized as:

at leading order. Since axion decay always takes place in a small region in e-fold inflation and is bounded by , then during the initial e-folding stage, under the Kähler-modulus/inflaton identification, the inflaton crosses at least once the period of the oscillatory term of:

and hence for large  and small  and , the Type-IIB 4-D SUGRA Kähler modulus under such inflaton-identification reproduces the ΛCDM Standard Model mainly because, as a Dirac variety, a D7-brane encircles a closed trajectory in Kähler moduli space enough times in the slow-roll scape to grow a contribution to the F-term potential that appears in the action of Type-IIB 4-D SUGRA. The phenomenological aspects of such claims can be computed from this source.

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