As we saw in my last post, the Standard ΛCDM Model of cosmology can be derived from Type-IIB SUGRA by identifying the inflaton with the Gukov-Vafa-Witten topologically twisted Kähler modulus embedded in a brane/anti-brane system. The advantages of the  system is that we can apriori embed anti-brane instantonic effects to allow de Sitter solutions, and by mirror symmetry, we get a Kaloper-Sorbo axion monodromy inflation, where the flatness of the inflaton potential is protected without dependence on a moduli stabilization mechanism. Noting that -branes probing Calabi-Yau 3-folds support 4-D  supersymmetric Yang-Mills gauge theories whose intersection-points generate the Standard Model chiral matter sector and generally the action of a -brane is given by a Dirac-Born-Infeld part coupled to a Chern-Simons WZ part:

with:

where is the worldvolume pullback with -orientifold action:

with:

and

where the pullback to the -worldvolume yields the 10-D SYM action:

with string coupling:

and the 10-D SUGRA dimensionally reduced Type-IIB action is:

with:

and in the string-frame, the type-IIB SUGRA action is given by:

with:

where the Calabi-Yau superpotential is:

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:

with the Large Volume Scenario -term is given by:

with:

and the Fayet-Illopoulos terms being:

where are the -brane charge-vectors. Then we saw that the Chern-Simons orientifold action gets a Calabi-Yau curvature correction in the form of:

with

due to the Gauss–Codazzi equations:

One can and should enhance such a Kähler scenario to one that involves a -brane inflationary theory mainly due to the fact that in this context, moduli stabilization entails the existence of a  de Sitter compactification that inherits the Lambda-CDM -CP gauge-bundle, yielding a non-truncated N=1 4D SYM on the bulk, deriving inflation on the -manifold orbifolded by the corresponding ALE singularities that give rise to the SM gauge structure and resolutions that yield the Dirac chiral-fermionic SM structure, giving rise to an isomorphism with a  Abrikosov-Nielsen-Olesen type manifold. Thus, we can identify the -brane system with an N = 2 gauge theory associated with the D-term inflation model I derived above, but with one hypermultiplet and one Fayet-Iliopoulos term, and dimensionally reduce to N=1. In this, part 1, I will defend such a  de Sitter brane inflationary scenario.

In a  brane system, a spontaneously broken gauge symmetry corresponds to a bound state where the  dissolves as a deformed Abelian instanton on the  Dirac-Born-Infeld theory of the  brane with Chern-Simons coupling:

where  gets a background NSNS  as well as  worldvolume gauge field contributions. Hence, the NSNS two-form becomes a non-commutative deformation parameter in this system. The brane system has an unbroken supersymmetry due to a -brane -symmetry. The vacuum endpoints is describable hence by a non-marginal bound state of  and -branes corresponding to the Higgs phase of the gauge theory of the -system. The LE-effective action of such a -system in flat space is:

with:

and

with the Chern-Simons part, with:

The doublet arises as the lightest d.o.f from strings stretched between the -brane and the parallel -brane, with covariant derivative:

and our metric has a Chern-Simons form:

We now must geometrically locate our system in a background that incorporates the field  and turn on a Ramond-Ramond field-strength  compatible with the orbifold and orientifold structure of the system with a metric of the form:

the 4D metric in the Einstein frame and  is a section on that factors in moduli stabilization involved in wrapping branes on cycles in the compact space .

Let us analyze the action. In a curved background given by our metric, this contribution to the LE-effective action:

reduces to:

with:

and we have integrated out the  fluctuation-modes in the  directions. The  brane covariant 2-form is composed of two terms:

with the field strength of the vector field living on the brane and  the pullback of the space-time NS-NS two-form field to the worldvolume of the -brane, with a Chern-Simons part induced by the RR field. Now let

be the volume of a fixed . Integrating over  gives us:

with:

and with coupling constants:

with our four-form given by:

thus the -Chern-Simons term becomes:

with:

Now since the invariant 5-form is self-dual in 10-D, there must be a 4-form field in all 10-dimensions. Hence, our action becomes:

with:

Adding coincident -branes forces us to generalize the connection  with corresponding Chan-Patton  gauge fields and a Yukawa-quiver gauge-theory describing the system.

After embedding the -brane in the same metric and Ramond-Ramond system, we get the following -action:

with:

Hence, the -modified total action is:

with:

Note that the hypermultiplet covariant derivative is still of the form:

hence, we can do the following gauge transformation:

consistent with:

Thus, our action now has the form:

Our string sectors and all our fields satisfy the required N = 1 chiral superfield-normalization condition and we have rigid N = 2 supersymmetry that gets naturally broken to N = 1 when coupled to gravity in D = 4. In part III of this series, we shall consider M5-brane effects and derive the P-term action.