In this, part III, of our series of deriving the Standard ΛCDM Model of cosmology from Type-IIB SUGRA by an identification of the inflaton with the Gukov-Vafa-Witten topologically twisted Kähler modulus, we recall that in part two, we derived the action of the  and the branes of our system. To complete the derivation, we need the -term, and to obtain it, we need an embedding in M-theory. Let us derive the action in a curved background given by our metric. The effective action is given by:

has a Kaloper-Sorbo reduction to:

with:

where 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. With:

the volume of a fixed , then 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 transformations:

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. Now we need an embedding in M-theory in order to derive our -term. Take M-theory with  parallel  branes spread along the orbifold , which preserves SUSY in 4-D, with the wrapped 6-D background along . Each  brane fills the 4-D non-compact spacetime and wraps the same holomorphic two-cycles  on the Calabi-Yau. The main terms of the 4-D SYM theory are the volume modulus of the Calabi-Yau:

the length modulus:
and the brane chiral superfields:
where stands for ‘open membrane’. Now we can start our derivation of the -term from the parallel brane system that supports the 5-branes. The   Lagrangian is:
with the covariant derivative:
and the Kähler potential, and the Chern-Simons term for the gauge potential is given by:
where define the brane transverse directions. The SUSY transformations are:
with gauge conditions:
and we must impose the Jacobi identity. Hence, we get the brane Lagrangian by Nambu-Poisson deformations defined in terms of:
which promote the system to a 6D SYM system with a Lagrangian:
where:
and  is given in terms of the kinetic terms for the ‘s:
with:
and  is given as such:
and where the relevant gauge field term is given by:
with the Hodge dual field strength is given by:
Thus, the equations of motion from  are:
Combining with the Bianchi identity:
gives us:
The term for the -field whose existence follows from the -Lagrangian, is:
Thus, we get:
giving us solutions of the form:
Integrating, we get the terms, which in our  system, satisfy:
as well as:
Plugging in the Kähler potential, we can derive the chiral super-field -term action:
with:
with the covariant derivative:
is the neutral gauge field charge, is the hypermultiplet charged under  gauged by the  vector multiplet  with superpotential and -term that drive inflation:
dynamically as a function of kinetic terms of type . The proof proceeds by plugging the RG-flow equation with the Hubble and inflaton term factored quadratically, with the -term potential:
Thus, the fermionic contributions to the inflaton field derive from the transformations:
and where the  gravitino connection  is:
which reduces to:
Now, by integrating the chiral super-field -term:
we get the -term:
with: