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:

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