M-Theory, Kähler Inflation, Type-IIB Branes, and the P-Term

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 D3 and the D7 branes of our system. To complete the derivation, we need the P-term, and to obtain it, we need an embedding in M-theory. Let us derive the D7 action in a curved background given by our metric. The effective action is given by:

\displaystyle {{F}^{{ST}}}\equiv -\frac{1}{{4g_{7}^{2}}}{{F}_{{ST}}}{{F}^{{ST}}}-\frac{1}{{2g_{7}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}\left( {{{{\left( {{{\partial }_{\mu }}{{X}^{i}}} \right)}}^{2}}+{{{\left( {{{\partial }_{m}}{{X}^{i}}} \right)}}^{2}}} \right)

has a Kaloper-Sorbo reduction to:

\displaystyle {{S}_{{D7}}}=\int_{\mathcal{W}}{{\tilde{\psi }}}\left[ {\tilde{\varphi }-\tilde{\beta }} \right]+\tilde{p}\int_{\mathcal{W}}{{\tilde{\alpha }}}

with:

\displaystyle \tilde{\psi }\equiv {{d}^{4}}x{{d}^{4}}y\sqrt{{-{{g}_{E}}}}{{R}^{{-12}}}\sqrt{{{{g}_{{K3}}}}}{{R}^{4}}

\displaystyle \tilde{\varphi }\equiv -\frac{1}{{4g_{7}^{2}}}\left( {{{F}_{{\mu \nu }}}{{F}_{{\rho \sigma }}}g_{E}^{{\mu \rho }}g_{E}^{{\nu \sigma }}{{R}^{{12}}}+{{{\tilde{F}}}^{D}}_{{mn}}{{{\tilde{F}}}^{D}}_{{rs}}g_{{K3}}^{{mr}}g_{{K3}}^{{ns}}{{R}^{{-4}}}} \right)

\displaystyle \tilde{\beta }\equiv -\frac{1}{{2g_{7}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{\partial }_{\mu }}{{X}^{i}}{{\partial }_{\nu }}{{X}^{j}}g_{E}^{{\mu \nu }}g_{{ij}}^{{{{\mathbb{R}}^{2}}}}{{R}^{8}}

\displaystyle \tilde{p}\int_{\mathcal{W}}{{\tilde{\alpha }}}\equiv {{\mu }_{7}}\frac{{{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{2!}}\int_{{D7}}{{{{C}_{{\left( 4 \right)}}}}}\wedge {{{\tilde{F}}}^{D}}\wedge {{{\tilde{F}}}^{D}}

where we have integrated out the {{X}_{i}} fluctuation-modes in the K3 directions. The D7 brane covariant 2-form is composed of two terms:

\displaystyle {{\tilde{F}}^{D}}\equiv F-B\equiv dA-B

with {{F}_{{mn}}} the field strength of the vector field {{A}_{m}} living on the brane and {{B}_{{mn}}} the pullback of the space-time NS-NS two-form field to the worldvolume of the D7-brane, with a Chern-Simons part induced by the RR field. With:

\displaystyle \int{{{{d}^{4}}}}y\sqrt{{{{g}_{{K3}}}}}={{V}_{{K3}}}

the volume of a fixed K3, then integrating over K3 gives us:

\displaystyle {{S}_{{D7}}}\int{{\left( {\Sigma +\Theta } \right)}}+\int{\Xi }\int_{{K3}}{\Upsilon }+{{\mu }_{{K3}}}\int{{{{{\tilde{F}}}^{D}}_{C}}}

with:

\displaystyle \Sigma \equiv {{d}^{4}}x\sqrt{{-{{g}_{E}}}}

\displaystyle \Theta \equiv -\frac{1}{{4\tilde{g}_{3}^{2}}}{{\left( {{{F}_{{\mu \nu }}}} \right)}^{2}}-\frac{{{{R}^{4}}}}{{2\tilde{g}_{3}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{\left( {{{\partial }_{\mu }}{{X}^{i}}} \right)}^{2}}

\displaystyle \Xi \equiv \text{Vo}{{\text{l}}_{{\left( 4 \right)}}}{{R}^{{-12}}}\left( {\frac{{-1}}{{4g_{7}^{2}}}} \right)

\displaystyle \Upsilon \equiv {{{\tilde{F}}}^{D}}\wedge *{{{\tilde{F}}}^{D}}

\displaystyle {{\mu }_{{K3}}}\int{{{{{\tilde{F}}}^{D}}_{C}}}\equiv \frac{1}{{4\pi {{g}_{s}}g_{7}^{2}}}\int{{{{C}_{{\left( 4 \right)}}}}}\int_{{K3}}{{{{{\tilde{F}}}^{D}}\wedge {{{\tilde{F}}}^{D}}}}

and with coupling constants:

\displaystyle \frac{{{{V}_{{K3}}}{{R}^{4}}}}{{g_{7}^{2}}}={{T}_{7}}{{\left( {2\pi {\alpha }'} \right)}^{2}}{{V}_{{K3}}}{{R}^{4}}=\frac{{{{V}_{{K3}}}{{R}^{4}}}}{{{{{\left( {2\pi } \right)}}^{4}}{{{{\alpha }'}}^{2}}}}\frac{1}{{g_{3}^{2}}}=\frac{1}{{\tilde{g}_{3}^{2}}}

with our four-form given by:

\displaystyle {{C}_{{\left( 4 \right)}}}={{g}_{s}}\pi {{R}^{{-12}}}\text{Vo}{{\text{l}}_{{\left( 4 \right)}}}/2

Thus the K3-Chern-Simons term becomes:

\displaystyle \int{{\text{Vo}{{\text{l}}_{{\left( 4 \right)}}}{{R}^{{-12}}}\left( {\frac{{-1}}{{8g_{7}^{2}}}} \right)\int_{{K3}}{{{{{\tilde{F}}}^{D}}^{-}}}\wedge *{{{\tilde{F}}}^{D}}^{-}}}

with:

\displaystyle {{\tilde{F}}^{D}}^{-}\equiv \left( {{{{\tilde{F}}}^{D}}-{{*}_{{K3}}}{{{\tilde{F}}}^{D}}} \right)

Now since the SL\left( {2,\mathbb{Z}} \right) invariant 5-form is self-dual in 10-D, there must be a 4-form field in all 10-dimensions. Hence, our action becomes:

\displaystyle {{S}_{{D7}}}=\int_{\mathcal{W}}{{\tilde{\lambda }}}\left( {\tilde{\Theta }-\tilde{\Omega }} \right)-\tilde{p}\int_{{K3}}{{{{{\tilde{V}}}_{\alpha }}}}

with:

\displaystyle \tilde{\lambda }\equiv {{d}^{4}}x\sqrt{{-{{g}_{E}}}}

\displaystyle \tilde{\Theta }\equiv -\frac{1}{{4\tilde{g}_{3}^{2}}}{{\left( {{{F}_{{\mu \nu }}}} \right)}^{2}}

\displaystyle \tilde{\Omega }\equiv -\frac{{{{R}^{{-4}}}}}{{2\tilde{g}_{3}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{\left( {{{\partial }_{\mu }}{{X}^{i}}} \right)}^{2}}

\displaystyle \tilde{p}\int_{{K3}}{{{{{\tilde{V}}}_{\alpha }}}}\equiv {{\int_{\mathcal{W}}{{\text{Vol}}}}_{{\left( 4 \right)}}}{{R}^{{-12}}}\frac{1}{{8g_{7}^{2}}}\int_{{K3}}{{{{{\tilde{F}}}^{D}}\wedge {{{\tilde{F}}}^{D}}}}

Adding coincident D7 -branes forces us to generalize the connection {{A}_{\mu }} with corresponding Chan-Patton U\left( {{{N}_{7}}} \right) gauge fields and a Yukawa