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 quiver gauge-theory describing the system.

After embedding the $D3$ -brane in the same metric and Ramond-Ramond system, we get the following $D3$ -action:

$\displaystyle {{S}_{{D3}}}\int{{{{G}_{E}}}}\left[ {\Xi -\xi } \right]$

with:

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

$\displaystyle \Xi \equiv \frac{{-1}}{{4g_{3}^{2}}}{{\left( {{{{{F}'}}_{{\mu \nu }}}} \right)}^{2}}$

$\displaystyle \xi \equiv \frac{{{{R}^{{-4}}}}}{{2g_{3}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}\left( {{{{\left( {{{\partial }_{\mu }}{{{{X}'}}^{n}}} \right)}}^{2}}+{{{\left( {{{\partial }_{\mu }}{{{{X}'}}^{m}}} \right)}}^{2}}} \right)$

Hence, the $D3$ -modified total action is:

$\displaystyle S=\int{{\wp \left[ {\Pi -{\mathrm Z}-\Omega -\Xi -{\mathrm T}+\Upsilon -\Theta -{\mathrm E}} \right]}}$

with:

$\displaystyle \wp \doteq {{d}^{4}}x\sqrt{{-{{g}_{E}}}}$

$\displaystyle \Pi \doteq \frac{1}{{4\tilde{g}_{3}^{2}}}{{\left( {{{F}_{{\mu \nu }}}} \right)}^{2}}$

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

$\displaystyle \Omega \doteq {{R}^{{12}}}\frac{1}{{8g_{7}^{2}}}\int_{{K3}}{{{{{\tilde{F}}}^{D}}\wedge {{{\tilde{F}}}^{D}}^{-}}}$

$\displaystyle \Xi \doteq \frac{1}{{4g_{3}^{2}}}{{\left( {{{F}_{{\mu \nu }}}} \right)}^{2}}$

$\displaystyle {\mathrm T}\doteq \frac{{{{R}^{{-4}}}}}{{2g_{3}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}\left( {{{{\left( {{{\partial }_{\mu }}{{{{X}'}}^{n}}} \right)}}^{2}}+{{{\left( {{{\partial }_{\mu }}{{{{X}'}}^{m}}} \right)}}^{2}}} \right)$

$\displaystyle \Upsilon \doteq {{R}^{6}}{{\left| {{{D}_{\mu }}\chi } \right|}^{2}}$

$\displaystyle \Theta \doteq {{R}^{{-10}}}{{\left( {\frac{{{{X}^{i}}-{{{{X}'}}^{i}}}}{{2\pi {\alpha }'}}} \right)}^{2}}{{\left| \chi \right|}^{2}}$

$\displaystyle {\mathrm E}\doteq \frac{{{{R}^{{12}}}\left( {g_{3}^{2}+\tilde{g}_{3}^{2}} \right)}}{2}{{\left( {{{\chi }^{\dagger }}{{\sigma }^{A}}\chi } \right)}^{2}}$

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

$\displaystyle {{D}_{\mu }}\chi =\left( {{{\partial }_{\mu }}+i{{A}_{\mu }}-i{{{{A}'}}_{\mu }}} \right)\chi$

hence, we can do the following gauge transformations:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {g{{W}_{\mu }}={{A}_{\mu }}-{{{{A}'}}_{\mu }}} \\ {g{{{{W}'}}_{\mu }}=\frac{{{{g}_{3}}}}{{{{{\tilde{g}}}_{3}}}}{{A}_{\mu }}+\frac{{{{{\tilde{g}}}_{3}}}}{{{{g}_{3}}}}{{{{A}'}}_{\mu }}} \\ {{{g}^{2}}=g_{3}^{2}+\tilde{g}_{3}^{2}} \end{array}} \right.$

consistent with:

$\displaystyle \frac{1}{{\tilde{g}_{3}^{2}}}{{F}^{2}}+\frac{1}{{g_{3}^{2}}}{{{F}'}^{2}}=F_{W}^{2}+{F}'_{W}^{2}$

Thus, our action now has the form:

$\displaystyle \begin{array}{l}S=\int{{{{d}^{4}}}}x\sqrt{{-{{g}_{E}}}}\left[ {-\frac{1}{4}} \right.{{\left( {{{F}_{W}}} \right)}^{2}}-\frac{1}{4}{{\left( {{{F}_{{{W}'}}}} \right)}^{2}}-\\{{\left| {\partial S} \right|}^{2}}-{{\left| {\partial {S}'} \right|}^{2}}-{{R}^{{-12}}}\frac{1}{{8g_{7}^{2}}}\int_{{K3}}{{{{{\tilde{F}}}^{D}}^{-}\wedge *{{{\tilde{F}}}^{D}}^{-}}}-\\{{\left| {{{D}_{\mu }}\chi } \right|}^{2}}-2{{g}^{2}}{{\left| S \right|}^{2}}{{\left| \chi \right|}^{2}}\left. {-\frac{{\left( {g_{3}^{2}+\tilde{g}_{3}^{2}} \right)}}{2}{{{\left( {{{\chi }^{\dagger }}{{\sigma }^{A}}\chi } \right)}}^{2}}} \right]\end{array}$

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 $P$ -term. Take M-theory with $N$ parallel $M5$ branes spread along the orbifold ${{S}^{1}}/{{\mathbb{Z}}_{2}}$ , which preserves $N=1$ SUSY in 4-D, with the wrapped 6-D background along ${{S}^{1}}/{{\mathbb{Z}}_{2}}$ . Each $M5$ brane fills the 4-D non-compact spacetime and wraps the same holomorphic two-cycles ${{\Sigma }_{2}}$ on the Calabi-Yau. The main terms of the 4-D $N=1$ SYM theory are the volume modulus of the Calabi-Yau:

$\displaystyle S={{\mathcal{V}}_{{OM}}}\sum\limits_{{i=1}}^{N}{{{{{\left( {\frac{{x_{i}^{{11}}}}{L}} \right)}}^{2}}}}+{{\sigma }_{s}}$

the length modulus:

$\displaystyle T={{\mathcal{V}}_{{OM}}}+i{{\sigma }_{L}}$

and the $M5$ brane chiral superfields:

$\displaystyle {{Y}_{i}}={{\mathcal{V}}_{{OM}}}\left( {\frac{{x_{i}^{{11}}}}{L}} \right)+{{\sigma }_{i}}\,\,,{{\ }_{{i=1...N}}}$

where $OM$ stands for ‘open membrane’. Now we can start our derivation of the $P$ -term from the $M2/M5$ parallel brane system that supports the $N$ 5-branes. The $\mathfrak{g}_{{Lie}}^{3}$ $M2$ Lagrangian is:

$\displaystyle \begin{array}{l}\mathcal{L}=-\frac{1}{2}\left\langle {{{D}^{\mu }}{{X}^{I}},{{D}_{\mu }}{{X}^{I}}} \right\rangle +\frac{i}{2}\left\langle {\bar{\Psi },{{\Gamma }^{\mu }}{{D}_{\mu }}\Psi } \right\rangle \\+\frac{i}{4}\left\langle {\bar{\Psi },{{\Gamma }_{{IJ}}}\left[ {{{X}^{I}},{{X}^{J}}\Psi } \right]} \right\rangle -{{V}_{X}}+{{\mathcal{L}}_{{CS}}}\end{array}$

with ${{{D}_{\mu }}}$ the covariant derivative:

$\displaystyle {{\left( {{{D}_{\mu }}{{X}^{I}}\left( x \right)} \right)}_{\alpha }}={{\partial }_{\mu }}X_{a}^{I}-{{f}^{{cdb}}}_{a}{{A}_{{\mu cd}}}\left( x \right)X_{b}^{I}$

and ${{V}_{X}}$ the Kähler potential, and the Chern-Simons term ${{\mathcal{L}}_{{CS}}}$ for the gauge potential is given by:

$\displaystyle \begin{array}{l}{{\mathcal{L}}_{{CS}}}=\frac{1}{2}{{\epsilon }^{{\mu \nu \lambda }}}\left( {{{f}^{{abcd}}}} \right.{{A}_{{\mu ab}}}{{\partial }_{\nu }}{{A}_{{\lambda cd}}}+\\\frac{2}{3}{{f}^{{cda}}}_{g}{{f}^{{efgb}}}{{A}_{{\mu ab}}}{{A}_{{\nu cd}}}\left. {{{A}_{{\lambda ed}}}} \right)\end{array}$

where $I,J,K{{,}_{{\ ,\ i...8}}}$ define the brane transverse directions. The SUSY transformations are:

$\displaystyle \delta X_{a}^{I}=i\bar{\epsilon }\,{{\Gamma }^{I}}{{\Psi }_{a}}$

$\displaystyle \delta {{\Psi }_{a}}={{D}_{\mu }}X_{a}^{I}{{\Gamma }^{\mu }}{{\Gamma }^{I}}\epsilon -\frac{1}{6}X_{b}^{I}X_{c}^{J}X_{d}^{K}{{f}^{{bcd}}}_{a}$

$\displaystyle {{\delta }^{a}}{{{\bar{A}}}_{\mu }}^{b}=i\bar{\epsilon }\,{{\Gamma }_{\mu }}{{\Gamma }_{I}}X_{C}^{I}{{\Psi }_{d}}{{f}^{{cdb}}}_{a}$

with gauge conditions:

$\displaystyle \delta X_{a}^{I}={{\Lambda }_{{cd}}}{{f}^{{cdb}}}_{a}X_{b}^{I}$

$\displaystyle {{\delta }^{a}}{{\tilde{A}}^{b}}={{\partial }_{\mu }}{{\tilde{\Lambda }}^{b}}_{a}-\tilde{\Lambda }_{c}^{b}{{\tilde{A}}^{c}}_{a}{{+}^{c}}{{\tilde{A}}_{\mu }}^{b}{{\tilde{\Lambda }}^{c}}_{a}$

and we must impose the Jacobi identity. Hence, we get the $M5$ brane Lagrangian by Nambu-Poisson deformations defined in terms of:

$\displaystyle {{X}^{I}}\left( {x,y} \right)=\sum\limits_{a}{{X_{a}^{I}}}\left( x \right){{\chi }^{a}}\left( y \right)$

$\displaystyle \Psi \left( {x,y} \right)=\sum\limits_{a}{{{{\Psi }_{a}}}}\left( x \right){{\chi }^{a}}\left( y \right)$

$\displaystyle {{A}_{{\mu b}}}\left( {x,y} \right)=\sum\limits_{a}{{{{A}_{{\mu ab}}}}}\left( x \right){{\chi }^{a}}\left( y \right)$

which promote the system to a 6D $\left( {2,0} \right)$ SYM system with a Lagrangian:

$\displaystyle \mathcal{L}=\mathcal{L}_{{CS}}^{Q}+{{\mathcal{L}}_{{kin}}}+{{\mathcal{L}}^{Q}}$

where:

$\displaystyle \begin{array}{l}\mathcal{L}_{{CS}}^{Q}=\frac{1}{2}{{\epsilon }^{{\mu \nu \lambda }}}\left( {{{f}^{{abcd}}}} \right.{{A}_{{\mu ab}}}{{\partial }_{\nu }}{{A}_{{\lambda cd}}}+\\\frac{2}{3}{{f}^{{cda}}}_{g}{{f}^{{cfgb}}}{{A}_{{\mu ab}}}{{A}_{{\nu cd}}}\left. {{{A}_{{\lambda cf}}}} \right)\end{array}$

and ${{\mathcal{L}}_{{kin}}}$ is given in terms of the kinetic terms for the ${{X}^{I}}$ ‘s:

$\displaystyle \begin{array}{l}{{\left( {{{D}_{\mu }}{{X}^{I}}} \right)}^{2}}={{\left( {{{D}_{\mu }}{{X}^{{\dot{\nu }}}}} \right)}^{2}}+{{\left( {{{D}_{\mu }}{{X}^{i}}} \right)}^{2}}=\\\frac{1}{2}{{\left( {{{F}_{{\mu \dot{\nu }\dot{\lambda }}}}} \right)}^{2}}+{{\left( {{{\partial }_{\mu }}{{X}^{i}}} \right)}^{2}}+...\end{array}$

with:

$\displaystyle {{F}_{{\mu \dot{\nu }\dot{\lambda }}}}\equiv {{\partial }_{\mu }}{{A}_{{\dot{\nu }\dot{\lambda }}}}-{{\partial }_{{\dot{\nu }}}}{{A}_{{\mu \dot{\lambda }}}}+{{\partial }_{{\dot{\nu }}}}{{A}_{{\mu \dot{\nu }}}}$

and ${{\mathcal{L}}^{Q}}$ is given as such:

$\displaystyle \begin{array}{l}{{\mathcal{L}}^{Q}}=-\frac{1}{2}\left[ {{{{\left( {{{\partial }_{\mu }}{{X}^{i}}} \right)}}^{2}}+{{{\left( {{{\partial }_{{\dot{\mu }}}}{{X}^{i}}} \right)}}^{2}}} \right]+\\\frac{i}{2}\left\langle {\bar{\Psi },\left( {{{\Gamma }^{\mu }}{{\partial }_{\mu }}+{{\Gamma }^{{\dot{\mu }}}}{{\partial }_{{\dot{\mu }}}}} \right)\Psi } \right\rangle -\\\frac{1}{4}{{F}_{{\mu \dot{\nu }\dot{\lambda }}}}^{2}-\frac{1}{{12}}{{F}_{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}^{2}-\frac{1}{2}{{\epsilon }^{{\mu \nu \lambda }}}{{\epsilon }^{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}{{\partial }_{\mu }}{{A}_{{\nu \dot{\mu }}}}{{\partial }_{{\dot{\nu }}}}{{A}_{{\lambda \dot{\lambda }}}}\end{array}$

and where the relevant gauge field term is given by:

$\displaystyle {{\mathcal{L}}_{G}}=-\frac{1}{4}{{F}_{{\mu \dot{\mu }\dot{\nu }}}}\left( {{{{\bar{F}}}_{{\mu \dot{\mu }\dot{\nu }}}}} \right)-\frac{1}{2}{{F}_{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}^{2}$

with the Hodge dual field strength is given by:

$\displaystyle {{\tilde{F}}_{{\underset{\cdots}{\mu }\underset{\cdots}{\nu }\underset{\cdots}{\lambda }}}}=\frac{1}{6}{{F}_{{\underset{\cdots}{\mu }\underset{\cdots}{\nu }\underset{\cdots}{\lambda }\underset{\cdots}{\kappa }\underset{\cdots}{\sigma }\underset{\cdots}{\rho }}}}{{F}_{{\underset{\cdots}{\kappa }\underset{\cdots}{\sigma }\underset{\cdots}{\rho }}}}$

Thus, the equations of motion from $\mathcal{L}_{G}^{Q}$ are:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{\partial }_{{\underset{\cdots}{\mu }}}}{{F}_{{\underset{\cdots}{\mu }\dot{\mu }\dot{\nu }}}}=0} \\ {{{\partial }_{{\dot{\nu }}}}{{F}_{{\dot{\nu }\underset{\cdots}{\mu }\dot{\mu }}}}+{{\partial }_{\nu }}{{{\tilde{F}}}_{{\nu \mu \dot{\mu }}}}=0} \end{array}} \right.$

Combining with the Bianchi identity:

$\displaystyle {{\partial }_{\nu }}{{\tilde{F}}_{{\nu \mu \dot{\mu }}}}+{{\partial }_{{\dot{\nu }}}}{{\tilde{F}}_{{\dot{\nu }\mu \dot{\mu }}}}=0$

gives us:

$\displaystyle {{\partial }_{{\dot{\nu }}}}\left( {{{F}_{{\mu \dot{\mu }\dot{\nu }}}}-{{{\tilde{F}}}_{{\dot{\nu }\mu \dot{\mu }}}}} \right)=0$

The term for the $B$ -field whose existence follows from the $M2$ -Lagrangian, is:

$\displaystyle {{F}_{{\mu \dot{\mu }\dot{\mu }}}}{{\tilde{F}}_{{\mu \dot{\mu }\dot{\nu }}}}={{\epsilon }_{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}{{\partial }_{{\dot{\lambda }}}}{{B}_{\mu }}$

Thus, we get:

$\displaystyle {{\partial }_{{\dot{\lambda }}}}\left( {{{F}_{{\dot{\kappa }\dot{\sigma }\dot{\rho }}}}+{{\epsilon }_{{\dot{\kappa }\dot{\sigma }\dot{\rho }}}}{{\partial }_{\lambda }}{{B}_{\lambda }}} \right)=0$

giving us solutions of the form:

$\displaystyle {{F}_{{\dot{\kappa }\dot{\sigma }\dot{\rho }}}}+{{\epsilon }_{{\dot{\kappa }\dot{\sigma }\dot{\rho }}}}{{\partial }_{\lambda }}{{B}_{\lambda }}=f\left( x \right){{\epsilon }_{{\dot{\kappa }\dot{\sigma }\dot{\rho }}}}$

Integrating, we get the $\left\{ {dB,H} \right\}$ terms, which in our $M2/M5$ system, satisfy:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{H}_{{\mu \nu \lambda }}}=\frac{1}{6}{{\epsilon }_{{\mu \nu \lambda }}}{{\epsilon }^{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}{{H}_{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}} \\ {{{H}_{{\mu \nu \dot{\mu }}}}=\frac{1}{2}{{\epsilon }_{{\mu \nu \lambda }}}{{\epsilon }^{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}{{H}_{{\lambda \dot{\nu }\dot{\lambda }}}}} \end{array}} \right.$

as well as:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{H}_{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}=\frac{1}{6}{{\epsilon }_{{\mu \nu \lambda }}}{{\epsilon }^{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}{{H}_{{\mu \nu \lambda }}}} \\ {{{H}_{{\mu \dot{\mu }\dot{\nu }}}}=\frac{1}{2}{{\epsilon }_{{\mu \nu \lambda }}}{{\epsilon }^{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}{{H}_{{\nu \lambda \dot{\lambda }}}}} \end{array}} \right.$

Plugging in the Kähler potential, we can derive the $N=1$ chiral super-field $P$ -term action:

$\displaystyle \begin{array}{l}\mathcal{L}=-\frac{1}{4}F_{{\mu \nu }}^{2}-{{\left| {{{D}_{\mu }}{{\Phi }_{+}}} \right|}^{2}}-{{\left| {{{D}_{\mu }}{{\Phi }_{-}}} \right|}^{2}}\\-2{{g}^{2}}\left( {S_{\xi }^{\Phi }} \right)-\frac{{{{g}^{2}}}}{2}{{\left( {{{{\left| {{{\Phi }_{+}}} \right|}}^{2}}-{{{\left| {{{\Phi }_{-}}} \right|}}^{2}}-{{\xi }_{3}}} \right)}^{2}}\end{array}$

with:

$\displaystyle S_{\xi }^{\Phi }\equiv {{\left| S \right|}^{2}}\left( {{{{\left| {{{\Phi }_{+}}} \right|}}^{2}}+{{{\left| {{{\Phi }_{-}}} \right|}}^{2}}} \right)+{{\left| {{{\Phi }_{+}}{{\Phi }_{-}}-{{\xi }_{{2,+}}}} \right|}^{2}}$

with the covariant derivative:

$\displaystyle {{D}_{\mu }}{{\Phi }_{\pm }}=\left( {{{\partial }_{\mu }}\pm ig{{W}_{{\dot{\mu }}}}} \right){{\Phi }_{\pm }}$

$S$ is the neutral $U\left( 1 \right)$ gauge field charge, $\left( {{{\Phi }_{+}},{{\Phi }_{-}}} \right)$ is the $N=2$ hypermultiplet charged under $U\left( 1 \right)$ gauged by the $N=2$ vector multiplet $\left( {{{W}_{\mu }},S} \right)$ with superpotential and $D$ -term that drive inflation:

$\displaystyle W=\sqrt{2}gS\left( {{{\Phi }_{+}}{{\Phi }_{-}}-{{\xi }_{+}}/2} \right)$

$\displaystyle D={{\left| {{{\Phi }_{+}}} \right|}^{2}}-{{\left| {{{\Phi }_{-}}} \right|}^{2}}-{{\xi }_{3}}$

dynamically as a function of kinetic terms of type ${{D}_{\mu }}\Phi {{D}^{\mu }}\Phi$ . The proof proceeds by plugging the RG-flow equation with the Hubble and inflaton term factored quadratically, with the $D$ -term potential:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{V}^{D}}=\frac{1}{2}{{D}^{2}}} \\ {D=g\left( {\xi -{{\varphi }^{i}}{{\varphi }_{i}}} \right)} \\ {{{F}_{{\mu \nu }}}\equiv {{\partial }_{\mu }}{{W}_{\nu }}-{{\partial }_{\nu }}{{W}_{\mu }}} \\ {{{D}_{\mu }}{{\varphi }_{i}}\equiv \left( {{{\partial }_{\mu }}-ig{{W}_{\mu }}} \right){{\varphi }_{i}}} \end{array}} \right.$

Thus, the fermionic contributions to the inflaton field derive from the transformations:

$\displaystyle \delta {{\psi }_{{\mu L}}}=\left( {{{\partial }_{\mu }}+\frac{1}{4}{{\omega }_{\mu }}^{{ab}}\left( e \right){{\gamma }_{{ab}}}+\frac{1}{2}\text{i}A_{\mu }^{B}} \right){{\epsilon }_{L}}$

$\displaystyle \delta {{\chi }_{i}}=\frac{1}{2}\left( {\not{\partial }-\text{i}g\not{W}} \right){{\varphi }_{i}}{{\epsilon }_{R}}$

$\displaystyle \delta \lambda =\frac{1}{4}{{\gamma }^{{\mu \nu }}}{{F}_{{\mu \nu }}}\epsilon +\frac{1}{2}\text{i}{{\gamma }_{5}}D\epsilon$

and where the $U{{\left( 1 \right)}_{{CP}}}$ gravitino connection $A_{\mu }^{B}$ is:

$\displaystyle A_{\mu }^{B}=\frac{1}{{2M_{{{{p}_{l}}}}^{2}}}\text{i}\left[ {{{\varphi }_{i}}{{\partial }_{\mu }}{{\varphi }^{i}}-{{\varphi }^{i}}{{\partial }_{\mu }}{{\varphi }_{i}}} \right]+\frac{1}{{M_{{{{p}_{l}}}}^{2}}}{{W}_{\mu }}D$

which reduces to:

$\displaystyle \frac{1}{{2M_{{{{p}_{l}}}}^{2}}}\text{i}\left[ {{{\varphi }_{i}}{{D}_{\mu }}{{\varphi }^{i}}-{{\varphi }^{i}}{{D}_{\mu }}{{\varphi }_{i}}} \right]+\frac{9}{{M_{{{{p}_{l}}}}^{2}}}{{W}_{\mu }}\xi$