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
Now, by integrating the N=1 chiral super-field P-term:
\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}
we get the P-term:
\displaystyle {{V}^{P}}=\frac{1}{2}{{D}^{2}}
with:
\displaystyle {{D}^{2}}=g{{S}_{\Phi }}{{\left( {\xi _{3}^{\dagger }+\left| {{{\Phi }_{+}}{{\Phi }_{-}}-{{\xi }_{{2,+}}}} \right|} \right)}^{2}}

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