Type-IIB String-Theory and Hybrid Inflation: Toward M-Theory Uplift

In part III, we completed the derivation of the Standard ΛCDM Model of cosmology from Type-IIB SUGRA with a Horava-Witten embedding in M-theory. However, we must also account for the de Sitter valley Coulomb gauge-phase as well as the Higgs waterfall gauge-phase leading to the ground state Einstein-Sasaki-Minkowski vacuum. As of yet, only M-theory can incorporate both phases. Here, we shall derive such phases based on a Type-IIB $D3/D7$ system. Let us recall the main mathematical results of part III. Starting with our action:

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

we 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’. Then we derived the $P$ -term from the $M2/M5$ parallel brane system that supports the $N$ 5-branes as such. 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 imposed 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:

we get the $P$ -term:

$\displaystyle {{V}^{P}}=\frac{1}{2}{{D}^{2}}$

with:

$\displaystyle {{D}^{2}}=g{{S}_{\Phi }}{{\left( {\xi _{3}^{\dagger }+\left| {{{\Phi }_{+}}{{\Phi }_{-}}-{{\xi }_{{2,+}}}} \right|} \right)}^{2}}$

Let us start our embedding of hybrid inflation. First, note that a $D3/D7$ brane system in the presence of Fayet-Illiopoulos parameter becomes unstable unless it is a completely coincident system. Take the $D7$ brane world volume action:

$\displaystyle S=-{{T}_{7}}\int{{{{\Phi }_{{\tilde{F},g,\phi }}}}}+{{T}_{7}}\int{{\sum{{{{\Psi }_{{CS,\tilde{F}}}}}}}}$

with:

$\displaystyle {{\Phi }_{{\tilde{F},g,\phi }}}\equiv {{d}^{8}}\sigma {{e}^{{-\phi }}}\sqrt{{-\det \left( {g+\tilde{F}} \right)}}$

$\displaystyle {{\Psi }_{{CS,\tilde{F}}}}\equiv {{C}_{{p+1}}}\wedge {{e}^{{\tilde{F}}}}$

$\displaystyle \tilde{F}=F-B$

where $B$ is the pull-back of the NS-NS 2-form and $F=dA$ is the Born-Infeld field strength. Now put the $D7$ brane in a $D3$ brane background. We get, for constant dilaton and metric:

$\displaystyle d{{s}^{2}}={{H}^{{-1/2}}}d{{s}^{2}}\left( {{{\text{E}}^{{3,1}}}} \right)+{{H}^{{1/2}}}d{{s}^{2}}\left( {{{\text{E}}^{6}}} \right)$

and for the self-dual RR form:

$\displaystyle {{{\tilde{F}}}^{{RR}}}={{\partial }_{i}}{{H}^{{-1}}}d{{x}^{i}}\wedge {{\epsilon }_{{{{\text{E}}^{{3,1}}}}}}+*\left( {{{\partial }_{i}}{{H}^{{-1}}}d{{x}^{i}}\wedge {{\epsilon }_{{{{\text{E}}^{{3,1}}}}}}} \right)$

where $H$ is the central Hodge harmonic function on ${{\text{E}}^{6}}$ with ${{\epsilon }_{{{{\text{E}}^{{3,1}}}}}}$ the volume form on ${{\text{E}}^{{3,1}}}$ , while factoring in the $D7$ brane worldvolume gauge fields. Thus, our effective potential is given by:

$\displaystyle V={{T}_{7}}{{\mathcal{V}}_{3}}\int{{d{{\sigma }^{i}}}}\left[ {\Theta -\Xi } \right]$

$\displaystyle \Theta \equiv \sqrt{{\left( {1+{{H}^{{-1}}}\tan {{\theta }_{1}}^{2}} \right)\left( {1+{{H}^{{-1}}}\tan {{\theta }_{2}}^{2}} \right)}}$

$\displaystyle \Xi \equiv \left( {{{H}^{{-1}}}-1} \right)\tan {{\theta }_{1}}^{2}\tan {{\theta }_{2}}^{2}$

If the angles $\displaystyle {{\theta }_{i}}$ are equal, the force between the $D3$ brane and the $D7$ brane vanishes, giving us a 4-D Euclidean self-dual system in the 6 and 9 directions. In polar coordinates, we hence have:

$\displaystyle V\simeq \mu _{{{{T}_{7}}}}^{Q}{{\mathcal{V}}_{3}}\int_{{{{d}^{2}}}}^{\Lambda }{{\frac{{d\lambda }}{\lambda }}}=\mu _{{{{T}_{7}}}}^{Q}{{\mathcal{V}}_{3}}\text{In}\frac{{{{d}^{2}}}}{\Lambda }$

$\displaystyle \mu _{{{{T}_{7}}}}^{Q}{{\mathcal{V}}_{3}}\equiv \frac{{{{\pi }^{2}}}}{2}Q{{T}_{7}}{{\mathcal{V}}_{3}}{{\left( {\sin {{\theta }_{1}}-\sin {{\theta }_{2}}} \right)}^{2}}$

where $\Lambda$ is the renormalization group cutoff, and $\kappa$ -symmetry allows us to deduce manifest supersymmetry breaking associated to the Yang-Mills field strength of the $D3/D7$ brane system. Our bosonic action is given by:

$\displaystyle S=-{{T}_{7}}\int{{{{\Phi }_{{\tilde{F},g,\phi }}}}}+{{T}_{7}}\int{{\sum{{{{\Psi }_{{CS,\tilde{F}}}}}}}}$

with:

$\displaystyle {{\Phi }_{{\tilde{F},g,\phi }}}\equiv {{d}^{8}}\sigma {{e}^{{-\phi }}}\sqrt{{-\det \left( {g+\tilde{F}} \right)}}$

$\displaystyle {{\Psi }_{{CS,\tilde{F}}}}\equiv {{C}_{{p+1}}}\wedge {{e}^{{\tilde{F}}}}$

$\displaystyle \tilde{F}=F-B$

and some solutions must have some unbroken SUSY since there exists solutions to the kappa-symmetry equation:

$\displaystyle \left( {1-{{\Gamma }_{\kappa }}} \right)\epsilon =0$

where ${{{\Gamma }_{\kappa }}}$ is the $\kappa$ -symmetry projection operator for a $D7$ brane in a $D3$ worldvolume background:

$\displaystyle \begin{array}{l}{{\Gamma }_{\kappa }}={{e}^{{-\frac{a}{2}}}}i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123456789}}}{{e}^{{\frac{a}{2}}}}\\={{e}^{{-a}}}i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123456789}}}\end{array}$

and ${{\epsilon }^{{\alpha I}}}$ is a Type IIB spinor with a chiral bi-Majorana spinor representation, and ${{\left( {{{\sigma }_{2}}} \right)}_{I}}^{J}$ is a Pauli matrix and in the absence of non-zero contorsion factor for $a$ , the Killing equation reproduces the $D7$ -brane projector:

$\displaystyle \left( {1-i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123456789}}}} \right)\epsilon =0$

corresponding to half of the unbroken supersymmetry. We hence have a skew-diagonal configuration with $\displaystyle F=dA$ on the worldvolume, and the matrix $\displaystyle \tilde{F}$ is antisymmetric and independent on the worldvolume coordinates:

$\displaystyle \begin{array}{l}a={{\sigma }_{3}}\otimes \left( {{{\theta }_{1}}{{\gamma }_{{67}}}+{{\theta }_{2}}{{\gamma }_{{89}}}} \right)\\={{\sigma }_{3}}\otimes {{H}^{{1/2}}}\left( {{{\theta }_{1}}{{\gamma }_{{67}}}+{{\theta }_{2}}{{\gamma }_{{89}}}} \right)\end{array}$

with:

$\displaystyle {{\gamma }_{i}}={{E}_{i}}^{a}{{\Gamma }_{a}}$

and the vielbeins are given by the $D3$ -brane metric and the $D3/D7$ brane-system Killing spinors condition is:

$\displaystyle \begin{array}{l}\exp \left\{ {-{{\sigma }_{3}}\otimes {{H}^{{1/2}}}\left( {{{\theta }_{1}}{{\Gamma }_{{\kappa ,}}}_{{67}}+{{\theta }_{2}}{{\Gamma }_{{\kappa ,}}}_{{89}}} \right)} \right\}i{{\sigma }_{2}}\\\otimes {{\Gamma }_{{\kappa ,0123456789}}}\epsilon =\epsilon \end{array}$

The Killing spinor satisfies, in the presence of a $D3$ background, the following two conditions:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\epsilon ={{H}^{{-1/4}}}{{\epsilon }_{0}}} \\ {i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123}}}{{\epsilon }_{0}}=0} \end{array}} \right.$

that break half the supersymmetry, which reduce:

to:

$\displaystyle \begin{array}{l}\exp \left\{ {-\frac{1}{2}} \right.{{\sigma }_{3}}\otimes {{H}^{{1/2}}}{{\Gamma }_{{\kappa ,67}}}\\\left. {\left[ {\left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)\left( {1+{{\Gamma }_{{\kappa ,6789}}}} \right)+\left. {\left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)\left( {1-{{\Gamma }_{{\kappa ,6789}}}} \right)} \right]} \right.} \right\}\\\otimes {{\Gamma }_{{\kappa ,6789}}}{{\epsilon }_{0}}=0\end{array}$

The $D3$ brane worldvolume Hodge-Dirac harmonic function at the $D7$ loci ${{d}^{2}}={{\left( {{{x}^{4}}} \right)}^{2}}+{{\left( {{{x}^{5}}} \right)}^{2}}=0$ is:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {H=1+Q/{{\rho }^{4}}} \\ {{{\rho }^{2}}={{{\left( {{{\sigma }^{6}}} \right)}}^{2}}+{{{\left( {{{\sigma }^{7}}} \right)}}^{2}}+{{{\left( {{{\sigma }^{8}}} \right)}}^{2}}+{{{\left( {{{\sigma }^{9}}} \right)}}^{2}}} \end{array}} \right.$

Hence, the Killing equation has solution of type:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{\epsilon }_{0}}={{\Gamma }_{{\kappa ,6789}}}{{\epsilon }_{0}}} \\ {{{{\tilde{F}}}^{{{{\nabla }^{{\omega ,-}}}}}}=0} \\ {\epsilon =i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123456789}}}\epsilon } \\ {\epsilon ={{\Gamma }_{{\kappa ,6789}}}\epsilon } \end{array}} \right.$

noting that in the Coulomb phase, unlike the Higgs phase, $H\left( \sigma \right)$ is a function of the $D$ -brane worldvolume coordinates and thus determined by the RR-RR and NS-NS forms. Any such configuration of $D3/D7$ branes must be unstable. To see why, consider a $D7$ brane probed by a $D3$ brane with $B$ field satisfying ${{B}^{-}}\ne 0$ . A SUSY solution gotten via mirror symmetry at the Hitchin holomorphic angles has the form:

$\displaystyle \begin{array}{l}ds_{{D7}}^{2}=Z_{7}^{{-1/2}}d{{s}^{2}}\left( {{{\text{E}}^{{3,1}}}} \right)+Z_{7}^{{1/2}}d{{s}^{2}}\left( {\text{E}_{{45}}^{2}} \right)+\\Z_{7}^{{-1/2}}{{H}_{1}}d{{s}^{2}}\left( {\text{E}_{{67}}^{2}} \right)+Z_{7}^{{-1/2}}{{H}_{2}}d{{s}^{2}}\left( {\text{E}_{{89}}^{2}} \right)\end{array}$

$\displaystyle {{e}^{{2\phi }}}=g_{s}^{2}Z_{7}^{{-2}}{{H}_{1}}{{H}_{2}}$

$\displaystyle {{B}_{{67}}}=-\tan {{\theta }_{1}}Z_{7}^{{-1}}{{H}_{1}}$

$\displaystyle {{B}_{{89}}}=-\tan {{\theta }_{2}}Z_{7}^{{-1}}{{H}_{2}}$

$\displaystyle {{C}_{4}}=\left( {Z_{7}^{{-1}}-1} \right)\sin {{\theta }_{1}}\sin {{\theta }_{2}}{{\epsilon }_{{{{\text{E}}^{{3,1}}}}}}$

$\displaystyle \begin{array}{*{20}{l}} {{{C}_{6}}=\left( {Z_{7}^{{-1}}-1} \right)\left[ {{{H}_{1}}\cos {{\theta }_{1}}\sin {{\theta }_{2}}d{{x}_{6}}\wedge d{{x}_{7}}} \right.} \\ {\left. {{{H}_{2}}\cos {{\theta }_{2}}\sin {{\theta }_{1}}d{{x}_{8}}\wedge d{{x}_{9}}} \right]\wedge {{\epsilon }_{{{{\text{E}}^{{3,1}}}}}}} \end{array}$

$\displaystyle {{C}_{8}}=\left( {Z_{7}^{{-1}}-1} \right){{H}_{1}}{{H}_{2}}\cos {{\theta }_{1}}\cos {{\theta }_{2}}{{\epsilon }_{{{{\text{E}}^{{3,1}}}}}}\wedge d{{x}_{6}}\wedge d{{x}_{7}}\wedge d{{x}_{8}}\wedge d{{x}_{9}}$

and we have:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{H}_{1}}\doteq {{{\left( {{{{\cos }}^{2}}{{\theta }_{1}}+{{{\sin }}^{2}}{{\theta }_{1}}Z_{7}^{{-1}}} \right)}}^{{-1}}}} \\ {{{H}_{2}}\doteq {{{\left( {{{{\cos }}^{2}}{{\theta }_{2}}+{{{\sin }}^{2}}{{\theta }_{2}}Z_{7}^{{-1}}} \right)}}^{{-1}}}} \\ {{{Z}_{7}}\doteq \underset{{\epsilon \to {{0}^{+}}}}{\mathop {\lim }}\,\left( {1+{{c}_{7}}\frac{{{{\Gamma }_{\kappa }}\left( {\epsilon /2} \right)}}{{{{r}^{\epsilon }}}}} \right)=1-2{{c}_{7}}\text{In}\left( {r/{{e}^{{1/\epsilon }}}} \right)} \end{array}} \right.$

We then find that the $D3$ brane action is given by:

$\displaystyle {{S}_{{D3}}}=-{{T}_{{D3}}}\int{{{{d}^{4}}}}\sigma {{e}^{{-\left( {\phi -{{\phi }_{0}}} \right)}}}\sqrt{{-\det \left( {g+\tilde{F}} \right)}}+{{T}_{{D3}}}\int{{{{C}_{4}}}}$

where we have implicitly defined ${{\phi }_{0}}$ by ${{e}^{{^{{{{\phi }_{0}}}}}}}={{g}_{s}}$ , and our potential is given by:

$\displaystyle V={{T}_{{D3}}}{{V}_{{D3}}}\left[ {1+{{c}_{7}}{{{\left( {\sin {{\theta }_{1}}-\sin {{\theta }_{2}}} \right)}}^{2}}\text{In}\left( {r/\Lambda } \right)} \right]$

in light of the gauge-invariance of:

$\displaystyle {{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}=\tilde{F}-B$

Now consider the $D7$ $\kappa$ -symmetric Dirac-Born-Infeld/WZ action above, and a $D3/D7$ supersymmetric bound state for a given ${\alpha }'$ and embed the $D7$ brane in the full Minkowski 10D space. The SUSY equation is then:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{\Gamma }_{\kappa }}\epsilon =\epsilon } \\ {{{\Gamma }_{\kappa }}={{e}^{{-a}}}i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123456789}}}} \\ {a=\frac{1}{2}{{Y}_{{ik}}}{{\Gamma }_{\kappa }}^{{ik}}\otimes {{\sigma }^{3}}} \end{array}} \right.$

In the Coulomb hybrid phase, we pick an everywhere skew diagonal basis for ${{\tilde{F}}^{{{{{\not{\nabla }}}^{\omega }}}}}$ , and in the Higgs phase, it can be allowed to be a function of the worldvolume coordinates. Hence $Y\left( \sigma \right)$ is a highly non-linear term given by:

$\displaystyle \begin{array}{l}{{e}^{{-\frac{1}{2}{{Y}_{{ik}}}{{\Gamma }_{\kappa }}^{{ik}}\otimes {{\sigma }^{3}}}}}=\frac{1}{{\sqrt{{\left| {\eta +{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}} \right|}}}}\left[ {1-\frac{1}{2}{{\sigma }_{3}}{{\Gamma }_{\kappa }}^{{ik}}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}{{\,}_{{ik}}}} \right.\\\left. {+\frac{1}{8}{{\Gamma }_{\kappa }}^{{ikml}}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}{{\,}_{{ik}}}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}{{\,}_{{ml}}}} \right]\end{array}$

Thus, the Killing spinor equation reduces to:

$\displaystyle \begin{array}{l}\exp \left\{ {-\frac{1}{4}{{\sigma }_{3}}{{\Gamma }_{\kappa }}^{{ik}}\left[ {Y_{{ik}}^{+}\left( {1-{{\Gamma }_{{\kappa ,6789}}}} \right)+Y_{{ik}}^{-}\left( {1+{{\Gamma }_{{\kappa ,6789}}}} \right)} \right]} \right\}\cdot \\i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123456789}}}\epsilon =\epsilon \end{array}$

with constant spinors. There are two ways to preserve SUSY. With an ${{\mathbb{R}}^{4}}$ chiral/anti-chiral spinor satisfying the conditions:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{Y}^{\pm }}\left( \sigma \right)-{{{\left( {{{Y}^{0}}} \right)}}^{\pm }}=0} \\ {\left( {1\pm {{\Gamma }_{{\kappa ,6789}}}} \right)\epsilon =0} \\ {\frac{{\partial {{Y}^{\pm }}}}{{\partial \sigma }}=0} \\ {{{{\hat{Y}}}^{\pm }}\left( \sigma \right)=0} \end{array}} \right.$

and with a spinor satisfying the equation:

$\displaystyle \exp \left\{ {-\frac{1}{2}{{\sigma }_{3}}\otimes {{\Gamma }_{\kappa }}^{{ik}}\left( {{{Y}^{0}}} \right)_{{ik}}^{\pm }} \right\}i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123456789}}}\epsilon =\epsilon$

Now since the supersymmetric configurations in the Higgs branch are given by:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123}}}\epsilon =\mp \epsilon } \\ {i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123456789}}}\epsilon =\epsilon } \\ {{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}^{\pm }=0} \end{array}} \right.$

the solution necessarily has chiral spinors in Minkowski 4D spacetime and our system is equivalent to a $\kappa$ -symmetric Euclideanized $D3$ brane dissolved into a $D7$ brane, as implied by the following relation:

$\displaystyle \frac{{{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}_{{ik}}^{-}}}{{1+\text{Pf}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}}}=-\frac{{B_{{ik}}^{-}}}{{1+\text{Pf}B}}$

Conjugation gives us the results in the Coulomb branch of hybrid inflation. We are now in a position to analyze a non-linear Seiberg-Witten solution to the above BPS equation. We put it in canonical Moriyama form:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\tilde{F}_{{-ab}}^{-}=\theta _{{-ab}}^{-}\text{Pf}{{{\tilde{F}}}_{-}}} \\ {{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}-}}}{{\,}_{{ij}}}={{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}{{\,}_{{ij}}}-\frac{1}{2}{{\epsilon }_{{ijkl}}}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}^{{kl}}} \\ {\text{Pf}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}=\frac{1}{8}{{\epsilon }^{{ijkl}}}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}{{\,}_{{ij}}}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}{{\,}_{{kl}}}} \\ {\tilde{F}_{{-ab}}^{-}={{{\tilde{F}}}_{{-ab}}}-\frac{1}{2}{{\epsilon }_{{abcd}}}\tilde{F}_{-}^{{cd}}} \\ {\text{Pf}{{{\tilde{F}}}_{-}}=\frac{1}{8}{{\epsilon }^{{abcd}}}{{{\tilde{F}}}_{{-ab}}}{{{\tilde{F}}}_{{-cd}}}} \end{array}} \right.$

and our frame metric is defined implicitly via the open string metric:

$\displaystyle {{G}_{{ij}}}={{\delta }_{{ij}}}-{{B}_{{ik}}}{{\delta }^{{km}}}{{B}_{{mj}}}$

and the vierbein and non-Abelian theta parameter are given by:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{e}^{a}}{{\,}_{i}}={{\delta }^{a}}{{\,}_{i}}-{{B}^{a}}{{\,}_{i}}={{{\left( {1-B} \right)}}^{a}}{{\,}_{i}}} \\ {{{G}_{{ij}}}={{\delta }_{{ab}}}{{e}^{a}}{{\,}_{i}}{{e}^{b}}{{\,}_{j}}={{{\left( {{{e}^{T}}e} \right)}}_{{ij}}}} \\ {{{\theta }^{{ij}}}=-{{{\left( {\frac{1}{{1-B}}} \right)}}^{{ik}}}{{B}_{{km}}}{{{\left( {\frac{1}{{1+B}}} \right)}}^{{mj}}}} \end{array}} \right.$

respectively and the frame-Pfaffian equations are given by:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\text{Pf}{{{\tilde{F}}}_{-}}={{{\left( {\det e} \right)}}^{{-1}}}\text{Pf}\tilde{F}} \\ {{{{\tilde{F}}}_{{-ab}}}={{{\left( {{{{\left( {{{e}^{T}}} \right)}}^{{-1}}}} \right)}}_{a}}^{i}{{{\tilde{F}}}_{{ij}}}{{{\left( {{{e}^{{-1}}}} \right)}}^{i}}_{b}} \\ {\theta _{-}^{{ab}}={{e}^{a}}_{i}{{\theta }^{{ij}}}{{{\left( {{{e}^{T}}} \right)}}_{j}}^{b}=-{{\delta }^{{ak}}}{{\delta }^{{mb}}}{{B}_{{km}}}=-{{B}^{{ab}}}} \end{array}} \right.$

We can now derive the identity:

$\displaystyle \begin{array}{l}\tilde{F}_{-}^{{--}}={{\left( {\det e} \right)}^{{-1}}}\left\{ {\left( {1+\text{Pf}B} \right)} \right.{{{\tilde{F}}}^{-}}-\frac{1}{4}{{\epsilon }^{{klmn}}}{{{\tilde{F}}}_{{kl}}}{{B}_{{mn}}}{{B}^{-}}\\-\frac{1}{2}\left. {\left( {{{B}^{-}}{{{\tilde{F}}}^{-}}-{{{\tilde{F}}}^{-}}{{B}^{-}}} \right)} \right\}\end{array}$

hence our BPS equation reduces to:

$\displaystyle \tilde{F}_{{-ab}}^{-}=-{{\left( {\det e} \right)}^{{-1}}}B_{{ij}}^{-}\delta _{a}^{i}\delta _{b}^{j}\text{Pf}\tilde{F}=\theta _{{-ab}}^{-}\text{Pf}{{\tilde{F}}_{-}}$

To solve, note that ${{\tilde{F}}_{-}}$ can be defined in terms of the frame coordinates and the gauge potential as such:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{x}^{a}}={{e}^{a}}_{i}{{x}^{i}}={{{\left( {1-B} \right)}}^{a}}_{i}{{x}^{i}}} \\ {{{A}_{{-,a}}}={{{\left( {{{{\left( {{{e}^{T}}} \right)}}^{{-1}}}} \right)}}_{a}}^{i}{{A}_{i}}={{{\left( {\frac{1}{{1+B}}} \right)}}_{a}}^{i}{{A}_{i}}} \end{array}} \right.$

hence, a solution to:

$\displaystyle \frac{{{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}_{{ik}}^{-}}}{{1+\text{Pf}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}}}=-\frac{{B_{{ik}}^{-}}}{{1+\text{Pf}B}}$

takes the form:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{A}_{{-,a}}}=\theta _{{-,a}}^{-}{{x}^{b}}h\left( R \right)} \\ {{{R}^{2}}={{\delta }_{{ab}}}{{x}^{a}}{{x}^{b}}} \end{array}} \right.$

with:

$\displaystyle h\left( R \right)=-\frac{1}{{2\text{Pf}{{\theta }_{-}}^{{{-}'}}}}\left( {1-\sqrt{{1+\frac{{4C\text{Pf}{{\theta }_{-}}^{{{-}'}}}}{{{{R}^{4}}}}}}} \right)$

In the presence of the RR field, the $U\left( 1 \right)$ -instanton gets a blow-up, and ceases to be singular and we get a UV non-linear Seiberg-Witten gauge equation:

$\displaystyle \frac{1}{{16{{\pi }^{2}}}}\int_{{{{\mathbb{R}}^{4}}}}{{\tilde{F}\wedge \tilde{F}=}}\frac{1}{{8{{\pi }^{2}}}}\int{{{{d}^{4}}}}x\text{Pf}{{\tilde{F}}_{-}}=-\frac{1}{8}\text{Pf}{{\theta }_{-}}^{-}\left( {{{R}^{4}}{{h}^{2}}} \right)=N$

Thus, the non-vanishing ${{\theta }_{-}}^{-}$ is our cosmological potential seed that also defines a positive vacuum energy. Thus we get a hybrid slow-roll inflation stage where our pocket-universe goes through a waterfall condensation stage, and eventually settles into the Minkowski vacuum described by a bound state of $D3/D7$ branes corresponding to the Higgs phase of the gauge theory with the FI term defined by ${{\theta }_{-}}^{-}$ . Since $D3$ living on $D7$ branes can be interpreted as instantons due to Chern-Simons gauge coupling:

$\displaystyle {{S}_{{CS}}}=\frac{1}{{16{{\pi }^{2}}}}\int_{{D7}}{{{{C}_{4}}}}\wedge {{\tilde{F}}^{{{{{\not{\nabla }}}^{\omega }}}}}\wedge {{\tilde{F}}^{{{{{\not{\nabla }}}^{\omega }}}}}=N\int_{{D3}}{{{{C}_{4}}}}$

the Higgs phase above is hence equivalent to a noncommutative Nekrasov-ADHM non-linear instanton in M-theory, and we have an intrinsic connection between the cosmological constant in 4D and the noncommutative ${{\theta }_{-}}^{-}$ parameter in internal space 6789. Next, we do a Type-IIB compactification and an uplift to M-theory.