Type-IIB SUGRA, M-Theory, Einstein-Sasaki-Minkowski Vacua, and Inflation

Recall we derived the Standard ‘ΛCDM’ Model of cosmology from Type-IIB SUGRA by identifying the inflaton with the Gukov-Vafa-Witten topologically hyper-twisted Kähler modulus with a Hořava-Witten uplift-embedding in M-theory. In this post, in the framework of Type-IIB $D3/D7$ system, I shall derive the two necessary conditions presupposed by such uplift, namely the de Sitter valley Coulomb gauge-phase and the Higgs waterfall gauge-phase that lead to the ground state Einstein-Sasaki-Minkowski vacuum. It is a remarkable feature of M-theory that it is the only quantum theory of gravity that can incorporate both phases. Let us recall some mathematical results I derived starting with the action:

$\displaystyle {S=\int{{{{d}^{4}}}}x\sqrt{{-{{g}_{E}}}}\left[ {-\frac{1}{4}} \right.{{{\left( {{{F}_{W}}} \right)}}^{2}}-\frac{1}{4}{{{\left( {{{F}_{{{W}'}}}} \right)}}^{2}}-}$

$\displaystyle {{{{\left| {\partial S} \right|}}^{2}}-{{{\left| {\partial {S}'} \right|}}^{2}}-{{R}^{{-12}}}\frac{1}{{8g_{7}^{2}}}\int\limits_{{K3}}{{{{{\tilde{F}}}^{D}}^{-}\wedge *{{{\tilde{F}}}^{D}}^{-}}}-}$

$\displaystyle {{{{\left| {{{D}_{\mu }}\chi } \right|}}^{2}}-2{{g}^{2}}{{{\left| S \right|}}^{2}}{{{\left| \chi \right|}}^{2}}\left. {-\frac{{\left( {g_{3}^{2}+\tilde{g}_{3}^{2}} \right)}}{2}{{{\left( {{{\chi }^{\dagger }}{{\sigma }^{A}}\chi } \right)}}^{2}}} \right]}$

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

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

and the length modulus:

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

with the Type-IIB Calabi-Yau superpotential:

$\displaystyle {{V}_{W}}=\int\limits_{X}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}+\sum\limits_{{i=1}}^{{{{h}^{{1,1}}}}}{{{{A}_{i}}}}\left( {\left( {{{e}^{{-\phi }}}+i{{C}_{0}}} \right),U} \right){{e}^{{-a\left( {{{e}^{{-\phi }}}{{\tau }_{i}}+i{{\rho }_{i}}} \right)}}}$

where the Ramond-Ramond gauge-coupling sector is given by the action:

$\displaystyle \mathcal{L}_{G}^{{Loc}}=\sum\limits_{{b=1}}^{{N-1}}{{\frac{1}{{2g_{b}^{2}}}}}{{\int{\text{d}}}^{2}}\theta {{W}^{\alpha }}{{W}_{\alpha }}{{\delta }^{2}}\left( {\left( {1-{{e}^{{ib\phi }}}} \right)z} \right)$

with the corresponding Chern-Simons action:

$\displaystyle {{S}_{{CS}}}=\frac{{{{T}_{p}}}}{2}\int\limits_{{{{\Sigma }_{{p+1}}}}}{{C\wedge \text{ch}}}\left( {\tilde{F}} \right)\wedge \sqrt{{\frac{{\hat{A}\left( {{{R}_{T}}} \right)}}{{\hat{A}\left( {{{R}_{N}}} \right)}}}}$

Via the F/M-theory duality, the M5 brane chiral superfields hence take the following form:

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

We then 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 takes the following form:

$\displaystyle {\mathcal{L}=-\frac{1}{2}\left\langle {{{D}^{\mu }}{{X}^{I}},{{D}_{\mu }}{{X}^{I}}} \right\rangle +\frac{i}{2}\left\langle {\bar{\Psi },{{\Gamma }^{\mu }}{{D}_{\mu }}\Psi } \right\rangle }$

$\displaystyle {+\frac{i}{4}\left\langle {\bar{\Psi },{{\Gamma }_{{IJ}}}\left[ {{{X}^{I}},{{X}^{J}}\Psi } \right]} \right\rangle -{{V}_{X}}+{{\mathcal{L}}_{{CS}}}}$

where $\displaystyle {{D}_{\mu }}$ is the covariant derivative:

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

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

$\displaystyle {{{\mathcal{L}}_{{CS}}}=\frac{1}{2}{{\epsilon }^{{\mu \nu \lambda }}}\left( {{{f}^{{abcd}}}} \right.{{A}_{{\mu ab}}}{{\partial }_{\nu }}{{A}_{{\lambda cd}}}+}$

$\displaystyle {\frac{2}{3}{{f}^{{cda}}}_{g}{{f}^{{efgb}}}{{A}_{{\mu ab}}}{{A}_{{\nu cd}}}\left. {{{A}_{{\lambda ed}}}} \right)}$

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

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

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

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

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 with the Jacobi identity satisfied. 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:

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

with:

$\displaystyle {\mathcal{L}_{{CS}}^{Q}=\frac{1}{2}{{\epsilon }^{{\mu \nu \lambda }}}\left( {{{f}^{{abcd}}}} \right.{{A}_{{\mu ab}}}{{\partial }_{\nu }}{{A}_{{\lambda cd}}}+}$

$\displaystyle {\frac{2}{3}{{f}^{{cda}}}_{g}{{f}^{{cfgb}}}{{A}_{{\mu ab}}}{{A}_{{\nu cd}}}\left. {{{A}_{{\lambda cf}}}} \right)}$

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

$\displaystyle {{{{\left( {{{D}_{\mu }}{{X}^{I}}} \right)}}^{2}}={{{\left( {{{D}_{\mu }}{{X}^{{\dot{\nu }}}}} \right)}}^{2}}+{{{\left( {{{D}_{\mu }}{{X}^{i}}} \right)}}^{2}}=}$

$\displaystyle {+{{{\left( {{{\partial }_{\mu }}{{X}^{i}}} \right)}}^{2}}+...}$

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:

$\displaystyle {{{\mathcal{L}}^{Q}}=-\frac{1}{2}\left[ {{{{\left( {{{\partial }_{\mu }}{{X}^{i}}} \right)}}^{2}}+{{{\left( {{{\partial }_{{\dot{\mu }}}}{{X}^{i}}} \right)}}^{2}}} \right]+}$

$\displaystyle {\frac{i}{2}\left\langle {\bar{\Psi },\left( {{{\Gamma }^{\mu }}{{\partial }_{\mu }}+{{\Gamma }^{{\dot{\mu }}}}{{\partial }_{{\dot{\mu }}}}} \right)\Psi } \right\rangle -}$

$\displaystyle {\frac{1}{4}{{F}_{{\mu \dot{\nu }\dot{\lambda }}}}^{2}-\frac{1}{{12}}{{F}_{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}^{2}-\frac{1}{2}{{\epsilon }^{{\mu \nu \lambda }}}{{\epsilon }^{{\dot{\mu }\dot{\nu }\dot{\lambda }}}}{{\partial }_{\mu }}{{A}_{{\nu \dot{\mu }}}}{{\partial }_{{\dot{\nu }}}}{{A}_{{\lambda \dot{\lambda }}}}}$

and where the relevant gauge field term is given as such:

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

with the Hodge dual field strength:

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

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

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

Combining with the Bianchi identity:

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

yields:

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

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

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

giving us:

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

yielding 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:

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

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

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

$\displaystyle \begin{array}{*{20}{l}} {\mathcal{L}=-\frac{1}{4}F_{{\mu \nu }}^{2}-{{{\left| {{{D}_{\mu }}{{\Phi }_{+}}} \right|}}^{2}}-{{{\left| {{{D}_{\mu }}{{\Phi }_{-}}} \right|}}^{2}}} \\ {-2{{g}^{2}}\left( {S_{\xi }^{\Phi }} \right)-\frac{{{{g}^{2}}}}{2}{{{\left( {{{{\left| {{{\Phi }_{+}}} \right|}}^{2}}-{{{\left| {{{\Phi }_{-}}} \right|}}^{2}}-{{\xi }_{3}}} \right)}}^{2}}} \end{array}$

where we have:

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

and 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, $\displaystyle \left( {{{\Phi }_{+}},{{\Phi }_{-}}} \right)$ 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:

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

and reduces to:

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

By integrating the N=1 chiral super-field P-term:

we get:

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

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

We can begin 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:

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

where:

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

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

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

and 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 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:

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

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

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 thus given by:

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

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

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

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

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:

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

$\displaystyle {{{e}^{{-a}}}i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123456789}}}}$

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 $F=dA$ on the worldvolume, and the matrix $\tilde{F}$ is antisymmetric and independent of the worldvolume coordinates:

$\displaystyle {a={{\sigma }_{3}}\otimes \left( {{{\theta }_{1}}{{\gamma }_{{67}}}+{{\theta }_{2}}{{\gamma }_{{89}}}} \right)=}$

$\displaystyle {{{\sigma }_{3}}\otimes {{H}^{{1/2}}}\left( {{{\theta }_{1}}{{\gamma }_{{67}}}+{{\theta }_{2}}{{\gamma }_{{89}}}} \right)}$

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:

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

$\displaystyle {\otimes {{\Gamma }_{{\kappa ,0123456789}}}\epsilon =\epsilon }$

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, and hence reductively yield:

$\displaystyle {\exp \left\{ {-\frac{1}{2}} \right.{{\sigma }_{3}}\otimes {{H}^{{1/2}}}{{\Gamma }_{{\kappa ,67}}}}$

$\displaystyle {\left. {\left[ {\left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)\left( {1+{{\Gamma }_{{\kappa ,6789}}}} \right)+\left. {\left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)\left( {1-{{\Gamma }_{{\kappa ,6789}}}} \right)} \right]} \right.} \right\}}$

$\displaystyle {\otimes {{\Gamma }_{{\kappa ,6789}}}{{\epsilon }_{0}}=0}$

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 deduced via mirror symmetry at the Hitchin holomorphic angles has the form:

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

$\displaystyle {Z_{7}^{{-1/2}}{{H}_{1}}d{{s}^{2}}\left( {\text{E}_{{67}}^{2}} \right)+Z_{7}^{{-1/2}}{{H}_{2}}d{{s}^{2}}\left( {\text{E}_{{89}}^{2}} \right)}$

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

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

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

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

$\displaystyle {{{C}_{6}}=\left( {Z_{7}^{{-1}}-1} \right)\left[ {{{H}_{1}}\cos {{\theta }_{1}}\sin {{\theta }_{2}}d{{x}_{6}}\wedge d{{x}_{7}}} \right.}$

$\displaystyle {\left. {{{H}_{2}}\cos {{\theta }_{2}}\sin {{\theta }_{1}}d{{x}_{8}}\wedge d{{x}_{9}}} \right]\wedge {{\epsilon }_{{{{\text{E}}^{{3,1}}}}}}}$

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

Thus 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 thus 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:

$\displaystyle {{{e}^{{-\frac{1}{2}{{Y}_{{ik}}}{{\Gamma }_{\kappa }}^{{ik}}\otimes {{\sigma }^{3}}}}}=\frac{1}{{\sqrt{{\left| {\eta +{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}} \right|}}}}\left[ {1-\frac{1}{2}{{\sigma }_{3}}{{\Gamma }_{\kappa }}^{{ik}}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}{{}_{{ik}}}} \right.}$

$\displaystyle {\left. {+\frac{1}{8}{{\Gamma }_{\kappa }}^{{ikml}}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}{{}_{{ik}}}{{{\tilde{F}}}^{{{{{\not{\nabla }}}^{\omega }}}}}{{}_{{ml}}}} \right]}$

Thus, the Killing spinor equation reduces to:

$\displaystyle {\exp \left\{ {-\frac{1}{4}{{\sigma }_{3}}{{\Gamma }_{\kappa }}^{{ik}}\left[ {Y_{{ik}}^{+}\left( {1-{{\Gamma }_{{\kappa ,6789}}}} \right)+Y_{{ik}}^{-}\left( {1+{{\Gamma }_{{\kappa ,6789}}}} \right)} \right]} \right\}}$

$\displaystyle {\cdot i{{\sigma }_{2}}\otimes {{\Gamma }_{{\kappa ,0123456789}}}\epsilon =\epsilon }$

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

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

$\displaystyle \otimes {{\Gamma }_{{\kappa ,0123456789}}}\epsilon =\epsilon$

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:

$\displaystyle {{{e}^{a}}{{}_{i}}={{\delta }^{a}}{{}_{i}}-{{B}^{a}}{{}_{i}}={{{\left( {1-B} \right)}}^{a}}{{}_{i}}}$

$\displaystyle {{{G}_{{ij}}}={{\delta }_{{ab}}}{{e}^{a}}{{}_{i}}{{e}^{b}}{{}_{j}}={{{\left( {{{e}^{T}}e} \right)}}_{{ij}}}}$

$\displaystyle {{{\theta }^{{ij}}}=-{{{\left( {\frac{1}{{1-B}}} \right)}}^{{ik}}}{{B}_{{km}}}{{{\left( {\frac{1}{{1+B}}} \right)}}^{{mj}}}}$

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:

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

$\displaystyle {-\frac{1}{2}\left. {\left( {{{B}^{-}}{{{\tilde{F}}}^{-}}-{{{\tilde{F}}}^{-}}{{B}^{-}}} \right)} \right\}}$

hence, our BPS equation reduces to:

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

$\displaystyle =\theta _{{-ab}}^{-}\text{Pf}{{{\tilde{F}}}_{-}}$

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:

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

has the following 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:

$\displaystyle \frac{1}{{16{{\pi }^{2}}}}\int\limits_{{{{\mathbb{R}}^{4}}}}{{\tilde{F}\wedge \tilde{F}=}}\frac{1}{{8{{\pi }^{2}}}}\int{{{{d}^{4}}}}x\text{Pf}{{{\tilde{F}}}_{-}}$

$\displaystyle =-\frac{1}{8}\text{Pf}{{\theta }_{-}}^{-}\left( {{{R}^{4}}{{h}^{2}}} \right)=N$

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 an Einstein-Sasaki-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:

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

the Higgs phase is hence equivalent to a non-commutative 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. An uplift to M-theory is achieved via Kovalev twisted-connected-sum constructed ${{G}_{2}}$ manifolds by gluing pairs of asymptotically cylindrical Calabi–Yau threefolds.

Type-IIB SUGRA, M-Theory, Einstein-Sasaki-Minkowski Vacua, and Inflation

George Shiber

M-Theory, Gauged SUGRA, Exceptional Field Theory, and Kovalev TCS G2-Manifolds

M-Theory, Exceptional Field Theory, U-Duality, and F-Theory

George Shiber

M-Theory, Gauged SUGRA, Exceptional Field Theory, and Kovalev TCS G2-Manifolds

M-Theory, Exceptional Field Theory, U-Duality, and F-Theory

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