• Type-IIB String-Theory, 4D N=1 SUGRA, D7-Branes and Kähler Inflation

    A natural and intuitive way of deriving the standard inflationary ΛCDM cosmological model is by identifying the Type-IIB Kähler modulus with the ΛCDM-inflaton. The D3/D7-brane interaction is of particular importance because the worldvolume theory on a stack of D3-branes probing a Calabi-Yau 3-fold is a 4-D N=1 supersymmetric gauge theory and the importance of a D7+1 worldvolume theory derives, via duality-relations, from F-theory compactification by availing ourselves to the heterotic/F-theory duality in eight dimensions. Generally, for a p-brane, p\ge 3, there naturally is a p+1-D N={{f}_{\#}}\left( {{{Q}^{{SUS{{Y}_{G}}}}}/{{\lambda }_{{Y{{C}_{{SM}}}}}}} \right) SYM theory associated with its p+1 worldvolume \mathcal{W}, where the Dirac-Born-Infeld action is given by:

    \displaystyle {{S}_{{DBI}}}=-{{T}_{p}}\int{{{{d}^{{p+1}}}}}\xi {{e}^{{-\Phi }}}\sqrt{{-\underset{{[a,b]}}{\mathop{{\det }}}\,\left( {{{g}_{{ab}}}+{{B}_{{ab}}}+2\pi {\alpha }'{{F}_{{ab}}}} \right)}}

    coupled to a Chern-Simons Wess-Zumino term:

    \displaystyle -{{T}_{p}}\int_{\mathcal{W}}{{\text{Tr}}}P\left[ {C\wedge {{e}^{B}}} \right]\wedge {{e}^{{2\pi {\alpha }'F}}}

    P the worldvolume \mathcal{W} pullback, and the corresponding p-orientifold action is:

    \displaystyle {{S}_{{{{O}_{p}}}}}={{2}^{{p-4}}}-{{T}_{p}}{{\int_{{{\mathcal{W}}'}}{\text{d}}}^{{p+1}}}\xi {{e}^{{-\phi }}}\sqrt{{-\det \left( {P\left[ {{{g}_{{\mu \nu }}}} \right]} \right)}}-{{2}^{{2-4}}}-{{T}_{p}}\int_{{{\mathcal{W}}'}}{{P\left[ {{{C}_{{p+1}}}} \right]}}

    with the string-coupling given by:

    \displaystyle \frac{1}{{{{g}_{s}}}}={{e}^{{-\Phi }}}

    {{g}_{{ab}}} and {{B}_{{ab}}} the pullbacks SUGRA fields to the Dp-brane worldvolume. Thus, we can derive the action:

    \displaystyle {{S}_{{DBI}}}=-\frac{{{{T}_{p}}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{4{{g}_{s}}}}\int{{{{d}^{{p+1}}}}}\left( {\hat{F}+\hat{X}} \right)-\frac{{{{T}_{p}}}}{{{{g}_{s}}}}{{V}_{\vartheta }}

    with:

    \displaystyle \hat{F}\equiv {{F}_{{ab}}}{{F}^{{ab}}}

    \displaystyle \hat{X}\equiv \frac{2}{{{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{\partial }_{a}}{{X}^{m}}{{\partial }^{a}}{{X}_{m}}

    \displaystyle {{V}_{\vartheta }}\equiv V_{{p+1}}^{{WV}}+\vartheta \left( {{{F}^{4}}} \right)

    which yields a 10-D SYM action:

    \displaystyle {{S}_{{YM}}}=\frac{1}{{4g_{{_{{YM}}}}^{2}}}\int{{{{d}^{{10}}}}}x\left[ {\text{Tr}\left( {{{F}_{{\mu \nu }}}{{F}^{{\mu \nu }}}} \right)+2\text{iTr}\left( {\bar{\psi }\,{{\Gamma }^{\mu }}{{D}_{\mu }}\psi } \right)} \right]

    with:

    \displaystyle g_{{_{{YM}}}}^{2}=\frac{{{{g}_{s}}}}{{\sqrt{{{\alpha }'}}}}{{\left( {2\pi \sqrt{{{\alpha }'}}} \right)}^{{p-2}}}

    and F the non-abelian field strength of the U\left( N \right) gauge field {{A}_{\mu }}:

    \displaystyle {{F}_{{\mu \nu }}}={{\partial }_{\mu }}{{A}_{\nu }}-{{\partial }_{\nu }}{{A}_{\mu }}-\text{i}{{\left[ {{{A}_{\mu }},{{A}_{\nu }}} \right]}_{P}}

    where the gauge covariant derivative is defined by:

    \displaystyle {{D}_{\mu }}\psi ={{\partial }_{\mu }}\psi -\text{i}\left[ {{{A}_{\mu }},\psi } \right]

    giving us the p+1-worldvolume SYM action:

    \displaystyle S_{{Dp}}^{{\text{wv}}}=\frac{{-{{T}_{p}}{{g}_{s}}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{4}\int{{{{d}^{{p+1}}}}}\xi \text{Tr}\left( {\tilde{F}+\tilde{\Phi }+\Sigma +\tilde{f}} \right)

    with:

    \displaystyle \tilde{F}\equiv {{F}_{{ab}}}{{F}^{{ab}}}

    \displaystyle \tilde{\Phi }\equiv 2{{D}_{a}}{{\Phi }^{m}}{{D}^{a}}{{\Phi }_{m}}

    \displaystyle \Sigma \equiv \sum\limits_{{m\ne n}}{{{{{\left[ {{{\Phi }^{m}},{{\Phi }^{n}}} \right]}}^{2}}}}

    \displaystyle \tilde{f}\equiv \text{fermions}

    with the potential:

    \displaystyle V\left( \Phi \right)=\sum\limits_{{m\ne n}}{{\text{Tr}}}{{\left[ {{{\Phi }^{m}},{{\Phi }^{n}}} \right]}^{2}}

    that will figure in the Kähler form that will characterize D7-brane cosmology. D7-branes are central not simply because of F-theory compactification due to p+1 backreaction and Type-IIB axio-dilaton, but also because the Kähler moduli space induced by the D7-brane F-theoretic backreaction has a modulus that can naturally be identified with the ΛCDM-inflaton. Thus, inflationary cosmology can be derived, up to isomorphism, from 6D F-theory compactification using the Type-IIB string-theory/11D SUGRA duality upon dimensional reduction to 4D, where the Type-IIB SUGRA action in the string frame is given by:

    \displaystyle {{S}_{{IIB}}}={{S}_{{NS}}}+{{S}_{R}}+{{S}_{{CS}}}

    with:

    \displaystyle {{S}_{{NS}}}=\frac{1}{{2k_{{10}}^{2}}}\int{{{{d}^{{10}}}}}x{{\left( {-{{G}_{{10}}}} \right)}^{{1/2}}}{{e}^{{-2\Phi }}}\left[ {{{R}_{{10}}}+4\left( {{{\partial }^{\mu }}\Phi } \right)\left( {{{\partial }_{\mu }}\Phi } \right)-\frac{1}{2}{{{\left| {{{H}_{{\left( 3 \right)}}}} \right|}}^{2}}} \right]

    \displaystyle {{S}_{R}}=-\frac{1}{{4k_{{10}}^{2}}}\int{{{{d}^{{10}}}}}x{{\left( {-{{G}_{{10}}}} \right)}^{{1/2}}}\left[ {{{{\left| {{{F}_{1}}} \right|}}^{2}}+{{{\left| {{{{\tilde{F}}}_{{\left( 3 \right)}}}} \right|}}^{2}}+\frac{1}{2}{{{\left| {{{{\tilde{F}}}_{{\left( 5 \right)}}}} \right|}}^{2}}} \right]

    \displaystyle {{S}_{{CS}}}=-\frac{1}{{4k_{{10}}^{2}}}\int{{{{C}_{{\left( 4 \right)}}}\wedge {{H}_{{\left( 3 \right)}}}\wedge {{F}_{{\left( 3 \right)}}}}}

    In light of flux moduli stabilization, this is a natural framework since the Kähler moduli potential apriori allows a Type-IIB-modulus/ΛCDM-inflaton identification. In such a SYM D7-brane scenario, we have an N-flaton model in the Large Volume Scenario that evades the \eta-problem, and periodicity of the modulus is achieved by defluxing the pullback of the H-term on the superstring worldsheet. This can be readily seen since the total supersymmetric worldsheet action in conformal gauge:

    \displaystyle {{S}_{{tot}}}={{S}_{{bos}}}+{{S}_{{fer}}}

    with:

    \displaystyle {{S}_{{bos}}}=\frac{1}{{4\pi {\alpha }'}}\int{{{{d}^{2}}}}z{{\partial }_{z}}{{X}^{\mu }}{{\partial }_{{\bar{z}}}}{{X}_{\mu }}-\text{i}\int\limits_{0}^{{2\pi }}{{d\theta {{{\dot{X}}}^{\mu }}}}{{A}_{\mu }}\left| {_{{r=1}}} \right.

    \displaystyle {{S}_{{fer}}}=\frac{\text{i}}{{4\pi {\alpha }'}}\int{{{{d}^{2}}}}z\left( {\bar{\psi }{{\partial }_{z}}\bar{\psi }+\psi {{\partial }_{{\bar{z}}}}\psi } \right)-\frac{1}{2}\int\limits_{0}^{{2\pi }}{{d\theta {{\psi }^{\mu }}}}{{F}_{{\mu \nu }}}{{\psi }^{\mu }}\left| {_{{r=1}}} \right.

    has modular-invariance. The skeleton of N=1 Type-IIB flux compactification on a Calabi-Yau threefold Y is constructed from a superpotential of the form:

    \displaystyle W=\int_{Y}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}+\sum\limits_{{i=1}}^{{{{h}^{{1,1}}}}}{{{{A}_{i}}}}\left( {S,U} \right){{e}^{{-{{a}_{i}}{{T}_{i}}}}}

    where:

    \displaystyle \int_{Y}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}

    is the Gukov-Vafa-Witten superpotential…