Type-IIB String-Theory and D4 N=1 Supersymmetric D3/D7 Kähler Inflation

As we saw in my last post, the Standard ΛCDM Model of cosmology can be derived from Type-IIB SUGRA by identifying the inflaton with the Gukov-Vafa-Witten topologically twisted Kähler modulus embedded in a D3/D7 brane/anti-brane system. The advantages of the D3/D7 system is that we can apriori embed anti-brane instantonic effects to allow de Sitter solutions, and by mirror symmetry, we get a Kaloper-Sorbo axion monodromy inflation, where the flatness of the inflaton potential is protected without dependence on a moduli stabilization mechanism. Noting that Dp-branes probing Calabi-Yau 3-folds support 4-D N=1 supersymmetric Yang-Mills gauge theories whose intersection-points generate the Standard Model chiral matter sector and generally the action of a Dp-brane is given by a Dirac-Born-Infeld part coupled to a Chern-Simons WZ part:

\displaystyle S_{{DBI}}^{{cs}}=-{{T}_{p}}\int_{{{\mathcal{W}}'}}{{{{d}^{{p+1}}}}}\xi {{e}^{{-\Phi }}}\Xi -{{T}_{p}}\int_{{{\mathcal{W}}'}}{{C_{F}^{B}}}

with:

\displaystyle \Xi \doteq \sqrt{{{{{\det }}_{{\left[ {a,b} \right]}}}\left( {{{g}_{{ab}}}+{{B}_{b}}+2\pi {\alpha }'{{F}_{{ab}}}} \right)}}

\displaystyle C_{F}^{B}\doteq \text{TrP}\left[ {C\wedge {{e}^{{-B}}}} \right]\wedge {{e}^{{2\pi {\alpha }'F}}}

where P is the worldvolume pullback with p-orientifold action:

\displaystyle {{S}_{{Op}}}={{2}^{{p-4}}}-{{T}_{p}}\int\limits_{{{{\mathcal{W}}_{\parallel }}}}{{{{d}^{{p+1}}}}}\xi {{e}^{{-\phi }}}\left( \Pi \right)-\Psi

with:

\displaystyle \Pi \doteq \sqrt{{-\det P\left[ {{{g}_{{\mu \nu }}}} \right]}}-{{2}^{{2-4}}}

and

\displaystyle \Psi \doteq -{{T}_{p}}\int_{{{{\mathcal{W}}_{\parallel }}}}{{P\left[ {{{C}_{{p+1}}}} \right]}}

where the pullback to the Dp-worldvolume yields the 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 string coupling:

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

and the 10-D SUGRA dimensionally reduced Type-IIB action is:

\displaystyle S_{{DBI}}^{{c{s}'}}=-\frac{{{{T}_{p}}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{4{{g}_{s}}}}\int_{{{\mathcal{W}}'}}{{{{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)

and in the string-frame, the type-IIB SUGRA action 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)}}}}}

where the Calabi-Yau superpotential is:

\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 stabilization complex term, as well as the axio-dilaton field:

\displaystyle S={{e}^{{-\phi }}}+i{{C}_{0}}

Given the presence of E3-brane instantons, {{T}_{i}} are of Kähler moduli Type-IIB-orbifold class:

\displaystyle {{T}_{i}}={{e}^{{-\phi }}}{{\tau }_{1}}+i{{\rho }_{i}}

with {{\tau }_{i}} being the volume of the divisor {{D}_{i}} and {{\rho }_{i}} the 4-form Ramond-Ramond axion field corresponding to:

\displaystyle {{\tau }_{i}}=\frac{1}{2}\int_{{{{D}_{i}}}}{{J\wedge J}}=\frac{1}{2}{{k}_{{ijk}}}+{{\,}^{j}}{{t}^{k}}

and:

\displaystyle {{\rho }_{i}}=\int_{{{{D}_{i}}}}{{{{C}_{4}}}}

where J is the Kähler form:

\displaystyle J=\sum\limits_{i}{{{{t}_{i}}}}{{\eta }_{i}}

and:

\displaystyle \left\{ {{{\eta }_{i}}} \right\}\in {{H}^{{1,1}}}\left( {Y,\mathbb{Z}} \right)

an integral-form basis and {{k}_{{ijk}}} the associated intersection coefficients. Hence, the Kähler potential is given by:

\displaystyle K=-2\text{In}\left( {\tilde{\mathcal{V}}+\frac{\xi }{{2g_{s}^{{3/2}}}}} \right)-\text{In}\left( {S+\tilde{S}} \right)-\text{In}\left( {-i\int_{Y}{{\Omega \wedge \tilde{\Omega }}}} \right)

with {\tilde{\mathcal{V}}} the Calabi-Yau volume, and in the Einstein frame, is given by:

\displaystyle \mathcal{V}=\frac{1}{{3!}}\int_{Y}{{J\wedge J\wedge J=}}\frac{1}{6}{{k}_{{ijk}}}{{t}^{i}}{{t}^{j}}{{t}^{k}}

The F-term is given by:

\displaystyle {{V}_{F}}={{e}^{K}}\left( {\sum\limits_{{i={{T}_{i}};j={{S}_{j}}}}{{{{K}^{{ij}}}}}{{D}_{i}}W{{D}_{j}}\tilde{W}-3{{{\left| {{{W}_{i}}} \right|}}^{2}}} \right)

with the Large Volume Scenario D-term is given by:

\displaystyle {{V}_{D}}=\sum\limits_{{i=1}}^{N}{{\frac{1}{{\operatorname{Re}\left( {{{f}_{i}}} \right)}}}}{{\left( {\sum\limits_{j}{{Q_{j}^{{\left( i \right)}}{{{\left| {{{\phi }_{j}}} \right|}}^{2}}-{{{\hat{\xi }}}_{i}}}}} \right)}^{2}}

with:

\displaystyle \operatorname{Re}\left( {{{f}_{i}}} \right)\doteq {{e}^{{-\phi }}}\frac{1}{2}\int_{{{{D}_{i}}}}{{J\wedge J-}}{{e}^{{-\phi }}}\int_{{{{D}_{i}}}}{{\text{c}{{\text{h}}_{2}}}}\left( {{{\mathcal{L}}_{i}}-B} \right)

and the Fayet-Illopoulos terms being:

\displaystyle {{\hat{\xi }}_{i}}=-\text{Im}\left( {\frac{1}{{\tilde{\mathcal{V}}}}\int_{Y}{{{{e}^{{-\left( {B+iJ} \right)}}}}}{{\Gamma }_{i}}} \right)

where {{\Gamma }_{i}} are the D7-brane charge-vectors. Then we saw that the Chern-Simons orientifold action gets a Calabi-Yau curvature correction in the form of:

\displaystyle {{S}_{{{{O}_{p}},CS}}}=-{{2}^{{p-4}}}-{{T}_{p}}\int\limits_{{{{\mathcal{W}}^{\prime }}}}{{P\left[ C \right]}}\wedge \Theta

with

\displaystyle \Theta \doteq \sqrt{{\frac{{L\left( {\frac{1}{4}{{{\left( {2\pi } \right)}}^{2}}{\alpha }'T{{\mathcal{W}}^{\prime }}} \right)}}{{L\left( {\frac{1}{4}{{{\left( {2\pi } \right)}}^{2}}{\alpha }'N{{\mathcal{W}}^{\prime }}} \right)}}}}

due to the Gauss–Codazzi equations:

\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]}}