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 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)

and in the large volume scenario (LVS) is:

\displaystyle \begin{array}{c}{{V}_{{{{F}^{L}}}}}=\frac{1}{{{{{\dot{\mathcal{V}}}}^{2}}}}\left( {-4\pi \text{Vol}\left( {{{D}_{{E3}}}\cap {{D}_{{E3}}}} \right)\tilde{\mathcal{V}}{{{\left| {{{A}_{{E3}}}} \right|}}^{2}}\cdot } \right.\\{{e}^{{-4\pi {{{\tilde{\tau }}}_{i}}E3}}}4\pi {{{\tilde{\tau }}}_{{E3}}}{{e}^{{-2\pi \tilde{\tau }E3}}}\left| {{{A}_{{E3}}}{{W}_{0}}} \right|\left. {+\frac{3}{4}\frac{{\hat{\xi }}}{{\tilde{\mathcal{V}}}}{{{\left| {{{W}_{0}}} \right|}}^{2}}} \right)\end{array}

Thus, the LVS 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. The Kähler LVS potential is now derivable, and takes the form:

\displaystyle \begin{array}{l}{{V}_{{LVS}}}=\sum\limits_{{i=2}}^{n}{{\frac{{8{{{\left( {{{a}_{i}}{{A}_{i}}} \right)}}^{2}}\sqrt{{{{\tau }_{i}}}}}}{{3\sqrt{{{{\lambda }_{i}}}}}}}}\,{{e}^{{-2{{a}_{i}}{{T}_{i}}}}}-\\\sum\limits_{{i=2}}^{n}{{\frac{{4{{a}_{i}}{{A}_{i}}{{W}_{0}}}}{{{{\mathcal{V}}^{2}}}}}}\,{{\tau }_{i}}{{e}^{{-{{a}_{i}}{{\tau }_{i}}}}}+\frac{{3\hat{\xi }W_{0}^{2}}}{{4{{\mathcal{V}}^{3}}}}+\frac{D}{{{{\mathcal{V}}^{\gamma }}}}\end{array}

One can now choose any {{\tau }_{n}} as the Kähler-modulus-inflaton, and inflation takes place in:

\displaystyle {{e}^{{{{a}_{n}}{{\tau }_{n}}}}}\gg {{\mathcal{V}}^{2}}

We can build our model now from the Kähler form and the superpotential. For a Witten-deformed D7-brane, the Kähler potential is thus derivable in the weak string-coupling limit of the F-theory Kähler potential arising from the elliptic fibration base Calabi-Yau four-fold:

\displaystyle K=-\text{In}\left( {\int{{{{\Omega }_{4}}\wedge {{{\tilde{\Omega }}}_{4}}}}} \right)

Expanded in the weak string coupling limit gives us:

\displaystyle {{K}_{{{{g}_{s}}\to 0}}}=-\log \left( {\left( {S-\bar{S}} \right)} \right){{\pi }_{A}}\left( u \right){{Q}^{{AB}}}{{\bar{\pi }}_{B}}\left( {\bar{u}} \right)+f\left( {u,\bar{u};c,\bar{c}} \right)+...

with {{\pi }_{A}}\left( u \right) the periods of integrals of the holomorphic (3,0)-form over the symplectic basis of 3-cycles with intersection matrix {{Q}^{{AB}}} and f\left( {u,\bar{u};c,\bar{c}} \right) is a function of symplectic and brane-moduli that figures in the flux-compactification moduli-stabilization but is independent of the axio-dilaton. Using mirror symmetry, the Kähler potential:

\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)

can be identified with the Kähler-moduli-potential of the mirror fourfold, which in the LVS-limit, involves the volume moduli of the elliptic fourfold, but not the corresponding axions due to shift-symmetry. Hence, the shift-invariant potential takes the form:

\displaystyle {{K}_{{SIP}}}=-\text{In}\left( {\frac{{{{k}_{{ijkl}}}}}{{4!}}\left( {{{z}^{i}}-{{{\bar{z}}}^{i}}} \right)\left( {{{z}^{j}}-{{{\bar{z}}}^{j}}} \right)\left( {{{z}^{k}}-{{{\bar{z}}}^{k}}} \right)\left( {{{z}^{l}}-{{{\bar{z}}}^{l}}} \right)+...} \right)

{{{k}_{{ijkl}}}} the self-intersecting matrix of the mirror four-fold divisors and {{{z}^{i}}} the complex structure of the moduli of the three-form. We can now identify any {{{z}^{i}}} with the axio-dilaton and the remaining identified with the D7-brane position-moduli {{c}^{p}} and the three-fold complex moduli {{u}^{a}}. Their interdependence can be expressed by the relation:

\displaystyle \begin{array}{l}S_{{{{g}_{s}}\to 0}}^{{SIP}}=-\text{In}\left( {\frac{{k_{{abc}}^{{\left( 1 \right)}}}}{{3!}}\left( {S-\bar{S}} \right)\left( {{{u}^{a}}-{{{\bar{u}}}^{a}}} \right)} \right.\left( {{{u}^{b}}-{{{\bar{u}}}^{b}}} \right)\left( {{{u}^{c}}-{{{\bar{c}}}^{c}}} \right)\\+\frac{{k_{{abpq}}^{{\left( 2 \right)}}}}{{4!}}\left( {{{u}^{a}}-{{{\bar{u}}}^{a}}} \right)\left( {{{u}^{b}}-{{{\bar{u}}}^{b}}} \right)\left( {{{c}^{p}}-{{{\bar{c}}}^{p}}} \right)\left( {{{c}^{q}}-{{{\bar{c}}}^{q}}} \right)\\\left. {+...} \right)\end{array}

Hence, any {{c}^{p}} can be identified with the deformation complex moduli term {{c}^{{D7}}} of a D7-brane, and Type-IIB N-flaton cosmology can reproduce ΛCDM by integrating out all the other moduli terms, and by substitution, the Kähler potential:

\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)

takes the following form:

\displaystyle K=-\text{In}\left( {A+iB\left( {c-\bar{c}} \right)-\frac{D}{2}{{{\left( {c-\bar{c}} \right)}}^{2}}} \right)

The F-theory backreacted counterpart Kähler potential hence takes the form:

\displaystyle W={{N}^{i}}{{\Pi }_{i}}\left( z \right)

where {{N}^{i}} are the flux compactification quantum numbers and {{\Pi }_{i}}\left( z \right) is the period four-fold Calabi-Yau vector. In terms of the brane modulus c, we can write the potential as:

\displaystyle W={{W}_{0}}+\alpha c+\frac{\beta }{2}{{c}^{2}}+.....

We now include the Kähler moduli term to the potential:

\displaystyle K=-2\text{In}\mathcal{V}-\text{In}\left( {A+iB\left( {c-\bar{c}} \right)-\frac{D}{2}{{{\left( {c-\bar{c}} \right)}}^{2}}} \right)

and the superpotential:

\displaystyle W={{W}_{0}}+\alpha c+\frac{\beta }{2}{{c}^{2}}+.....

Then, instanton correction terms from the four-fold blow-up cycles yield:

\displaystyle W={{W}_{0}}+\alpha c+\frac{\beta }{2}{{c}^{2}}+{{e}^{{-2\pi {{T}_{s}}}}}

from which the F-term potential:

\displaystyle {{V}_{F}}={{e}^{K}}\left( {{{K}^{{T\gamma {{{\bar{T}}}_{s}}}}}{{D}_{{{{T}_{\gamma }}}}}\overline{{W{{D}_{{{{T}_{\delta }}}}}W}}-3{{{\left| W \right|}}^{2}}+{{K}^{{c\bar{c}}}}{{{\left| {{{D}_{c}}W} \right|}}^{2}}} \right)

can be derived. {{T}_{\gamma }} the complexified Kähler moduli that measure the 3-fold four-cycles when restricted to the reals, in string-length unit measure and as standard, \mathcal{V} the Calabi-Yau 3-fold volume. Now since the Kähler metric is block-diagonal in the total space, no mixed derivatives in c and {{T}_{\gamma }} occur in:

\displaystyle {{V}_{F}}={{e}^{K}}\left( {{{K}^{{T\gamma {{{\bar{T}}}_{s}}}}}{{D}_{{{{T}_{\gamma }}}}}\overline{{W{{D}_{{{{T}_{\delta }}}}}W}}-3{{{\left| W \right|}}^{2}}+{{K}^{{c\bar{c}}}}{{{\left| {{{D}_{c}}W} \right|}}^{2}}} \right)

hence the term \displaystyle {{K}^{{c\bar{c}}}}{{\left| {{{D}_{c}}W} \right|}^{2}} in the F-term potential dominates and stabilizes c in a supersymmetric minimum:

\displaystyle {{D}_{c}}W=0

Thus, we have a classic moduli stabilization and applies to our Large Volume Scenario straightforwardly and yields the AdS minimum:

\displaystyle \approx -{{\left| {{{{\bar{W}}}_{0}}} \right|}^{2}}/{{\mathcal{V}}^{3}}

that can be uplifted to a Minkowski minimum. Since inflation occurs in \operatorname{Re}\left( c \right) and the terms:

\displaystyle -{{\left| {{{{\bar{W}}}_{0}}} \right|}^{2}}/{{\mathcal{V}}^{3}}

are subleading in the inverse of \mathcal{V} with respect to -3{{\left| W \right|}^{2}}, it follows that the leading inflaton mass term is part of:

\displaystyle {{e}^{K}}{{K}^{{c\bar{c}}}}{{\left| {{{D}_{c}}W} \right|}^{2}}

Since the LVS large-scale structure of the Kähler moduli space guarantees that loop-corrections are stably subleading with respect to \mathcal{V} moduli-stabilization {\alpha }'-corrections, the Type-IIB axio-dilaton and brane moduli are integrated out at higher scales. Central is the c-stabilization, since from:

\displaystyle {{D}_{c}}W=0

we can derive:

\displaystyle \frac{{\alpha +\beta c}}{{{{W}_{0}}+\alpha c+\frac{\beta }{2}{{c}^{2}}}}=\frac{{iB-D\left( {c-\bar{c}} \right)}}{{A+iB\left( {c-\bar{c}} \right)-\frac{D}{2}{{{\left( {c-\bar{c}} \right)}}^{2}}}}

with c=x+iy. At the ground level,

\displaystyle \frac{{\alpha +\beta c}}{{{{W}_{0}}+\alpha c+\frac{\beta }{2}{{c}^{2}}}}

vanishes and we have:

\displaystyle {{y}_{0}}=\frac{B}{{2D}}

and imposing the condition that the real part of:

\displaystyle \frac{{iB-D\left( {c-\bar{c}} \right)}}{{A+iB\left( {c-\bar{c}} \right)-\frac{D}{2}{{{\left( {c-\bar{c}} \right)}}^{2}}}}

at first-order in the \alpha and \beta terms vanishes gives us:

\displaystyle {{x}_{0}}\frac{{\operatorname{Im}\left( {\beta {{{\bar{W}}}_{0}}} \right){{y}_{0}}-\operatorname{Re}\left( {\alpha {{{\bar{W}}}_{0}}} \right)}}{{\operatorname{Re}\left( {\beta {{{\bar{W}}}_{0}}} \right)}}

We are now in position to derive the inflaton mass which arises from:

\displaystyle {{\left| {{{D}_{c}}W} \right|}^{2}}

Moreover, in the minimum, we have:

\displaystyle {{D}_{c}}W=0

and given that the Chern-Simons orientifold action gets the following Calabi-Yau curvature-correction:

\displaystyle {{S}_{{{{O}_{p}},CS}}}=-{{2}^{{p-4}}}-{{T}_{p}}\int_{{{\mathcal{W}}'}}{{P\left[ C \right]}}\wedge \sqrt{{\frac{{L\left( {\frac{1}{4}{{{\left( {2\pi } \right)}}^{2}}{\alpha }'T{\mathcal{W}}'} \right)}}{{L\left( {\frac{1}{4}{{{\left( {2\pi } \right)}}^{2}}{\alpha }'N{\mathcal{W}}'} \right)}}}}

we hence only need to expand it in the leading order with respect to the real part \delta x and moduli stabilization enforces {{K}_{c}}={{W}_{c}}/W which scales linearly. Hence we have:

\displaystyle \delta {{D}_{c}}W=\delta \left( {{{K}_{c}}W+{{W}_{c}}} \right)\simeq \delta {{W}_{c}}=\beta \delta x

and:

\displaystyle {{e}^{K}}{{K}^{{c\bar{c}}}}{{\left| \beta \right|}^{2}}\delta {{x}^{2}}+\text{h}\text{.o}\text{.t}\left( {\alpha ,\beta ,\delta x} \right)

Thus, the SM renormalization parameter connects the inflaton to {\delta x}. Since the kinetic term for {\delta x} is contained in:

\displaystyle {{K}_{{c\bar{c}}}}{{\left| {\partial c} \right|}^{2}}

then from:

\displaystyle {{K}_{{c\bar{c}}}}\sim \operatorname{Im}{{\left( z \right)}^{2}}

and:

\displaystyle {{e}^{K}}\sim \operatorname{Im}{{\left( z \right)}^{4}}

we can deduce:

\displaystyle m_{\varphi }^{2}\sim \frac{1}{\mathcal{V}}\frac{1}{{\operatorname{Im}{{{\left( z \right)}}^{4}}}}\operatorname{Im}{{\left( z \right)}^{2}}\operatorname{Im}{{\left( z \right)}^{2}}{{\left| \beta \right|}^{2}}=\frac{{{{{\left| \beta \right|}}^{2}}}}{{{{\mathcal{V}}^{2}}}}

In the ΛCDM Standard Model, quadratic inflation has a potential V={{m}^{2}}{{\varphi }^{2}} with slow-roll parameters obeying the following relations:

\displaystyle \left\{ {\begin{array}{*{20}{c}} {\epsilon =\frac{1}{2}{{{\left( {\frac{{{V}'}}{V}} \right)}}^{2}}=\frac{2}{{{{\varphi }^{2}}}}} \\ {\eta =\frac{{{V}''}}{V}=\frac{2}{{{{\varphi }^{2}}}}} \end{array}} \right.

with spectral index:

\displaystyle {{n}_{s}}-1=-6\epsilon +2\eta =-\frac{8}{{{{\varphi }^{2}}}}

Phenomenologically, the ΛCDM Standard Model sets this term at \simeq \,\sim -0.04 with e-fold field displacement \sim 60 and {{\varphi }^{2}}\simeq 200. Moreover, the amplitude of the curvature perturbations (chapter 2/2.11) yields \sqrt{{\frac{V}{{2\epsilon }}}}=5.1\cdot {{10}^{{-4}}} and hence we have m\simeq 0.5\cdot {{10}^{{-5}}}.

In Type-IIA/B string theories in the LVS, this remarkably yields:

\displaystyle {{m}_{\varphi }}=\frac{{\left| \beta \right|}}{\mathcal{V}}=0.5\cdot {{10}^{{-5}}}

In order not to destroy the moduli stabilization, we impose the harmless condition:

\displaystyle m_{\varphi }^{2}{{\varphi }^{2}}\simeq 0.5\cdot {{10}^{{-8}}}\ll \frac{{{{{\left| {{{W}_{0}}} \right|}}^{2}}}}{{{{\mathcal{V}}^{3}}}}

along the inflaton field flow. Now during the inflationary phase, we have:

\displaystyle \operatorname{Re}\left( c \right)\in \left[ {\Theta \left| {_{{LVS}}} \right.} \right]

and the imaginary part does not destroy moduli stability since:

\displaystyle {{K}_{{c\bar{c}}}}{{\left( {\partial \delta x} \right)}^{2}}\sim \frac{{{{{\left( {\partial \delta x} \right)}}^{2}}}}{{\operatorname{Im}{{{\left( z \right)}}^{2}}}}

holds, giving:

\displaystyle \left\{ {\begin{array}{*{20}{c}} {\delta {{x}_{N}}\sim 14\cdot \operatorname{Im}\left( z \right)} \\ {{{\varphi }_{N}}\sim 14} \end{array}} \right.

and since the LVS F-term is given by:

\displaystyle \begin{array}{c}{{V}_{{{{F}^{L}}}}}=\frac{1}{{{{{\dot{\mathcal{V}}}}^{2}}}}\left( {-4\pi \text{Vol}\left( {{{D}_{{E3}}}\cap {{D}_{{E3}}}} \right)\tilde{\mathcal{V}}{{{\left| {{{A}_{{E3}}}} \right|}}^{2}}\cdot } \right.\\{{e}^{{-4\pi {{{\tilde{\tau }}}_{i}}E3}}}4\pi {{{\tilde{\tau }}}_{{E3}}}{{e}^{{-2\pi \tilde{\tau }E3}}}\left| {{{A}_{{E3}}}{{W}_{0}}} \right|\left. {+\frac{3}{4}\frac{{\hat{\xi }}}{{\tilde{\mathcal{V}}}}{{{\left| {{{W}_{0}}} \right|}}^{2}}} \right)\end{array}

we have:

\displaystyle \left\{ {\begin{array}{*{20}{c}} {\frac{{14\left| \beta \right|\operatorname{Im}{{{\left( z \right)}}^{2}}}}{{\left| {{{W}_{0}}} \right|}}\ll 1} \\ {\left| {\delta {{y}_{0}}+\delta y} \right|\ll {{y}_{0}}} \end{array}} \right.

To handle the cubic and quartic terms in \delta x, we expand:

\displaystyle {{V}_{F}}={{e}^{K}}\left( {{{K}^{{T\gamma {{{\bar{T}}}_{s}}}}}{{D}_{{{{T}_{\gamma }}}}}\overline{{W{{D}_{{{{T}_{\delta }}}}}W}}-3{{{\left| W \right|}}^{2}}+{{K}^{{c\bar{c}}}}{{{\left| {{{D}_{c}}W} \right|}}^{2}}} \right)

in \text{ }\!\!~\!\!\text{ }\delta x where at the minimum we have:

\displaystyle \begin{array}{l}\mathcal{V}\sim \frac{{{{{\left| \beta \right|}}^{2}}\delta {{x}^{2}}}}{{{{\mathcal{V}}^{2}}\operatorname{Im}{{{\left( z \right)}}^{2}}}}\left\{ {1+\vartheta \left( {\frac{{{{\alpha }^{3}}}}{{\beta W_{0}^{2}}},\frac{{{{\alpha }^{2}}{{c}_{0}}}}{{W_{0}^{2}}},\frac{{\alpha \beta c_{0}^{2}}}{{W_{0}^{2}}},\frac{{{{\beta }^{2}}c_{0}^{3}}}{{W_{0}^{2}}}} \right)} \right.\\+\vartheta \left( {\frac{{{{\alpha }^{2}}}}{{W_{0}^{2}}},\frac{{\alpha \beta {{c}_{0}}}}{{W_{0}^{2}}},\frac{{{{\beta }^{2}}c_{0}^{2}}}{{W_{0}^{2}}}} \right)\left. {\delta {{x}^{2}}} \right\}\end{array}

Solving:

\displaystyle {{D}_{c}}W=0

with respect to y gives us:

\displaystyle V\supset \frac{{{{{\left| \beta \right|}}^{2}}}}{{{{\mathcal{V}}^{2}}\operatorname{Im}{{{\left( z \right)}}^{2}}}}\vartheta \left( {\frac{{{{\beta }^{2}}c_{0}^{2}}}{{W_{0}^{2}}}} \right)\delta {{x}^{4}}

and:

\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.

with respect to the kappa symmetric shift Type-IIB parameter, we get the Type-IIA dual instanton alpha-corrections to the Kähler potential thus yielding the correct Hilbert inflaton spectrum correction to the F-term:

\displaystyle \sim {{e}^{{-2\pi {{y}_{0}}}}}\frac{{{{{\left| {{{W}_{0}}} \right|}}^{2}}}}{{{{\mathcal{V}}^{2}}}}

Now, given:

\displaystyle {{y}_{0}}=\frac{B}{{2D}}

the Kaluza-Klein inter-brane mode loop corrections give us the correct periodicity condition:

\displaystyle \sim \left\{ {\alpha ,\beta } \right\}\cdot \frac{{{{{\left| {{{W}_{0}}} \right|}}^{2}}}}{{{{\mathcal{V}}^{{8/3}}}}}

due to the Gauss–Codazzi equations for:

\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]}}

and are parametrized as:

\displaystyle V={{m}^{2}}{{\varphi }^{2}}+\gamma \cos \left( {\frac{\varphi }{f}+\delta } \right)

at leading order. Since axion decay always takes place in a small region in e-fold inflation and is bounded by f\le 1/4\pi, then during the initial e-folding stage, under the Kähler-modulus/inflaton identification, the inflaton crosses at least once the period of the oscillatory term of:

\displaystyle V={{m}^{2}}{{\varphi }^{2}}+\gamma \cos \left( {\frac{\varphi }{f}+\delta } \right)

and hence for large \operatorname{Im}\left( z \right) and small \left| \alpha \right| and \left| \beta \right|, the Type-IIB 4-D SUGRA Kähler modulus under such inflaton-identification reproduces the ΛCDM Standard Model mainly because, as a Dirac variety, a D7-brane encircles a closed trajectory in Kähler moduli space enough times in the slow-roll scape to grow a contribution to the F-term potential that appears in the action of Type-IIB 4-D SUGRA. The phenomenological aspects of such claims can be computed from this source.

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