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

By George Shiber Monday, January 1, 2018Friday, October 26, 2018 In Mathematics, Physics, Super String Theory Kähler cosmology, Type-IIB string theory

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:

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

One can and should enhance such a Kähler scenario to one that involves a $D3/D7$ -brane inflationary theory mainly due to the fact that in this context, moduli stabilization entails the existence of a $\bar{D}3$ – ${{\Lambda }^{{++}}}$ de Sitter compactification that inherits the Lambda-CDM $U\left( 1 \right)$ -CP gauge-bundle, yielding a non-truncated N=1 4D SYM on the bulk, deriving inflation on the ${{G}_{2}}$ -manifold orbifolded by the corresponding ALE singularities that give rise to the SM gauge structure and resolutions that yield the Dirac chiral-fermionic SM structure, giving rise to an isomorphism with a ${{G}_{2}}$ Abrikosov-Nielsen-Olesen type manifold. Thus, we can identify the $D3/D7$ -brane system with an N = 2 gauge theory associated with the D-term inflation model I derived above, but with one hypermultiplet and one Fayet-Iliopoulos term, and dimensionally reduce to N=1. In this, part 1, I will defend such a $D3/D7$ $\bar{D}3$ – ${{\Lambda }^{{++}}}$ de Sitter brane inflationary scenario.

In a $D3/D7$ brane system, a spontaneously broken $U\left( 1 \right)$ gauge symmetry corresponds to a bound state where the $D3$ dissolves as a deformed Abelian instanton on the $U\left( 1 \right)$ Dirac-Born-Infeld theory of the $D7$ brane with Chern-Simons coupling:

$\displaystyle {{S}_{{SC}}}=\frac{1}{{16{{\pi }^{2}}}}\int_{{D7}}{{{{C}_{4}}\wedge {{{\tilde{F}}}^{D}}}}\wedge {{{\tilde{F}}}^{D}}=\int_{{D3}}{{{{C}_{4}}}}$

where ${{{\tilde{F}}}^{D}}=dA-B$ gets a background NSNS $D{{3}_{\bot }}$ as well as $D7$ worldvolume gauge field contributions. Hence, the NSNS two-form becomes a non-commutative deformation parameter in this system. The $D3/D7$ brane system has an unbroken supersymmetry due to a $D7$ -brane $\kappa$ -symmetry. The vacuum endpoints is describable hence by a non-marginal bound state of $D3$ and $D7$ -branes corresponding to the Higgs phase of the gauge theory of the $D3/D7$ -system. The LE-effective action of such a $D3/D7$ -system in flat space is:

$\displaystyle S=\int{{{{d}^{8}}}}x{{F}^{{ST}}}+\int{{{{d}^{4}}}}x\Upsilon {{{\tilde{g}}}_{s}}+\int{{{{d}^{4}}}}xD_{\chi }^{\dagger }+{{S}_{{CS}}}$

with:

$\displaystyle {{F}^{{ST}}}\equiv -\frac{1}{{4g_{7}^{2}}}{{F}_{{ST}}}{{F}^{{ST}}}-\frac{1}{{2g_{7}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}\left( {{{{\left( {{{\partial }_{\mu }}{{X}^{i}}} \right)}}^{2}}+{{{\left( {{{\partial }_{m}}{{X}^{i}}} \right)}}^{2}}} \right)$

$\displaystyle \Upsilon {{{\tilde{g}}}_{s}}\equiv -\frac{1}{{4g_{7}^{2}}}{{{{F}'}}_{{\mu \nu }}}{{{{F}'}}^{{\mu \nu }}}-\frac{1}{{2g_{3}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}\left( {{{{\left( {{{\partial }_{\mu }}{{{{X}'}}^{i}}} \right)}}^{2}}+{{{\left( {{{\partial }_{\mu }}{{{{X}'}}^{m}}} \right)}}^{2}}} \right)$

and

$\displaystyle D_{\chi }^{\dagger }\equiv -{{\left| {{{D}_{\mu }}\chi } \right|}^{2}}-{{\left( {\frac{{{{X}^{i}}-{{{{X}'}}^{i}}}}{{2\pi {\alpha }'}}} \right)}^{2}}{{\left| \chi \right|}^{2}}-\frac{{g_{3}^{2}}}{2}{{\left( {{{\chi }^{\dagger }}{{\sigma }^{A}}\chi } \right)}^{2}}$

with ${{S}_{{CS}}}$ the Chern-Simons part, with:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\mu \in \left\{ {0,...3} \right\}} \\ {i\in \left\{ {\begin{array}{*{20}{c}} {\to 4} \\ {\to 5} \end{array}} \right.} \\ {m\in \left\{ {\begin{array}{*{20}{c}} {\to 6} \\ {\to 9} \end{array}} \right.} \\ {\left\{ {\mu ,m} \right\}\supset \left\{ {S,T} \right\}} \end{array}} \right.$

The doublet arises as the lightest d.o.f from strings stretched between the $D3$ -brane and the parallel $D7$ -brane, with covariant derivative:

$\displaystyle {{D}_{\mu }}\chi =\left( {{{\partial }_{\mu }}+i{{A}_{\mu }}-i{{{{A}'}}_{\mu }}} \right)\chi$

and our metric has a Chern-Simons form:

$\displaystyle g_{p}^{2}={{\left( {2\pi } \right)}^{{p-2}}}{{g}_{s}}{{{\alpha }'}^{{\left( {p-3} \right)/2}}}$

We now must geometrically locate our system in a background that incorporates the field $B$ and turn on a Ramond-Ramond field-strength ${{F}_{5}}=d{{C}_{{\left( 4 \right)}}}$ compatible with the orbifold and orientifold structure of the system with a metric of the form:

$\displaystyle d{{s}^{2}}={{R}^{{-6}}}\left( x \right)g_{{\mu \nu }}^{E}d{{x}^{\mu }}d{{x}^{\nu }}+{{R}^{2}}\left( x \right)\left( {ds_{{{{\mathbb{R}}^{2}}/{{\mathbb{Z}}_{2}}}}^{2}+ds_{{{{K}_{3}}}}^{2}} \right)$

$g_{{\mu \nu }}^{E}$ the 4D metric in the Einstein frame and $R$ is a section on $K3$ that factors in moduli stabilization involved in wrapping branes on cycles in the compact space $K3\times {{T}^{2}}/{{\mathbb{Z}}_{2}}$ .

Let us analyze the $D7$ action. In a curved background given by our metric, this contribution to the LE-effective action:

reduces to:

$\displaystyle {{S}_{{D7}}}=\int_{\mathcal{W}}{{\tilde{\psi }}}\left[ {\tilde{\varphi }-\tilde{\beta }} \right]+\tilde{p}\int_{\mathcal{W}}{{\tilde{\alpha }}}$

with:

$\displaystyle \tilde{\psi }\equiv {{d}^{4}}x{{d}^{4}}y\sqrt{{-{{g}_{E}}}}{{R}^{{-12}}}\sqrt{{{{g}_{{K3}}}}}{{R}^{4}}$

$\displaystyle \tilde{\varphi }\equiv -\frac{1}{{4g_{7}^{2}}}\left( {{{F}_{{\mu \nu }}}{{F}_{{\rho \sigma }}}g_{E}^{{\mu \rho }}g_{E}^{{\nu \sigma }}{{R}^{{12}}}+{{{\tilde{F}}}^{D}}_{{mn}}{{{\tilde{F}}}^{D}}_{{rs}}g_{{K3}}^{{mr}}g_{{K3}}^{{ns}}{{R}^{{-4}}}} \right)$

$\displaystyle \tilde{\beta }\equiv -\frac{1}{{2g_{7}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{\partial }_{\mu }}{{X}^{i}}{{\partial }_{\nu }}{{X}^{j}}g_{E}^{{\mu \nu }}g_{{ij}}^{{{{\mathbb{R}}^{2}}}}{{R}^{8}}$

$\displaystyle \tilde{p}\int_{\mathcal{W}}{{\tilde{\alpha }}}\equiv {{\mu }_{7}}\frac{{{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{2!}}\int_{{D7}}{{{{C}_{{\left( 4 \right)}}}}}\wedge {{{\tilde{F}}}^{D}}\wedge {{{\tilde{F}}}^{D}}$

and we have integrated out the ${{X}_{i}}$ fluctuation-modes in the $K3$ directions. The $D7$ brane covariant 2-form is composed of two terms:

$\displaystyle {{\tilde{F}}^{D}}\equiv F-B\equiv dA-B$

with ${{F}_{{mn}}}$ the field strength of the vector field ${{A}_{m}}$ living on the brane and ${{B}_{{mn}}}$ the pullback of the space-time NS-NS two-form field to the worldvolume of the $D7$ -brane, with a Chern-Simons part induced by the RR field. Now let

$\displaystyle \int{{{{d}^{4}}}}y\sqrt{{{{g}_{{K3}}}}}={{V}_{{K3}}}$

be the volume of a fixed $K3$ . Integrating over $K3$ gives us:

$\displaystyle {{S}_{{D7}}}\int{{\left( {\Sigma +\Theta } \right)}}+\int{\Xi }\int_{{K3}}{\Upsilon }+{{\mu }_{{K3}}}\int{{{{{\tilde{F}}}^{D}}_{C}}}$

with:

$\displaystyle \Sigma \equiv {{d}^{4}}x\sqrt{{-{{g}_{E}}}}$

$\displaystyle \Theta \equiv -\frac{1}{{4\tilde{g}_{3}^{2}}}{{\left( {{{F}_{{\mu \nu }}}} \right)}^{2}}-\frac{{{{R}^{4}}}}{{2\tilde{g}_{3}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{\left( {{{\partial }_{\mu }}{{X}^{i}}} \right)}^{2}}$

$\displaystyle \Xi \equiv \text{Vo}{{\text{l}}_{{\left( 4 \right)}}}{{R}^{{-12}}}\left( {\frac{{-1}}{{4g_{7}^{2}}}} \right)$

$\displaystyle \Upsilon \equiv {{{\tilde{F}}}^{D}}\wedge *{{{\tilde{F}}}^{D}}$

$\displaystyle {{\mu }_{{K3}}}\int{{{{{\tilde{F}}}^{D}}_{C}}}\equiv \frac{1}{{4\pi {{g}_{s}}g_{7}^{2}}}\int{{{{C}_{{\left( 4 \right)}}}}}\int_{{K3}}{{{{{\tilde{F}}}^{D}}\wedge {{{\tilde{F}}}^{D}}}}$

and with coupling constants:

$\displaystyle \frac{{{{V}_{{K3}}}{{R}^{4}}}}{{g_{7}^{2}}}={{T}_{7}}{{\left( {2\pi {\alpha }'} \right)}^{2}}{{V}_{{K3}}}{{R}^{4}}=\frac{{{{V}_{{K3}}}{{R}^{4}}}}{{{{{\left( {2\pi } \right)}}^{4}}{{{{\alpha }'}}^{2}}}}\frac{1}{{g_{3}^{2}}}=\frac{1}{{\tilde{g}_{3}^{2}}}$

with our four-form given by:

$\displaystyle {{C}_{{\left( 4 \right)}}}={{g}_{s}}\pi {{R}^{{-12}}}\text{Vo}{{\text{l}}_{{\left( 4 \right)}}}/2$

thus the $K3$ -Chern-Simons term becomes:

$\displaystyle \int{{\text{Vo}{{\text{l}}_{{\left( 4 \right)}}}{{R}^{{-12}}}\left( {\frac{{-1}}{{8g_{7}^{2}}}} \right)\int_{{K3}}{{{{{\tilde{F}}}^{D}}^{-}}}\wedge *{{{\tilde{F}}}^{D}}^{-}}}$

with:

$\displaystyle {{\tilde{F}}^{D}}^{-}\equiv \left( {{{{\tilde{F}}}^{D}}-{{*}_{{K3}}}{{{\tilde{F}}}^{D}}} \right)$

Now since the $SL\left( {2,\mathbb{Z}} \right)$ invariant 5-form is self-dual in 10-D, there must be a 4-form field in all 10-dimensions. Hence, our action becomes:

$\displaystyle {{S}_{{D7}}}=\int_{\mathcal{W}}{{\tilde{\lambda }}}\left( {\tilde{\Theta }-\tilde{\Omega }} \right)-\tilde{p}\int_{{K3}}{{{{{\tilde{V}}}_{\alpha }}}}$

with:

$\displaystyle \tilde{\lambda }\equiv {{d}^{4}}x\sqrt{{-{{g}_{E}}}}$

$\displaystyle \tilde{\Theta }\equiv -\frac{1}{{4\tilde{g}_{3}^{2}}}{{\left( {{{F}_{{\mu \nu }}}} \right)}^{2}}$

$\displaystyle \tilde{\Omega }\equiv -\frac{{{{R}^{{-4}}}}}{{2\tilde{g}_{3}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{\left( {{{\partial }_{\mu }}{{X}^{i}}} \right)}^{2}}$

$\displaystyle \tilde{p}\int_{{K3}}{{{{{\tilde{V}}}_{\alpha }}}}\equiv {{\int_{\mathcal{W}}{{\text{Vol}}}}_{{\left( 4 \right)}}}{{R}^{{-12}}}\frac{1}{{8g_{7}^{2}}}\int_{{K3}}{{{{{\tilde{F}}}^{D}}\wedge {{{\tilde{F}}}^{D}}}}$

Adding coincident $D7$ -branes forces us to generalize the connection ${{A}_{\mu }}$ with corresponding Chan-Patton $U\left( {{{N}_{7}}} \right)$ gauge fields and a Yukawa-quiver gauge-theory describing the system.

After embedding the $D3$ -brane in the same metric and Ramond-Ramond system, we get the following $D3$ -action:

$\displaystyle {{S}_{{D3}}}\int{{{{G}_{E}}}}\left[ {\Xi -\xi } \right]$

with:

$\displaystyle {{G}_{E}}\equiv {{d}^{4}}x\sqrt{{-{{g}_{E}}}}$

$\displaystyle \Xi \equiv \frac{{-1}}{{4g_{3}^{2}}}{{\left( {{{{{F}'}}_{{\mu \nu }}}} \right)}^{2}}$

$\displaystyle \xi \equiv \frac{{{{R}^{{-4}}}}}{{2g_{3}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}\left( {{{{\left( {{{\partial }_{\mu }}{{{{X}'}}^{n}}} \right)}}^{2}}+{{{\left( {{{\partial }_{\mu }}{{{{X}'}}^{m}}} \right)}}^{2}}} \right)$

Hence, the $D3$ -modified total action is:

$\displaystyle S=\int{{\wp \left[ {\Pi -{\mathrm Z}-\Omega -\Xi -{\mathrm T}+\Upsilon -\Theta -{\mathrm E}} \right]}}$

with:

$\displaystyle \wp \doteq {{d}^{4}}x\sqrt{{-{{g}_{E}}}}$

$\displaystyle \Pi \doteq \frac{1}{{4\tilde{g}_{3}^{2}}}{{\left( {{{F}_{{\mu \nu }}}} \right)}^{2}}$

$\displaystyle {\mathrm Z}\doteq \frac{{{{R}^{{-4}}}}}{{2\tilde{g}_{3}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}{{\left( {{{\partial }_{\mu }}{{X}^{i}}} \right)}^{2}}$

$\displaystyle \Omega \doteq {{R}^{{12}}}\frac{1}{{8g_{7}^{2}}}\int_{{K3}}{{{{{\tilde{F}}}^{D}}\wedge {{{\tilde{F}}}^{D}}^{-}}}$

$\displaystyle \Xi \doteq \frac{1}{{4g_{3}^{2}}}{{\left( {{{F}_{{\mu \nu }}}} \right)}^{2}}$

$\displaystyle {\mathrm T}\doteq \frac{{{{R}^{{-4}}}}}{{2g_{3}^{2}{{{\left( {2\pi {\alpha }'} \right)}}^{2}}}}\left( {{{{\left( {{{\partial }_{\mu }}{{{{X}'}}^{n}}} \right)}}^{2}}+{{{\left( {{{\partial }_{\mu }}{{{{X}'}}^{m}}} \right)}}^{2}}} \right)$

$\displaystyle \Upsilon \doteq {{R}^{6}}{{\left| {{{D}_{\mu }}\chi } \right|}^{2}}$

$\displaystyle \Theta \doteq {{R}^{{-10}}}{{\left( {\frac{{{{X}^{i}}-{{{{X}'}}^{i}}}}{{2\pi {\alpha }'}}} \right)}^{2}}{{\left| \chi \right|}^{2}}$

$\displaystyle {\mathrm E}\doteq \frac{{{{R}^{{12}}}\left( {g_{3}^{2}+\tilde{g}_{3}^{2}} \right)}}{2}{{\left( {{{\chi }^{\dagger }}{{\sigma }^{A}}\chi } \right)}^{2}}$

Note that the hypermultiplet covariant derivative is still of the form:

$\displaystyle {{D}_{\mu }}\chi =\left( {{{\partial }_{\mu }}+i{{A}_{\mu }}-i{{{{A}'}}_{\mu }}} \right)\chi$

hence, we can do the following gauge transformation:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {g{{W}_{\mu }}={{A}_{\mu }}-{{{{A}'}}_{\mu }}} \\ {g{{{{W}'}}_{\mu }}=\frac{{{{g}_{3}}}}{{{{{\tilde{g}}}_{3}}}}{{A}_{\mu }}+\frac{{{{{\tilde{g}}}_{3}}}}{{{{g}_{3}}}}{{{{A}'}}_{\mu }}} \\ {{{g}^{2}}=g_{3}^{2}+\tilde{g}_{3}^{2}} \end{array}} \right.$

consistent with:

$\displaystyle \frac{1}{{\tilde{g}_{3}^{2}}}{{F}^{2}}+\frac{1}{{g_{3}^{2}}}{{{F}'}^{2}}=F_{W}^{2}+{F}'_{W}^{2}$

Thus, our action now has the form:

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

Our string sectors and all our fields satisfy the required N = 1 chiral superfield-normalization condition and we have rigid N = 2 supersymmetry that gets naturally broken to N = 1 when coupled to gravity in D = 4. In part III of this series, we shall consider M5-brane effects and derive the P-term action.