Deriving 4D De Sitter Space from T-Branes via D7-Brane Action

By George Shiber Friday, December 16, 2016Sunday, August 18, 2019 In M Theory, Mathematical Physics, Mathematics, Physics, Super String Theory, Supersymmetry Calabi-Yau Manifolds, De Sitter Spacetime

Building on my earlier work on T-branes and F-theory, here I will show how De Sitter space emerges from an expansion of the D7-brane action around a T-brane background in the presence of 3-form supersymmetry breaking fluxes: this is crucial since de Sitter space is unstable in quantum gravity, while a D7-brane action resolves the instability. “T-branes are a non-abelian generalization of intersecting branes in which the matrix of normal deformations is nilpotent along some subspace”, and it is truly remarkable that “the simplest heterotic string compactifications are dual to T-branes in F-theory.” Recalling that the pull-back on the D7-brane worldvolume is given by:

${\rm{P}}{\left[ {{V_\mu }{\rm{d}}{z^\mu }} \right]_\alpha } = {V_\alpha } + \lambda {V_i}{\partial _\alpha }{\Phi ^i}$

where $\alpha$ is a coordinate on $\tilde S$ and the second quantized integral of the D-brane partition function for closed strings is given by:

$P_{{\rm{int}}}^{Dp} \equiv \not Z = \sum\limits_{\gamma = 0}^\infty {\underbrace {\int {{D^K}\gamma {{D'}^K}X{e^{S_{cld}^s}}} }_{{\rm{Topologies}}}}$

with a non-Abelian D-term:

$D_{\hat A}^K = \int_{\tilde S} {\rm{P}} \left[ {{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}} \wedge \sqrt {{{\rm{A}}^\prime }(\Gamma )/{{\overline {\rm{A}} }^\prime }({\rm{N}})} {\rm{ }}$

and

$\sqrt {{\rm A}'(\Gamma )/\bar {\rm A}'({\rm N})}$

is the first Pontryagin class-term, and $J$ is the flat space Kähler form:

$J = \underbrace {\frac{i}{2}{\rm{dx}} \wedge {\rm{d\bar x + }}\frac{i}{2}{\rm{dy}} \wedge {\rm{d\bar y}}}_{ = :\omega } + 2i{\rm{dz}} \wedge {\rm{d\bar z}}$

where $S_{cld}^s$ in:

$P_{{\rm{int}}}^{Dp} \equiv \not Z = \sum\limits_{\gamma = 0}^\infty {\underbrace {\int {{D^K}\gamma {{D'}^K}X{e^{S_{cld}^s}}} }_{{\rm{Topologies}}}}$

is given by:

$S_{cld}^s = - \frac{1}{{4\pi {\alpha ^\dagger }}}\int_{\partial E_S^5} {{d^2}\sigma d} \Omega {\left( {{\phi _{INST}}} \right)^2}\sigma \sqrt { - \gamma } \left( {\phi \left( {{{\bar X}^\mu }} \right)} \right.{R_{icci}} + {\gamma ^{\alpha \beta }}{\not \partial _\alpha }{X^\mu }{g_{\mu \nu }}\left( {{{\bar X}^\nu }} \right) + \frac{1}{{\sqrt { - \gamma } }}{\varepsilon ^{ - {c_{2n}}/{Y_k}\left( {{{\cos }^2}\varphi } \right)}}{\not \partial _\beta }{\bar X^\nu }{b_{\mu \nu }}{\left( {\bar X} \right)^2}$

The non-Abelian profiles for $\Phi$ and $A$ must satisfy the 7-brane functional equations of motion. Non-Abelian generalisation of:

${W^O} = \int_{{\Sigma _5}} {\rm{P}} \left[ {{\Omega _0} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}$

${D^K} = \int_{\tilde S} {\rm{P}} \left[ {{\mathop{\rm Im}\nolimits} {e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}$

are built up as follows. Write locally:

${\Omega _0} \wedge {e^B} = d\gamma$

and localize the integral in:

${W^O} = \int_{{\Sigma _5}} {\rm{P}} \left[ {{\Omega _0} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}$

as:

$\int_{\tilde S} {P\left[ \gamma \right]} \wedge {e^{\lambda F}}$

thus,

the non-Abelian generalisation of ${W^0}$ and ${D^K}$ have both the form of the D7-brane Chern-Simons action and hence satisfy the T-brane equation of motion

So effectively, we have a Kähler-equivalence of the derivatives in the pull-back with gauge-covariant ones, yielding:

${W^0} = \int_{\tilde S} {S{\rm{Tr}}} \left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}} \right] \wedge {e^{\lambda F}}} \right\}$

${D^K} = \int_{\tilde S} {S\left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}{\mathop{\rm Im}\nolimits} {e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}} \right\}}$

with $\iota \Phi$ the inclusion of the complex Higgs field $\Phi$ , and $S$ represents the symmetrization over gauge indices.

In this local description, the Higgs field is given by:

$\Phi \equiv \phi \frac{\partial }{{\bar \partial z}} + \bar \phi \frac{{\bar \partial }}{{\partial \bar z}}$

where $\phi$ is a matrix in the complexified adjoint representation of $G$ and $\bar \phi$ its Hermitian conjugate. Thus, locally, we have:

$\gamma \equiv z{\rm{d}}x \wedge {\rm{d}}y$

with:

$\iota \Phi \gamma = 0$

a Kähler coordinate expansion of $\gamma$ and gives us, after inserting it in:

${W^0} = \int_{\tilde S} {S{\rm{Tr}}} \left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}} \right] \wedge {e^{\lambda F}}} \right\}$

the following:

$\begin{array}{l}{W^0} = {\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\phi dx \wedge dy \wedge F} \right\} = \\{\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\iota \Phi \Omega \wedge F} \right\}\end{array}$

which is the exact 7-brane superpotential for F-theory and the integrand is independent of $\lambda$ , entailing that the F-term conditions are purely topological and in no need for $\alpha '$ -corrections

Let me now derive the T-brane/De Sitter emergence relation, which underlies a T/dS duality relation that plays a fundamental role in compactifications in F-theory

Hidden-sector D7-branes are properties of globally consistent compact Calabi-Yau manifolds due to tadpole cancellation. The orientifold involution thus generates $O$ -7-plane fixed points that possess a Ramond-Ramond D7-brane charge and is measured by $- 8\left[ {O7} \right]$ , where $\left[ {O7} \right]$ is the homology class of the four-cycle wrapped by the $O$ -7-plane. If we place 4 D7- branes on top of the $O$ -7-plane locus, we get charge-cancellation. Therefore, the gauge group of the D7-wordvolume is $SO(8)$ and by recombining the 4 D7-branes into a D7-brane wrapping the invisible-sector invariant divisor ${D_h}$ in the homology class $4\left[ {O7} \right]$ , we get an

$SU(2) \cong USp(2)$

stack whose branes support a flux $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over F}$ along the Cartan generator of $SU(2)$ :

$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over F} = \left( {2\pi \alpha '} \right)F - {\iota ^ * }B$

with ${\iota ^ * }B$ the pullback of the NS-NS $B$ -field on ${D_h}$ . The Freed-Witten anomaly cancels since the gauge flux $\left( {2\pi \alpha '} \right)F$ satisfies the quantization condition:

$\left( {2\pi \alpha '} \right)F + \frac{{{c_1}\left( {{D_h}} \right)}}{2} \in {H^2}\left( {{D_h},\mathbb{Z}} \right)$

and since ${c_1}\left( {{D_h}} \right)$ is an even Calabi-Yau integral two-form, the gauge flux is integrally quantized. Moreover, the moduli stabilisation anomaly can be cancelled by the Green-Schwarz mechanism. Moving on to string modulation, the closed string modulus ${T_j}$ with its real part parametrizing the Einstein-frame volume of ${D_j}$ in units of ${\ell _s} = 2\pi \sqrt {\alpha '}$ gets a flux-dependent $U(1)$ -charge:

${q_{hj}} = \frac{1}{{\ell _s^4}}\int_{{D_h}} {{{\hat D}_j}} \wedge \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over F}$

with ${\hat D_j}$ the two-form Poincaré dual to ${D_h}$ . Now, since $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over F}$ yields a moduli-dependent Fayet-Iliopoulos term:

$\begin{array}{l}\frac{{{\xi _h}}}{{M_P^2}} = \frac{{e - \phi /2}}{{4\pi \hat V}}\frac{1}{{\ell _s^4}}\int_{{D_h}} {J \wedge \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over F} = } \\\frac{1}{{2\pi }}\sum\limits_j {\frac{{{q_{hj}}}}{2}} \frac{{{t_j}}}{{\hat V}} = \frac{1}{{2\pi }}\sum\limits_j {{q_{hj}}\frac{{\partial K}}{{\partial {T_j}}}} \end{array}$

with $\phi$ the dilaton, $\hat V$ the Einstein-frame Calabi-Yau volume in units of ${\ell _s} = M_s^{ - 1}$ , and the Planck scale and string scale are related via

${M_s} = g_s^{1/4}{M_P}/\sqrt {4\pi \hat V}$

and the Kähler form expanded in a basis of (1,1)-forms ${\hat D_j}$ is given by $J = {t_i}{\hat D_j}$ , and let $K/M_P^2 = - 2{\rm{In}}\hat V$ be the tree-level Kähler potential for the $T$ -moduli.

Moreover, the number of $U(1)$ -D7-branes chiral fields with charge +2 is:

${I_{U(1)}} = \frac{2}{{\ell _s^2}}\int_{{D_h} \cap {D_h}} {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over F} }$

The resulting D-term potential is then given by:

and given the identity: ${f_h} = {T_h}/\left( {2\pi } \right)$ , the Kähler metric is then:

$K \supset \tilde K{\sum\limits_j {\left| {{\phi _j}} \right|} ^2} = \sum\limits_j {\frac{{{{\left| {{\phi _j}} \right|}^2}}}{s}}$

The vanishing D-term-condition is given by:

$\sum\limits_j {\frac{{{q_{{\phi _j}}}}}{s}} {\left| {{\phi _j}} \right|^2} = \xi \left( {{\tau _i}} \right)$

since the Kähler moduli is ${T_i} = {\tau _i} + i{\psi _i}$ . Hence, the anomalous $U(1)$ acquires a mass term:

$M_{U(1)}^2 \simeq \frac{{M_P^2}}{{{\mathop{\rm Re}\nolimits} \left( {{f_{D7}}} \right)}}\left( {f_\theta ^2 + f_\psi ^2} \right)$

Both, $f_\theta ^2$ and $f_\psi ^2$ are proportional to the open and closed string axion decay constants ${f_\phi }$ and ${f_\psi }$ given by:

$f_\theta ^2 = {\left| \phi \right|^2} \simeq \xi \simeq \frac{{\partial K}}{{\partial \tau }}M_P^2$

and

$f_\psi ^2 \simeq \frac{{{\partial ^2}K}}{{\partial {\tau ^2}}}M_P^2 \simeq {\xi ^2}$

Now, since the relation:

$\sum\limits_j {\frac{{{q_{{\phi _j}}}}}{s}} {\left| {{\phi _j}} \right|^2} = \xi \left( {{\tau _i}} \right)$

fixes $\left| \phi \right|$ while leaving $\tau$ as a flat direction, the mass of the anomalous $U(1)$ is of order of the Kaluza-Klein scale:

$\begin{array}{c}{M_{U(1)}} \simeq \frac{{{f_\theta }}}{{{\tau ^{1/2}}}}{M_P}\\ \simeq \\\frac{{{M_P}}}{{{{\hat V}^{2/3}}}}\\ \simeq \\\frac{{{M_s}}}{{{{\hat V}^{1/6}}}}\\ \simeq \\\frac{1}{{{\rm{Vo}}{{\rm{l}}^{1/6}}}}\end{array}$

given that the Planck scale ${M_P}$ , via dimensional reduction, is related to the string scale ${M_s}$ as ${M_P} \simeq {M_s}{\hat V^{1/2}}$ . An easy substitution of:

$\frac{{{{\left| \phi \right|}^2}}}{s} = \frac{\xi }{{{q_\phi }}} = \frac{{{q_T}}}{{{q_\phi }}}\frac{t}{{4\pi \hat V}}M_P^2$

into the F-term scalar potential for the matter field $\phi$ yields a moduli-dependent positive definite contribution to the total scalar potential, and is an uplifting term: actually, the F-term potential for $\phi$ comes from supersymmetry breaking that generates scalar soft masses of the form:

$m_0^2 = m_{3/2}^2 - {F^I}{F^{\bar J}}{\partial _I}{\partial _{\bar J}}{\rm{In}}\tilde K$

with ${m_{3/2}} = {e^{K/2}}\left| W \right|$ the gravitino mass given in terms of the 4D Kähler potential $K$ and superpotential $W$ . Since the Kähler metric for D7 matter fields depends only on the axio-dilaton $S$ that is fixed supersymmetrically by turning on 3-form background fluxes ${H_3}$ and ${F_3}$ , with ${F^S} = 0$ , we have the following relation:

$m_0^2 = m_{3/2}^2 > 0$

Now, define $\varphi \equiv {s^{ - 1/2}}\phi$ , and we get the uplifting term:

${c_{Up}} = {e^{{K_{cs}}}}{\left( {\frac{6}{k}} \right)^{1/3}}\frac{{{q_T}}}{{{q_\phi }}}\frac{{{{\left| W \right|}^2}}}{{8\pi s}} > 0$

We need now to show how the uplifting term can be combined with the effects which determine the Kähler moduli $T$ to get in the end a viable dS vacuum. The key in freezing the $T$ -moduli is the presence of a non-perturbative superpotential ${W_{np}}$ generated by gaugino condensation on D7-branes: equivalently, E3-instantons.

If an E3-instanton wraps the divisor ${D_{E3}}$ with no gauge flux, then the number of these $E3$ zero-modes is given by:

${I_{D7 - E3}} = \frac{1}{{\ell _s^2}}\int_{{D_h} \cap {D_{E3}}} {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over F} }$

Considering the simple situation, without loss of generality, of a matter field $\phi$ and a Kähler modulus $T$ with the following U(1) transformations:

$\left\{ {\begin{array}{*{20}{c}}{\delta \phi = {\rm{i}}{q_\phi }\phi }\\{\delta T = {\rm{i}}\frac{{{q_T}}}{{2\pi }}}\end{array}} \right.$

The non-perturbative superpotential is then given by:

${W_{np}} = A{\phi ^n}{e^{ - \frac{{2\pi }}{N}T}}\quad ;\,\;\quad {M_P} = 1$

with $U(1)$ -transformation:

$\begin{array}{c}\delta {W_{np}} = {W_{np}}\left( {n\frac{{\delta \phi }}{\phi } - \frac{{2\pi }}{N}\delta T} \right)\\ = {\rm{i}}{W_{np}}\left( {n{q_\phi } - \frac{{{q_T}}}{N}} \right)\end{array}$

Therefore, ${W_{{\rm{np}}}}$ is $U(1)$ -invariant iff:

$n{q_\phi } = \frac{{{q_T}}}{N}$

which is a condition that allows us to obtain a dS vacuum in models where $T$ is fixed by non-perturbative effects, as in typically, KKLT models. A KKLT-like $F$ -term-based vacua using hidden matter fields that are non-zero due to $D$ -term stabilisation faces the following problem:

since

$n{q_\phi } = \frac{{{q_T}}}{N}$

is equivalent to:

${q_\phi }\frac{{\partial W}}{{\partial \phi }}\phi + \frac{{{q_T}}}{{2\pi }}\frac{{\partial W}}{{\partial T}} = 0$

which, given the following:

$\left\{ {\begin{array}{*{20}{c}}{{D_I}W = {W_I} + W{K_I}}\\{{V_D} = {D^2}/\left( {2{\mathop{\rm Re}\nolimits} \left( {{f_i}} \right)} \right)}\end{array}} \right.$

reduces to:

$\begin{array}{l}{q_\phi }\phi \frac{{{D_\phi }W}}{W} + \frac{{{q_T}}}{{2\pi }}\frac{{{D_T}W}}{W}\\ = \left( {{q_\phi }\phi \frac{{\partial K}}{{\partial \phi }} + \frac{{{q_T}}}{{2\pi }}\frac{{\partial K}}{{\partial T}}} \right)\\ = D\end{array}$

exhibiting the proportionality of the $D$ -term and $F$ -terms.

Hence, the following conditions characterize the KKLT vacua:

$\left\{ {\begin{array}{*{20}{c}}{{D_T}W = 0}\\{D = 0}\\{{D_\phi }W = 0}\end{array}} \right.$

However, in the large volume scenario (LVS), where we have: ${D_T}W \ne 0$ , ${D_\phi }W$ is an uplifting term even if $D = 0$ is at leading order.

dS LVS vacua

Take the LVS vacua QCD hidden sector $D$ 7-branes and $E$ 3-instantons where $D$ and $E$ are given by:

$K = - 2{\rm{In}}\left( {\tilde V + \frac{{\xi {s^{3/2}}}}{2}} \right) + c\frac{{\phi \bar \phi }}{s}$

and

$W = {W_0} + {A_s}{e^{ - {a_s}{T_s}}}$

with:

$\zeta = - \frac{{\chi \left( {CY} \right)\zeta \left( 3 \right)}}{{2{{\left( {2\pi } \right)}^3}}}$

a constant dominating the leading order $\alpha '$ -correction and the CY volume is a function of the big cycle ${\tau _b}$ and the small cycle ${\tau _s}$ as:

$\tilde V = {\lambda _b}\tau _b^{3/2} - {\lambda _s}\tau _s^{3/2}$

with ${\lambda _b}$ and ${\lambda _s}$ depending on the Kähler triple intersection numbers.

Now, to generate a non-vanishing ${T_s}$ -dependent contribution to $W$ , the $B$ -field must be picked as to cancel the Freed-Witten anomaly on the ${T_s}$ , which yields a nonvanishing gauge flux on the stack of hidden sector $D$ 7-branes wrapping the big cycle ${T_b}$ . Hence, giving ${T_b}$ a non-zero $U(1)$ -charge ${q_b}$ generating a moduli-dependent FI-term whose $D$ -term potential is:

${\tilde V_D} = \frac{\pi }{{{\tau _b}}}{\left( {\frac{{{q_\phi }}}{s}{{\left| \phi \right|}^2} - {\xi _b}} \right)^2}$

with the FI-term being:

${\xi _b} = - \frac{{{q_b}}}{{2\pi }}\frac{{\partial K}}{{\partial {T_b}}} = \frac{{3{q_b}}}{{4\pi {\tau _b}}}$

Hence, the total scalar potential is given by:

$\begin{array}{l}{{\tilde V}_{{\rm{tot}}}} = {{\tilde V}_D} + {{\tilde V}_F} = \frac{\pi }{{{\tau _b}}}{\left( {\frac{{{q_\phi }}}{s}{{\left| \phi \right|}^2} - \frac{{3{q_\phi }}}{{4\pi {\tau _b}}}} \right)^2}\\ + \frac{1}{s}m_{3/2}^2{\left| \phi \right|^2} + \frac{{{{\tilde V}_F}}}{{2s}}\end{array}$

with:

$m_{3/2}^2 = {e^K}\left| W \right| \simeq W_0^2/\left( {2s{{\tilde V}^2}} \right)$

being the gravitino mass and ${V_F}\left( T \right)$ is given by:

${\tilde V_F}\left( T \right) = \frac{8}{{3{\lambda _s}}}{\left( {{a_s}{A_s}} \right)^2}\sqrt {{\tau _s}} \frac{{{e^{ - 2{a_s}{\tau _s}}}}}{{\tilde V}}$

Minimizing with respect to $\phi$ gives us:

$\frac{{{q_\phi }}}{s}{\left| \phi \right|^2} = {\xi _b} - \frac{{m_{3/2}^2{\tau _b}}}{{2\pi {q_\phi }}}$

thus yielding:

${\tilde V_{{\rm{tot}}}} = \frac{{{g_s}}}{2}\left[ {{c_{\rm{L}}}\frac{{W_0^2}}{{{{\tilde V}^{8/3}}}}\left( {1 - \frac{{{c_{{\rm{sb}}}}}}{{{V^{2/3}}}}} \right) + {{\tilde V}_F}\left( T \right)} \right]$

with:

$\left\{ {\begin{array}{*{20}{c}}{{c_{\rm{L}}} = \frac{{3{q_b}\lambda _b^{2/3}}}{{4\pi {q_\phi }}}}\\{{c_{{\rm{sb}}}} = \frac{{W_0^2{g_s}}}{{6{q_\phi }{q_b}\lambda _b^{4/3}}}}\end{array}} \right.$

and the first term in:

is the uplifting term:

${\tilde V_{\rm{L}}} = \frac{{{g_s}{c_{\rm{L}}}}}{2}\frac{{W_0^2}}{{{{\tilde V}^{8/3}}}}$

and modifies the scalar potential in the sense that it admits a dS minimum, which is expressed, in terms of ${\tau _s}$ , as:

${V^{vol}} = \frac{{3{\lambda _s}{W_0}\sqrt {{\tau _s}} }}{{4{a_s}{A_s}}}{\varepsilon ^{{a_s}{\tau _s}}}\frac{{\left( {1 - \varepsilon } \right)}}{{\left( {1 - \varepsilon /4} \right)}}$

with:

$\varepsilon \equiv \frac{1}{{{a_s}{\tau _s}}} \sim \vartheta \left( {\frac{1}{{{\rm{In}}\tilde V}}} \right) \ll 1$

Substituting into:

the effective volume-potential reduces to:

$\begin{array}{l}{{\tilde V}_{{\rm{tot}}}}\left( {{V^{vol}}} \right) = \frac{{{g_s}W_0^2}}{{2{V^{vol}}^3}}\left[ {{c_{\rm{L}}}{V^{vol}}^{1/3}} \right. - \\\frac{{3{\lambda _3}}}{{2a_s^{3/2}}}{\varepsilon ^{ - 3/2}}\left( {1 - \left( {\frac{{3\varepsilon }}{4}} \right)} \right) + \left. {\frac{{3\zeta }}{{4g_s^{3/2}}}} \right]\end{array}$

with:

$\begin{array}{l}{\varepsilon ^{ - 1}} = {\rm{In}}\left( {{c_{{\rm{sb}}}}\frac{{{V^{vol}}}}{{{W_0}}}} \right) - {\rm{In}}\sqrt {{\tau _s}} + {\rm{In}}\frac{{\left( {1 - \varepsilon } \right)}}{{\left( {1 - \varepsilon /4} \right)}}\\ = {\rm{In}}\left( {{c_{{\rm{sb}}}}\frac{{{V^{vol}}}}{{{W_0}}}} \right)\left[ {1 - \frac{\varepsilon }{2}{\rm{In}}{\tau _s} + \vartheta \left( {{\varepsilon ^2}} \right)} \right]\end{array}$

and ${c_{\rm{L}}} = 4{a_s}{A_{s/}}\left( {3{\lambda _s}} \right)$ .

Now, since we have ${\rm{In}}{\tau _s} \sim \vartheta (1)$ , we get a re-scaling:

$\begin{array}{c}{\varepsilon ^{ - 1}} = {\rm{In}}\left( {{c_{{\rm{sb}}}}\frac{{{V^{vol}}}}{{{W_0}}}} \right) - \frac{1}{2}{\rm{In}}\left[ {\frac{1}{{{a_s}}}{\rm{In}}\left( {{c_{{\rm{sb}}}}\frac{{{V^{vol}}}}{{{W_0}}}} \right)} \right]\\ + \vartheta (1)\end{array}$

Let us see how all this fits in together. Turn now to the vacuum energy tuning. Minimising the potential:

with respect to ${V^{vol}}$ yields:

$\begin{array}{l}\frac{{3\zeta }}{{4g_s^{3/2}}} = \frac{{3{\lambda _s}}}{2}\tau _s^{3/2}\left[ {1 - \frac{\varepsilon }{2}\vartheta \left( {{\varepsilon ^2}} \right)} \right]\\ - \frac{8}{9}{c_{\rm{L}}}{V^{vo{l^{1/3}}}}\end{array}$

and when we substitute it in ${\tilde V_{{\rm{tot}}}}\left( {{V^{vol}}} \right)$ above, gives the vacuum energy expectation value of the form:

$\left\langle {{{\tilde V}_{{\rm{tot}}}}} \right\rangle = \frac{{{g_s}}}{{18}}\frac{{W_0^2}}{{{V^{vol}}}}\left[ {{c_{\rm{L}}}{V^{vol}}^{1/3} - \frac{{27{\lambda _s}}}{{4a_s^{3/2}}}\left( {1 - \frac{9}{8}\varepsilon + \vartheta \left( {{\varepsilon ^2}} \right)} \right)} \right]$

Setting it to zero and substituting the result back in:

one gets:

${\tau _s} = {\left( {\frac{\zeta }{{2{\lambda _s}}}} \right)^{2/3}}\frac{1}{{{g_s}}}\left[ {1 + 3\varepsilon + \vartheta \left( {{\varepsilon ^2}} \right)} \right]$

Localizing the dS vacuum for ${c_{{\rm{sb}}}} = 0$ , we find that the shift of the vacuum expectation value (VEV) of ${\tau _s}$ is proportional to $\varepsilon$ :

$\frac{{\Delta {\tau _s}}}{{{\tau _s}\left| {_{{c_{\rm{L}}} = 0}} \right.}}\frac{{8\varepsilon }}{3}\left( {1 + \vartheta \left( \varepsilon \right)} \right)$

Now, plugging into:

${V^{vol}} = \frac{{3{\lambda _s}{W_0}\sqrt {{\tau _s}} }}{{4{a_s}{A_s}}}{\varepsilon ^{{a_s}{\tau _s}}}\frac{{\left( {1 - \varepsilon } \right)}}{{\left( {1 - \varepsilon /4} \right)}}$

the volume vacuum expectation value is then:

$\frac{{\Delta {V^{vol}}}}{{{V^{vol}}\left| {_{{c_{\rm{L}}} = 0}} \right.}} = {\varepsilon ^{8/3}}\left( {1 + \vartheta \left( \varepsilon \right)} \right) - 14.4\left( {1 + \vartheta \left( \varepsilon \right)} \right) - 1$

thus matching the dS vacuum 4D effective action as factored in the T-brane worldvolume, as I derived it here:

$\begin{array}{*{20}{l}}{^{\rm{T}}{W^0} = {\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\phi dx \wedge dy \wedge F} \right\} = }\\{{\lambda ^2}\int_{{{\tilde S}^{\rm{T}}}} {{\rm{Tr}}} \left\{ {\iota \Phi \Omega \wedge F} \right\}}\end{array}$

and solving the D7-brane’s action:

$^7{D^K} = \int_{{{\tilde S}^7}} {S\left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}} \right\}}$

by Kählericity, the dS VEV can be derived from the RR VEV of the T-brane‘s worldvolume, where the uplift term in the 4D potential is:

${\tilde V_{{c_{\rm{L}}}}} = \frac{{{C_{{c_{\rm{L}}}}}}}{{{V^{vo{l^{8/3}}}}}}M_P^4$

Deriving 4D De Sitter Space from T-Branes via D7-Brane Action

$\int_{\tilde S} {P\left[ \gamma \right]} \wedge {e^{\lambda F}}$

the non-Abelian generalisation of ${W^0}$ and ${D^K}$ have both the form of the D7-brane Chern-Simons action and hence satisfy the T-brane equation of motion

which is the exact 7-brane superpotential for F-theory and the integrand is independent of $\lambda$ , entailing that the F-term conditions are purely topological and in no need for $\alpha '$ -corrections

${q_{hj}} = \frac{1}{{\ell _s^4}}\int_{{D_h}} {{{\hat D}_j}} \wedge \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over F}$

${I_{U(1)}} = \frac{2}{{\ell _s^2}}\int_{{D_h} \cap {D_h}} {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over F} }$

$\sum\limits_j {\frac{{{q_{{\phi _j}}}}}{s}} {\left| {{\phi _j}} \right|^2} = \xi \left( {{\tau _i}} \right)$

$\sum\limits_j {\frac{{{q_{{\phi _j}}}}}{s}} {\left| {{\phi _j}} \right|^2} = \xi \left( {{\tau _i}} \right)$

${I_{D7 - E3}} = \frac{1}{{\ell _s^2}}\int_{{D_h} \cap {D_{E3}}} {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over F} }$

$n{q_\phi } = \frac{{{q_T}}}{N}$

$n{q_\phi } = \frac{{{q_T}}}{N}$

dS LVS vacua

The Lindblad Collapse Equation and Quantum Decoherence

The Cosmological Quantum State from Deformation Quantization

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the non-Abelian generalisation of and have both the form of the D7-brane Chern-Simons action and hence satisfy the T-brane equation of motion

which is the exact 7-brane superpotential for F-theory and the integrand is independent of , entailing that the F-term conditions are purely topological and in no need for -corrections

dS LVS vacua

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the non-Abelian generalisation of ${W^0}$ and ${D^K}$ have both the form of the D7-brane Chern-Simons action and hence satisfy the T-brane equation of motion

which is the exact 7-brane superpotential for F-theory and the integrand is independent of $\lambda$ , entailing that the F-term conditions are purely topological and in no need for $\alpha '$ -corrections