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In flux compactifications to 4D, e.g. Type IIB on a CY orientifold $X$, one uses fluxes to stabilize the axio-dilaton $\tau$ and the complex structure moduli $z_a$ - the periods of the holomorphic three-form $\Omega$ over the basis three-cycles: $z_a=\int_{\alpha_a}\Omega$, by solving a system of equations for a supersymmetric extremum: $\partial_{\tau}W+W\partial_{\tau}K=0$, $\partial_{z_a}W+W\partial_{z_a}K=0$, where $W$ is the flux superpotential $W=\int_X G_3\wedge\Omega$ and $K$ is the Kahler potential. Since in Type IIB the supergravity scalar potential has a no-scale structure, i.e. $V=e^KD^iW{\bar D_i \bar W}$ where $i$ runs over the complex structure and the axio-dilaton, the potential is positive-definite and therefore the supersymmetric extremum appears as an actual minimum, although the Kahler moduli are still unfixed. However, once we include the non-perturbative corrections to $W$ to stabilize the Kahler moduli and break supersymmetry by e.g. an anti-D3 brane, the scalar potential is no longer positive-definite. So, my question is: is there a proof that the eigenvalues of the Hessian of the scalar potential with respect to all the moduli are positive-definite once all the moduli are fixed by the combination of fluxes and non-perturbative terms? In particular, there seems to be no good reason to believe that there are no tachyons in any of the complex structure directions.

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Dear stringpheno, in the KKLT model, the uplifting by the anti-D3-brane means an addition of a positive term $D/\sigma^3$ to the potential. Here, $\sigma$ is what appears in $\exp(-K/3)$.

It creates a new minimum for very small value of $\sigma$ because the coefficient $D$ is also expected (or required) to be small, and because the original second derivative near the zero was positively definite (a supersymmetric AdS vacuum can't have tachyons by supersymmetry) and $D/\sigma^3$ also has a positive second derivative, the second derivative at the new minimum - which is near the original $\sigma=0$ point - remains positively definite, too.

Of course, if one allows $D$ to become large, the sketched derivation above won't necessarily hold. There can be local maxima somewhere in the configuration space - like the electroweak-symmetry-preserving point of the Higgs potential - which have tachyonic directions but they're not considered "uplifting by anti-D3-branes".

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Dear @Luboš Motl, thank you very much for your answer. However, my question was more about the possibility of tachyons in the complex structure directions. –  stringpheno Mar 4 '11 at 18:38
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I think this is already implicit in Lubos's discussion, but to try to be more explicit: the assumption that's being made here is essentially just the usual logic of effective field theory and decoupling. The complex structure moduli will get large masses (which are maybe roughly of order the string scale times some small couplings), and those masses are not tachyonic to the extent that breaking of no-scale structure is small, by the argument you gave in your question. So you should imagine that you can supersymmetrically integrate them out in a consistent way, and then ignore them. The KKLT discussion proceeds after this point: it begins with a superpotential for the remaining Kähler moduli, assuming that there are small numbers involved ($W_0 \ll 1$ by tuning, and $A e^{-aT} \ll 1$ at the minimum dynamically, with similar smallness assumptions on SUSY breaking as Lubos mentioned). It is only to the extent that these remaining moduli fields are light compared to the complex structure moduli that you can consistently assume the CS moduli have been integrated out. (You can always play with a toy model where you keep a heavy modulus around and study how its VEV and mass change after turning on the no-scale breaking $Ae^{-aT}$ term; you will find that, provided you have a large hierarchy of scales in your model, the changes are small.)

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Dear @Matt Reece, thank you very much for your answer. What you say is basically the standard folklore but this is not as straightforward as it sounds. Let me give you a direct quote from iopscience.iop.org/1126-6708/2005/03/007: The scalar potential is $V=\frac{e^{K_{cs}}}{V^2}\left(G^{\mu\bar\nu}D_{\mu}W\bar D_{\bar\nu}\bar W-3|W|^2\right)$. If $D_{\mu}W=0$ for all moduli, the potential is negative at $O(1/V^{\,\,2})$. –  stringpheno Mar 11 '11 at 14:21
However, if we move one modulus, for concreteness the dilaton, away from its stabilised value, the resulting positive definite contribution $\frac{e^{K_{cs}}}{V^2}G^{\tau\bar\tau}D_{\tau}W\bar D_{\bar\tau}\bar W$ is only of the same order as the minimum. Moving the dilaton alters the value of $e^{K_{cs}}$ and thus may increase the numerator of the negative term. As the positive and negative contributions are of the same order, we see that depending on the magnitude of $G^{\tau\bar\tau}D_{\tau}W\bar D_{\bar\tau}\bar W$, this may in general decrease the overall value of the potential. –  stringpheno Mar 11 '11 at 14:22
Therefore it is necessary to check explicitly for each choice of fluxes that the resulting potential has a minimum. –  stringpheno Mar 11 '11 at 14:22
If you have a concrete model for $W$ and $K_{cs}$ that you're concerned about, we could calculate. But I maintain that if the complex structure directions are not minima at the point where the Kähler modulus is stabilized, it means they're being strongly influenced by the no-scale breaking $e^{-aT}$ terms and thus you never had a hierarchy of scales. –  Matt Reece Mar 11 '11 at 15:37
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The question is also dependent upon supersymmetry. For the case of $E_6$, relevant for the current “mini-twistor revolution,” positive condition appears straight forwards. The first Kahler form $K_1~=~ln(W(z)$ is computed by a three-form (or three $1,1$) forms from $h_{1,1}$ in the Hodge diamond, one element for a ${\bf\bar 27}$ of the $E_6$ $$ W(z)~=~\int\omega\wedge\omega\wedge\omega $$ The holomorphic metric on the $E_6$ is given by this potential $$ g_{a{\bar b}}~=~-\partial_{z^a}\partial_{{\bar z}^b}W(z) $$ The $h_{1,2}$ and $h_{2,1}$, given by $\bf 27$ of $E_6$ determine a second Kahler form $$ e^{-iK_2}~=~\int\Omega_3\wedge\Omega^*_3 $$ and the $\bf 27$ metric in generality is $g’_{a{\bar b}}~=~exp((k_2-K_1)/3) g_{a{\bar b}}$. The $3,0$ forms determine a superpotential as well, which is related to $K_2$ as the third Betti number is given by $h_{2,1}$, which is the cycle $int\Omega_{3,0}~=~w$ $K_2$ may be computed as well as $log(Im(z^a\partial_aw(z)))$, since $b_3~=~2h_{2,1}~+~2$. The superpotential is then the holomorphic function $$ W(z)~=~\phi^a\phi^b\phi^c\frac{\partial^3w(z)}{\partial z^a\partial z^b\partial z^c} $$ Supersymmetry in its unbroken phase will have a zero energy, which will hold for any topology of the CY manifold. Broken supersymmetry will have positive energy, which is then mapped to the CY manifold. The holomorphic functions are then given by the topology of the CY manifold. However, in the Hodge diamond elements are chiral fields of $E_6$ which are positive.

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