Musings

The KKLT story proceeds in three stages

1. The fluxes generate a nontrivial superpotential for the complex structure moduli and the dilaton. (Giddings, Kachru and Polchinski work near the orientifold limit, where there is a globally-defined “dilaton”). The no-scale structure of the tree-level Kähler potential leaves leaves the Kähler moduli, $\rho_i$ as flat directions in the scalar potential. At this stage we have a supersymmetric solution with 4D flat space.

2. D3-brane instantons or gaugino condensation generate a superpotential for the $\rho_i$, ruining the no-scale structure. This fixes the remaining moduli. But, since the vacuum energy is generically negative, we only have a supersymmetric solution in 4D anti-de Sitter space.

   $\sigma$-model ($\alpha'$) perturbative corrections to the Kähler potential could also ruin the no-scale structure and generate a potential for the $\rho_i$. But, almost by definition, since that trades-off different orders in perturbation theory, the minimum of such a potential will not be at large radius, where the supergravity approximation to the 10D geometry is valid. To stabilize the Kähler moduli at large radius, we do need to generate a superpotential for them.

3. Finally, KKLT introduce a supersymmetry-breaking effect, a mismatch in the flux-induced D3-brane charge, which necessitates the introduction of some anti-D3 branes to cancel the net charge. This makes a positive contribution to the vacuum energy, possibly leading to a (meta)stable 4D de Sitter space, or possibly to a runaway.

   Since this contribution to the vacuum energy is perturbative, it would ordinarily swamp the aforementioned nonperturbative contribution and lead to a runaway. To avoid this, KKLT assume that the complex structure is stabilized near a conifold locus, with the flux on the shrinking cycle leading to a large warp factor. The anti-D3 brane is located far down the “throat,” on the minimal $S^3$. Its contribution to the vacuum energy is suppressed by a power of the warp factor, and can hopefully be tuned to be comparable to the vacuum energy of the AdS.

One need not take the supersymmetry-breaking mechanism too seriously. There are lots of other effects which might break supersymmetry. This particular one has the advantage of being more-or-less calculable.

The hard part is generating the superpotential for the Kähler moduli. This can come from D3-brane instantons wrapped on a divisor on the base, $B$ of the F-theory 4-fold $X\overset{\pi}{\longrightarrow} B$. Or, more generally, it could come from gaugino condensation¹, again associated with a divisor on $B$. The trouble is, you need to generate a superpotential that lifts all the flat directions.

The only divisors which can generate such a superpotential must be non-NEF². To lift all the flat directions, you need the Kähler cone to be spanned by non-NEF hypersurfaces.

Several people have thought about this, including my student Jae Park and me. It’s rather hard. One parameter models won’t do it. With a single divisor class, the Kähler class must be a multiple $k=a D$. Since the volume of $D$ must be positive and the volume of $B$ must be positive, we have $$0 \lt \int_D k^2 = a^2 D^3$$ and $$0 \lt \int_B k^3 = a^3 D^3$$ from which we conclude $a\gt 0$. But $k$ is effective (all curves have positive area), and hence so must be $D$.

You can go down the list of Fano³ 3-folds, and it’s easy to eliminate most of them, as having a NEF divisor which is not in the span of the non-NEF hypersurfaces. For instance, $B= \mathcal{P}^1\times S$, where $S$ is a del Pezzo surface has a divisor of the form $\text{pt}\times S$ which is NEF and is not in the span of the non-NEF divisors, which are all of the form $\mathcal{P}^1\times C$, where $C$ is a curve in $S$.

But there’s still a fairly decent list of choices of $B$ for which a superpotential which lifts all the flat directions could be generated. But they’re all formidably complicated.

Jae and I weren’t strong enough to compute the superpotential for one of these examples, much less to show that the moduli were actually stabilized in the regime of validity of the supergravity analysis.

Luckily, other, more capable people were thinking along similar lines. Denef, Douglas and Florea came out with a paper today, in which they carry through the computation for two examples in which $B$ is a toric variety, one with $h^{1,1}=3$ and one with $h^{1,1}=5$.

The analysis is very involved. But the upshot is that they can, indeed stabilize all the moduli in these examples. But it wasn’t particularly easy, and it’s not clear that, having succeeded at step 2. of the above program, one has enough remaining freedom to break supersymmetry and achieve a metastable de Sitter vacuum.