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May 3, 2004

Digging up the Landscape

Much excitement has been generated by the work of KKLT. At least for one class of compactifications down to 4 dimensions (F-theory backgrounds with fluxes) we seem to have the physics which lifts the degeneracy of the moduli space under good control.

I say seem, because there are some important gaps, one of which got filled today.

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, ρ i\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 ρ i\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.

    σ\sigma-model (α\alpha') perturbative corrections to the Kähler potential could also ruin the no-scale structure and generate a potential for the ρ i\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 3S^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, BB of the F-theory 4-fold XπBX\overset{\pi}{\longrightarrow} B. Or, more generally, it could come from gaugino condensation1, again associated with a divisor on BB. 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=aDk=a D. Since the volume of DD must be positive and the volume of BB must be positive, we have 0< Dk 2=a 2D 3 0 \lt \int_D k^2 = a^2 D^3 and 0< Bk 3=a 3D 3 0 \lt \int_B k^3 = a^3 D^3 from which we conclude a>0a\gt 0. But kk is effective (all curves have positive area), and hence so must be DD.

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=𝒫 1×SB= \mathcal{P}^1\times S, where SS is a del Pezzo surface has a divisor of the form pt×S\text{pt}\times S which is NEF and is not in the span of the non-NEF divisors, which are all of the form 𝒫 1×C\mathcal{P}^1\times C, where CC is a curve in SS.

But there’s still a fairly decent list of choices of BB 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 BB is a toric variety, one with h 1,1=3h^{1,1}=3 and one with h 1,1=5h^{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.


1 If π 1(D)X\pi^{-1}(D)\in X has arithmetic genus 1, then instantons on DD generate a superpotential. If it has arithmetic genus >1\gt 1, then one does not get an instanton-induced superpotential, but presumably one does get one from gaugino condensation (sometimes known as “fractional instantons”).

2 A NEF divisor on BB has non-negative intersection with every curve on BB.

3 A Fano variety is one whose anticanonical bundle is ample. In two complex dimensions, Fano varieties are called del Pezzo surfaces.

Posted by distler at May 3, 2004 6:28 PM

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Re: Digging up the Landscape

What of the “barren landscape” of Robbins and Sethi? Does this just mean that they didn’t work quite as hard as Denef et al?

Not that I’m able to follow all the details of “DDF”… e.g. “The large complex structure prepotential of the original Calabi-Yau is given by this expression with the x^i replaced by complex variable, and takes the form F = t_1^3 - t_2^2t_3 + 780 more terms.” or
“… this involves running over 118C88 ~ 10^23 candidate generators. It would take about the age of the Universe to complete this task on a PC.”

But they make it clear in the conclusions that there is still some distance to go before the program is realized.

Posted by: Thomas Dent on May 5, 2004 4:47 AM | Permalink | Reply to this

Barren

What of the “barren landscape” of Robbins and Sethi?

Well, lessee … (flips through the paper which he hasn’t had a chance to look through yet) … they have

  • a proof for the one-parameter case (the above 2-line proof, stretched out over several pages)
  • some words about bases with h 1,1>1h^{1,1}\gt 1, but no list

Jae and I, at least, went through the list of Fano 3-folds of Mori and Mukai to see which ones could potentially generate a superpotential. But, when we saw Denef et al, we realized that even that was already done in an old paper by Antonella Grassi.

Sethi and Robbins don’t appear to have done even that much.

But they [Denef et al] make it clear in the conclusions that there is still some distance to go before the program is realized.

That’s certainly true …

Posted by: Jacques Distler on May 5, 2004 8:14 AM | Permalink | PGP Sig | Reply to this

Re: Digging up the Landscape

Someone just informed me about your comments on our recent paper. On briefly looking at your comments, I suggest you reread our paper.

First, there are no examples of compactifications with all moduli stabilized at large volume. No-one has come even remotely close. That is the burden to be met by anyone claiming to have a construction. Your discussion of the superpotential is wrong. You omit two key points. First, you cannot wrap random divisors in the presence of flux. There is a constraint (see, our paper for an explanation). Second, you cannot trivially wrap non-spin (but spin_c) divisors without turning on a gauge-bundle. Many of the examples of attempted Kahler stabilization to which you allude involve non-spin divisors. This changes the instanton analysis. Gaugino condensation is largely irrelevant. It can be analyzed in the same way as the abelian instanton story (again see, our paper).

We derived a necessary (but not sufficient) expression for a divisor on B to contribute (for 4-folds with h^{1,1} \geq 2 but h^{1,1}(B)=1, there is no possible contribution). It is extremely hard to find any example with all moduli stabilized (our claim is that any such examples, should any exist, are highly non-generic). If you can construct one example with all moduli stabilized, we would be happy to examine it.

- Savdeep Sethi

Posted by: Savdeep Sethi on May 10, 2004 6:00 PM | Permalink | Reply to this

Barren

First, there are no examples of compactifications with all moduli stabilized at large volume. No-one has come even remotely close.

I would call the Rutgers paper “remotely close.” You may disagree.

First, you cannot wrap random divisors in the presence of flux.

I believe you are referring to the well-known constraint, most easily expressed in the M-theory language, that the restriction of the 4-form flux to the divisor DD (here I mean a divisor on the CY 4-fold, XX) be cohomologically trivial: [G]| D=0[G]|_{D}=0.

Second, you cannot trivially wrap non-spin (but spinc) divisors without turning on a gauge-bundle.

The world-volume spinors are spinors of the normal bundle as well. Since the total 10-dimensional space is spin, there is no additional constraint from whether the divisor on BB (or its inverse-image in XX) is spin.

This is also a familiar fact.

[F]or 4-folds with h 1,12h^{1,1} \geq 2 but h 1,1(B)=1h^{1,1}(B)=1, there is no possible contribution.

See the 2-line proof above.

Doubtless, I’ve missed some essential point. In which case, I hope you’ll straighten me out.

Posted by: Jacques Distler on May 10, 2004 9:11 PM | Permalink | PGP Sig | Reply to this

Re: Barren

You are right, you have missed an essential point. There is a global anomaly both in M theory (hep-th/9912086) and string
theory (hep-th/9907189) for branes wrapping non-spin spaces. All my prior comments apply. If you think a construction is remotely close to realization, why don’t you go ahead and build one.

I think you will find it much harder than you realize to put together all of the constraints in a model that exists at large volume. I spent a great deal of time studying and constructing the first F-theory compactifications with flux (hep-th/9908088), and I am not sanguine (although I am still thinking about it).

I find it interesting that all the recent attempts to construct examples (and most discussions about this proposal) prior to our paper and Denef et. al. have assumed that Kahler stabilization is possible for 1 Kahler class models, either by gaugino condensation or by abelian instantons. Let me close (and this is my final comment) by noting that there are no new ingredients involved in this proposal, only a new claim. The burden is to actually realize the claim.

Posted by: Savdeep Sethi on May 11, 2004 1:02 PM | Permalink | Reply to this

Global anomalies

You are right, you have missed an essential point. There is a global anomaly both in M theory (hep-th/9912086) and string theory (hep-th/9907189) for branes wrapping non-spin spaces.

I don’t see the relevance of these papers to the matter at hand.

There is no anomaly associated to DD not being spin. If DD is not spin, then S S_- does not exist, and S(ν)S(\nu) does not exist (ν\nu is the normal bundle), but S S(ν)S_-\otimes S(\nu) always exists and makes sense. That’s all one needs because, as I said, the world-volume fermions are sections of S S(ν)S_-\otimes S(\nu).

The Freed-Witten anomaly is a global anomaly when the submanifold, QQ on which you wish to wrap the D-brane does not admit a spinc structure. The obstruction to defining a spinc structure is W 3(Q)W_3(Q), and the condition of Freed and Witten is that [H]| Q=W 3(Q)[H]|_Q= W_3(Q).

The Freed-Witten anomaly is irrelevant to the matter at hand, because the divisors on BB on which we wish to wrap D3-branes are complex surfaces, and W 3=0W_3=0 for any compact, oriented Euclidean 4-manifold. There is never an obstruction to defining a spinc structure for these D-instantons (neither is there a corresponding problem in M-theory).

Posted by: Jacques Distler on May 11, 2004 3:24 PM | Permalink | PGP Sig | Reply to this

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