**Moduli stabilization of F-theory flux vacua again**

There have been too many points in the talk to describe all of them here. They studied, among other things, all possible orientifolded and simultaneously orbifolded toroidal (T^6) vacua of type IIB string theory, their resolution, description in terms of toric geometry, flops, and especially the stabilization of the moduli. One of the unexpected insights was that one can't stabilize the Kähler moduli and the dilaton after the uplift to the de Sitter space if there are no complex structure moduli to start with; rigid stabilized anti de Sitter vacua may be found but can't be promoted to the positive cosmological constant case. Some possibilities are eliminated, some possibilities survive, if you require all moduli to be stabilized.

Recall that the complex structure moduli and the dilaton superfield are normally stabilized by the Gukov-Vafa-Witten superpotential - the integral of the holomorphic 3-form wedged with a proper combination of the 3-form field strengths - while the Kähler moduli are stabilized by forces that are not necessarily supernatural but they are non-perturbative which is pretty similar. The latter nonperturbative processes used to stabilize the Kähler moduli include either D3-brane instantons or gaugino condensation in D7-branes.

At this level, one obtains supersymmetric AdS4 vacua. Semirealistic dS4 vacua may be obtained by adding anti-D3-branes, but Susanne et al. do not deal with these issues.

**Nonperturbative superpotential: generated or not?**

One of the interesting yet controversial points was their usage of the non-perturbative terms in the superpotential to modify the Dirac equations for the fermionic zero modes. Recall that D3-brane instantons wrapped on holomorphic 4-cycles - or, in a dual M-theoretical description, M5-brane instantons wrapped on some six-cycles - modify dynamics as nonperturbative effects.

But they only contribute to the superpotential W if you can find exactly two fermionic zero modes - let's say zero modes of the goldstino or the modulino - which means two solutions of the massless Dirac equation defined on the worldvolume of the D3-brane or M5-brane instanton. This is typically the case if you start with a vacuum that is supersymmetric at the tree level. Such a condition of supersymmetry requires that the 4-form d=11 field strength is of the type (2,2).

However, funny things happen. If you now accept that this instanton exists, you will generate a new term in the superpotential, W_{np}. When you solve the new equations of motion that include W_{np}, you will see that it is no longer true that the four-form is of the type (2,2). Other four-forms appear, too. The appearance of other components of the 4-form field strength than (2,2) modifies the Dirac equation for the fermionic zero modes; in fact you find out that all the zero modes are lifted. What does it mean? It means that the instanton does not generate a superpotential. That contradicts the assumptions that started this paragraph.

What does this contradiction mean? According to some people, it just means that the instanton calculation was done incorrectly. The non-perturbative corrections to the equations of motion arising from an instanton XY should not be included when you calculate the instanton XY itself. That would be like pushing a cart in front of the horse, and indeed, a colleague of mine claims that this is what our German friends are doing.

**Superpotential survives**

According to Susanne and her collaborators, the contradiction is resolved if one also modifies the supersymmetry variations by a term corresponding to W_{np}. A happy end is that their conclusion seems to be that there are two fermionic zero modes after all, and an agreement with the papers by Saulina and another paper by Kallosh et al. is achieved. But the exact assessment of the steps that lead to the conclusion remains a source of some slight controversy.

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