RECUPERACION ETICA

As has been indicated already in previous chapters, the function W has actually an interpretation in terms of the effective spacetime superpotential.

11.4 Superpotentials 153

It therefore encodes information about the different vacua of the model (at least in the part of the moduli space under consideration).

In the above we have seen explicitly that the RG flow is a gradient flow of a potential. This potential is precisely the contribution to the effective superpotential W that is first order in the bulk field Φ and exact in the boundary fieldψ1. To see this we simply note that the term that appears on

the right hand side of (11.41) is the bulk-boundary coefficient that involves one insertion of the bulk field Φ and one insertion of the boundary field ψ1 (that couples to µ). This bulk-boundary correlator was evaluated at

anarbitrary point in the brane moduli space; if we start around any given point of the brane moduli space, the above expression therefore involves an arbitrary number of insertions of ψ1 (that allow one to move around this

brane moduli space). Thus the right-hand-side of (11.41) is the generating function describing symmetrised correlators involving an arbitrary number of boundary fieldsψ1, together with one insertion of the boundary fieldψ1

and one insertion of the bulk field Φ. We can produce the insertion of the boundary fieldψ1 by taking a derivative with respect to the corresponding

boundary coupling constant. It thus follows that the function W (that we obtained by integrating up the right hand side of (11.41)) is precisely the generating function of one bulk field Φ with an arbitrary number of boundary fields. It therefore defines the corresponding contribution of the effective superpotential.

It is also clear from this argument that this method can be applied to calculate the corresponding terms of the effective superpotential for an ar- bitrary bulk deformation, not just one of the form (11.38). For the other cases, the result is however trivial: the complex structure deformations (11.38) are the only monomials (instead ofx3

1 we may also allow for an ar-

bitrary third order polynomial in x1 and x2) for which the bulk-boundary

OPE coefficient with ψ1 is non-zero. Thus to first order in the bulk per-

turbation the above terms are the only terms that appear in the effective superpotential. It should also be obvious how to perform the same analysis for the other (45) families of D2-branes.

It should be noted that a priori only the minima of the effective su- perpotential do have a physical interpretation. The function itself may be subject to field redefinitions, so that it not clear if W contains definite off-shell information. Nevertheless it is interesting to learn how the open

154 11. Open-closed superpotential

string moduli spaces change under bulk deformations, and to see how they are connected.

Note also that for the investigation of the open-closed moduli space it was only necessary to deal with the boundary part of the BRST operator. This is very reminiscent of bulk deformations which have been considered in chapter 8 for bosonic BSFT. In fact, the restriction to q instead of Q is possible because in the topological theory the bulk-boundary map is almost trivial. It is given by the trivialisation Φ _"→ Φ 1, where Φ is a bulk field. This map involves no other modes than the constant one. This has the effect that OPEs between any fields are always regular and do not contain singularities. Therefore there is a trivial map from bulk to boundary fields which makes it possible to view the bulk fields naturally as a subset of the boundary fields. Once the projection on constant modes is abandoned, extra singularities will appear when moving bulk operators to the boundary, as explained in previous sections. These were seen to lead finally to non- local excitations, thus in a non-topological setting one would again expect the appearance of non-local terms.

Chapter 12

Conclusions

The main goal of the work presented in this thesis was to develop a version of BSFT which is valid in curved backgrounds, to find a way to isolate closed string deformations in this language and to find support for the idea that the open string field theory is indeed capable of describing closed string deformations.

In the bosonic case it has been possible to achieve these goals and fortify the approach by concrete calculations. The paths taken rested on an ex- tension of BSFT suitable for curved target spaces. While the factorisation conjecture, which enabled this extensions, has only been proven for WZW targets, and therefore for a large and important class of target spaces, it does not seem too farfetched to put this forward also for general targets.

Further tests of the constructed BSFT action have been provided. In fact it has been shown that tachyon condensation on D-branes yields the expected results. For this we have applied the open-closed string corre- spondence developed in chapter 7 [22] to a specific example, where the qualitative features observed should be rather generic. Apart from the nu- merical values not much depended on the details of the group manifold in question. Given the highly symmetric set-up one might hope that some of the phenomena discussed in chapter 8 and in [21] within perturbation theory could be established exactly at least for some simple processes.

In particular within the perturbative approximation utilised here we are not able to see all D2-branes corresponding to conjugacy classes of the

156 12. Conclusions

group. Rather we only see the ‘biggest’ 2-brane. This should be related to the fact that we worked in the large radius regime. Pushing the perturba- tion inλfurther it is conceivable that additional fixed points appear which describe ‘smaller’ conjugacy classes, but in order to see these much more powerful methods are needed. More interestingly it would be worthwhile in- vestigating, if non-symmetry preserving branes exist in these models. Also, although we have observed the absence of divergences in the 2-brane the- ory by brute force computation, there may well be symmetry arguments that imply finiteness of the loop correction. It would be interesting to know if such a symmetry exists, in particular in view of a non-perturbative approach to these models.

With chapter 11 a first step has been taken to repeat the bosonic ap- proach in a supersymmetric setting. The starting point for this investiga- tion has been a spacetime much more complicated than flat space, namely the quintic. Most remarkably, it has been possible to derive exact results on the open string moduli space of this Calabi-Yau and its behaviour under closed string deformations. The observations made are in agreement with the philosophy put forward in chapters 7 and 8. The fact that closed string deformations can be treated completely by looking only at the boundary BRST operator is, from this point of view, to be expected. If this is a relict of the immense simplification achieved by projecting on the topological sec- tor of the theory can only be decided once the factorisation conjecture (7.3) has been proven for supersymmetric theories on Calabi-Yau. In addition we have worked exclusively in the B-model, because there a description of D-branes which is close to the worldsheet formalism is available. It is not clear how to conduct similar calculations in the A-model, or how to consider even situations, where A- and B-branes are considered simultaneously.

While a supersymmetric version of (7.3) for arbitrary target spaces is an important further step in the investigation of open-closed correspondence, the results of chapter 11 do have immediate application to other problems, too. Prominent among them is the existence of open-closed Picard-Fuchs equations [85, 139]. This in turn opens up the door for an investigation of mirror symmetry when both, open and closed string moduli are included. This is an interesting question in itself, but it should be investigated with the far aim of shedding light on a general understanding of open-closed correspondence in string field theory.

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