Similarly, using the inversion formula (C.4.3) presented in appendix we can write:

where {Bαβ , Bα } are the connection and vielbein of the internal coset manifold P3. Relying once again on the inversion formulae discussed in Appendix C.4 we

conclude that we can rewrite (9.4.51)–(9.4.55) as follows:

where the objects introduced above are the Maurer Cartan forms on the supercoset (9.4.2) according to:

Σ = L−1 dL

(9.4.78)

Consequently the gauge completion of the B[2] form becomes:

9.5 Conclusions

As we stressed in the introduction to the present very long chapter, the topics we might still address in this context are both numerous, relevant and challenging. We might discuss instanton solutions, compactifications on Calabi-Yau manifolds, the generic strategy of harmonic analysis to derive the spectra of given compactifications, toroidal compactifications with brane wrapping, D-brane solutions with conifolds sitting in the transverse dimensions to the brane and much more. Obviously there is neither room nor enough mathematical background in order to develop such topics and therefore it is time to stop.

We just hope that our reader has been able to follow our arguments up to this point. If this has happened, starting from the first intuitions about Lorentz symmetry in the first chapter of the first volume he has made a long and adventurous trip to

the frontiers of current research in gravitational theories, little by little absorbing a remarkable wealth of geometrical lore and of consciousness about its physical meaning.

Hopefully our reader should by now be convinced that the geometrical seeds first implanted in the XIXth century by Gauss and Riemann, not only inspired Einstein and, through his mind, produced a beautiful and so far fully verified theory, but have still a lot to say about Gravity. Whether supersymmetric or not, Gravitation is certainly the most fundamental interaction among the fundamental ones, it governs the structure of the Universe and it is a manifestation of Geometry. Which geometry the humans will still debate for a long time.

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