Instantonic Branes, Atiyah-Hirzebruch Spectral Sequence, and SL(2,Z) duality of N=4 SYM

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We study the equivalence among orientifold three-planes in the context of the Atiyah-Hirzebruch Spectral Sequence. This equivalence refers to the fact that two different cohomology classes in the same cohomology group, which classify the orientifolds, are lifted to the same trivial class in K-theory. The physical interpretation of this mathematical algorithm is given by the role of D-brane instantons. By following some recent ideas, we extend the sequence to include a classification of NS-NS fluxes. We find that such equivalences, in the low energy limit of the dynamics on the worldvolume of type IIB D3-branes on top of the orientifolds, are interpreted as the SL(2,Z) duality in four dimensional N=4 SYM theories.