The extended M-theory supersymmetry algebra is, more or less, known in the literature to be the algebra of supersymmetry Noether currents of the M5-brane sigma-model, which receives the central extension due to the fact that the kappa-symmetry WZW term in the M5 Lagrangian is preserved by supersymmetry only up to a divergence. For the M2-brane this was famously argued in
and the extension of this argument to the M5 was indicated in
Now, these articles consider infinitesimal (super-)symmetries only, and already for this case the argument that it is the de Rham cohomology classes of the brane currents, instead of their differential form representatives that drop out of the Noether procedure, is, I believe this is fair to say, a littel vague.
I did now (or so I think) an analysis of the full M-theory group (in fact it is a super 6-group), i.e. of the global object whose elements are the finite supersymmetries of the full globally defined M5 brane WZW term on curved supermanifolds in the presence of a globally defined M2-brane condensate, equipped with the finite gauge transformations that relate the full transformed WZW term to itself. This reveals a global effect that is not otherwise visible, and my question here is if some incarnation of this effect has surfaced elsewhere, and if so, if anyone could provide me with some pointers, thanks.
Namely I find that the "naive" extension of the superymmetry group on an 11-dimensional spacetime \(X\) is not in general just the expected \(H^5(X) \oplus H^2(X)\) (5-brane charges and 2-brane charges, including their time components which dualise to the KK-monopole charges and the "M9" brane charges, thus accounting for all the species of M-branes) but that this is just the \(E_2\)page of a spectral sequence whose relevant non-trivial differential is a \(d_4 : H^1(X) \to H^5(X)\) and \(d_4 : H^2(X) \to H^6(X)\) in direct analogy to the familiar \(d_3\)in the Atiyah-Hirzebruch spectral sequence for D-brane charges down in 10d .
So intuitively this says that part of the charge of wrapped M5-branes which naively seems to contribute to the extended M-theory supersymmetry may potentially get cancelled by some kind of 1-brane charge, and similarly that part of the naively present M2-brane charge is actually absorbed by some kind of 6-brane charge.
For the first statement there seems to be an immediate physical interpretation, by the charges of self-dual strings inside the 5-branes which are the boundaries of M2s ending on the M5s. For the second statement I am not sure about the physics interpretation, would be grateful for any suggestions, it looks like it should come from M2s binding with KK-monopoles.. (Though I should emphasize that while the above two differentials are the only two that are potentially relevant, they may happen to not contribute after all.)
A directly analogous differential to the \(d_4 \) that I find here, but being a \(d_7\), has been argued in
to encode M-brane charges as \(d_3\)is well-known to encode D-brane charges. Possibly the \(d_4\)that I am seeing is an electric-magnetic dual of that \(d_7\), but there seem to be some differences. Not sure yet.
My question is if anything related to these \(d_4\)-corrections have appeared anywhere before in the literature.