There is a curious gap in the supergravity literature concerning the extensions of the susy algebra by those brane charges. I had checked for the literature recently [1] and it seems clear that indeed this had not been discussed properly before:
the seminal article [2] first derives the central super Lie algebra extension of the susy translation Lie algebra by differential forms from computing the Lie algebra of conserved currents of the super p-brane sigma models.
Then, and that's the gap, hands are waved a little and it is argued that some of these differential forms just found are not to count, namely that the exact forms are to be discarded, that the extension is not in fact by differential forms (as just computed) but just by their de Rham cohomology classes.
Physically this is "clear", since these forms are, indeed, currents, whose integral over cycles (under which the exact forms drop out) computes the corresponding charges, and from the physics of branes one expects there to be effects by these net charges.
But what's the systematic rigorous way to pass from computing a super Lie algebra extension to then discarding some of the extending elements?
I'll tell you what it is: it's homotopy Lie algebras of higher gauge symmetries. The point is that those currents of super p-branes which arise via the generalized Noether theorem from weak symmetries of the kappa-symmetry WZW-action term have themselves higher order gauge transformations between them, by higher order currents (given by lower degree forms). Here two currents are higher gauge equivalent when they differ by the differential of a higher currents. So THAT's why the exact pieces in the extension drop out: while they are present in the super Lie 1-algebra of the symmetries, instead really there is a super Lie n-algebra of higher symmetries, and in there these spurious currents are indeed -- while not on the nose zero -- gauge equivalent to zero.
There is a systematic rigorous way to compute the super Lie n-algebras of higher symmetries of the kappa-symmetry WZW terms (or any other action functional), and it has precisely all these properties. Moreover, we have a theorem that shows that these super Lie n-algebras of higher currents are higher extension by the de Rham cocycle homotopy n-type of the given spacetime. This means that after truncating the super Lie n-algebra down to its 0th Postnikov stage, then it becomes an extension of the plain symmetry algebra by de Rham cohomology. And this is exactly what traditional literature argues for, without, I think, giving a genuine derivation of.
I have now written this out in [3], sections 1.2.11.3 and 1.2.15.3.3
[1] https://plus.google.com/+UrsSchreiber/posts/CYt43HDR5cZ
[2] José de Azcárraga, Jerome Gauntlett, J.M. Izquierdo, Paul Townsend, Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys.Rev.Lett. 63 (1989) 2443ncatlab.org/nlab/show/super Poincare+Lie+algebra#AGIT89
[3] https://dl.dropboxusercontent.com/u/12630719/dcct.pdf
section 1.2.11.3 and 1.2.15.3.3
the seminal article [2] first derives the central super Lie algebra extension of the susy translation Lie algebra by differential forms from computing the Lie algebra of conserved currents of the super p-brane sigma models.
Then, and that's the gap, hands are waved a little and it is argued that some of these differential forms just found are not to count, namely that the exact forms are to be discarded, that the extension is not in fact by differential forms (as just computed) but just by their de Rham cohomology classes.
Physically this is "clear", since these forms are, indeed, currents, whose integral over cycles (under which the exact forms drop out) computes the corresponding charges, and from the physics of branes one expects there to be effects by these net charges.
But what's the systematic rigorous way to pass from computing a super Lie algebra extension to then discarding some of the extending elements?
I'll tell you what it is: it's homotopy Lie algebras of higher gauge symmetries. The point is that those currents of super p-branes which arise via the generalized Noether theorem from weak symmetries of the kappa-symmetry WZW-action term have themselves higher order gauge transformations between them, by higher order currents (given by lower degree forms). Here two currents are higher gauge equivalent when they differ by the differential of a higher currents. So THAT's why the exact pieces in the extension drop out: while they are present in the super Lie 1-algebra of the symmetries, instead really there is a super Lie n-algebra of higher symmetries, and in there these spurious currents are indeed -- while not on the nose zero -- gauge equivalent to zero.
There is a systematic rigorous way to compute the super Lie n-algebras of higher symmetries of the kappa-symmetry WZW terms (or any other action functional), and it has precisely all these properties. Moreover, we have a theorem that shows that these super Lie n-algebras of higher currents are higher extension by the de Rham cocycle homotopy n-type of the given spacetime. This means that after truncating the super Lie n-algebra down to its 0th Postnikov stage, then it becomes an extension of the plain symmetry algebra by de Rham cohomology. And this is exactly what traditional literature argues for, without, I think, giving a genuine derivation of.
I have now written this out in [3], sections 1.2.11.3 and 1.2.15.3.3
[1] https://plus.google.com/+UrsSchreiber/posts/CYt43HDR5cZ
[2] José de Azcárraga, Jerome Gauntlett, J.M. Izquierdo, Paul Townsend, Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys.Rev.Lett. 63 (1989) 2443ncatlab.org/nlab/show/super Poincare+Lie+algebra#AGIT89
[3] https://dl.dropboxusercontent.com/u/12630719/dcct.pdf
section 1.2.11.3 and 1.2.15.3.3
'via Blog this' The Honourable Schoolboy
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