# Extended SuSy from the kappa-Symmetry WZW terms

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it was famously observed that the central brane-charge extensions of the super-translation Lie algebras may be understood as the current algebras of the Green-Schwarz super p-branes, the central extension being due to the fact that the corresponding kappa-symmetry super-WZW term is supersymmetric only up to a divergence, so that the Noether theorem for "weak" symmetries applies.

That's great. But following this argument further, there are then gauge-of-gauge symmetries of the divergence term. Have these higher order gauge transformations been discussed anywhere?

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It seems to me that this is a real gap in the supergravity literature, but here is what I think the answer is.

So recall that the seminal article

• José de Azcárraga, Jerome Gauntlett, J.M. Izquierdo, Paul Townsend, Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys.Rev.Lett. 63 (1989) 2443 (spire)

first derives the central extension by differential forms of the super Lie algebra extension of the susy translation Lie algebra 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.

From the physics 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? What's the way to really derive this from the computation of conserved currents of super p-branes sigma-models?

I'll tell you what it is: it's Lie n-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 more detail, sections 1.2.11.3 and 1.2.15.3.3 of "Differential cohomology in a cohesive topos" (pdf)

answered Feb 19, 2015 by (6,025 points)

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