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The nLab presently gives me an application error when trying to open anything, so I’ll record some things here.
The following needs to be added to the References-section of the entry M-theory super Lie algebra:
The M-theory super Lie algebra as first considered in
Jan-Willem van Holten, Antoine Van Proeyen, $N=1$ supersymmetry algebras in $d=2,3,4 \,mod\, 8$ J.Phys. A15, 3763 (1982).
Paul Townsend, p-Brane Democracy (arXiv:hep-th/9507048)
Discussion of its formulation in terms of octonions (see also at division algebra and supersymmetry) includes
- A. Anastasiou, L. Borsten, Michael Duff, L. J. Hughes, S. Nagy, An octonionic formulation of the M-theory algebra (arXiv:1402.4649)
It worked for me just now.
By the way, where it says
The M-theory Lie algebra was named as such in…
I took a look, and I only see it called ’M-algebra’.
Thanks for adding it! I have edited a bit more.
(The problem persists for me via Telekom HotSpot on this ICE train. With my O2 surfstick all works fine as usual.)
Regaring terminology: there is a troubling habit in some corners of dropping the “Lie” qualifier. One shouldn’t follow this. But I have re-adjusted the text around the citations now.
One thought:
given that the M-theory Lie algebra is the automorphisms of the supergravity Lie 3-algebra $\hat {\mathbb{R}}^{10,1|32}$, and that the latter is the space on which the superspacetimes are modeled on which the M2-brane propagates, this means that the frame bundles of these superspacetimes are $GL(\hat {\mathbb{R}}^{10,1|32})$-principal bundles. Here $GL(\hat {\mathbb{R}}^{10,1|32})$ contains (receives a map from) the super $\infty$-group to which the M-theory super Lie algebra integrates, the latter being the group of those linear automorphisms which also preserves the Lie structure.
Maybe this is relevant for something…
I have the suspicion that there is a better statement: the M-theory super Lie algebra is the degree-0 part of the super-Heisenberg Lie 3-algebra of $\mathbb{R}^{10,1|32}$ equipped with its its super 3-plectic form $\bar \psi \wedge \Gamma^{a b} \psi \wedge E_a \wedge E_b$.
This is in degree 0 an extension by constant 2-forms, and so their components are just the central charges traditionally written $Z^{a b} \coloneqq \mathbf{d}x^a \wedge \mathbf{d}x^b$ etc.
The formula for the Heisenberg Lie 3-algebra bracket in degree 0 gives for instance for the super-commutator of the supercharges
$\begin{aligned} [Q^\alpha, Q^\beta]_{Heis} & = [Q^\alpha, Q^\beta] + \iota_{Q^\alpha}\iota_{Q^\beta} \bar \psi \wedge \Gamma^{a b} \psi \wedge E_a \wedge E_b \\ & = \Gamma^a_{\alpha \beta} P_a + \Gamma^{a b}_{\alpha \beta} E_a \wedge E_b \end{aligned}$Now the key is that the left invariant 1-form $E^a$ is not constant but is $E^a = \mathbf{d}x^a + \theta \Gamma^a \mathbf{d}\theta$. So there is a constant piece
$\Gamma_{a b\alpha \beta} \mathbf{d}x^a \wedge \mathbf{d}x^b = \Gamma_{a b\alpha \beta} Z^{a b}$in the above, as in the M-theory super Lie algebra, plus a non-constant piece. One has to check that the non-constant piece (hence also its sum with the constant piece) is really a Hamiltonian form for the translation $\Gamma^a_{\alpha \beta}P_a$, hence that the action of $\mathbb{R}^{10,1|32}$ on itself is really Hamiltonian with respect to this super-3-plectic structure. I think this follows with the 11d Fierz identity, but I should write it down carefully now.
Ah, of course this works. And the main part of the answer is secretly already in
reviewed in section 8.8. of
Notice how the classification of the brane charges comes out exactly right under this identification:
Given a supergravity-spacetime $X$ modeled on $\mathbb{R}^{d-1,1|N}$ equipped with the closed super $(p+2)$-form $\omega_{WZW}$ definite on $\bar \psi E^p \psi \in \Omega^{p+2}(\mathbb{R}^{d-1,1|N})$, then the corresponding $L_\infty$-algebra of local observables of this super $(p+1)$-plectic geometry is an extension of the super-Lie algebra of supersymmetries by $\mathbf{H}(X,\flat \mathbf{B}^p \mathbb{R})$, hence by the real cohomology cocycle $\infty$-groupoid in degree $p$. Truncating down this is hence an extension by the real cohomology $H^p(X,\mathbb{R})$, saying that there are central charges extending the susy algebra for each degree-$p$ cohomology class.
This is precisely the expected result for these brane charges,e.g. p. 8 of the original AGIT 89.
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