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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 25th 2015
    • (edited Feb 11th 2015)

    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

    Discussion of its formulation in terms of octonions (see also at division algebra and supersymmetry) includes

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 25th 2015

    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’.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 25th 2015

    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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 27th 2015

    One thought:

    given that the M-theory Lie algebra is the automorphisms of the supergravity Lie 3-algebra ^ 10,1|32\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(^ 10,1|32)GL(\hat {\mathbb{R}}^{10,1|32})-principal bundles. Here GL(^ 10,1|32)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…

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2015
    • (edited Feb 11th 2015)

    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 10,1|32\mathbb{R}^{10,1|32} equipped with its its super 3-plectic form ψ¯Γ abψE aE b\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 abdx adx bZ^{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

    [Q α,Q β] Heis =[Q α,Q β]+ι Q αι Q βψ¯Γ abψE aE b =Γ αβ aP a+Γ αβ abE aE b \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 aE^a is not constant but is E a=dx a+θΓ adθE^a = \mathbf{d}x^a + \theta \Gamma^a \mathbf{d}\theta. So there is a constant piece

    Γ abαβdx adx b=Γ abαβZ ab \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 Γ αβ aP a\Gamma^a_{\alpha \beta}P_a, hence that the action of 10,1|32\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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2015

    Ah, of course this works. And the main part of the answer is secretly already in

    reviewed in section 8.8. of

    • José de Azcárraga, José M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics , Cambridge monographs of mathematical physics, (1995)
    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2015
    • (edited Feb 12th 2015)

    Notice how the classification of the brane charges comes out exactly right under this identification:

    Given a supergravity-spacetime XX modeled on d1,1|N\mathbb{R}^{d-1,1|N} equipped with the closed super (p+2)(p+2)-form ω WZW\omega_{WZW} definite on ψ¯E pψΩ p+2( d1,1|N)\bar \psi E^p \psi \in \Omega^{p+2}(\mathbb{R}^{d-1,1|N}), then the corresponding L L_\infty-algebra of local observables of this super (p+1)(p+1)-plectic geometry is an extension of the super-Lie algebra of supersymmetries by H(X,B p)\mathbf{H}(X,\flat \mathbf{B}^p \mathbb{R}), hence by the real cohomology cocycle \infty-groupoid in degree pp. Truncating down this is hence an extension by the real cohomology H p(X,)H^p(X,\mathbb{R}), saying that there are central charges extending the susy algebra for each degree-pp cohomology class.

    This is precisely the expected result for these brane charges,e.g. p. 8 of the original AGIT 89.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 26th 2015

    Added a paragraph on the higher Lie-theoretic interpretation of the original derivation by D’Auri-Fre of the M-theory supersymmetry algebra (here) and added in the References-section pointers to literature suggesting relation to E11