∞-Lie theory (higher geometry)
superalgebra and (synthetic ) supergeometry
A super Lie algebra which is a polyvector extension of the super Poincaré Lie algebra (supersymmetry) in $D = 10+1$ for $N=1$ supersymmetry by charges corresponding to the M2-brane and the M5-brane (“extended supersymmetry”).
In (AGIT 89) it is shows that the M-algebra-like polyvector extensions arise as the algebras of conserved currents of the Green-Schwarz super p-brane sigma-models.
By the discussion at conserved current – In higher prequantum geometry this means that this is the degree-0 piece in the Heisenberg Lie n-algebra which is induced by regarding the WZW-curvature terms as super n-plectic forms on $\mathbb{R}^{10,1|32}$.
In (Castellani 05) it is implicitly shown, (FSS 13), that the M-extension arises as the derivations/automorphisms of the supergravity Lie 3-algebra/supergravity Lie 6-algebra (see there for the details).
The original construction in (D’Auria-Fre 82) asks for a super Lie algebra extension $\mathbb{R}^{10,1\vert 32} \rtimes \mathfrak{g}$ of super Minkowski spacetime $\mathbb{R}^{10,1\vert 32}$ such that the 4-cocycle $\mu_4 = \overline{\psi} \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b$ for the M2-brane trivializes when pulled back to this:
(In the language of local prequantum field theory this identifies a boundary condition for the WZW term of the M2-brane.)
They find, see also (Bandos-Azcarraga-Izquierdo-PiconVarela 04) that a solution for $\mathfrak{g}$ includes a fermionic extension of the M-theory super Lie algebra.
The Polyvector extensions of $\mathfrak{Iso}(\mathbb{R}^{10,1|32})$ were first considered in
with more comprehensive analysis in
where (a further fermionic extension of it) is derived as a super Lie algebra extension of 11d super Minkowski spacetime on which the M2-brane 4-cocycle trivializes.
See also
That a limiting case of this is given by the orthosymplectic super Lie algebra $\mathfrak{osp}(1\vert 32)$ is due to
That the underlying bosonic body of this super Lie algebra happens to be the typical fiber of what would be the 11-d exceptional generalized tangent bundle, namely the level-2 truncation of the l1-representation of E11 according to (West 04) was highlighted in the review
From a different perspective the M-theory algebra extensions were (apparently independently) introduced 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)
with further amplification including
Paul Townsend, M(embrane) theory on $T^0$, Nucl.Phys.Proc.Suppl.68:11-16,1998 (arXiv:hep-th/9708034)
Paul Townsend, M-theory from its superalgebra, Cargese lectures 1997 (arXiv:hep-th/9712004)
In their global form, where differential forms are replaced by their de Rham cohomology classes on curved superspacetimes, these algebras were identified (for the case including the 2-form piece but not the 5-form piece) as the algebras of conserved currents of the Green-Schwarz super p-brane sigma-models in
reviewed in
The generalization of this including also the contribution of the M5-brane was considered in
Further detailed discussion along these lines producing also the type II supersymmetry algebras is in
The full extension was named “M-algebra” in
In (D’Auria-Fré 82) the motivation is from the formulation of the fields of 11-dimensional supergravity as connections with values in the supergravity Lie 3-algebra, see at D'Auria-Fré formulation of supergravity. Realization of the M-theory super Lie algebra as the algebra of derivations of the supergravity Lie 3-algebra is in
with amplification in
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields (2013)
Discussion of a formulation in terms of octonions (see also at division algebra and supersymmetry) includes
Arguments that the charges of the M-theory super Lie algebra may be identified inside E11 are given in
Peter West, $E_{11}$, $SL(32)$ and Central Charges, Phys.Lett.B575:333-342,2003 (arXiv:hep-th/0307098v2)
Paul Cook, around p. 75 of Connections between Kac-Moody algebras and M-theory (arXiv:0711.3498)