∞-Lie theory (higher geometry)
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superalgebra and (synthetic ) supergeometry
A super Lie algebra which is a polyvector extension of the super Poincaré Lie algebra (supersymmetry) in for 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 .
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-Fré 82) asks for a super Lie algebra extension of super Minkowski spacetime such that the 4-cocycle 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 includes a fermionic extension of the M-theory super Lie algebra.
Two versions of a fermionic extension of the Polyvector extensions of on which the M2-brane 4-cocycle trivializes were first found in
and then a 1-parameter family of such was discovered in
Igor Bandos, José de Azcárraga, J.M. Izquierdo, M. Picon, O. Varela, On the underlying gauge group structure of D=11 supergravity, Phys.Lett.B596:145-155,2004 (arXiv:hep-th/0406020)
Igor Bandos, José de Azcárraga, Moises Picon, Oscar Varela, On the formulation of supergravity and the composite nature of its three-from field, Annals Phys. 317 (2005) 238-279 (arXiv:hep-th/0409100)
That a limiting case of this is given by the orthosymplectic super Lie algebra is due to
Further discussion is in
Laura Andrianopoli, Riccardo D'Auria, Lucrezia Ravera, Hidden Gauge Structure of Supersymmetric Free Differential Algebras, JHEP 1608 (2016) 095 (arXiv:1606.07328)
Laura Andrianopoli, Riccardo D'Auria, Lucrezia Ravera, More on the Hidden Symmetries of 11D Supergravity (arXiv:1705.06251)
where the algebra is referred to as the DF-algebra, in honor of D’Auria-Fré 82.
All this is reviewed in
also
Another, alternative “weak decomposition” of the M2-brane extended super-Minkowski spacetime was found in
with the interesting difference that for this splitting super Lie algebra is non-abelian, in fact an extension of the Lie algebra of the Spin group (Ravera 18, (3.5)-(3-6)).
See also:
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
For analogous discussion in 7d supergravity and 4d supergravity, see the references there.
From a different perspective the M-theory algebra extensions were (apparently independently) introduced in
Jan-Willem van Holten, Antoine Van Proeyen, supersymmetry algebras in J.Phys. A15, 3763 (1982).
Paul Townsend, p-Brane Democracy (arXiv:hep-th/9507048)
with further amplification including
Paul Townsend, M(embrane) theory on , 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, , 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)
Last revised on October 26, 2023 at 18:07:42. See the history of this page for a list of all contributions to it.