∞-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”).
Concretely, where the non-trivial super Lie bracket of the super-Minkowski Lie algebra is
in the M-theory super-Lie algebra this equation is extended to
(D’Auria & Fré 1982, Table 4, Townsend 1995 (13) 1997a (1) 1997b (24), BdAPV 2005 (1.2-4); the term “M-algebra” for a further extension is due to Sezgin 1997.)
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 .
Castellani 2005 implicitly shows (cf. 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, Picon & Varela 2004) 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, José M. Izquierdo, Moises Picon, Oscar Varela: On the underlying gauge group structure of D=11 supergravity, Phys. Lett.B 596 (2004) 145-155 [doi:10.1016/j.physletb.2004.06.079, 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 [doi:10.1016/j.aop.2004.11.016, arXiv:hep-th/0409100]
That a limiting case of this is given by the orthosymplectic super Lie algebra :
Review and further discussion:
Oscar Varela: Symmetry and holonomy in M Theory, PhD thesis, Valencia (2006) [arXiv:hep-th/0607088, hdl:10550/15484, spire:1397691]
Laura Andrianopoli, Riccardo D'Auria, Lucrezia Ravera, Hidden Gauge Structure of Supersymmetric Free Differential Algebras, JHEP 1608 (2016) 095 [arXiv:1606.07328, doi:10.1007/JHEP08(2016)095]
Laura Andrianopoli, Riccardo D'Auria, Lucrezia Ravera, More on the Hidden Symmetries of 11D Supergravity, Phys. Lett. B 772 (2017) 578 [doi:10.1016/j.physletb.2017.07.016, arXiv:1705.06251]
where the algebra is referred to as the DF-algebra, in honor of D’Auria & Fré 1982.
All this is reviewed in
also
Lucrezia Ravera, On the hidden symmetries of supergravity, Springer Proceedings in Mathematics & Statistics 396 Springer (2021) [doi:10.1007/978-981-19-4751-3_15, arXiv:2112.00445]
Laura Andrianopoli, Riccardo D'Auria, Supergravity in the Geometric Approach and its Hidden Graded Lie Algebra [arXiv:2404.13987]
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:
For analogous discussion in 7d supergravity and 4d supergravity, see the references there.
Alternatively, from considerations of brane charges, the M-theory algebra extensions were (apparently independently) introduced in
Jan-Willem van Holten, Antoine Van Proeyen, supersymmetry algebras in J. Phys. A 15 (1982) 3763 [doi:10.1088/0305-4470/15/12/028, spire:177060]
Paul Townsend, -Brane Democracy, in: Particles, strings and cosmology Proceedings, 19th Johns Hopkins Workshop and 5th PASCOS Interdisciplinary Symposium, Baltimore, USA, March 22-25 (1995) [arXiv:hep-th/9507048, spire:397058]
reprinted in: Michael Duff (ed.), The World in Eleven Dimensions, IoP (1999) 375-389
with further amplification including
Paul Townsend, M(embrane) theory on , Nucl. Phys. Proc. Suppl. 68 (1998) 11-16 [doi:10.1016/S0920-5632(98)00136-4, arXiv:hep-th/9708034]
Paul Townsend: M-theory from its superalgebra, in Strings, Branes and Dualities, NATO ASI Series 520, Springer (1999) [arXiv:hep-th/9712004, doi:10.1007/978-94-011-4730-9_5]
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)
Relation to exceptional field theory:
Last revised on October 17, 2024 at 16:57:53. See the history of this page for a list of all contributions to it.