nLab M-theory supersymmetry algebra

Idea

A super Lie algebra which is a polyvector extension of the super Poincaré Lie algebra (supersymmetry) in $D = 10+1$ for $\mathcal{N}=1$ 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

$$\big[ Q_\alpha ,\, Q_\beta \big] \;=\; -2 \, \Gamma^a_{\alpha \beta} \, P_a$$

in the M-theory super-Lie algebra this equation is extended to

$$\big[ Q_\alpha ,\, Q_\beta \big] \;=\; -2 \, \Gamma^a_{\alpha \beta} \, P_a \,-\,2 \Gamma^{a_1 a_2}_{\alpha \beta} \, Z_{a_1 a_2} \,-\,2 \Gamma^{a_1 \cdots a_5}_{\alpha \beta} \, Z_{a_1 \cdots a_5}$$

(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.)

Properties

As the algebra of conserved currents of the M-branes

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 $\mathbb{R}^{10,1|32}$.

As the Lie algebra of derivations of the SuGra Lie 3-algebra

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).

As an 11-dimensional boundary condition for the M2-brane

The original construction in (D’Auria-Fré 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:

$$\array{ & & \mathbb{R}^{10,1\vert 32} \rtimes \mathfrak{g} \\ & \swarrow && \searrow \\ \ast && \swArrow_{\simeq} && \mathbb{R}^{10,1\vert 32} \\ & \searrow && \swarrow_{\mathrlap{\mu_4}} \\ && B^{3}\mathbb{R} } \,.$$

(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 $\mathfrak{g}$ includes a fermionic extension of the M-theory super Lie algebra.

Related concepts

• type II super Lie algebra

• BPS state

• supergravity Lie 3-algebra

• supergravity Lie 6-algebra

• M2-brane, M5-brane

• M-theory

References

From the M2-Cocycle

Two versions of a fermionic extension of the Polyvector extensions of $\mathfrak{Iso}(\mathbb{R}^{10,1|32})$ on which the M2-brane 4-cocycle trivializes were first found in

• Riccardo D'Auria, Pietro Fré: Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B 201 (1982) 101-140 [doi:10.1016/0550-3213(82)90376-5, errata]

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 $D=11$ 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 $\mathfrak{osp}(1\vert 32)$:

• J. J. Fernandez, José M. Izquierdo, M. A. del Olmo, Contractions from $osp(1|32) \oplus osp(1|32)$ to the M-theory superalgebra extended by additional fermionic generators, Nuclear Physics B 897 (2015) 87-97 [arXiv:1504.05946, doi:10.1016/j.nuclphysb.2015.05.018]

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

• José de Azcárraga, section 5 of: Superbranes, $D=11$ CJS supergravity and enlarged superspace coordinates/fields correspondence, AIP Conf. Proc. 767 (2005) 243-267 [doi:10.1063/1.1923338, arXiv:hep-th/0501198]

also

• Lucrezia Ravera, On the hidden symmetries of $D=11$ 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, Expositiones Mathematicae, [arXiv:2404.13987 10.1016/j.exmath.2024.125631]

Another, alternative “weak decomposition” of the M2-brane extended super-Minkowski spacetime was found in

• Lucrezia Ravera, section 3: of Hidden Role of Maxwell Superalgebras in the Free Differential Algebras of D=4 and D=11 Supergravity, Eur. Phys. J. C 78 3 (2018) 211 [doi:10.1140/epjc/s10052-018-5673-8, arXiv:1801.08860]

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 $Spin(10,1)$ (Ravera 18, (3.5)-(3-6)).

See also:

• Lucrezia Ravera, Group Theoretical Hidden Structure of Supergravity Theories in Higher Dimensions (arXiv:1802.06602)

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:

• Silvia Vaula, On the underlying $E_{11}$ symmetry of the $D = 11$ Free Differential Algebra, JHEP 0703:010 (2007) [arXiv:hep-th/0612130, doi:10.1088/1126-6708/2007/03/010]

For analogous discussion in 7d supergravity and 4d supergravity, see the references there.

Alternative

Alternatively, from considerations of brane charges, 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. A 15 (1982) 3763 [doi:10.1088/0305-4470/15/12/028, spire:177060]

• Paul Townsend, $p$-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 $T^9$, 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]

• Chris Hull: pp 5 in: Gravitational Duality, Branes and Charges, Nucl. Phys. B 509 (1998) 216-251 [arXiv:hep-th/9705162, doi:10.1016/S0550-3213(97)00501-4]

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

• José de Azcárraga, Jerome Gauntlett, José M. Izquierdo, Paul Townsend: Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys. Rev. Lett. 63 (1989) 2443 [doi:10.1103/PhysRevLett.63.2443, spire:26393]

reviewed in

• José de Azcárraga, José M. Izquierdo, section 8.8 of: Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics, Cambridge monographs of mathematical physics, (1995)

The generalization of this including also the contribution of the M5-brane was considered in

• Dmitri Sorokin, Paul Townsend, M-theory superalgebra from the M-5-brane, Phys.Lett. B412 (1997) 265-273 (arXiv:hep-th/9708003)

Further detailed discussion along these lines producing also the type II supersymmetry algebras is in

• Hanno Hammer, Topological Extensions of Noether Charge Algebras carried by D-p-branes, Nucl.Phys. B521 (1998) 503-546 (arXiv:hep-th/9711009)

The full extension was named “M-algebra” in

• Ergin Sezgin, The M-Algebra, Phys. Lett. B 392 (1997) 323-331 [doi:10.1016/S0370-2693(96)01576-6, arXiv:hep-th/9609086]

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

• Leonardo Castellani, Lie derivatives along antisymmetric tensors, and the M-theory superalgebra, J. Phys. Math. 3 (2011) 1-7 [arXiv:hep-th/0508213, euclid:jpm/1359468398]

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)

• Hisham Sati, Urs Schreiber, Lie n-algebras of BPS charges

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

• A. Anastasiou, Leron Borsten, Michael Duff, L. J. Hughes, S. Nagy, An octonionic formulation of the M-theory algebra (arXiv:1402.4649)

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)

Relation to exceptional field theory:

• Igor Bandos: Exceptional field theories, superparticles in an enlarged 11D superspace and higher spin theories, Nucl. Phys. B 925 (2017) 28-62 [arXiv:1612.01321, doi:10.1016/j.nuclphysb.2017.10.001]