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 [arXiv:2404.13987]

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]

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, J.M. Izquierdo, Paul Townsend, Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys.Rev.Lett. 63 (1989) 2443 (spire)

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]