nLab M-theory supersymmetry algebra

Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Super-Geometry

String theory

Contents

Idea

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

[Q α,Q β]=2Γ αβ aP a \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

[Q α,Q β]=2Γ αβ aP a2Γ αβ a 1a 2Z a 1a 22Γ αβ a 1a 5Z a 1a 5 \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 10,1|32\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 10,1|32𝔤\mathbb{R}^{10,1\vert 32} \rtimes \mathfrak{g} of super Minkowski spacetime 10,1|32\mathbb{R}^{10,1\vert 32} such that the 4-cocycle μ 4=ψ¯Γ abψe ae b\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:

10,1|32𝔤 * 10,1|32 μ 4 B 3. \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 𝔤\mathfrak{g} includes a fermionic extension of the M-theory super Lie algebra.

References

From the M2-Cocycle

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

and then a 1-parameter family of such was discovered in

That a limiting case of this is given by the orthosymplectic super Lie algebra 𝔬𝔰𝔭(1|32)\mathfrak{osp}(1\vert 32):

Review and further discussion:

where the algebra is referred to as the DF-algebra, in honor of D’Auria & Fré 1982.

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 Spin(10,1)Spin(10,1) (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.

Alternative

Alternatively, from considerations of brane charges, the M-theory algebra extensions were (apparently independently) introduced in

with further amplification including

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

  • 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

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

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

Relation to exceptional field theory:

Last revised on November 25, 2024 at 07:04:12. See the history of this page for a list of all contributions to it.