nLab D=11 supergravity Lie 3-algebra

Definition

The supergravity Lie 3-algebra $\mathfrak{sugra}(10,1)$ or M2-brane extension $\mathfrak{m}2\mathfrak{brane}$ is a super L-∞ algebra that is a shifted extension

$$0 \to b^2 \mathbb{R} \to \mathfrak{sugra}(10,1) \to \mathfrak{siso}(10,1) \to 0$$

of the super Poincare Lie algebra $\mathfrak{siso}(10,1)$ in 10+1 dimensions induced by the exceptional degree 4-super Lie algebra cocycle

$$\mu_4 = \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b \;\; \in CE(\mathfrak{siso}(10,1)) \,.$$

This is the same mechanism by which the String Lie 2-algebra is a shifted central extension of $\mathfrak{so}(n)$.

Properties

The Chevalley-Eilenberg algebra

Proposition

The Chevalley-Eilenberg algebra $CE(\mathfrak{sugra}(10,1))$ is generated on

• elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$

• a single element $c$ of degree $(3,even)$

• and elements $\{\psi^\alpha\}$ of degree $(1,odd)$

with the differential defined by

$$d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}$$

$$d_{CE} \, e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi$$

$$d_{CE} \, \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi$$

$$d_{CE} \, c = \frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b \,.$$

(fill in details)

Hidden super Lie 1-algebra

At the end of (D’Auria-Fre 82) the authors ask for a super Lie 1-algebra $\mathfrak{g}$, equipped with a degree-3 element $A$ in its Chevalley-Eilenberg algebra, and equipped with a homomorphism $p\colon \mathfrak{g}\longrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}$ such that the pullback of the 4-cocycle $\mu_4$ along $p$ is trivialized by $A$:

$$p^\ast \mu_4 = d_{CE}A \,.$$

In the homotopy theory of L-infinity algebras? this means that

$$\array{ \mathfrak{g} &\longrightarrow& \ast \\ {}^{\mathllap{p}} \downarrow &\swArrow_{\mathrlap{A}}& \downarrow \\ \mathbb{R}^{10,1\vert \mathbf{32}} &\underset{\mu_4}{\longrightarrow}& \mathbf{B}^3 \mathbb{R} } \,.$$

Compare this to the characterization of $\mathfrak{sugra}(10,1)$ as the homotopy fiber of $\mu_4$, hence as the universal solution to this situation

$$\array{ \mathfrak{sugra}(10,1) &\longrightarrow& \ast \\ \downarrow &\swArrow_{}& \downarrow \\ \mathbb{R}^{10,1\vert \mathbf{32}} &\underset{\mu_4}{\longrightarrow}& \mathbf{B}^3 \mathbb{R} } \,.$$

In any case, in (D’Auria-Fre 82) possible choices for $p \colon \mathfrak{g} \to \mathbb{R}^{10,1\vert\mathbf{32}}$ are found.

Curiously, the bosonic body of $\mathfrak{g}$ is such that when adapted to a compactification to 4d, then it is the exceptional tangent bundle on which the U-duality group E7 has a canonical action.

In (BAIPV 04) these solutions are shown to extend to a 1-parameter family of solutions. Further comments are in (Andrianopoli-D’Auria-Ravera 16).

Relation to M5-brane action functional

The supergravity Lie 3-algebra carries a 7-cocycle (the one that induces the supergravity Lie 6-algebra-extension of it). The corresponding WZW term is that of the M5-brane in its Green-Schwarz action functional-like formulation.

The brane scan.

The Green-Schwarz type super $p$-brane sigma-models (see at table of branes for further links and see at The brane bouquet for the full classification):

$\stackrel{d}{=}$	$p =$	1	2	3	4	5	6	7	8	9
11			M2			M5
10	D0	F1, D1	D2	D3	D4	NS5, D5	D6	D7	D8	D9
9					*
8				*
7			M2${}_{top}$
6		F1${}_{little}$, S1${}_{sd}$		S3
5			*
4		*	*
3		*

(The first columns follow the exceptional spinors table.)

The corresponding exceptional super L-∞ algebra cocycles (schematically, without prefactors):

$\stackrel{d}{=}$	$p =$	1	2	3	4	5	6	7	8	9
11			$\Psi^2 E^2$ on sIso(10,1)			$\Psi^2 E^5 + \Psi^2 E^2 C_3$ on m2brane
10		$\Psi^2 E^1$ on sIso(9,1)	$B_2^2 + B_2 \Psi^2 + \Psi^2 E^2$ on StringIIA	$\cdots$ on StringIIB	$B_2^3 + B_2^2 \Psi^2 + B_2 \Psi^2 E^2 + \Psi^2 E^4$ on StringIIA	$\Psi^2 E^5$ on sIso(9,1)	$B_2^4 + \cdots + \Psi^2 E^6$ on StringIIA	$\cdots$ on StringIIB	$B_2^5 + \cdots + \Psi^2 E^8$ in StringIIA	$\cdots$ on StringIIB
9					$\Psi^2 E^4$ on sIso(8,1)
8				$\Psi^2 E^3$ on sIso(7,1)
7			$\Psi^2 E^2$ on sIso(6,1)
6		$\Psi^2 E^1$ on sIso(5,1)		$\Psi^2 E^3$ on sIso(5,1)
5			$\Psi^2 E^2$ on sIso(4,1)
4		$\Psi^2 E^1$ on sIso(3,1)	$\Psi^2 E^2$ on sIso(3,1)
3		$\Psi^2 E^1$ on sIso(2,1)

The Brane molecule

Furthermore, there exists a more general classification of possible supermembranes in spacetime with $S$ spatial dimensions and $T$ time dimensions, appearing in (Blencowe-Duff 88). In this sense, the brane scan is but the $T=1$ branch of the brane molecule. The objects appearing here are expected to be related to other generalizations of string theory. See D=12 supergravity and bosonic M-theory.

Compare:

• Miles Blencowe, Mike Duff, Supermembranes and the Signature of Space-time, Nucl. Phys. B310 (1988) 387-404 (spire:262142, 10.1016/0550-3213(88)90155-1, pdf)

• brane bouquet

Relation to the 11-dimensional polyvector super Poincaré-algebra

Via derivations

Proposition

Let $\mathfrak{der}(\mathfrak{sugra}(10,1))$ be the automorphism ∞-Lie algebra of $\mathfrak{sugra}(10,1)$. This is a dg-Lie algebra. Write $\mathfrak{der}(\mathfrak{sugra}(10,1))_0$ for the ordinary Lie algebra in degree 0.

This is isomorphic to the polyvector-extension of the super Poincaré Lie algebra (see there) in $d = 10+1$ – the “M-theory super Lie algebra” – with “2-brane central charge”: the Lie algebra spanned by generators $\{P_a, Q_\alpha, J_{a b}, Z^{a b}\}$ and graded Lie brackets those of the super Poincaré Lie algebra as well as

$$[Q_\alpha, Q_\beta] = i (C \Gamma^a)_{\alpha \beta} P_a + (C \Gamma_{a b})Z^{a b}$$

$$[Q_\alpha, Z^{a b}] = 2 i (C \Gamma^{[a})_{\alpha \beta}Q^{b]\beta}$$

etc.

This observation appears implicitly in (Castellani 05, section 3.1), see (FSS 13).

Proof

With the presentation of the Chevalley-Eilenberg algebra $CE(\mathfrak{sugra}(10,1))$ as in prop. above, the generators are identified with derivations on $CE(\mathfrak{sugra}(10,1))$ as

$$P_a = [d_{CE}, \frac{\partial}{\partial e^a} ]$$

and

$$Q_\alpha = [d_{CE}, \frac{\partial}{\partial \psi^\alpha} ]$$

and

$$J_{a b} = [d_{CE}, \frac{\partial}{\partial \omega^{a b}} ]$$

and

$$Z^{a b} = [d_{CE}, e^a \wedge e^b \wedge \frac{\partial}{\partial c}]$$

etc. With this it is straightforward to compute the commutators. Notably the last term in

$$[Q_\alpha, Q_\beta] = i (C \Gamma^a)_{\alpha \beta} P_a + (C \Gamma_{a b})Z^{a b}$$

arises from the contraction of the 4-cocycle $\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b$ with $\frac{\partial}{\partial \psi^\alpha}\wedge \frac{\partial}{\partial \psi^\beta}$.

Via the Heisenberg Lie 3-algebras

(…)

Applications

The field configurations of 11-dimensional supergravity may be identified with ∞-Lie algebra-valued forms with values in $\mathfrak{sugra}(10,1)$. See D'Auria-Fre formulation of supergravity.

Related concepts

supergravity Lie 6-algebra$\to$ supergravity Lie 3-algebra $\to$ super Poincaré Lie algebra

• D=4 supergravity Lie 2-algebra

• extended supersymmetry

• type II supergravity Lie 2-algebra

• type II supersymmetry algebra

• M-theory supersymmetry algebra

References

The Chevalley-Eilenberg algebra of $\mathfrak{sugra}(10,1)$ first appears in (3.15) of

• Riccardo D'Auria, Pietro Fré, Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982)

and later in the textbook

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective

The manifest interpretation of this as a Lie 3-algebra and the supergravity field content as ∞-Lie algebra valued forms with values in this is mentioned in

• Hisham Sati, Urs Schreiber, Jim Stasheff, L-∞ algebra connections

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields

A systematic study of the super-Lie algebra cohomology involved is in

• John Baez, John Huerta, Division algebras and supersymmetry I (arXiv:0909.0551)

• John Baez, John Huerta, Division algebras and supersymmetry II (arXiv:1003.34360)

See also division algebra and supersymmetry.

Further discussion of its “hidden” super Lie algebra includes

• 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 [arXiv;hep-th/0406020]

• Igor Bandos, Jose de Azcarraga, 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 [arXiv:hep-th/0409100]

• Laura Andrianopoli, Riccardo D'Auria, Lucrezia Ravera, Hidden Gauge Structure of Supersymmetric Free Differential Algebras [arXiv:1606.07328]

Further review:

• Lucrezia Ravera, On the hidden symmetries of $D=11$ supergravity [arXiv:2112.00445]

The computation of the automorphism Lie algebra of $\mathfrak{sugra}(10,1)$ 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]

A similar argument with more explicit use of the Lie 3-algebra as underlying the Green-Schwarz-like action functional for the M5-brane is in

• Dmitri Sorokin, Paul Townsend, M-theory superalgebra from the M-5-brane, Phys. Lett. B 412 (1997) 265-273 [arXiv:hep-th/9708003]