nLab 11d supergravity Lie 3-algebra



\infty-Lie theory

∞-Lie theory (higher geometry)


Smooth structure

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\infty-Lie groupoids

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The supergravity Lie 3-algebra 𝔰𝔲𝔤𝔯𝔞(10,1)\mathfrak{sugra}(10,1) or M2-brane extension 𝔪2𝔟𝔯𝔞𝔫𝔢\mathfrak{m}2\mathfrak{brane} is a super L-∞ algebra that is a shifted extension

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

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

μ 4=ψ¯Γ abψe ae bCE(𝔰𝔦𝔰𝔬(10,1)). \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 𝔰𝔬(n)\mathfrak{so}(n).


The Chevalley-Eilenberg algebra


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

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

  • a single element cc of degree (3,even)(3,even)

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

with the differential defined by

d CEω ab=ω a bω bc d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}
d CEe a=ω a be b+i2ψ¯Γ aψ d_{CE} \, e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi
d CEψ=14ω abΓ abψ d_{CE} \, \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi
d CEc=12ψ¯Γ abψe ae b. 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 𝔤\mathfrak{g}, equipped with a degree-3 element AA in its Chevalley-Eilenberg algebra, and equipped with a homomorphism p:𝔤 10,1|32p\colon \mathfrak{g}\longrightarrow \mathbb{R}^{10,1\vert \mathbf{32}} such that the pullback of the 4-cocycle μ 4\mu_4 along pp is trivialized by AA:

p *μ 4=d CEA. p^\ast \mu_4 = d_{CE}A \,.

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

𝔤 * p A 10,1|32 μ 4 B 3. \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 𝔰𝔲𝔤𝔯𝔞(10,1)\mathfrak{sugra}(10,1) as the homotopy fiber of μ 4\mu_4, hence as the universal solution to this situation

𝔰𝔲𝔤𝔯𝔞(10,1) * 10,1|32 μ 4 B 3. \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:𝔤 10,1|32p \colon \mathfrak{g} \to \mathbb{R}^{10,1\vert\mathbf{32}} are found.

Curiously, the bosonic body of 𝔤\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 pp-brane sigma-models (see at table of branes for further links and see at The brane bouquet for the full classification):

=d\stackrel{d}{=}p=p =123456789
10D0F1, D1D2D3D4NS5, D5D6D7D8D9
7M2 top{}_{top}
6F1 little{}_{little}, S1 sd{}_{sd}S3

(The first columns follow the exceptional spinors table.)

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

=d\stackrel{d}{=}p=p =123456789
11Ψ 2E 2\Psi^2 E^2 on sIso(10,1)Ψ 2E 5+Ψ 2E 2C 3\Psi^2 E^5 + \Psi^2 E^2 C_3 on m2brane
10Ψ 2E 1\Psi^2 E^1 on sIso(9,1)B 2 2+B 2Ψ 2+Ψ 2E 2B_2^2 + B_2 \Psi^2 + \Psi^2 E^2 on StringIIA\cdots on StringIIBB 2 3+B 2 2Ψ 2+B 2Ψ 2E 2+Ψ 2E 4B_2^3 + B_2^2 \Psi^2 + B_2 \Psi^2 E^2 + \Psi^2 E^4 on StringIIAΨ 2E 5\Psi^2 E^5 on sIso(9,1)B 2 4++Ψ 2E 6B_2^4 + \cdots + \Psi^2 E^6 on StringIIA\cdots on StringIIBB 2 5++Ψ 2E 8B_2^5 + \cdots + \Psi^2 E^8 in StringIIA\cdots on StringIIB
9Ψ 2E 4\Psi^2 E^4 on sIso(8,1)
8Ψ 2E 3\Psi^2 E^3 on sIso(7,1)
7Ψ 2E 2\Psi^2 E^2 on sIso(6,1)
6Ψ 2E 1\Psi^2 E^1 on sIso(5,1)Ψ 2E 3\Psi^2 E^3 on sIso(5,1)
5Ψ 2E 2\Psi^2 E^2 on sIso(4,1)
4Ψ 2E 1\Psi^2 E^1 on sIso(3,1)Ψ 2E 2\Psi^2 E^2 on sIso(3,1)
3Ψ 2E 1\Psi^2 E^1 on sIso(2,1)

Relation to the 11-dimensional polyvector super Poincaré-algebra

Via derivations


Let 𝔡𝔢𝔯(𝔰𝔲𝔤𝔯𝔞(10,1))\mathfrak{der}(\mathfrak{sugra}(10,1)) be the automorphism ∞-Lie algebra of 𝔰𝔲𝔤𝔯𝔞(10,1)\mathfrak{sugra}(10,1). This is a dg-Lie algebra. Write 𝔡𝔢𝔯(𝔰𝔲𝔤𝔯𝔞(10,1)) 0\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+1d = 10+1 – the “M-theory super Lie algebra” – with “2-brane central charge”: the Lie algebra spanned by generators {P a,Q α,J ab,Z ab}\{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 α,Q β]=i(CΓ a) αβP a+(CΓ ab)Z ab [Q_\alpha, Q_\beta] = i (C \Gamma^a)_{\alpha \beta} P_a + (C \Gamma_{a b})Z^{a b}
[Q α,Z ab]=2i(CΓ [a) αβQ b]β [Q_\alpha, Z^{a b}] = 2 i (C \Gamma^{[a})_{\alpha \beta}Q^{b]\beta}


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


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

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


Q α=[d CE,ψ α] Q_\alpha = [d_{CE}, \frac{\partial}{\partial \psi^\alpha} ]


J ab=[d CE,ω ab] J_{a b} = [d_{CE}, \frac{\partial}{\partial \omega^{a b}} ]


Z ab=[d CE,e ae bc] 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 α,Q β]=i(CΓ a) αβP a+(CΓ ab)Z ab [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 ψ¯Γ abψe ae b\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b with ψ αψ β\frac{\partial}{\partial \psi^\alpha}\wedge \frac{\partial}{\partial \psi^\beta}.

Via the Heisenberg Lie 3-algebras



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

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


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

and later in the textbook

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

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

See also division algebra and supersymmetry.

Further discussion of its “hidden” super Lie algebra includes

Further review:

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

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

Last revised on December 2, 2021 at 01:32:51. See the history of this page for a list of all contributions to it.