nLab
supergravity Lie 3-algebra

Definition

The supergravity Lie 3-algebra $\mathrm{𝔰𝔲𝔤𝔯𝔞}(10,1)$ or M2-brane extension $𝔪2\mathrm{𝔟𝔯𝔞𝔫𝔢}$ is a super L-∞ algebra that is a shifted extension

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

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

$$\mu =\overline{\psi }\wedge {\Gamma }^{ab}\psi \wedge e_a\wedge e_b\in \mathrm{CE}(\mathrm{𝔰𝔦𝔰𝔬}(10,1)).$$\mu = \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 extensiono of $\mathrm{𝔰𝔬}(n)$.

Properties

The Chevalley-Eilenberg algebra

Proposition

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

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

• a single element $c$ of degree $(3,\mathrm{even})$

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

with the differential defined by

$$d_{\mathrm{CE}}{\omega }^{ab}={\omega }^a{}_b\wedge {\omega }^{bc}$$d_{CE} \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}

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

$$d_{\mathrm{CE}}\psi =\frac{1}{4}{\omega }^{ab}{\Gamma }_{ab}\psi$$d_{CE} \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi

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

(fill in details)

Relation to the 11-dimensional polyvector super Poincaré-algebra

Proposition

Let $\mathrm{𝔡𝔢𝔯}(\mathrm{𝔰𝔲𝔤𝔯𝔞}(10,1))$ be the automorphism ∞-Lie algebra of $\mathrm{𝔰𝔲𝔤𝔯𝔞}(10,1)$. This is a dg-Lie algebra. Write $\mathrm{𝔡𝔢𝔯}(\mathrm{𝔰𝔲𝔤𝔯𝔞}(10,1){)}_0$ 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-Lie algebra” – with “2-brane central charge”: the Lie algebra spanned by generators $\{P_a,Q_{\alpha },J_{ab},Z^{ab}\}$ and graded Lie brackets those of the super Poincaré Lie algebra as well as

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

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

etc.

This observation appears in (Castellani05, section 3.1).

Proof

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

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

and

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

and

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

and

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

$$[Q_{\alpha },Q_{\beta }]=i(C{\Gamma }^a{)}_{\alpha \beta }{P}_a+(C{\Gamma }_{ab})Z^{ab}$$[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 $\overline{\psi }{\Gamma }^{ab}\wedge \psi \wedge e_a\wedge e_b$ with $\frac{\partial }{\partial {\psi }^{\alpha }}\wedge \frac{\partial }{\partial {\psi }^{\beta }}$.

Applications

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

Related concepts

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

References

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

• 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 Staasheff, $L_{\mathrm{\infty }}$-algebra valued connections (web)

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.

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

• Leonardo Castellani, Lie derivatives along antisymmetric tensors and the M-theory superalgebra (arXiv:hep-th/0508213)