∞-Lie theory

superalgebra

and

supergeometry

# Contents

## Definition

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

$0\to {b}^{2}ℝ\to \mathrm{𝔰𝔲𝔤𝔯𝔞}\left(10,1\right)\to \mathrm{𝔰𝔦𝔰𝔬}\left(10,1\right)\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 $\mathrm{𝔰𝔦𝔰𝔬}\left(10,1\right)$ in 10+1 dimensions induced by the exceptional degree 4-super Lie algebra cocycle

$\mu =\overline{\psi }\wedge {\Gamma }^{ab}\psi \wedge {e}_{a}\wedge {e}_{b}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in \mathrm{CE}\left(\mathrm{𝔰𝔦𝔰𝔬}\left(10,1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $\mathrm{𝔰𝔬}\left(n\right)$.

## Properties

### The Chevalley-Eilenberg algebra

###### Proposition

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

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

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

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

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
${d}_{\mathrm{CE}}c=\frac{1}{2}\overline{\psi }{\Gamma }^{ab}\wedge \psi \wedge {e}_{a}\wedge {e}_{b}\phantom{\rule{thinmathspace}{0ex}}.$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 $\mathrm{𝔡𝔢𝔯}\left(\mathrm{𝔰𝔲𝔤𝔯𝔞}\left(10,1\right)\right)$ be the automorphism ∞-Lie algebra of $\mathrm{𝔰𝔲𝔤𝔯𝔞}\left(10,1\right)$. This is a dg-Lie algebra. Write $\mathrm{𝔡𝔢𝔯}\left(\mathrm{𝔰𝔲𝔤𝔯𝔞}\left(10,1\right){\right)}_{0}$ 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 $\left\{{P}_{a},{Q}_{\alpha },{J}_{ab},{Z}^{ab}\right\}$ and graded Lie brackets those of the super Poincaré Lie algebra as well as

$\left[{Q}_{\alpha },{Q}_{\beta }\right]=i\left(C{\Gamma }^{a}{\right)}_{\alpha \beta }{P}_{a}+\left(C{\Gamma }_{ab}\right){Z}^{ab}$[Q_\alpha, Q_\beta] = i (C \Gamma^a)_{\alpha \beta} P_a + (C \Gamma_{a b})Z^{a b}
$\left[{Q}_{\alpha },{Z}^{ab}\right]=2i\left(C{\Gamma }^{\left[a}{\right)}_{\alpha \beta }{Q}^{b\right]\beta }$[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 $\mathrm{CE}\left(\mathrm{𝔰𝔲𝔤𝔯𝔞}\left(10,1\right)\right)$ as in prop. 1 above, the generators are identified with derivations on $\mathrm{CE}\left(\mathrm{𝔰𝔲𝔤𝔯𝔞}\left(10,1\right)\right)$ as

${P}_{a}=\left[{d}_{\mathrm{CE}},\frac{\partial }{\partial {e}^{a}}\right]$P_a = [d_{CE}, \frac{\partial}{\partial e^a} ]

and

${Q}_{\alpha }=\left[{d}_{\mathrm{CE}},\frac{\partial }{\partial {\psi }^{\alpha }}\right]$Q_\alpha = [d_{CE}, \frac{\partial}{\partial \psi^\alpha} ]

and

${J}_{ab}=\left[{d}_{\mathrm{CE}},\frac{\partial }{\partial {\omega }^{ab}}\right]$J_{a b} = [d_{CE}, \frac{\partial}{\partial \omega^{a b}} ]

and

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

$\left[{Q}_{\alpha },{Q}_{\beta }\right]=i\left(C{\Gamma }^{a}{\right)}_{\alpha \beta }{P}_{a}+\left(C{\Gamma }_{ab}\right){Z}^{ab}$[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 $\mathrm{𝔰𝔲𝔤𝔯𝔞}\left(10,1\right)$. See D'Auria-Fre formulation of supergravity.

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

## References

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

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

• Hisham Sati, Urs Schreiber, Jim Staasheff, ${L}_{\infty }$-algebra valued connections (web)

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

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