∞-Lie theory

superalgebra

and

supergeometry

# Contents

## 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

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

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

## 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)

### 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}{=}$$p =$123456789
11M2M5
10D0F1, D1D2D3D4NS5, D5D6D7D8D9
9$\ast$
8$\ast$
7M2${}_{top}$
6F1${}_{little}$, S1${}_{sd}$S3
5$\ast$
4$\ast$$\ast$
3$\ast$

(The first colums follow the exceptional spinors table.)

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

$\stackrel{D}{=}$$p =$123456789
11$\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)

### Relation to the 11-dimensional polyvector super Poincaré-algebra

###### 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 (Castellani05, section 3.1), see (FSS 13).

###### Proof

With the presentation of the Chevalley-Eilenberg algebra $CE(\mathfrak{sugra}(10,1))$ as in prop. 1 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 $\frac{\partial}{\partial \psi^\alpha}\wedge \frac{\partial}{\partial \psi^\beta}$.

## 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.

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

## References

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

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