∞-Lie theory (higher geometry)
and
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
of the super Poincare Lie algebra $\mathfrak{siso}(10,1)$ in 10+1 dimensions induced by the exceptional degree 4-super Lie algebra cocycle
This is the same mechanism by which the String Lie 2-algebra is a shifted central extensiono of $\mathfrak{so}(n)$.
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
(fill in details)
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 $A$:
In the homotopy theory of L-infinity algebra? this means that
Compare this to the characterization of $\mathfrak{sugra}(10,1)$ as the as the homotopy fiber of $\mu_4$, hence as the universal solution to this situation
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).
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 | $\ast$ | |||||||||
8 | $\ast$ | |||||||||
7 | M2${}_{top}$ | |||||||||
6 | F1${}_{little}$, S1${}_{sd}$ | S3 | ||||||||
5 | $\ast$ | |||||||||
4 | $\ast$ | * | ||||||||
3 | * |
(The first colums 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) |
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
etc.
This observation appears implicitly in (Castellani 05, section 3.1), see (FSS 13).
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
and
and
and
etc. With this it is straightforward to compute the commutators. Notably the last term in
arises from the contraction of the 4-cocycle $\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b$ with $\frac{\partial}{\partial \psi^\alpha}\wedge \frac{\partial}{\partial \psi^\beta}$.
(…)
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
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
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, J.M. Izquierdo, M. Picon, O. Varela, On the underlying gauge group structure of D=11 supergravity, Phys.Lett.B596:145-155,2004 (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)
L. Andrianopoli, Riccardo D'Auria, L. Ravera, Hidden Gauge Structure of Supersymmetric Free Differential Algebras (arXiv:1606.07328)
The computation of the automorphism Lie algebra of $\mathfrak{sugra}(10,1)$ 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