∞-Lie theory (higher geometry)
Formal Lie groupoids
The supergravity Lie 3-algebra or M2-brane extension is a super L-∞ algebra that is a shifted extension
of the super Poincare Lie algebra 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 .
The Chevalley-Eilenberg algebra
The Chevalley-Eilenberg algebra is generated on
elements and of degree
a single element of degree
and elements of degree
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 , equipped with a degree-3 element in its Chevalley-Eilenberg algebra, and equipped with a homomorphism such that the pullback of the 4-cocycle along is trivialized by :
In the homotopy theory of L-infinity algebra? this means that
Compare this to the characterization of as the as the homotopy fiber of , hence as the universal solution to this situation
In any case, in (D’Auria-Fre 82) possible choices for are found.
Curiously, the bosonic body of 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 -brane sigma-models (see at table of branes for further links and see at The brane bouquet for the full classification):
(The first colums follow the exceptional spinors table.)
The corresponding exceptional super L-∞ algebra cocycles (schematically, without prefactors):
|11|| on sIso(10,1)|| on m2brane|
|10|| on sIso(9,1)|| on StringIIA|| on StringIIB|| on StringIIA|| on sIso(9,1)|| on StringIIA|| on StringIIB|| in StringIIA|| on StringIIB|
|9|| on sIso(8,1)|
|8|| on sIso(7,1)|
|7|| on sIso(6,1)|
|6|| on sIso(5,1)|| on sIso(5,1)|
|5|| on sIso(4,1)|
|4|| on sIso(3,1)|| on sIso(3,1)|
|3|| on sIso(2,1)|
Relation to the 11-dimensional polyvector super Poincaré-algebra
Let be the automorphism ∞-Lie algebra of . This is a dg-Lie algebra. Write for the ordinary Lie algebra in degree 0.
This is isomorphic to the polyvector-extension of the super Poincaré Lie algebra (see there) in – the “M-theory super Lie algebra” – with “2-brane central charge”: the Lie algebra spanned by generators and graded Lie brackets those of the super Poincaré Lie algebra as well as
This observation appears implicitly in (Castellani 05, section 3.1), see (FSS 13).
With the presentation of the Chevalley-Eilenberg algebra as in prop. 1 above, the generators are identified with derivations on as
etc. With this it is straightforward to compute the commutators. Notably the last term in
arises from the contraction of the 4-cocycle with .
Via the Heisenberg Lie 3-algebras
The field configurations of 11-dimensional supergravity may be identified with ∞-Lie algebra-valued forms with values in . See D'Auria-Fre formulation of supergravity.
supergravity Lie 6-algebra supergravity Lie 3-algebra super Poincaré Lie algebra
The Chevalley-Eilenberg algebra of 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
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 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 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