∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
superalgebra and (synthetic ) supergeometry
The supergravity Lie 6-algebra is a super L-∞ algebra such that ∞-connections with values in it encode
a vielbein field and a spin connection, hence the first order formulation of gravity for a graviton field in 11-dimensions;
a gravitino field;
the supergravity C-field in degree 3 and its magnetic dual.
This is such that the field strengths and Bianchi identities of these fields are governed by certain fermionic super L-∞ algebraic cocycles as suitable for 11-dimensional supergravity.
The supergravity Lie 3-algebra carries an L-∞ algebra cocycle of degree 7, given in the standard generators (vielbein), (spin connection) (gravitino) and (supergravity C-field) by
where
is the 4-cocycle which defines as an extension of , and where is the generator that cancels the class of this cocycle, .
This appears in (DAuria-Fre, page 18) and Castellani-DAuria-Fre, III.8.3.
Hence if we write
and
then
This is the structure of the equations of motion of the field strength of the supergravity C-field and its Hodge dual in 11-dimensional supergravity.
The supergravity Lie 6-algebra is the super Lie 7-algebra that is the -extension of classified by the cocycle from def. .
This means that the Chevalley-Eilenberg algebra is generated from
(vielbein) in degree
(spin connection) in degree ;
(gravitino) in degree
(supergravity C-field) in degree
(magnetic dual supergravity C-field) in degree
with differential defined by
This appears as (Castellani-DAuria-Fre, (III.8.18)).
According to (Castellani-DAuria-Fre, comment below (III.8.18)): “no further extension is possible”.
The supergravity Lie 6-algebra is something like the gauge -algebra of 11-dimensional supergravity, in the sense discussed at D'Auria-Fre formulation of supergravity .
Write for the Weil algebra of the supergravity Lie 6-algebra.
Write and for the shifted generators of the Weil algebra corresponding to and , respectively.
Define a modified Weil algebra by declaring it to have the same generators and differential as before, except that the differential for is modified to
and hence the differential of is accordingly modified in the unique way that ensures (yielding the modified Bianchi identity for ).
This ansatz appears as (CastellaniDAuriaFre, (III.8.24)).
Note that this amounts simply to a redefinition of curvatures
A field configuration of 11-dimensional supergravity is given by L-∞ algebra valued differential forms with values in . Among all of these the solutions to the equations of motion (the points in the covariant phase space) can be characterized as follows:
A field configuration
solves the equations of motion precisely if
all curvatures sit in the ideal of differential forms spanned by the 1-form fields (vielbein) and (gravitino);
more precisely if we have
(super-torsion);
(dual field strength)
(Dirac operator applied to gravitino)
such that the coefficients of terms containing s are polynomials in the coefficients of the terms containing no s. (“rheonomy”).
This is the content of (CastellaniDAuriaFre, section III.8.5).
In particular this implies that on-shell the 4- and 7-field strength are indeed dual of each other
This is the content of (CastellaniDAuriaFre, equation (III.8.52)).
supergravity Lie 6-algebra supergravity Lie 3-algebra super Poincaré Lie algebra
The supergravity Lie 6-algebra appears first on page 18 of
and is recalled in section 4 of
Volume 146, Issue 1, March 1983, Pages 35–77
A textbook discussion is in section III.8.3 of
The same is being recalled for instance in section 3 of
Then it is rediscovered around equation (8.8) in
which gives a detailed and comprehensive discussion.
A discussion in the context of smooth super ∞-groupoids is in
in the last section of
Last revised on May 27, 2020 at 06:30:16. See the history of this page for a list of all contributions to it.