Formal Lie groupoids
Critical string models
The supergravity Lie 6-algebra is a super L-∞ algebra such that ∞-connections with values in it encode
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
is the 4-cocycle which defines as an extension of m 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.
This expression vanishes due to the Fierz identities
The supergravity Lie 6-algebra is the super Lie 7-algebra that is the -extension of classified by the cocycle from def. 1.
This appears as (Castellani-DAuria-Fre, (III.8.18)).
Relation to supergravity
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
(field strength of supergravity C-field)
(dual field strength)
(Dirac operator applied to gravitino)
(Riemann tensor: field strength of gravity)
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
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