∞-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 (D’Auria & Fré 1982, p. 18) is (with hindsight, following FSS15) a super L-∞ algebra such that -algebra valued differential forms (local ∞-connections) with values in it encode field histories of
the vielbein field with a spin connection,
hence the first order formulation of gravity for a graviton field
in 10+1 dimensions;
the gravitino field;
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.
Our normalization conventions entirely follow Castellani, D’Auria & Fré 1991, §III.8.3, but we denote the super-vielbein 1-forms by instead of by .
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, .
In other words, we have
which has the same structure as the equations of motion of the field strength of the supergravity C-field and its Hodge dual in 11-dimensional supergravity (to which it is related below)
This appears (in the dual language of Chevalley-Eilenberg algebras) in DAuria & FrFré 1982, page 18 and Castellani, D’Auria & Fré 1991, §III.8.3.
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 in Castellani, D’Auria & Fré, (III.8.18).
According to Castellani, D’Auria & Fré 1991, 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-Fré 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 an adjusted 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 in Castellani, D’Auria & Fré 1991 (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, in paraphrase, the content of Castellani, D’Auria & Fré, section III.8.5.
In particular, the Bianchi identity for the super-form enhancement of the supergravity C-field flux density is (by III.8.23j, 34, 35, 36 and using hupf):
and its rheonomic solution implies for the “actual” flux densities and that
the CJS higher Maxwell equation holds
(III.8.53)
the Hodge duality holds
(III.8.52)
supergravity Lie 6-algebra supergravity Lie 3-algebra super Poincaré Lie algebra
The supergravity Lie 6-algebra originates in:
Brief review is in:
Leonardo Castellani, Pietro Fré, F. Giani, Krzysztof Pilch, Peter van Nieuwenhuizen, §4 of: Gauging of supergravity?, Annals of Physics 146 1 (1983) 35-77 [spire:11998, doi:10.1016/0003-4916(83)90052-0]
Pietro Fré, Pietro Antonio Grassi, §3 of: Free Differential Algebras, Rheonomy, and Pure Spinors [arXiv:0801.3076, inspire:777785]
Pietro Fré, §6.4.1 in: Gravity, a Geometrical Course, Volume 2: Black Holes, Cosmology and Introduction to Supergravity, Springer (2013) [doi:10.1007/978-94-007-5443-0]
A monograph account is in:
It was (apparently) rediscovered in:
where a detailed discussion is given.
These authors all consider the Chevalley-Eilenberg algebra (“FDA”) of the actual super -algebra. The latter and its relation to smooth super ∞-groupoids is considered in:
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, Int. J. Geom. Meth. Modern Physics 12 02 (2015) 1550018 [arXiv:1308.5264, doi:10.1142/S0219887815500188]
Urs Schreiber, last section of: differential cohomology in a cohesive topos
Last revised on February 16, 2024 at 11:43:29. See the history of this page for a list of all contributions to it.