superalgebra and (synthetic ) supergeometry
The notion of superdifferential form is the generalization of the notion of differential form from manifolds to supermanifolds.
(differential forms on super Cartesian space)
For , the de Rham algebra of super-differential forms on the super Cartesian space is the free differential graded-commutative superalgebra over the supercommutative algebra
on
generators in bi-degree (the canonical bosonic 1-forms)
generators in bi-degree
hence:
with differential having the evident definition on generators, and extended from there as a derivation of bi-degree
(sign rule for differential forms on super Cartesian spaces)
For , the generators of the differential graded-commutative superalgebra of differential forms on the super Cartesian space (Def. ) have the following bi-degree
generator | bi-degree |
---|---|
(0,even) | |
(0,odd) | |
(1,even) | |
(1,odd) |
and satisfy the following graded-commutation relations, depending on one of the two equivalent (see here) sign rules:
sign rule | Deligne’s | Bernstein’s |
---|---|---|
(pullback over super Cartesian spaces)
Let
be a morphism of super Cartesian spaces, hence formally dually a algebra homomorphism of supercommutative superalgebras.
By the fact that (Def. ) is free over on generators , (1) this extends to a unique homomorphism on the de Rham dgc-superalgebra
subject to the condition that :
This operation is called pullback of differential forms along maps of super Cartesian spaces.
(classifing super formal smooth set of super differential forms)
The operation of pullback of differential forms (Def. ) over super Cartesian spaces respects identity morphisms and composition. Hence the assignment of differential forms on super Cartesian spaces (Def. ) is a presheaf on SuperCartSp:
with values in differential graded-commutative superalgebras, in fact a sheaf and hence a differential graded algebra internal to the sheaf topos over SuperCartSp:
The construction generalizes in an evident way also to sheaves over super formal Cartesian spaces, hence to super formal smooth sets:
We may now proceed as in the discussion of differential forms on smooth sets (hereets#DifferentialForms)):
(super differential forms on general super formal smooth sets)
Let be a supermanifold or more generally a super formal smooth set
Then super differential forms on are morphisms
If a choice of integral top-forms is made, needed for a notion of integration over supermanifolds, then there is an additional grading by “picture number” (Belopolsky 97b, Witten 12), see (Catenacci-Grassi-Noja 18 (5.8) to (5.12)).
Let . The superalgebra of functions on is the exterior algebra that is generated over from a single generator in odd degree (the canonical odd coordinate).
The algebra of superdifferential forms on is the exterior algebra generated over from
a generator in odd degree (the canonical odd coordinate);
a generator in odd degree (the differential of the canonical even coordinate);
a generator in even degree (the differential of the canonical odd coordinate).
Notice in particular that while the wedge product is non-vanishing, since is in even degree. In fact all higher wedge powers of with itself exist.
Being a -graded locally free algebra itself, one can regard itself (even for a usual manifold!) as the “algebra of functions” (more precisely inner hom, i.e. mapping space into the line) on another supermanifold. That supermanifold is called , the shifted tangent bundle of . By definition we have . From this point of view, the existence of the differential on the graded algebra translates into the existence of a special odd vector field on . This is a homological vector field in that it is odd and the super Lie bracket of it with itself vanishes: .
In the context of L-infinity algebroids, where one may regard as the Chevalley-Eilenberg algebra of an -algebroid it is useful to notice that is the corresponding Weil algebra. If is a Lie -algebroid then is a Lie -algebroid.
Original articles:
See also:
In the context of algebraic geometry:
Discussion in view of Lie algebra cohomology of super Lie algebras:
Discussion with an eye towards quantization of the superstring is in
Geometric discussion of picture number appearing in the context of integration over supermanifolds (and originally seen in the quantization of the NSR superstring, crucial in superstring field theory) is due to
and further amplified in
Edward Witten, appendix D of Notes On Super Riemann Surfaces And Their Moduli (arXiv:1209.2459)
Roberto Catenacci, Pietro Antonio Grassi, Simone Noja, Superstring Field Theory, Superforms and Supergeometry (arXiv:1807.09563)
Last revised on September 6, 2024 at 15:16:06. See the history of this page for a list of all contributions to it.