nLab differential form on a supermanifold

Idea

The notion of superdifferential form is the generalization of the notion of differential form from manifolds to supermanifolds.

Definition

Definition

(differential forms on super Cartesian space)

For $n,q \in \mathbb{N}$, the de Rham algebra $\Omega^\bullet(\mathbb{R}^{n\vert q})$ of super-differential forms on the super Cartesian space $\mathbb{R}^{n\vert q}$ is the free differential graded-commutative superalgebra over the supercommutative algebra

$$C^\infty(\mathbb{R}^{n\vert q}) \;=\; \underset{ even }{ \underbrace{ C^\infty(\mathbb{R}^n) }} \otimes_{\mathbb{R}} \underset{ odd }{ \underbrace{ \wedge^\bullet \mathbb{R}^q }}$$

on

1. $n$ generators $\mathbf{d} x^a$ in bi-degree $(1,even)$ (the canonical bosonic 1-forms)

2. $q$ generators $\mathbf{d} \theta^\alpha$ in bi-degree $(1,odd)$

hence:

(1)$$\Omega^\bullet( \mathbb{R}^{n\vert q} ) \;\coloneqq\; C^\infty(\mathbb{R}^{n\vert q}) \left[ \underset{deg = (1,even)}{\underbrace{ \left\langle \mathbf{d} x^a \right\rangle_{a = 1}^{n} }},\;\;\;\;\; \underset{ deg = (1,odd) }{ \underbrace{ \left\langle \mathbf{d} \theta^\alpha \right\rangle_{\alpha = 1}^q }} \right]$$

with differential having the evident definition on generators, and extended from there as a derivation of bi-degree $(1, even) \in \mathbb{Z} \times (\mathbb{Z}/2)$

Example

(sign rule for differential forms on super Cartesian spaces)

For $n,q \in \mathbb{N}$, the generators of the differential graded-commutative superalgebra $\Omega^\bullet(\mathbb{R}^{n\vert q})$ of differential forms on the super Cartesian space (Def. ) have the following bi-degree

$\phantom{A}$generator$\phantom{A}$	$\phantom{A}$bi-degree$\phantom{A}$
$\phantom{A}$$x^a$$\phantom{A}$	$\phantom{A}$(0,even)$\phantom{A}$
$\phantom{A}$$\theta^\alpha$$\phantom{A}$	$\phantom{A}$(0,odd)$\phantom{A}$
$\phantom{A}$$\mathbf{d}x^a$$\phantom{A}$	$\phantom{A}$(1,even)$\phantom{A}$
$\phantom{A}$$\mathbf{d}\theta^\alpha$$\phantom{A}$	$\phantom{A}$(1,odd)$\phantom{A}$

and satisfy the following graded-commutation relations, depending on one of the two equivalent (see here) sign rules:

$\phantom{A}$sign rule$\phantom{A}$	$\phantom{A}$Deligne’s$\phantom{A}$	Bernstein’s$\phantom{A}$
$\phantom{A}$$x^{a} \; x^{b} =$	$\phantom{A}$$+ x^{b} \; x^{a}$$\phantom{A}$	$\phantom{A}$$+ x^{b} \; x^{a}$$\phantom{A}$
$\phantom{A}$$x^a \;\theta^\alpha =$$\phantom{A}$	$\phantom{A}$$+ \theta^\alpha \; x^a$$\phantom{A}$	$\phantom{A}$$+ \theta^\alpha \; x^a$$\phantom{A}$
$\phantom{A}$$\theta^{\alpha} \; \theta^{\beta} =$$\phantom{A}$	$\phantom{A}$$- \theta^{\beta} \; \theta^{\alpha}$$\phantom{A}$	$\phantom{A}$$- \theta^{\beta} \; \theta^{\alpha}$$\phantom{A}$
$\phantom{A}$$x^{a} (\mathbf{d}x^{a}) =$$\phantom{A}$	$\phantom{A}$$+ (\mathbf{d}x^{b}) x^{a}$$\phantom{A}$	$\phantom{A}$$+ (\mathbf{d}x^{b}) x^{a}$$\phantom{A}$
$\phantom{A}$$\theta^\alpha (\mathbf{d}x^a) =$$\phantom{A}$	$\phantom{A}$$+ (\mathbf{d}x^a) \theta^\alpha$$\phantom{A}$	$\phantom{A}$${\color{blue}{-}} (\mathbf{d}x^a) \theta^\alpha$$\phantom{A}$
$\phantom{A}$$\theta^{\alpha} (\mathbf{d}\theta^{\beta}) =$$\phantom{A}$	$\phantom{A}$$- (\mathbf{d}\theta^{\beta}) \theta^{\alpha}$$\phantom{A}$	$\phantom{A}$${\color{blue}{+}} (\mathbf{d}\theta^{\beta}) \theta^{\alpha}$$\phantom{A}$
$\phantom{A}$$(\mathbf{d}x^{a}) (\mathbf{d} x^{b}) =$$\phantom{A}$	$\phantom{A}$$- (\mathbf{d} x^{b}) (\mathbf{d} x^{a})$$\phantom{A}$	$\phantom{A}$$- (\mathbf{d} x^{b}) (\mathbf{d} x^{a})$$\phantom{A}$
$\phantom{A}$$(\mathbf{d}x^a) (\mathbf{d} \theta^{\alpha}) =$$\phantom{A}$	$\phantom{A}$$- (\mathbf{d}\theta^{\alpha}) (\mathbf{d} x^a)$$\phantom{A}$	$\phantom{A}$${\color{blue}{+}} (\mathbf{d}\theta^{\alpha}) (\mathbf{d} x^a)$$\phantom{A}$
$\phantom{A}$$(\mathbf{d}\theta^{\alpha}) (\mathbf{d} \theta^{\beta}) =$	$\phantom{A}$$+ (\mathbf{d}\theta^{\beta}) (\mathbf{d} \theta^{\alpha})$$\phantom{A}$	$\phantom{A}$$+ (\mathbf{d}\theta^{\beta}) (\mathbf{d} \theta^{\alpha})$$\phantom{A}$

Definition

(pullback over super Cartesian spaces)

Let

$$\array{ \mathbb{R}^{n_1 \vert q_1} &\overset{f}{\longrightarrow}& \mathbb{R}^{n_2 \vert q_2} \\ C^\infty(\mathbb{R}^{n_1\vert q_1}) &\overset{f^\ast}{\longleftarrow}& C^\infty(\mathbb{R}^{n_2\vert q_2}) }$$

be a morphism of super Cartesian spaces, hence formally dually a algebra homomorphism $f^\ast$ of supercommutative superalgebras.

By the fact that $\Omega^\bullet(\mathbb{R}^{n \vert q})$ (Def. ) is free over $C^\infty(\mathbb{R}^{n\vert q})$ on generators $\mathbf{d}x^a$, $\mathbf{d}\theta^\alpha$ (1) this extends to a unique homomorphism on the de Rham dgc-superalgebra

$$\Omega^\bullet(\mathbb{R}^{n_1\vert q_1}) \overset{f^\ast}{\longleftarrow} \Omega^\bullet(\mathbb{R}^{n_2\vert q_2})$$

subject to the condition that $f^\ast \circ \mathbf{d}_2 = \mathbf{d}_1 \circ f^\ast$:

$$\array{ \Omega^\bullet(\mathbb{R}^{n_1\vert q_1}) &\overset{\phantom{AA}f^\ast\phantom{AA}}{\longleftarrow}& \Omega^\bullet(\mathbb{R}^{n_2\vert q_2}) \\ {}^{\mathbf{d}_1}\Big\downarrow && \Big\downarrow{}^{\mathbf{d}_2} \\ \Omega^\bullet(\mathbb{R}^{n_1\vert q_1}) &\overset{\phantom{AA}f^\ast\phantom{AA}}{\longleftarrow}& \Omega^\bullet(\mathbb{R}^{n_2\vert q_2}) }$$

This operation is called pullback of differential forms along maps of super Cartesian spaces.

Proposition

(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:

$$\mathbf{\Omega}\bullet \;\colon\; SuperCartSp^{op} \longrightarrow dgcsAlg$$

with values in differential graded-commutative superalgebras, in fact a sheaf and hence a differential graded algebra internal to the sheaf topos over SuperCartSp:

$$\mathbf{\Omega}^n \;\in\; Sh(SuperCartSp) \,.$$

The construction generalizes in an evident way also to sheaves over super formal Cartesian spaces, hence to super formal smooth sets:

$$\mathbf{\Omega}^n \;\in\; SuperFormalSmoothSet \;\coloneqq\; Sh(SuperFormalCartSp) \,.$$

We may now proceed as in the discussion of differential forms on smooth sets (hereets#DifferentialForms)):

Definition

(super differential forms on general super formal smooth sets)

Let $X$ be a supermanifold or more generally a super formal smooth set

$$X \;\in\; SuperFormalSmoothSet$$

Then super differential forms on $X$ are morphisms

$$X \longrightarrow \mathbf{\Omega}^\bullet$$

into the classifying sheaf $\mathbf{\Omega}^\bullet$ from Def. .

Properties

Integral top-forms and Picture number

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)).

Examples

Let $\mathbf{X} = \mathbb{R}^{1|1}$. The superalgebra of functions on $\mathbf{X}$ is the exterior algebra that is generated over $C^\infty(\mathbf{R})$ from a single generator $\theta$ in odd degree (the canonical odd coordinate).

The algebra of superdifferential forms on $\mathbb{R}^{1|1}$ is the exterior algebra generated over $C^\infty(\mathbb{R})$ from

• a generator $\theta$ in odd degree (the canonical odd coordinate);

• a generator $d x$ in odd degree (the differential of the canonical even coordinate);

• a generator $d \theta$ in even degree (the differential of the canonical odd coordinate).

Notice in particular that while $d x \wedge d x = 0$ the wedge product $d \theta \wedge d\theta$ is non-vanishing, since $d \theta$ is in even degree. In fact all higher wedge powers of $d \theta$ with itself exist.

Remarks

• Being a $\mathbb{Z}_2$-graded locally free algebra itself, one can regard $\Omega^\bullet(X)$ itself (even for $X$ 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 $T[1] X$, the shifted tangent bundle of $X$. By definition we have $C^\infty(T[1]X) = \Omega^\bullet(X)$. From this point of view, the existence of the differential $d$ on the graded algebra $\Omega^\bullet(X)$ translates into the existence of a special odd vector field on $T[1]X$. This is a homological vector field in that it is odd and the super Lie bracket of it with itself vanishes: $[d,d] = 0$.

• In the context of L-infinity algebroids, where one may regard $C^\infty(X)$ as the Chevalley-Eilenberg algebra of an $L_\infty$-algebroid it is useful to notice that $\Omega^\bullet(X)$ is the corresponding Weil algebra. If $X$ is a Lie $n$-algebroid then $T[1]X$ is a Lie $(n+1)$-algebroid.

Related concepts

• signs in supergeometry

• picture changing operator

• super Lie algebroid

References

Original articles:

• Joseph Bernstein, Dimitry Leites: Integral forms and the Stokes formula on supermanifolds, Functional Analysis and its Applications 11 (1977) 45–47 [doi:10.1007/BF01135531]

See also:

• Simone Noja, On the Geometry of Forms on Supermanifolds, Differential Geometry and its Applications, 88 (2023) 101999 [arXiv:2111.12841, doi:10.1016/j.difgeo.2023.101999]

In the context of algebraic geometry:

• Alexander Polishchuk: De Rham cohomology for supervarieties [arXiv:2305.01858]

Discussion in view of Lie algebra cohomology of super Lie algebras:

• Roberto Catenacci, C. A. Cremonini, Pietro Antonio Grassi, Simone Noja, Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces (arXiv:2012.05246)

Discussion with an eye towards quantization of the superstring is in

• Alexander Belopolsky, De Rham Cohomology of the Supermanifolds and Superstring BRST Cohomology, Phys.Lett. B403 (1997) 47-50 (arXiv:hep-th/9609220)

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

• Alexander Belopolsky, Picture changing operators in supergeometry and superstring theory (arXiv:hep-th/9706033)

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)