nLab
superdifferential form

Idea

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

Definition

Ordinary differential forms on a manifold $X$ may be regarded as the functions on the supermanifold called the shifted tangent bundle

$${\Omega }^{•}(X)=C^{\mathrm{\infty }}(T[1]X).$$\Omega^\bullet(X) = C^\infty(T[1] X) \,.

The notion of shifted tangent bundle makes sense also when $X$ itself was already a supermanifold. Superdifferential forms on a supermanifold $X$ are similarly the algebra of functions on the shifted tangent bundle $T[1]X$.

Another way to think of superdifferential forms is using the perspective of Lie theory:

For $X$ a supermanifold with function algebra $C^{\mathrm{\infty }}(X)$, the qDGCA ${\Omega }^{•}(X)$ of differential forms on $X$ is the Weil algebra of $C^{\mathrm{\infty }}(X)$, (regarded as a ${\mathbb{Z}}_2$-graded dg-algebra).

Examples

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

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

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

• a generator $dx$ 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 $dx\wedge dx=0$ the wedge product $d\theta \wedge d\theta$ is non-vanishing, since $d\theta$ is in even degree. In fact al higher wedge powers of $d\theta$ with itself exist.

Remarks

• Being a ${\mathbb{Z}}_2$-graded locally free algebra itself, one can regard ${\Omega }^{•}(X)$ 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 $C^{\mathrm{\infty }}(T[1]X)={\Omega }^{•}(X)$. From this point of view, the existence of the differential $d$ on the graded algebra ${\Omega }^{•}(X)$ 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 NQ-supermanifolds, where one may regard $C^{\mathrm{\infty }}(X)$ as the Chevalley-Eilenberg algebra of an $L_{\mathrm{\infty }}$-algebroid it is useful to notice that ${\Omega }^{•}(X)$ is the corresponding Weil algebra. If $X$ is a Lie $n$-algebroid then $T[1]X$ is a Lie $(n+1)$-algebroid.