superalgebra

and

supergeometry

Contents

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

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

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

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

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

For more see at super Cartesian space and at signs in supergeometry.

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.

References

Revised on January 23, 2015 22:16:53 by Urs Schreiber (88.100.66.95)