nLab super tangent bundle

Idea

The tangent bundle of an ordinary manifold is a vector bundle whose sheaf of sections? is given by the derivations of the structure sheaf. The same idea applies to a supermanifold to produce a super vector bundle.

Definition

The super tangent bundle $T X$ of a supermanifold $X$ is given by the sheaf $U \mapsto Der O_X(U)$.

So a super tangent vector is a global section of this sheaf of derivations.

Example

On the supermanifold $\mathbb{R}^{1|1}$ with its canonical coordinates

$$t \in C^\infty(\mathbb{R}^{1|1})^{ev}$$

$$\theta \in C^\infty(\mathbb{R}^{1|1})^{odd}$$

there is the odd vector field

$$D \coloneqq \partial_\theta + \theta \cdot \partial_{t}$$

whose super Lie bracket with itself vanishes

$$[D, D] = 0 \,.$$

=–

Claim

This odd vector field $D$ is left invariant with respect to the super translation group structure on $\mathbb{R}^{1|1}$.

This means that $Lie(\mathbb{R}^{1|1})$ is free on one odd generator.