nLab super tangent bundle

Super tangent bundles

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.

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.

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 \,.

=–

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.

Last revised on December 7, 2011 at 02:06:32. See the history of this page for a list of all contributions to it.