nLab
super tangent bundle

Super tangent bundles

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 TXT X of a supermanifold XX is given by the sheaf UDerO X(U)U \mapsto Der O_X(U).

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

Example

On the supermanifold 11\mathbb{R}^{1|1} with its canonical coordinates

tC ( 11) ev t \in C^\infty(\mathbb{R}^{1|1})^{ev}
θC ( 11) odd \theta \in C^\infty(\mathbb{R}^{1|1})^{odd}

there is the odd vector field

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

whose super Lie bracket with itself vanishes

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

This odd vector field DD is left invariant with respect to the super translation group structure on 11\mathbb{R}^{1|1}.

This means that Lie( 11)Lie(\mathbb{R}^{1|1}) is free on one odd generator.

Revised on December 7, 2011 02:06:32 by Toby Bartels (64.89.53.157)