nLab
super vector bundle

Contents

Idea

An ordinary smooth vector bundle on a manifold XX may be identified with its sheaf of sections, which is a locally free sheaf of modules over its structure sheaf and all such locally free module sheaves arise this way.

The definition of vector bundles in terms of sheaves of sections therefore immediately generalizes to every ringed space and in particular to supermanifolds.

Definition

A super vector bundle over a supermanifold XX is a locally free sheaf on the category of open subsets of X|X| of modules for the structure sheaf O XO_X.

Examples

Super tangent bundles

The tangent bundle of an ordinary manifold has the sheaf of sections given by the derivations of the structure sheaf. The same definition works here:

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 field 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 := \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 April 5, 2013 13:42:24 by Anonymous Coward (134.76.82.221)