nLab
differential form on a supermanifold

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 XX may be regarded as the functions on the supermanifold called the shifted tangent bundle

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

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

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

For X\mathbf{X} a supermanifold with function algebra C (X)C^\infty(\mathbf{X}), the qDGCA Ω (X)\Omega^\bullet(X) of differential forms on XX is the Weil algebra of C (X)C^\infty(\mathbf{X}), (regarded as a 2\mathbb{Z}_2-graded dg-algebra).

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

Examples

Let X= 1|1\mathbf{X} = \mathbb{R}^{1|1}. The superalgebra of functions on X\mathbf{X} is the exterior algebra that is generated over C (R)C^\infty(\mathbf{R}) from a single generator θ\theta in odd degree (the canonical odd coordinate).

The algebra of superdifferential forms on 1|1\mathbb{R}^{1|1} is the exterior algebra generated over C ()C^\infty(\mathbb{R}) from

  • a generator θ\theta in odd degree (the canonical odd coordinate);

  • a generator dxd x in odd degree (the differential of the canonical even coordinate);

  • a generator dθd \theta in even degree (the differential of the canonical odd coordinate).

Notice in particular that while dxdx=0 d x \wedge d x = 0 the wedge product dθdθd \theta \wedge d\theta is non-vanishing, since dθd \theta is in even degree. In fact all higher wedge powers of dθd \theta with itself exist.

Remarks

  • Being a 2\mathbb{Z}_2-graded locally free algebra itself, one can regard Ω (X)\Omega^\bullet(X) itself (even for XX 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]XT[1] X, the shifted tangent bundle of XX. By definition we have C (T[1]X)=Ω (X)C^\infty(T[1]X) = \Omega^\bullet(X). From this point of view, the existence of the differential dd on the graded algebra Ω (X)\Omega^\bullet(X) translates into the existence of a special odd vector field on T[1]XT[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[d,d] = 0.

  • In the context of L-infinity algebroids, where one may regard C (X)C^\infty(X) as the Chevalley-Eilenberg algebra of an L L_\infty-algebroid it is useful to notice that Ω (X)\Omega^\bullet(X) is the corresponding Weil algebra. If XX is a Lie nn-algebroid then T[1]XT[1]X is a Lie (n+1)(n+1)-algebroid.

References

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