nLab
integration over supermanifolds

Idea

There are different ways to define a differential volume element on a smooth manifold. Some of these definitions can be carried over to supergeometry, others cannot. The possibly most familiar way of talking about differential volume elements, in terms of top-degree differential forms, does not carry over to supermanifolds.

No top-degree forms in supergeometry

In supergeometry the notion of top-degree form does not in general make sense, since there are no top-degree wedge powers of “odd 1-forms”: if for instance ${\theta }_1$ and ${\theta }_2$ are odd functions on some supermanifold and $d{\theta }_1$ and $d{\theta }_2$ are their differential 1-forms, then the wedge product of these is symmetric in that

$$d{\theta }_1\wedge d{\theta }_2=+d{\theta }_2\wedge d{\theta }_1.$$d\theta_1 \wedge d \theta_2 = + d\theta_2 \wedge d \theta_1 \,.

Notice the plus sign on the right, which is the product of one minus sign for interchanging ${\theta }_1$ and ${\theta }_2$, and another minus sign for interchanging the two differentials.

Accordingly, the wedge product of the differential of an odd function $\theta$ with itself does not in general vanish:

$$(d\theta \wedge d\theta =0)\iff (\theta =0).$$(d \theta \wedge d\theta = 0) \Leftrightarrow (\theta = 0) \,.

On the cartesian supermanifold ${ℝ}^{n\mid m}$ with canonical even coordinate functions $\{x^i{\}}_1^n$ and canonical odd coordinate functions $\{{\theta }^j{\}}_1^m$ the differential form which one would want to regard as the canonical volume form is

$$\omega :=d{x}^1\wedge \cdots \wedge {\mathrm{dx}}^n\wedge d{\theta }^1\wedge \cdots \wedge d{\theta }^m.$$\omega := d x^1 \wedge \cdots \wedge dx^n \wedge d\theta^1 \wedge \cdots \wedge d\theta^m \,.

Due to the above, this is not a top form, since for instance

$$\omega \wedge d{\theta }^1\ne 0.$$\omega \wedge d\theta^1 \neq 0 \,.

But this example also indicates the solution: apparently for integration it is not really essential that a form is a top power, what is rather essential is that it is, locally, the wedge product of a basis of 1-forms. This perspective then does lead to a sensible definition of volume forms (and more generally “integrable forms”) on supermanifolds, described below.

Solution

Therefore the naïve identification of differential volume measures with top degree forms has to be refined. The idea is to characterize a volume form by other means, in particular as an equivalence class of choices of bases for the space of 1-forms, and then to define integrable forms to be pairs consisting of such a generalized volume form and a multivector: this pair is supposed to represent the differential form one would obtain could one contract the multivector with the volume form, as in ordinary differential geometry.

The definition of integration of integrable forms in supergeometry in terms of multivectorfields leads, in the case that the supermanifold in an NQ-supermanifold to the BV formalism.

Related concepts

• integration

• Berezin integral

References

An exposition of the standard lore is in

• Urs Schreiber, Integration over supermanifolds

A general abstract discussion in terms of D-module theory is in

• Frédéric Paugam, Homotopical Poisson Reduction of gauge theories (pdf)