Differential forms as superfunctions

[ajb]

I wanted to share with you some useful results concerning differential forms on a manifold. For those that know, the coordinate free definitions of differential forms and their derivations (de Rham, interior product and Lie derivative) are I feel quite complicated. I have very little feeling for them.

 

If we turn to a "super" description in terms of local coordinates then the definitions are much more workable. What I am about to say is not new.

 

Definition(ish) A supermanifold is a "manifold" with both even and odd coordinates.

 

Locally, we mean that a supermanifold can be endowed with coordinates [math]z^{A}=\{x^{a}, \theta^{\alpha}\}[/math] such that [math]x^{a}x^{b} = x^{b}x^{a}[/math], [math]\theta^{\alpha}\theta^{\beta} = -\theta^{\beta}\theta^{\alpha}[/math] and [math]x^{a}\theta^{\alpha} = \theta^{\alpha}x^{a}[/math]. Collectively, we can write this as [math]z^{A}z^{B}= (-1)^{\tilde{A}\tilde{B}}z^{B}z^{A}[/math], where we have [math]\tilde{A}= 0[/math] for "even coordinates" [math]x[/math] and [math]\tilde{A}= 1[/math] for "odd coordinates" [math]\theta[/math]. Note [math]\theta^{A}\theta^{A}=0[/math]. A supermanifold [math]SM[/math] is said to be off dimension [math]n|m[/math] if there are [math]n[/math] even coordinates and [math]m[/math] odd coordinates.

 

Remark Functions on a supermanifold are polynomials of degree [math]m[/math] in [math]\theta[/math]. This is due to [math]\theta^{2}=0[/math]. So locally we write

 

[math]f(x,\theta) = f(x) + \theta^{\alpha}f_{\alpha}(x) + \frac{1}{2!} \theta^{\alpha}\theta^{\beta}f_{\beta \alpha}(x)+ \cdots + \frac{1}{m!} \theta^{\alpha_{1}}\cdots \theta^{\alpha_{m}}f_{\alpha_{m}\cdots \alpha_{1}}(x) \in C^{\infty}(SM) \simeq \Omega^{*}(M)[/math].f's are smooth functions of [math]x[/math] that are antisymmetric in their indices.

 

Proposition Pseudo-differential forms (don't keep [math]\mathbb{Z}[/math] grading) on a manifold [math]M[/math] are isomorphic to functions on the supermanifold [math]\Pi TM[/math].

 

This of course needs some explaining. [math]\Pi[/math] is the reverse parity functor, it shifts the parity of the fibres of a vector bundle. So, if we give [math]TM[/math] local coordinates [math] \{x^{a}, v^{a}\}[/math] with [math]\tilde{x} = \tilde{v}=0[/math], then [math]\Pi TM[/math] has local coordinates [math]\{x^{a}, dx^{a}\}[/math] such that [math]\tilde{dx} = \tilde{x}+1=1[/math]. As a graded vector bundle, the bundle automorphisms are identical to that of [math]TM[/math], linear transformations are identical in the graded and non-graded cases. Thus [math]dx[/math] does indeed transform like a vector, but it is now odd.

 

Thus, a function on [math]\Pi TM[/math] locally looks like

 

[math]\omega(x,\theta) = \omega(x) + dx^{a}\omega_{a}(x) + \frac{1}{2!} dx^{a}dx^{b}\omega_{b a}(x)+ \cdots + \frac{1}{n!} dx^{a_{1}}\cdots dx^{a_{n}}\omega_{a_{n}\cdots a_{1}}(x) \in C^{\infty}(\Pi TM)[/math],

 

i.e. a pseudo-differential form.

 

Proposition The de Rham differential [math]d[/math], the interior product [math]i_{X}[/math] and the Lie derivative [math]L_{X}[/math] are all understood as vector fields on the supermanifold [math]\Pi TM[/math].

 

This is easily "justified". The only derivations of functions are vector fields.

 

In local coordinates we have

 

[math]d = dx^{a}\frac{\partial}{\partial x^{a}}[/math],

[math]i_{X} = X^{a} \frac{\partial}{\partial dx^{a}}[/math],

[math]L_{X} = X^{a}\frac{\partial}{\partial x^{a}} + dX^{a}\frac{\partial}{\partial dx^{a}}[/math],

 

with [math]X \in \mathfrak{X}(M)[/math]

 

Anyway, I thought it was pretty neat when I "re-discovered" it. The earliest reference I can find is Th. Voronov Geometric Integration Theory on Supermanifolds 1991, but I am sure people knew this before.

 

Hope this is useful to someone else on here...