Sign convention for derivatives in a $\mathbb{Z}_2$ graded space

Question

Suppose $V=V_0\oplus\theta V_1$ is a $\mathbb{Z}_2$ graded super vector space.

Note: Since $\theta^2=0$, this implies $\theta\mathrm{d}\theta=-\mathrm{d}\theta\cdot\theta$.

However, I wish to know if the following equation is correct -- $$ \frac{\partial}{\partial\theta}\left(\theta\frac{\partial}{\partial t}\right)=\frac{\partial}{\partial t}-\theta\frac{\partial^2}{\partial\theta\partial t} $$

Answer 1

The article on superspace on Wikipedia has some simple discussions. Maybe also look at Supersolutions by Daniel Freed and Pierre Deligne.

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An arbitrary function on $\mathbb{R}^{1|1}$ is $f(t) + \theta g(t)$ and the operator $\tfrac{d}{d\theta}$ is going to take the 1-st order term in theta, so

$$ \frac{d}{d\theta} \big[ f(t) + \theta g(t) \big] = g(t)$$

We can see what happens when we apply your differential operator:

$$ \frac{d}{d\theta} \left( \theta \frac{d}{dt} \right) \big[ f(t) + \theta g(t) \big] = \frac{d}{d\theta} \big[ \theta \frac{df}{dt} \big] = \frac{df}{dt} $$

Which sign does that correspond to?