The odd line is the supermanifold – a super Cartesian space and in particular a superpoint – characterized by the fact that its -graded algebra of functions is the algebra free on a single odd generator : .
This algebra is essentially the ring of dual numbers, but with the single generator in odd degree.
which is the semidirect product group of the multiplicative group (the group of units, hence when working over the real numbers) with the additive group shifted into odd degree. (See at References – Automorphism group for the origin of this observation.)
In the topos over superpoints this is seen over the test space itself with canonical odd coordinate by taking the canonical odd coordinate of the odd line that we are taking automorphism of to be and observing that maps
with in even degree and arbitrary in odd degree. Notice that the inverse of this map exists and is given by
Moreover, observe that an action of on a supermanifold corresponds to a choice of grading and a choice of differential. A clean account of this statement is in (Carchedi-Roytenberg 12). (At least for the grading this is essentially the classical statement about graded rings, see at affine line the section Properties – Grading).
That an action of the endomorphism/automorphism supergroup of the odd line on a supermanifold is equivalent to a choice of grading and a differential first observed in
It was later amplified in section 3.2 of
The same mechanism is amplified further in the discussion of derived differential geometry in
An interpretation of an -action on a supermanifold of local quantum observables of a supersymmetric field theory as the formalization of the concept of topologically twisted super Yang-Mills theory is in section 15 of
For more on this see at topologically twisted super Yang-Mills theory – Formalization.