odd line

and

The **odd line** is the supermanifold $\mathbb{R}^{0|1}$ – a super Cartesian space and in particular a superpoint – characterized by the fact that its $\mathbb{Z}_2$-graded algebra of functions is the algebra free on a single odd generator $\theta$: $C^\infty(\mathbb{R}^{0|1}) = \mathbb{R}[\theta] = \mathbb{R} \oplus \theta\cdot \mathbb{R}$.

This algebra is essentially the ring of dual numbers, but with the single generator in odd degree.

The interal automorphism group of the odd line in the topos of smooth super spaces is the supergroup

$\mathbf{Aut}(\mathbb{A}^{0|1})
\simeq
\mathbb{G}_m \ltimes (\Pi \mathbb{G}_{ad})$

which is the semidirect product group of the multiplicative group (the group of units, hence $\mathbb{R}^\times$ 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 $\mathbb{R}^{0|q}$ this is seen over the test space $\mathbb{R}^{0|1}$ itself with canonical odd coordinate $\theta$ by taking the canonical odd coordinate of the odd line that we are taking automorphism of to be $\epsilon$ and observing that maps

$\mathbb{R}^{0|1} \to \mathbf{Aut}(\mathbb{R}^{0|1})$

are then given, under the evaluation map-isomorphism and via the Yoneda lemma by Grassmann algebra homomorphisms of the form

$\langle \epsilon, \theta\rangle
\leftarrow
\langle \epsilon\rangle$

that send

$\epsilon \mapsto x \epsilon + y \theta$

with $x \neq 0$ in even degree and arbitrary $y$ in odd degree. Notice that the inverse of this map exists and is given by

$\epsilon \mapsto x^{-1}\epsilon - y x^{-1} \theta
\,.$

Moreover, observe that an action of $\mathbf{Aut}(\mathbb{R}^{0|1})$ 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

- Maxim Kontsevich,
*Deformation quantization of Poisson manifolds, I*, Lett. Math. Phys. 66:157-216,2003 (arXiv:q-alg/9709040)

It was later amplified in section 3.2 of

- Denis Kochan, Pavol Ševera,
*Differential gorms, differential worms*(arXiv:math/0307303),

where it is used to exhibit the canonical de Rham differential action on the odd tangent bundle? $Maps(\mathbb{R}^{0|1}, X)$ of a supermanifold $X$.

The same mechanism is amplified further in the discussion of derived differential geometry in

- David Carchedi, Dmitry Roytenberg,
*Homological Algebra for Superalgebras of Differentiable Functions*(arXiv:1212.3745)

An interpretation of an $\mathbb{G}_m \ltimes \Pi \mathbb{G}_{ad}$-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

- Kevin Costello,
*Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4*(arXiv:1111.4234)

For more on this see at *topologically twisted super Yang-Mills theory – Formalization*.

Revised on October 9, 2013 11:36:56
by Urs Schreiber
(89.204.139.155)