nLab odd line

Definition

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.

Properties

The automorphism super-group

The internal 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).

Related concepts

• Cartesian space

• infinitesimal space

• synthetic differential supergeometry

References

General

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Automorphism group

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 (2003) 157-216 [arXiv:q-alg/9709040]

It was later amplified in

• Denis Kochan, Pavol Ševera, section 3.2 of: Differential gorms, differential worms, Mathematical Physics, World Scientific (2005) 128-130 [arXiv:math/0307303, doi:10.1142/9789812701862_0034]

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 dimension 2 and 4, talk at Geometric and Algebraic Structures in Mathematics, Stony Brook (2011) [arXiv:1111.4234]

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