odd line



The odd line is the supermanifold 0|1\mathbb{R}^{0|1} – a super Cartesian space and in particular a superpoint – characterized by the fact that its 2\mathbb{Z}_2-graded algebra of functions is the algebra free on a single odd generator θ\theta: C ( 0|1)=[θ]=θC^\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 automorphism super-group

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

Aut(𝔸 0|1)𝔾 m(Π𝔾 ad) \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 0|q\mathbb{R}^{0|q} this is seen over the test space 0|1\mathbb{R}^{0|1} itself with canonical odd coordinate θ\theta by taking the canonical odd coordinate of the odd line that we are taking automorphism of to be ϵ\epsilon and observing that maps

0|1Aut( 0|1) \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

ϵ,θϵ \langle \epsilon, \theta\rangle \leftarrow \langle \epsilon\rangle

that send

ϵxϵ+yθ \epsilon \mapsto x \epsilon + y \theta

with x0x \neq 0 in even degree and arbitrary yy in odd degree. Notice that the inverse of this map exists and is given by

ϵx 1ϵyx 1θ. \epsilon \mapsto x^{-1}\epsilon - y x^{-1} \theta \,.

Moreover, observe that an action of Aut( 0|1)\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).


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

It was later amplified in section 3.2 of

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

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

An interpretation of an 𝔾 mΠ𝔾 ad\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

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

Revised on December 27, 2016 11:56:13 by David Corfield (