odd line



The odd line is the supermanifold 01\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 ( 01)=[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 interal automorphism group of the odd line in the topos of smooth super spaces is the supergroup

Aut(𝔸 01)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 0q\mathbb{R}^{0|q} this is seen over the test space 01\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

01Aut( 01) \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( 01)\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( 01,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 October 9, 2013 11:36:56 by Urs Schreiber (