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\begin{document}

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\section*{odd line}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry}

[[!include supergeometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{TheAutomorphismSuperGroup}{The automorphism super-group}\dotfill \pageref*{TheAutomorphismSuperGroup} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{ReferencesAutomorphismGroup}{Automorphism group}\dotfill \pageref*{ReferencesAutomorphismGroup} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The \textbf{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.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{TheAutomorphismSuperGroup}{}\subsubsection*{{The automorphism super-group}}\label{TheAutomorphismSuperGroup}

The internal [[automorphism group]] of the odd line in the [[topos]] of [[smooth super infinity-groupoids|smooth super spaces]] is the [[supergroup]]

\begin{displaymath}
\mathbf{Aut}(\mathbb{A}^{0|1})
  \simeq
  \mathbb{G}_m \ltimes (\Pi \mathbb{G}_{ad})
\end{displaymath}
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 \emph{\hyperlink{ReferencesAutomorphismGroup}{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

\begin{displaymath}
\mathbb{R}^{0|1} \to \mathbf{Aut}(\mathbb{R}^{0|1})
\end{displaymath}
are then given, under the [[evaluation map]]-isomorphism and via the [[Yoneda lemma]] by [[Grassmann algebra]] [[homomorphisms]] of the form

\begin{displaymath}
\langle \epsilon, \theta\rangle
  \leftarrow
   \langle \epsilon\rangle
\end{displaymath}
that send

\begin{displaymath}
\epsilon \mapsto x \epsilon + y \theta
\end{displaymath}
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

\begin{displaymath}
\epsilon \mapsto x^{-1}\epsilon - y x^{-1} \theta
  \,.
\end{displaymath}
Moreover, observe that an [[action]] of $\mathbf{Aut}(\mathbb{R}^{0|1})$ on a [[supermanifold]] corresponds to a choice of [[graded object|grading]] and a choice of [[differential]]. A clean account of this statement is in (\hyperlink{CarchediRoytenberg12}{Carchedi-Roytenberg 12}). (At least for the grading this is essentially the classical statement about graded rings, see at \emph{[[affine line]]} the section \emph{\href{affine+line#Grading}{Properties -- Grading}}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Cartesian space]]


\item [[infinitesimal space]]


\item [[synthetic differential supergeometry]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{ReferencesAutomorphismGroup}{}\subsubsection*{{Automorphism group}}\label{ReferencesAutomorphismGroup}

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

\begin{itemize}%
\item [[Maxim Kontsevich]], \emph{Deformation quantization of Poisson manifolds, I}, Lett. Math. Phys. 66:157-216,2003 (\href{http://arxiv.org/abs/q-alg/9709040}{arXiv:q-alg/9709040})

\end{itemize}
It was later amplified in section 3.2 of

\begin{itemize}%
\item Denis Kochan, [[Pavol Ševera]], \emph{Differential gorms, differential worms} (\href{http://arxiv.org/abs/math/0307303}{arXiv:math/0307303}),

\end{itemize}
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

\begin{itemize}%
\item [[David Carchedi]], [[Dmitry Roytenberg]], \emph{Homological Algebra for Superalgebras of Differentiable Functions} (\href{http://arxiv.org/abs/1212.3745}{arXiv:1212.3745})

\end{itemize}
An interpretation of an $\mathbb{G}_m \ltimes \Pi \mathbb{G}_{ad}$-[[action]] on a [[supermanifold]] of local [[quantum observables]] of a [[supersymmetry|supersymmetric]] [[field theory]] as the formalization of the concept of [[topologically twisted super Yang-Mills theory]] is in section 15 of

\begin{itemize}%
\item [[Kevin Costello]], \emph{Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4} (\href{http://arxiv.org/abs/1111.4234}{arXiv:1111.4234})

\end{itemize}
For more on this see at \emph{\href{topologically+twisted+D%3D4+super+Yang-Mills+theory#Formalization}{topologically twisted super Yang-Mills theory -- Formalization}}.



\end{document}