\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{mathtools} \usepackage{graphicx} \usepackage{color} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{xparse} \usepackage{hyperref} %----Macros---------- % % Unresolved issues: % % \righttoleftarrow % \lefttorightarrow % % \color{} with HTML colorspec % \bgcolor % \array with options (without options, it's equivalent to the matrix environment) % Of the standard HTML named colors, white, black, red, green, blue and yellow % are predefined in the color package. Here are the rest. \definecolor{aqua}{rgb}{0, 1.0, 1.0} \definecolor{fuschia}{rgb}{1.0, 0, 1.0} \definecolor{gray}{rgb}{0.502, 0.502, 0.502} \definecolor{lime}{rgb}{0, 1.0, 0} \definecolor{maroon}{rgb}{0.502, 0, 0} \definecolor{navy}{rgb}{0, 0, 0.502} \definecolor{olive}{rgb}{0.502, 0.502, 0} \definecolor{purple}{rgb}{0.502, 0, 0.502} \definecolor{silver}{rgb}{0.753, 0.753, 0.753} \definecolor{teal}{rgb}{0, 0.502, 0.502} % Because of conflicts, \space and \mathop are converted to % \itexspace and \operatorname during preprocessing. % itex: \space{ht}{dp}{wd} % % Height and baseline depth measurements are in units of tenths of an ex while % the width is measured in tenths of an em. \makeatletter \newdimen\itex@wd% \newdimen\itex@dp% \newdimen\itex@thd% \def\itexspace#1#2#3{\itex@wd=#3em% \itex@wd=0.1\itex@wd% \itex@dp=#2ex% \itex@dp=0.1\itex@dp% \itex@thd=#1ex% \itex@thd=0.1\itex@thd% \advance\itex@thd\the\itex@dp% \makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}} \makeatother % \tensor and \multiscript \makeatletter \newif\if@sup \newtoks\@sups \def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}% \def\reset@sup{\@supfalse\@sups={}}% \def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else% \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}% \else \append@sup#2 \@suptrue \fi% \expandafter\mk@scripts\fi} \def\tensor#1#2{\reset@sup#1\mk@scripts#2_/} \def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2% \reset@sup\mk@scripts#3_/} \makeatother % \slash \makeatletter \newbox\slashbox \setbox\slashbox=\hbox{$/$} \def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$} \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa \copy\slashbox \kern-\@tempdima \box\@tempboxa} \def\slash{\protect\itex@pslash} \makeatother % math-mode versions of \rlap, etc % from Alexander Perlis, "A complement to \smash, \llap, and lap" % http://math.arizona.edu/~aprl/publications/mathclap/ \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\mathllap{\mathpalette\mathllapinternal} \def\mathrlap{\mathpalette\mathrlapinternal} \def\mathclap{\mathpalette\mathclapinternal} \def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}} \def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}} \def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}} % Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2} \let\oldroot\root \def\root#1#2{\oldroot #1 \of{#2}} \renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}} % Manually declare the txfonts symbolsC font \DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n} \SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n} \DeclareFontSubstitution{U}{txsyc}{m}{n} % Manually declare the stmaryrd font \DeclareSymbolFont{stmry}{U}{stmry}{m}{n} \SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n} % Manually declare the MnSymbolE font \DeclareFontFamily{OMX}{MnSymbolE}{} \DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n} \SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n} \DeclareFontShape{OMX}{MnSymbolE}{m}{n}{ <-6> MnSymbolE5 <6-7> MnSymbolE6 <7-8> MnSymbolE7 <8-9> MnSymbolE8 <9-10> MnSymbolE9 <10-12> MnSymbolE10 <12-> MnSymbolE12}{} % Declare specific arrows from txfonts without loading the full package \makeatletter \def\re@DeclareMathSymbol#1#2#3#4{% \let#1=\undefined \DeclareMathSymbol{#1}{#2}{#3}{#4}} \re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46} \re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12} \re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64} \re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6} \re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77} \re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77} \makeatother % \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE \makeatletter \def\Decl@Mn@Delim#1#2#3#4{% \if\relax\noexpand#1% \let#1\undefined \fi \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}} \def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}} \def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}} \Decl@Mn@Open{\llangle}{mnomx}{'164} \Decl@Mn@Close{\rrangle}{mnomx}{'171} \Decl@Mn@Open{\lmoustache}{mnomx}{'245} \Decl@Mn@Close{\rmoustache}{mnomx}{'244} \makeatother % Widecheck \makeatletter \DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}} \def\@widecheck#1#2{% \setbox\z@\hbox{\m@th$#1#2$}% \setbox\tw@\hbox{\m@th$#1% \widehat{% \vrule\@width\z@\@height\ht\z@ \vrule\@height\z@\@width\wd\z@}$}% \dp\tw@-\ht\z@ \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@ \setbox\tw@\hbox{% \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box \tw@}}}% {\ooalign{\box\tw@ \cr \box\z@}}} \makeatother % \mathraisebox{voffset}[height][depth]{something} \makeatletter \NewDocumentCommand\mathraisebox{moom}{% \IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{% \IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}% }{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}% \mathpalette\@temp{#4}} \makeatletter % udots (taken from yhmath) \makeatletter \def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.} \mkern2mu\raise4\p@\hbox{.}\mkern1mu \raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}} \makeatother %% Fix array \newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}} %% \itexnum is a noop \newcommand{\itexnum}[1]{#1} %% Renaming existing commands \newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}} \newcommand{\widevec}{\overrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\nearr}{\nearrow} \newcommand{\nwarr}{\nwarrow} \newcommand{\searr}{\searrow} \newcommand{\swarr}{\swarrow} \newcommand{\curvearrowbotright}{\curvearrowright} \newcommand{\uparr}{\uparrow} \newcommand{\downuparrow}{\updownarrow} \newcommand{\duparr}{\updownarrow} \newcommand{\updarr}{\updownarrow} \newcommand{\gt}{>} \newcommand{\lt}{<} \newcommand{\map}{\mapsto} \newcommand{\embedsin}{\hookrightarrow} \newcommand{\Alpha}{A} \newcommand{\Beta}{B} \newcommand{\Zeta}{Z} \newcommand{\Eta}{H} \newcommand{\Iota}{I} \newcommand{\Kappa}{K} \newcommand{\Mu}{M} \newcommand{\Nu}{N} \newcommand{\Rho}{P} \newcommand{\Tau}{T} \newcommand{\Upsi}{\Upsilon} \newcommand{\omicron}{o} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\Union}{\bigcup} \newcommand{\Intersection}{\bigcap} \newcommand{\Oplus}{\bigoplus} \newcommand{\Otimes}{\bigotimes} \newcommand{\Wedge}{\bigwedge} \newcommand{\Vee}{\bigvee} \newcommand{\coproduct}{\coprod} \newcommand{\product}{\prod} \newcommand{\closure}{\overline} \newcommand{\integral}{\int} \newcommand{\doubleintegral}{\iint} \newcommand{\tripleintegral}{\iiint} \newcommand{\quadrupleintegral}{\iiiint} \newcommand{\conint}{\oint} \newcommand{\contourintegral}{\oint} \newcommand{\infinity}{\infty} \newcommand{\bottom}{\bot} \newcommand{\minusb}{\boxminus} \newcommand{\plusb}{\boxplus} \newcommand{\timesb}{\boxtimes} \newcommand{\intersection}{\cap} \newcommand{\union}{\cup} \newcommand{\Del}{\nabla} \newcommand{\odash}{\circleddash} \newcommand{\negspace}{\!} \newcommand{\widebar}{\overline} \newcommand{\textsize}{\normalsize} \renewcommand{\scriptsize}{\scriptstyle} \newcommand{\scriptscriptsize}{\scriptscriptstyle} \newcommand{\mathfr}{\mathfrak} \newcommand{\statusline}[2]{#2} \newcommand{\tooltip}[2]{#2} \newcommand{\toggle}[2]{#2} % Theorem Environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{differential form on a supermanifold} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry} [[!include supergeometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{integral_topforms_and_picture_number}{Integral top-forms and Picture number}\dotfill \pageref*{integral_topforms_and_picture_number} \linebreak \noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak \noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The notion of superdifferential form is the generalization of the notion of [[differential form]] from [[manifolds]] to [[supermanifolds]]. \hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition} \begin{defn} \label{DifferentialFormsOnSuperCartesianSpace}\hypertarget{DifferentialFormsOnSuperCartesianSpace}{} \textbf{([[differential forms]] on [[super Cartesian space]])} For $n,q \in \mathbb{N}$, the \emph{de Rham algebra} $\Omega^\bullet(\mathbb{R}^{n\vert q})$ of super-differential forms on the [[super Cartesian space]] $\mathbb{R}^{n\vert q}$ is the [[free construction|free]] [[differential graded-commutative superalgebra]] over the [[supercommutative algebra]] \begin{displaymath} C^\infty(\mathbb{R}^{n\vert q}) \;=\; \underset{ even }{ \underbrace{ C^\infty(\mathbb{R}^n) }} \otimes_{\mathbb{R}} \underset{ odd }{ \underbrace{ \wedge^\bullet \mathbb{R}^q }} \end{displaymath} on \begin{enumerate}% \item $n$ [[generators and relations|generators]] $\mathbf{d} x^a$ in \href{chain+complex+in+super+vector+spaces#ChainComplexesOfSuperVectorSpaces}{bi-degree} $(1,even)$ (the canonical bosonic 1-forms) \item $q$ generators $\mathbf{d} \theta^\alpha$ in \href{chain+complex+in+super+vector+spaces#ChainComplexesOfSuperVectorSpaces}{bi-degree} $(1,odd)$ \end{enumerate} hence: \begin{equation} \Omega^\bullet( \mathbb{R}^{n\vert q} ) \;\coloneqq\; C^\infty(\mathbb{R}^{n\vert q}) \left[ \underset{deg = (1,even)}{\underbrace{ \left\langle \mathbf{d} x^a \right\rangle_{a = 1}^{n} }},\;\;\;\;\; \underset{ deg = (1,odd) }{ \underbrace{ \left\langle \mathbf{d} \theta^\alpha \right\rangle_{\alpha = 1}^q }} \right] \label{SuperCartSpDiffForms}\end{equation} with [[nLab:differential]] having the evident definition on generators, and extended from there as a [[derivation]] of bi-degree $(1, even) \in \mathbb{Z} \times (\mathbb{Z}/2)$ \end{defn} \begin{example} \label{SignRulesForDifferentialFormsOnSuperCartesianSpaces}\hypertarget{SignRulesForDifferentialFormsOnSuperCartesianSpaces}{} \textbf{([[signs in supergeometry|sign rule]] for [[differential forms]] on [[super Cartesian spaces]])} For $n,q \in \mathbb{N}$, the generators of the [[differential graded-commutative superalgebra]] $\Omega^\bullet(\mathbb{R}^{n\vert q})$ of differential forms on the [[super Cartesian space]] (Def. \ref{DifferentialFormsOnSuperCartesianSpace}) have the following \href{chain+complex+in+super+vector+spaces#ChainComplexesOfSuperVectorSpaces}{bi-degree} \begin{tabular}{l|l} $\phantom{A}$generator$\phantom{A}$&$\phantom{A}$\href{chain+complex+in+super+vector+spaces#ChainComplexesOfSuperVectorSpaces}{bi-degree}$\phantom{A}$\\ \hline $\phantom{A}$$x^a$$\phantom{A}$&$\phantom{A}$(0,even)$\phantom{A}$\\ $\phantom{A}$$\theta^\alpha$$\phantom{A}$&$\phantom{A}$(0,odd)$\phantom{A}$\\ $\phantom{A}$$\mathbf{d}x^a$$\phantom{A}$&$\phantom{A}$(1,even)$\phantom{A}$\\ $\phantom{A}$$\mathbf{d}\theta^\alpha$$\phantom{A}$&$\phantom{A}$(1,odd)$\phantom{A}$\\ \end{tabular} and satisfy the following graded-commutation relations, depending on one of the two equivalent (see \href{chain+complex+in+super+vector+spaces#EquivalenceTwoSymmetricMonoidalStructuresOnChSuperVect}{here}) [[signs in supergeometry|sign rules]]: \newline | $\phantom{A}$$x^{a} \; x^{b} =$ | $\phantom{A}$$+ x^{b} \; x^{a}$$\phantom{A}$ | $\phantom{A}$$+ x^{b} \; x^{a}$$\phantom{A}$ | | $\phantom{A}$$x^a \;\theta^\alpha =$$\phantom{A}$ | $\phantom{A}$$+ \theta^\alpha \; x^a$$\phantom{A}$ | $\phantom{A}$$+ \theta^\alpha \; x^a$$\phantom{A}$ | | $\phantom{A}$$\theta^{\alpha} \; \theta^{\beta} =$$\phantom{A}$ | $\phantom{A}$$- \theta^{\beta} \; \theta^{\alpha}$$\phantom{A}$ | $\phantom{A}$$- \theta^{\beta} \; \theta^{\alpha}$$\phantom{A}$ | | $\phantom{A}$$x^{a} (\mathbf{d}x^{a}) =$$\phantom{A}$ | $\phantom{A}$$+ (\mathbf{d}x^{b}) x^{a}$$\phantom{A}$ | $\phantom{A}$$+ (\mathbf{d}x^{b}) x^{a}$$\phantom{A}$ | | $\phantom{A}$$\theta^\alpha (\mathbf{d}x^a) =$$\phantom{A}$ | $\phantom{A}$$+ (\mathbf{d}x^a) \theta^\alpha$$\phantom{A}$ | $\phantom{A}$${\color{blue}{-}} (\mathbf{d}x^a) \theta^\alpha$$\phantom{A}$ | | $\phantom{A}$$\theta^{\alpha} (\mathbf{d}\theta^{\beta}) =$$\phantom{A}$ | $\phantom{A}$$- (\mathbf{d}\theta^{\beta}) \theta^{\alpha}$$\phantom{A}$ | $\phantom{A}$${\color{blue}{+}} (\mathbf{d}\theta^{\beta}) \theta^{\alpha}$$\phantom{A}$ | | $\phantom{A}$$(\mathbf{d}x^{a}) (\mathbf{d} x^{b}) =$$\phantom{A}$ | $\phantom{A}$$- (\mathbf{d} x^{b}) (\mathbf{d} x^{a})$$\phantom{A}$ | $\phantom{A}$$- (\mathbf{d} x^{b}) (\mathbf{d} x^{a})$$\phantom{A}$ | | $\phantom{A}$$(\mathbf{d}x^a) (\mathbf{d} \theta^{\alpha}) =$$\phantom{A}$ | $\phantom{A}$$- (\mathbf{d}\theta^{\alpha}) (\mathbf{d} x^a)$$\phantom{A}$ | $\phantom{A}$${\color{blue}{+}} (\mathbf{d}\theta^{\alpha}) (\mathbf{d} x^a)$$\phantom{A}$ | | $\phantom{A}$$(\mathbf{d}\theta^{\alpha}) (\mathbf{d} \theta^{\beta}) =$ | $\phantom{A}$$+ (\mathbf{d}\theta^{\beta}) (\mathbf{d} \theta^{\alpha})$$\phantom{A}$ | $\phantom{A}$$+ (\mathbf{d}\theta^{\beta}) (\mathbf{d} \theta^{\alpha})$$\phantom{A}$ | \end{example} \begin{defn} \label{SuperCartSpaPullbackOfSuperDifferentialForms}\hypertarget{SuperCartSpaPullbackOfSuperDifferentialForms}{} \textbf{([[pullback of differential forms|pullback]] over [[super Cartesian spaces]])} Let \begin{displaymath} \itexarray{ \mathbb{R}^{n_1 \vert q_1} &\overset{f}{\longrightarrow}& \mathbb{R}^{n_2 \vert q_2} \\ C^\infty(\mathbb{R}^{n_1\vert q_1}) &\overset{f^\ast}{\longleftarrow}& C^\infty(\mathbb{R}^{n_2\vert q_2}) } \end{displaymath} be a morphism of [[super Cartesian spaces]], hence [[formal duality|formally dually]] a [[algebra homomorphism]] $f^\ast$ of [[supercommutative superalgebras]]. By the fact that $\Omega^\bullet(\mathbb{R}^{n \vert q})$ (Def. \ref{DifferentialFormsOnSuperCartesianSpace}) is [[free construction|free]] over $C^\infty(\mathbb{R}^{n\vert q})$ on generators $\mathbf{d}x^a$, $\mathbf{d}\theta^\alpha$ \eqref{SuperCartSpDiffForms} this extends to a unique homomorphism on the de Rham [[dgc-superalgebra]] \begin{displaymath} \Omega^\bullet(\mathbb{R}^{n_1\vert q_1}) \overset{f^\ast}{\longleftarrow} \Omega^\bullet(\mathbb{R}^{n_2\vert q_2}) \end{displaymath} subject to the condition that $f^\ast \circ \mathbf{d}_2 = \mathbf{d}_1 \circ f^\ast$: \begin{displaymath} \itexarray{ \Omega^\bullet(\mathbb{R}^{n_1\vert q_1}) &\overset{\phantom{AA}f^\ast\phantom{AA}}{\longleftarrow}& \Omega^\bullet(\mathbb{R}^{n_2\vert q_2}) \\ {}^{\mathbf{d}_1}\Big\downarrow && \Big\downarrow{}^{\mathbf{d}_2} \\ \Omega^\bullet(\mathbb{R}^{n_1\vert q_1}) &\overset{\phantom{AA}f^\ast\phantom{AA}}{\longleftarrow}& \Omega^\bullet(\mathbb{R}^{n_2\vert q_2}) } \end{displaymath} This operation is called \emph{[[pullback of differential forms]]} along maps of [[super Cartesian spaces]]. \end{defn} \begin{prop} \label{SuperDifferentialFormClassifying}\hypertarget{SuperDifferentialFormClassifying}{} \textbf{(classifing [[super formal smooth set]] of [[super differential forms]])} The operation of [[pullback of differential forms]] (Def. \ref{SuperCartSpaPullbackOfSuperDifferentialForms}) over [[super Cartesian spaces]] respects [[identity morphisms]] and [[composition]]. Hence the assignment of [[differential forms]] on [[super Cartesian spaces]] (Def. \ref{DifferentialFormsOnSuperCartesianSpace}) is a [[presheaf]] on [[SuperCartSp]]: \begin{displaymath} \mathbf{\Omega}\bullet \;\colon\; SuperCartSp^{op} \longrightarrow dgcsAlg \end{displaymath} with values in [[differential graded-commutative superalgebras]], in fact a [[sheaf]] and hence a [[differential graded algebra]] [[internalization|internal]] to the [[sheaf topos]] over [[SuperCartSp]]: \begin{displaymath} \mathbf{\Omega}^n \;\in\; Sh(SuperCartSp) \,. \end{displaymath} The construction generalizes in an evident way also to sheaves over [[super formal Cartesian spaces]], hence to [[super formal smooth sets]]: \begin{displaymath} \mathbf{\Omega}^n \;\in\; SuperFormalSmoothSet \;\coloneqq\; Sh(SuperFormalCartSp) \,. \end{displaymath} \end{prop} We may now proceed as in the discussion of [[differential forms]] on [[smooth sets]] (hereets\#DifferentialForms)): \begin{defn} \label{SuperDifferentialFormsOnGeneralSuperFormalSmoothSets}\hypertarget{SuperDifferentialFormsOnGeneralSuperFormalSmoothSets}{} \textbf{([[super differential forms]] on general [[super formal smooth sets]])} Let $X$ be a [[supermanifold]] or more generally a [[super formal smooth set]] \begin{displaymath} X \;\in\; SuperFormalSmoothSet \end{displaymath} Then \emph{[[super differential forms]]} on $X$ are morphisms \begin{displaymath} X \longrightarrow \mathbf{\Omega}^\bullet \end{displaymath} into the classifying sheaf $\mathbf{\Omega}^\bullet$ from Def. \ref{SuperDifferentialFormClassifying}. \end{defn} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{integral_topforms_and_picture_number}{}\subsubsection*{{Integral top-forms and Picture number}}\label{integral_topforms_and_picture_number} If a choice of \emph{integral top-forms} is made, needed for a notion of \emph{[[integration over supermanifolds]]}, then there is an additional grading by ``[[picture number]]'' (\hyperlink{Belopolsky97b}{Belopolsky 97b}, \hyperlink{Witten12}{Witten 12}), see (\hyperlink{CatenacciGrassiNoja18}{Catenacci-Grassi-Noja 18 (5.8) to (5.12)}). \hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples} Let $\mathbf{X} = \mathbb{R}^{1|1}$. The [[superalgebra]] of functions on $\mathbf{X}$ is the [[exterior algebra]] that is generated over $C^\infty(\mathbf{R})$ from a single generator $\theta$ in odd degree (the canonical odd coordinate). The algebra of superdifferential forms on $\mathbb{R}^{1|1}$ is the [[exterior algebra]] generated over $C^\infty(\mathbb{R})$ from \begin{itemize}% \item a generator $\theta$ in odd degree (the canonical odd coordinate); \item a generator $d x$ in odd degree (the differential of the canonical even coordinate); \item a generator $d \theta$ in even degree (the differential of the canonical odd coordinate). \end{itemize} Notice in particular that while $d x \wedge d x = 0$ the wedge product $d \theta \wedge d\theta$ is non-vanishing, since $d \theta$ is in even degree. In fact all higher wedge powers of $d \theta$ with itself exist. \hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks} \begin{itemize}% \item Being a $\mathbb{Z}_2$-graded locally free algebra itself, one can regard $\Omega^\bullet(X)$ itself (even for $X$ a usual manifold!) as the ``algebra of functions'' (more precisely [[inner hom]], i.e. mapping space into the line) on another supermanifold. That supermanifold is called $T[1] X$, the \textbf{[[shifted tangent bundle]]} of $X$. By definition we have $C^\infty(T[1]X) = \Omega^\bullet(X)$. From this point of view, the existence of the differential $d$ on the graded algebra $\Omega^\bullet(X)$ translates into the existence of a special odd vector field on $T[1]X$. This is a \textbf{homological vector field} in that it is odd and the super Lie bracket of it with itself vanishes: $[d,d] = 0$. \item In the context of [[L-infinity algebroids]], where one may regard $C^\infty(X)$ as the [[Chevalley-Eilenberg algebra]] of an $L_\infty$-[[Lie infinity-algebroid|algebroid]] it is useful to notice that $\Omega^\bullet(X)$ is the corresponding [[Weil algebra]]. If $X$ is a Lie $n$-algebroid then $T[1]X$ is a Lie $(n+1)$-algebroid. \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[signs in supergeometry]] \item [[picture changing operator]] \item [[super Lie algebroid]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Discussion with an eye towards [[quantization]] of the [[superstring]] is in \begin{itemize}% \item [[Alexander Belopolsky]], \emph{De Rham Cohomology of the Supermanifolds and Superstring BRST Cohomology}, Phys.Lett. B403 (1997) 47-50 (\href{http://arxiv.org/abs/hep-th/9609220}{arXiv:hep-th/9609220}) \end{itemize} Geometric discussion of [[picture number]] appearing in the context of [[integration over supermanifolds]] (and originally seen in the [[quantization]] of the [[NSR superstring]], crucial in [[superstring field theory]]) is due to \begin{itemize}% \item [[Alexander Belopolsky]], \emph{Picture changing operators in supergeometry and superstring theory} (\href{https://arxiv.org/abs/hep-th/9706033}{arXiv:hep-th/9706033}) \end{itemize} and further amplified in \begin{itemize}% \item [[Edward Witten]], appendix D of \emph{Notes On Super Riemann Surfaces And Their Moduli} (\href{http://arxiv.org/abs/1209.2459}{arXiv:1209.2459}) \item R. Catenacci, P.A. Grassi, S. Noja, \emph{Superstring Field Theory, Superforms and Supergeometry} (\href{https://arxiv.org/abs/1807.09563}{arXiv:1807.09563}) \end{itemize} [[!redirects differential forms on a supermanifold]] [[!redirects differential form on supermanifold]] [[!redirects differential forms on supermanifolds]] [[!redirects super differential form]] [[!redirects super differential forms]] [[!redirects superdifferential form]] [[!redirects superdifferential forms]] \end{document}