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\begin{document}

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\section*{integration over supermanifolds}

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

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

[[!include supergeometry - contents]]

\hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory}

[[!include integration theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{no_topdegree_forms_in_supergeometry}{No top-degree forms in supergeometry}\dotfill \pageref*{no_topdegree_forms_in_supergeometry} \linebreak
\noindent\hyperlink{solution}{Solution}\dotfill \pageref*{solution} \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}

There are different ways to define a [[volume form|differential volume element]] on a [[smooth manifold]]. Some of these definitions can be carried over to [[supergeometry]], others cannot. The possibly most familiar way of talking about differential volume elements, in terms of top-degree [[differential form]]s, does \emph{not} carry over to [[supermanifold]]s.

\hypertarget{no_topdegree_forms_in_supergeometry}{}\subsubsection*{{No top-degree forms in supergeometry}}\label{no_topdegree_forms_in_supergeometry}

In supergeometry the notion of \emph{top-degree form} does not in general make sense, since there are no top-degree wedge powers of ``[[superdifferential form|odd 1-forms]]'': if for instance $\theta_1$ and $\theta_2$ are odd functions on some [[super Cartesian space]] $\mathbb{R}^{p|q}$ and $d \theta_1$ and $d \theta_2$ are their [[differential 1-forms]], then the wedge product of these is symmetric in that

\begin{displaymath}
d\theta_1 \wedge d \theta_2 = + 
  d\theta_2 \wedge d \theta_1
  \,.
\end{displaymath}
Notice the plus sign on the right, which is the product of one minus sign for interchanging $\theta_1$ and $\theta_2$, and another minus sign for interchanging the two differentials. See at \emph{[[signs in supergeometry]]} for more on this.

Accordingly, the wedge product of the differential of an odd function $\theta$ with itself does not in general vanish:

\begin{displaymath}
(d \theta \wedge d\theta = 0)
  \Leftrightarrow
  (\theta = 0)
  \,.
\end{displaymath}
On the cartesian supermanifold $\mathbb{R}^{n|m}$ with canonical even coordinate functions $\{x^i\}_1^n$ and canonical odd coordinate functions $\{\theta^j\}_1^m$ the differential form which one would want to regard as the canonical [[volume form]] is

\begin{displaymath}
\omega :=
  d x^1 \wedge \cdots \wedge dx^n \wedge 
  d\theta^1 \wedge \cdots \wedge d\theta^m
  \,.
\end{displaymath}
Due to the above, this is not a \emph{top} form, since for instance

\begin{displaymath}
\omega \wedge d\theta^1 \neq 0
  \,.
\end{displaymath}
But this example also indicates the solution: apparently for integration it is not really essential that a form is a top power, what is rather essential is that it is, locally, the wedge product of a \emph{basis} of 1-forms. This perspective then does lead to a sensible definition of volume forms (and more generally ``integrable forms'') on supermanifolds, described below.

\hypertarget{solution}{}\subsubsection*{{Solution}}\label{solution}

Therefore the na\"i{}ve identification of differential volume measures with top degree forms has to be refined. The idea is to characterize a volume form by other means, in particular as an equivalence class of choices of [[bases]] for the space of 1-forms, and then to define \textbf{integrable forms} to be pairs consisting of such a generalized volume form and a \textbf{multivector}: this pair is supposed to represent the differential form one would obtain could one contract the multivector with the volume form, as in ordinary differential geometry.

The definition of integration of integrable forms in supergeometry in terms of multivectorfields leads, in the case that the supermanifold in an [[NQ-supermanifold]] to the [[BV theory|BV formalism]].

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

\begin{itemize}%
\item [[integration]]


\item [[Berezin integral]]



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

An exposition of the standard lore is in

\begin{itemize}%
\item [[Urs Schreiber]], \href{http://www.math.uni-hamburg.de/home/schreiber/sin.pdf}{\emph{Integration over supermanifolds}}

\end{itemize}
A general abstract discussion in terms of [[D-module]] theory is in

\begin{itemize}%
\item [[Frédéric Paugam]], \emph{Homotopical Poisson Reduction of gauge theories} (\href{http://people.math.jussieu.fr/~fpaugam/documents/homotopical-poisson-reduction-of-gauge-theories.pdf}{pdf})

\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}


\end{document}