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\section*{correlator as differential form on configuration space of points}

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

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{higher_chernsimons_theory}{Higher Chern-Simons theory}\dotfill \pageref*{higher_chernsimons_theory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[Euclidean field theory]], an alternative to regarding [[propagators]]/[[correlators]] as [[distributions of several variables]] with [[singularities]] on the [[fat diagonal]], is to [[pullback of distributions|pull-back]] these distributions to [[smooth functions]]/[[differential forms]] on ([[Fulton-MacPherson compactifications]] of) [[configuration spaces of points]] and regard them in this incarnation, in particular discuss their [[renormalization]] from this perspective.

Analogous to the perspective of [[wavefront sets]] for distributions, this perspective amounts to recording around each potentially singular point an [[n-sphere|(d-1)-sphere]] worth of extra [[direction of a vector|directional]] information carried by the [[correlator]]/[[Feynman amplitude]] in the vicinity of the point.

This approach goes back to \hyperlink{AxelrodSinger93}{Axelrod-Singer 93} in the discussion of [[perturbative quantum field theory|perturbative]] [[quantization of Chern-Simons theory]]. Here the [[graph complex]] of \hyperlink{Kontsevich94}{Kontsevich 94} (full details due to \hyperlink{LambrechtsVolic14}{Lambrechts-Volić 14}) shows that the [[de Rham algebra]] of the [[configuration space of points]] is actuall [[quasi-isomorphism|quais-isomorphic]] to all possible [[Feynman amplitudes]] for free [[Chern-Simons theory|Chern-Simons]]/[[AKSZ theory]].

A general and systematic discussion of [[perturbative quantum field theory]] and its [[renormalization]] from this perspective is offered in \hyperlink{Berghoff14a}{Berghoff 14a}, \hyperlink{Berghoff14b}{Berghoff 14b} (albeit presently only for [[Euclidean quantum field theory]], not for [[relativistic quantum field theory]]).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{higher_chernsimons_theory}{}\subsubsection*{{Higher Chern-Simons theory}}\label{higher_chernsimons_theory}

This approach to [[pQFT]] was originally considered specifically for the [[Chern-Simons propagator]] in [[quantization of 3d Chern-Simons theory]] in \hyperlink{AxelrodSinger93}{Axelrod-Singer 93}, see also \hyperlink{BottCattaneo97}{Bott-Cattaneo 97, Remark 3.6} and \hyperlink{CattaneoMnev10}{Cattaneo-Mnev 10, Remark 11}. The analysis applies verbatim to [[higher Chern-Simons theory]] such as notably the [[AKSZ sigma-models]], too, since the [[Feynman propagator]] depends only on the [[free field theory]]-[[equations of motion]], which is $d A = 0$ in all these cases.

Here the [[Chern-Simons propagator]] regarded as a non-singular [[differential form]] on the [[Fulton-MacPherson compactification|compactification]] of the [[configuration space of points]] serves to exhibit its [[Feynman amplitudes]] as providing [[graph complex]]-models for the [[de Rham cohomology]] of these [[Fulton-MacPherson compactification|compactified]] [[configuration spaces of points]], a point due to \hyperlink{Kontsevich94}{Kontsevich 94}, \hyperlink{Kontsevich93}{Kontsevich 93, 5}:

\begin{equation}
\underset{
    \color{blue}
    \itexarray{
      \phantom{A}
      \\
      \text{graph complex}
      \\
      \text{of n-point Feynman diagrams}
      \\
      \text{for Chern-Simons theory}
      \\
      \text{on} \; \Sigma 
    }
  }{
    Graphs_n(\Sigma)
  }
  \underoverset{
    \simeq_{\mathrlap{qi}}
  }
  {
    \color{blue}
    \itexarray{
      \text{assign Feynman amplitudes}
      \\
      \text{of Chern-Simons theory}
      \\
      \phantom{A}
    }
  }
  {
    \longrightarrow
  }
  \underset{
    \color{blue}
    \itexarray{
      \phantom{A}
      \\
      \text{de Rham algebra}
      \\
      \text{of semi-algebraic differential forms}
      \\
      \text{on the FM-compactification}
      \\
      \text{of the configuration space of n points}
      \\
      \text{in}\; \Sigma
    }
  }{
  \Omega^\bullet_{PA}
  \big(
    Conf_n\big(  \Sigma \big)
  \big)
  }
  \,.
\label{TheQuasiIsomorphism}\end{equation}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The approach was originally considered specifically for [[Chern-Simons theory]] in

\begin{itemize}%
\item [[Scott Axelrod]], [[Isadore Singer]], \emph{Chern--Simons Perturbation Theory II}, J. Diff. Geom. 39 (1994) 173-213 (\href{http://arxiv.org/abs/hep-th/9304087}{arXiv:hep-th/9304087})

\end{itemize}
which was re-amplified in

\begin{itemize}%
\item [[Raoul Bott]], [[Alberto Cattaneo]], Remark 3.6 in \emph{Integral invariants of 3-manifolds}, J. Diff. Geom., 48 (1998) 91-133 (\href{https://arxiv.org/abs/dg-ga/9710001}{arXiv:dg-ga/9710001})


\item [[Alberto Cattaneo]], [[Pavel Mnev]], Remark 11 in \emph{Remarks on Chern-Simons invariants}, Commun.Math.Phys.293:803-836,2010 (\href{https://arxiv.org/abs/0811.2045}{arXiv:0811.2045})


\item [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], appendix B of \emph{Perturbative quantum gauge theories on manifolds with boundary}, Communications in Mathematical Physics, January 2018, Volume 357, Issue 2, pp 631–730 (\href{https://arxiv.org/abs/1507.01221}{arXiv:1507.01221}, \href{https://doi.org/10.1007/s00220-017-3031-6}{doi:10.1007/s00220-017-3031-6})



\end{itemize}
and highlighted as a means to obtain [[graph complex]]-models for the [[de Rham cohomology]] of [[configuration spaces of points]] in

\begin{itemize}%
\item [[Maxim Kontsevich]], \emph{Vassiliev's knot invariants}, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 (\href{http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf}{pdf})


\item [[Maxim Kontsevich]], pages 11-12 of \emph{Feynman diagrams and low-dimensional topology}, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics \textbf{120}, Birkh\"a{}user (1994), 97--121 (\href{http://www.ihes.fr/~maxim/TEXTS/Feynman%20%20diagrams%20and%20low-dimensional%20topology.pdf}{pdf})



\end{itemize}
with full details and proofs in

\begin{itemize}%
\item [[Pascal Lambrechts]], [[Ismar Volić]], sections 6 and 7 of \emph{Formality of the little N-disks operad}, Memoirs of the American Mathematical Society no. 1079, 2014 (\href{https://arxiv.org/abs/0808.0457}{arxiv:0808.0457}, \href{http://dx.doi.org/10.1090/memo/1079}{doi:10.1090/memo/1079})

\end{itemize}
see also

\begin{itemize}%
\item [[Ricardo Campos]], Najib Idrissi, [[Pascal Lambrechts]], [[Thomas Willwacher]], \emph{Configuration Spaces of Manifolds with Boundary} (\href{https://arxiv.org/abs/1802.00716}{arXiv:1802.00716})

\end{itemize}
A systematic development of [[Euclidean field theory|Euclidean]] [[perturbative quantum field theory]] with [[n-point functions]] considered as [[smooth functions]] on [[Fulton-MacPherson compactifications]]/[[wonderful compactifications]] of [[configuration spaces of points]] and more generally of subspace arrangements is due to

\begin{itemize}%
\item [[Christoph Bergbauer]], [[Romeo Brunetti]], [[Dirk Kreimer]], \emph{Renormalization and resolution of singularities}, ESI preprint 2010 (\href{https://arxiv.org/abs/0908.0633}{arXiv:0908.0633}, \href{https://mat.univie.ac.at/~esiprpr/esi2244.pdf}{ESI:2244})


\item [[Christoph Bergbauer]], \emph{Renormalization and resolution of singularities}, talks as IHES and Boston, 2009 (\href{http://www.emg.uni-mainz.de/Dateien/bergbauer.pdf}{pdf})


\item [[Marko Berghoff]], \emph{Wonderful renormalization}, 2014 (\href{http://www2.mathematik.hu-berlin.de/%7Ekreimer/wp-content/uploads/Berghoff-Marko.pdf}{pdf}, \href{https://doi.org/10.18452/17160}{doi:10.18452/17160})


\item [[Marko Berghoff]], \emph{Wonderful compactifications in quantum field theory}, Communications in Number Theory and Physics Volume 9 (2015) Number 3 (\href{https://arxiv.org/abs/1411.5583}{arXiv:1411.5583})



\end{itemize}
[[!redirects correlator as differential form on configuration space of points]] [[!redirects correlators as differential forms on configuration space of points]]

[[!redirects correlator as differential form on a configuration space of points]] [[!redirects correlator as a differential form on a configuration space of points]] [[!redirects correlators as differential forms on a configuration space of points]] [[!redirects correlators as differential forms on configuration spaces of points]]

[[!redirects Feynman amplitudes as differential forms on configuration spaces of points]]

[[!redirects Feynman amplitude on compactified configuration space of points]] [[!redirects Feynman amplitude on compactified configuration spaces of points]] [[!redirects Feynman amplitudes on compactified configuration spaces of points]]

[[!redirects propagators regarded as differential forms on configuration spaces]]



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