configuration space of points

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###Context###
#### Topology
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[[!include topology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[mathematics]], the term "configuration space" of a [[topological space]] $X$ typically refers by default to the topological space of pairwise distinct [[element|points]] in $X$, also called _[[Fadell's configuration space]]_, for emphasis.

In principle many other kinds of configurations and the spaces these form may be referred to by "configuration space", notably in [[physics]] the usage is in a broader sense, see at _[[configuration space (physics)]]_.

## Definition
  {#Definition}

Several variants of configuration spaces of points are of interest. They differ in whether

1. points are linearly ordered or not;

1. points are labeled in some labelling space;

1. points vanish on some subspace or if their labels are in some subspace.

Here are some of these variant definitions:

### Ordered unlabeled points

+-- {: .num_defn #OrderedUnlabeledConfigurations}
###### Definition
**(ordered unlabeled configurations of a fixed number of points)**

Let $X$ be a [[closed manifold|closed]] [[smooth manifold]].
For $n \in \mathbb{N}$ write

$$
  \underset{ {}^{\{1,\cdots, n\}} }{ Conf }
  \big(
    X
  \big)
  \;\coloneqq\;
  \big(
    X
  \big)^n
  \setminus
  \mathbf{\Delta}^n_X
$$

for the [[complement]] of the [[fat diagonal]] inside the $n$-fold [[Cartesian product]] of $X$ with itself.

This is the space of ordered but otherwise unlabeled configurations of $n$ points_ in $X$.

=--

\linebreak

### Unordered unlabeled points

+-- {: .num_defn #UnorderedUnlabeledConfigurations}
###### Definition
**(unordered unlabeled configurations of a fixed number of points)**

Let $X$ be a [[closed manifold|closed]] [[smooth manifold]],
For $n \in \mathbb{N}$ write

\[
  \label{UnorderedUnlabelednPointConfigurationSpaces}
  \begin{aligned}
    Conf_n
    \big(
      X
    \big)
    &
    \coloneqq
    \;
    \Big(
      \underset{{}^{1,\cdots,n}}{Conf}
      \big(
        X
      \big)
    \Big) / Sym(n)
    \\
    &
    =\;
    \Big(
      \big(
        X
      \big)^n
      \setminus
      \mathbf{\Delta}^n_X
    \Big) / Sym(n)
  \end{aligned}
\]

for the [[quotient space]] of the ordered configuration space (Def. \ref{OrderedUnlabeledConfigurations}) by the evident [[action]] of the [[symmetric group]] $Sym(n)$ via [[permutation]] of the [[ordering]] of the points.

This is the space of unordered and unlabeled configurations of $n$ points_ in $X$.

We write

\[
  \label{UnorderedUnlabeledConfigurationSpace}
  Conf(X)
  \;\coloneqq\;
  \underset{n \in \mathbb{N}}{\sqcup}
  Conf_n\big( X\big)
\]

for the unordered unlabeled configuration space of any [[finite number]] of points, being the [[disjoint union]] of these spaces (eq:UnorderedUnlabelednPointConfigurationSpaces) over all [[natural numbers]] $n$.

=--

+-- {: .num_remark #MonoidStructureAndItsGroupCompletion}
###### Remark
**([[monoid]]-[[mathematical structure|structure]] on [[configuration space of points]])

For $X = \mathbb{R}^D$ a [[Euclidean spaces]] the configuration space of points $Conf\big( \mathbb{R}^D \big)$ (eq:UnorderedUnlabeledConfigurationSpace) carries the [[mathematical structure|structure]] of a [[topological monoid]] with product operation being the [[disjoint union]] of point configurations, after a suitable shrinking to put them next to each other ([Segal 73, p. 1-2](#Segal73)).

For emphasis, we write $B_{{}_{\sqcup}\!} Conf(\mathbb{R}^D)$ for the [[delooping]] ("[[classifying space]]") with respect to this [[topological monoid]]-[[structure]]. The corresponding [[based loop space]] is then the [[group completion]] of the configuration space, with respect to disjoint union of points:

\[
  \label{GroupCompletionOfConfigurationSpaceMonoid}
  Conf
  \big(
    \mathbb{R}^D
  \big)
  \overset{\color{blue}\text{group completion}}{\longrightarrow}
  \Omega B_{{}_{\sqcup}\!} Conf(\mathbb{R}^D)
  \,.
\]

=--

+-- {: .num_remark }
###### Remark

The configuration space of unordered unlabeled configurations of $n$ points (Def. \ref{OrderedUnlabeledConfigurations}) is naturally a [[topological subspace]] of the [[space of finite subsets]] of [[cardinality]] $\leq n$

\[
  \label{InjectionOfUnorderedConfigurationSpaceOfPoints}
  Conf_n(X)
  \hookrightarrow
  \exp^n(X)
\]

=--

+-- {: .num_prop #UnorderedConfigurationSpaceInSpaceOfFiniteSubsets}
###### Proposition

Let $X$ be an [[inhabited set|non-empty]] [[regular topological space]] and $n \geq 2 \in \mathbb{N}$.

Then the injection (eq:InjectionOfUnorderedConfigurationSpaceOfPoints)

\[
  \label{UnorderedConfigurationSpaceOpenSubsetInQuotientOfSpacesOfFiniteSubsets}
  Conf_n(X)
  \hookrightarrow
  \exp^n(X)/\exp^{n-1}(X)
\]

of the [[unordered configuration space of n points]] of $X$ (Def. \ref{OrderedUnlabeledConfigurations}) into the [[quotient space]] of the [[space of finite subsets]] of cardinality $\leq n$ by its subspace of subsets of cardinality $\leq n-1$ is an [[open subspace]]-inclusion.

Moreover, if $X$ is [[compact topological space|compact]], then so is $\exp^n(X)/\exp^{n-1}(X)$ and the inclusion (eq:UnorderedConfigurationSpaceOpenSubsetInQuotientOfSpacesOfFiniteSubsets) exhibits the [[one-point compactification]] $\big(  Conf_n(X) \big)^{+} $ of the configuration space:

$$
  \big(
    Conf_n(X)
  \big)^{+}
  \;\simeq\;
  \exp^n(X)/\exp^{n-1}(X)
  \,.
$$

=--

([Handel 00, Prop. 2.23](#Handel00), see also [Félix-Tanré 10](#FelixTanre10))

### Unordered labeled points
 {#UnorderedLabeledPoints}

+-- {: .num_defn #UnorderedLabeledFixedn}
###### Definition

For $X$ a [[smooth manifold]] and $k \in \mathbb{N}$, the space of _unordered configurations of points in $X$ with labels in $S^k$_ is

\[
  \label{UnorderedLabeledFixednCOnfigurationSpace}
  Conf_n\big(X, S^k \big)
  \;\coloneqq\;
  \underset{ {}^{\{1,\cdots, n\}} }{ Conf }
  \big(
    X
  \big) \underset{Sym(n)}{\times}
  \big(
    S^k
  \big)^n
\]

=--

For $k \in \mathbb{N}$, consider the [[n-sphere|k-sphere]] as a [[pointed topological space]], with the base point regarded as the "vanishing label".

+-- {: .num_defn #UnorderedLabeledConfigurationsVanishingWithVanishingLabel}
###### Definition
**(unordered labeled configurations vanishing with vanishing label)**

For $X$ a [[smooth manifold]] and $k \in \mathbb{N}$, the space of _unordered configurations of points in $X$ with labels in $S^k$ and vanishing at vanishing label value_ is the [[quotient space]]

\[
  \label{UnorderedLabeledCOnfigurationSpace}
  Conf
  \big(
    X,
    S^k
  \big)
  \;\coloneqq\;
  \Big(
    \underset{n \in \mathbb{N}}{\sqcup}
    Conf_n\big(X,S^k \big)
  \Big)/\sim
\]

of the [[disjoint union]] of all unordered labeled $n$-point configuration spaces (eq:UnorderedLabeledFixednCOnfigurationSpace) by the [[equivalence relation]] which regards points with vanishing label as absent.

=--

+-- {: .num_defn #ConfigurationSpacesOfnPoints}
###### Definition
**(unordered labeled configurations of a fixed number of points)**

Let $X$ be a [[manifold]], possibly with [[manifold with boundary|boundary]].
For $n \in \mathbb{N}$, the
 _**configuration space of $n$ unordered points** in $X$ disappearing at the boundary_ is the [[topological space]]
  $$
    \mathrm{Conf}_{n}(X)
    \;\coloneqq\;
    \Big(
      \big(
        X^n \setminus \mathbf{\Delta}_X^n
      \big)
      / \partial(X^n)
    \Big)
    /\Sigma(n)
    \,,
  $$
  where
  $\mathbf{\Delta}_X^n : = \{(x^i) \in X^n | \underset{i,j}{\exists} (x^i = x^j) \}$  is the [[fat diagonal]] in $X^n$
  and where $\Sigma(n)$ denotes the evident [[action]] of the [[symmetric group]] by [[permutation]] of factors of $X$ inside $X^n$.

  More generally, let $Y$ be another [[manifold]], possibly with [[manifold with boundary|boundary]]. For $n \in \mathbb{N}$, the _**configuration space of $n$ points** in $X \times Y$ vanishing at the boundary and distinct as points in $X$_ is the [[topological space]]
  $$
    \mathrm{Conf}_{n}(X,Y)
    \;\coloneqq\;
    \Big(
    \big(
      (
        X^n \setminus \mathbf{\Delta}_X^n
      )
      \times
      Y^n
    \big)
    /\Sigma(n)
    \Big)
    / \partial(X^n \times Y^n)
  $$
  where now $\Sigma(n)$ denotes the evident [[action]] of the [[symmetric group]] by [[permutation]] of factors of $X \times Y$ inside
  $X^n \times Y^n \simeq (X \times Y)^n$.

  This more general definition reduces to the previous case for $Y = \ast \coloneqq \mathbb{R}^0$ being the point:
  $$
    \mathrm{Conf}_n(X)
    \;=\;
    \mathrm{Conf}_n(X,\ast)
    \,.
  $$

Finally the _**configuration space of an arbitrary number of points** in $X \times Y$ vanishing at the boundary and distinct already as points of $X$_ is the [[quotient topological space]] of the [[disjoint union space]]

$$
  Conf\left( X, Y\right)
  \;\coloneqq\;
  \left(
  \underset{n \in \mathbb{n}}{\sqcup}
      \big(
      (
        X^n \setminus \mathbf{\Delta}_X^n
      )
      \times
      Y^k
    \big)
    /\Sigma(n)
  \right)/\sim
$$

by the [[equivalence relation]] $\sim$ given by

$$
  \big(
    (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}), (x_n, y_n)
  \big)
  \;\sim\;
  \big(
    (x_1, y_1), \cdots, (x_{n-1}, y_{n-1})
  \big)
  \;\;\;\; \Leftrightarrow
  \;\;\;\; (x_n, y_n) \in \partial (X \times Y)
  \,.
$$

This is naturally a [[filtered topological space]] with filter stages

$$
  Conf_{\leq n}\left( X, Y\right)
  \;\coloneqq\;
  \left(
  \underset{k \in \{1, \cdots, n\}}{\sqcup}
      \big(
      (
        X^k \setminus \mathbf{\Delta}_X^k
      )
      \times
      Y^k
    \big)
    /\Sigma(k)
  \right)/\sim
  \,.
$$

The corresponding [[quotient topological spaces]] of the filter stages reproduces the above configuration spaces of a fixed number of points:

$$
  Conf_n(X,Y)
  \;\simeq\;
  Conf_{\leq n}(X,Y) / Conf_{\leq (n-1)}(X,Y)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark
**(comparison to notation in the literature)**

The above Def. \ref{ConfigurationSpacesOfnPoints} is less general but possibly more suggestive than what is considered for instance in [Bödigheimer 87](#Boedigheimer87). Concretely, we have the following translations of notation:

  $$
    \array{
      \text{ here: }
        &&
      \array{ \text{ Segal 73,} \\ \text{ Snaith 74}: }
        &&
      \text{ Bödigheimer 87: }
      \\
      \\
      Conf(\mathbb{R}^d,Y)
        &=&
      C_d( Y/\partial Y )
        &=&
      C( \mathbb{R}^d, \emptyset; Y )
      \\
      \mathrm{Conf}_n\left( \mathbb{R}^d \right)
      & = &
      F_n C_d( S^0 ) / F_{n-1} C_d( S^0 )
      & = &
      D_n\left( \mathbb{R}^d, \emptyset; S^0  \right)
      \\
      \mathrm{Conf}_n\left( \mathbb{R}^d, Y \right)
      & = &
      F_n C_d( Y/\partial Y ) / F_{n-1} C_d( Y/\partial Y )
      & = &
      D_n\left( \mathbb{R}^d, \emptyset; Y/\partial Y  \right)
      \\
      \mathrm{Conf}_n( X ) && &=& D_n\left( X, \partial X; S^0  \right)
      \\
      \mathrm{Conf}_n( X, Y  ) && &=& D_n\left( X, \partial X; Y/\partial Y  \right)
    }
  $$
  Notice here that when $Y$ happens to have [[empty space|empty]] [[boundary]], $\partial Y = \emptyset$, then the [[pushout]]

$$
  X / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast
$$

is $Y$ with a [disjoint basepoint attached](pointed+topological+space#ForgettingAndAdjoiningBasepoints). Notably for $Y =\ast$ the [[point space]], we have that

$$
  \ast/\partial \ast = S^0
$$

is the [[0-sphere]].

=--

A slight variation of the definition is sometimes useful:

+-- {: .num_defn #ConfigurationSpaceOfDisks}
###### Definition
**(configuration space of $dim(X)$-disks)**

In the situation of Def. \ref{ConfigurationSpaceOfParticleswithLabels}, with $X$ a [[manifold]] of [[dimension]] $dim(X) \in \mathbb{N}$

$$
  DiskConf(X,A)
  \longrightarrow
  Conf(X,A)
$$

be, on the left, the labeled configuration space of joint [[embedding of smooth manifolds|embeddings]] of [[tuples]]

$$
  \left(
    D^{dim(X)}
      \overset{ \iota_i }{\hookrightarrow}
    X
  \right)
$$

of $dim(X)$-dimensional disks/[[closed balls]] into $X$, with identifications as in Def. \ref{ConfigurationSpaceOfParticleswithLabels} (in particular the disks centered at the basepoint are quotiented out) and with the comparison map sending each disk to its center.

This map is evidently a [[deformation retraction]] hence in particular a [[homotopy equivalence]].

=--

## Properties

### Basic properties

\begin{proposition}\label{ForgettingPointsIsAFibration}
**(forgetting points is a fibration)**
\linebreak
  Let $X$ be a [[topological manifold]].
  For $n, N \,\in\, \mathbb{N}$, the continuous map
  $$
    \underset{\{1,\cdots, n+N\}}{Conf}(X)
    \xrightarrow{\;\;}
    \underset{\{1,\cdots, N\}}{Conf}(X)
  $$
  which forgets the first $n$ points is a locally trivial [[fiber bundle]] with [[typical fiber]] $X \setminus \{x_1, \cdots, x_N\}$. In particular it is a [[Hurewicz fibration]].
\end{proposition}
(e.g. [Cohen 09, Thm. 3.3 with proof in §7](#Cohen09))

Using this one may deduce that:

\begin{proposition}\label{CnfSpaceOfPointsInPlaneIsEMSpace}
**(configuration space of points in plane is EM-space of braid group)**
\linebreak
  The [[homotopy type]] of a configuration space of points in the [[plane]] is that of an [[Eilenberg-MacLane space]] of the [[braid group]] $Br(n)$ in degree 1:

$$
  \underset{\{1,\cdots,n\}}{Conf}\big(\mathbb{R}^2\big)
  \;\;
  \underset{whe}{\simeq}
  \;\;
  K\big(
    Br(n)
    ,
  1
  \big)
$$
\end{proposition}
([Fadell & Neuwirth 1962, p. 118](#FadellNeuwirth62), [Fox & Neuwirth 1962, §7](#FoxNeuwirth62), reviewed in [Williams 2020, pp. 9](#Williams20))

\begin{proposition} The [[topological complexity]] of a configuration space is
\[
  \label{TopologicalComplexityOfConfigurationSpace}
    TC
    \big(
      Conf(\mathbb{R}^m,n)
    \big)
    \;=\;
    \left\{
    \begin{array}{ll}
       2n-1 & m \; odd
       \\
       2n-2 & m \; even
    \end{array}
    \right.
\]
(with convention $TC(*)=1$).
\end{proposition}

([Farber & Grant 08, Theorem 1](#FarberGrant08))

### Ordered unlabeled configurations from unordered labeled configurations
 {#OrderedUnlabeledConfigurationsFromUnorderedLabeledConfigurations}

> under construction

(...)

(...)

### Cohomotopy charge map
 {#CohomotopyChargeMap}

The _Cohomotopy charge map_ is the [[function]] that assigns to a [[configuration space of points|configuration of points]] their total [[charge]] as measured in [[Cohomotopy]]-[[generalized cohomology|cohomology theory]].

This is alternatively known as the "electric field map" ([Salvatore 01](#Salvatore01) following [Segal 73, Section 1](#Segal73), see also [Knudsen 18, p. 49](#Knudsen18)) or the "scanning map" ([Kallel 98](#Kallel98)).

For $D \in \mathbb{N}$ the _Cohomotopy charge map_ is the  [[continuous function]]

\[
  \label{CohomotopyChargeMapOnEuclideanSpace}
  Conf\big( \mathbb{R}^D \big)
  \overset{cc}{\longrightarrow}
  \mathbf{\pi}^D
  \Big(
    \big( \mathbb{R}^D \big)^{cpt}
  \Big)
  =
  Maps^{\ast/\!}\Big( \big(\mathbb{R}^D\big)^{cpt} , S^D\big)
  =
  \Omega^{D} S^D
\]

from the [[configuration space of points]] in the [[Euclidean space]] $\mathbb{R}^D$ to the $D$-[[Cohomotopy]] [[cocycle space]] [[vanishing at infinity]] on the [[Euclidean space]], which is equivalently the [[space of maps|space of pointed maps]] from the [[one-point compactification]] $S^D \simeq \big( \mathbb{R}^D \big)$ to itself, and hence equivalently the $D$-fold [[iterated based loop space]] of the [[n-sphere|D-sphere]]), which sends a configuration of points in $\mathbb{R}^D$, each regarded as carrying unit [[charge]] to their total [[charge]] as measured in  [[Cohomotopy]]-[[generalized cohomology|cohomology theory]] ([Segal 73, Section 3](#Segal73)).

The construction has evident generalizations to other manifolds than just Euclidean spaces, to spaces of labeled configurations and to [[equivariant Cohomotopy]]. The following graphics illustrates the Cohomotopy charge map on [[G-space]] [[tori]] for $G = \mathbb{Z}_2$ with values in $\mathbb{Z}_2$-[[equivariant Cohomotopy]]:

> graphics grabbed from [SS 19](Cohomotpy+charge+map#SatiSchreiber19)

#### Relation to iterated loop spaces of iterated suspensions
 {#LoopSpacesOfSuspensions}

In some situations the [[Cohomotopy charge map]] is a [[weak homotopy equivalence]] and hence exhibits, for all purposes of [[homotopy theory]], the [[Cohomotopy]] [[cocycle space]] of Cohomotopy charges as an equivalent reflection of the [[configuration space of points]]:

+-- {: .num_prop #GroupCompletionOfConfigurationSpaceIsIteratedBasedLoopSpace}
###### Proposition
**([[group completion]] on [[configuration space of points]] is [[iterated based loop space]])**

The [[Cohomotopy charge map]] (eq:CohomotopyChargeMapOnEuclideanSpace)

$$
  Conf
  \big(
    \mathbb{R}^D
  \big)
  \overset{
    cc
  }{\longrightarrow}
  \Omega^D S^D
$$

from the full unordered and unlabeled configuration space (eq:UnorderedUnlabeledConfigurationSpace) of [[Euclidean space]] $\mathbb{R}^D$  to the $D$-fold [[iterated based loop space]] of the [[n-sphere|D-sphere]], exhibits the [[group completion]] (eq:GroupCompletionOfConfigurationSpaceMonoid) of the configuration space monoid

$$
  \Omega B_{{}_{\sqcup}\!}
  Conf
  \big(
    \mathbb{R}^D
  \big)
  \overset{
    \simeq
  }{\longrightarrow}
  \Omega^D S^D
$$

=--

([Segal 73, Theorem 1](#Segal73))

+-- {: .num_prop #CohomotopyChargeMapIsEquivalenceOnSPhereLabeledConfihgurationSpace}
###### Proposition
**([[Cohomotopy charge map]] is [[weak homotopy equivalence]] on sphere-labeled [[configuration space of points]])**

For $D, k \in \mathbb{N}$ with $k \geq 1$, the [[Cohomotopy charge map]] (eq:CohomotopyChargeMapOnEuclideanSpace)

$$
  Conf
  \big(
    \mathbb{R}^D,
    S^k
  \big)
  \underoverset{\simeq}{\;\;cc\;\;}{\longrightarrow}
  \Omega^D S^{D + k}
  \;\simeq\;
  \mathbf{\pi}^{D+ k}\Big( \big( \mathbb{R}^{D}\big)^{cpt} \Big)
$$

is a [[weak homotopy equivalence]]

* from the configuration space (eq:UnorderedLabeledCOnfigurationSpace) of unordered points with labels in $S^k$ and vanishing at the base point of the label space

* to the $D$-fold [[iterated loop space]] of the [[n-sphere|D+k-sphere]]

hence equivalently

* to the [[cocycle space]] of [[Cohomotopy]] [[generalized cohomology theory|cohomology theory]] in degree $D + k$ [[vanishing at infinity]] on [[Euclidean space]] of [[dimension]] $D$.

=--

([Segal 73, Theorem 3](#Segal73))

This statement generalizes to [[equivariant homotopy theory]], with equivariant configurations carrying charge in [[equivariant Cohomotopy]]:

Let $G$ be a [[finite group]] and $V \in RO(G)$ an [[orthogonal group|orthogonal]] $G$-[[linear representation]], with its induced [[pointed topological space|pointed]] [[topological G-spaces]]:

1. the corresponding [[representation sphere]] $S^V \in G TopSpaces$,

1. the corresponding [[Euclidean G-space]] $\mathbb{R}^V \in G TopSpaces$.

For $X \in G TopSpaces$ any [[pointed topological space|pointed]] [[topological G-space]], consider

1. the equivariant $V$-[[suspension]], given by the [[smash product]] with the $V$-[[representation sphere]]:

   $\Sigma^V X \;\coloneqq\; X \wedge S^V \;\in G TopSpaces\;$

1. the equivariant $V$-[[iterated based loop space]], given by the $G$-[[fixed point subspace]] inside the [[space of maps]] out of the [[representation sphere]]:

   $\Omega^V X \;\coloneqq\; Maps^{\ast/}\big( S^V, X\big)^G$.

+-- {: .num_defn #EquivariantUnorderedLabeledConfigurationsVanishingWithVanishingLabel}
###### Definition
**(equivariant unordered labeled configurations vanishing with vanishing label)**

Write

$$
  Conf\big( \mathbb{R}^V , X \big)^G
  \;\hookrightarrow\;
  Conf\big( \mathbb{R}^V , X \big)
$$

for the $G$-[[fixed point subspace]] in the unordered $X$-labelled configuration space of points (Def. \ref{UnorderedLabeledConfigurationsVanishingWithVanishingLabel}), with respect to the [[diagonal action]] on $\mathbb{R}^V \times X$.

=--

+-- {: .num_prop #EquivariantCohomotopyChargeMapEquivalence}
###### Proposition
**([[Cohomotopy charge map]]-equivalence for configurations on [[Euclidean G-spaces]])**

Let

1. $G$ be a [[finite group]],

1. $V$ an [[orthogonal group|orthogonal]] $G$-[[linear representation]]

1.  $X$ a [[topological G-space]]

If $X$ is $G$-connected, in that for all [[subgroups]] $H \subset G$ the $H$-[[fixed point subspace]] $X^H$ is a [[connected topological space]], then the [[Cohomotopy charge map]]

\[
  \label{RourkeSandersonEquivalence1}
  Conf
  \big(
    \mathbb{R}^V,
    X
  \big)^G
  \underoverset{\simeq}{\;cc\;}{\longrightarrow}
  \Omega^V \Sigma^V X
  \phantom{AAA}
  \text{if X is G-connected}
\]

from the equivariant un-ordered $X$-labeled configuration space of points (Def. \ref{EquivariantUnorderedLabeledConfigurationsVanishingWithVanishingLabel}) in the corresponding [[Euclidean G-space]] to the based $V$-loop space of the $V$-suspension of $X$, is a [[weak homotopy equivalence]].

If $X$ is not necessarily $G$-connected, then this still holds for the [[group completion]] of the configuration space, under disjoint union of configurations

$$
  \Omega
  B_{{}_{\sqcup}\!}
  Conf
  \big(
    \mathbb{R}^V,
    X
  \big)
  \underoverset{\simeq}{\;cc\;}{\longrightarrow}
  \Omega^{V+1} \Sigma^{V+1} X
  \,.
$$

=--

([Rourke-Sanderson 00, Theorem 1, Theorem 2](#RourkeSanderson00))

More generally:

+-- {: .num_prop #ScanningMapEquivalenceOverCartesianSpace}
###### Proposition
**([[iterated loop spaces]] equivalent to [[configuration spaces of points]])**

For

1. $d \in \mathbb{N}$, $d \geq 1$ a [[natural number]] with $\mathbb{R}^d$ denoting the [[Cartesian space]]/[[Euclidean space]] of that [[dimension]],

1. $Y$ a [[manifold]], with [[inhabited set|non-empty]] [[manifold with boundary|boundary]] so that  $Y / \partial Y$ is [[connected topological space|connected]],

the [[Cohomotopy charge map]] constitutes a [[homotopy equivalence]]

$$
  Conf\left(
    \mathbb{R}^D, Y
  \right)
  \overset{cc}{\longrightarrow}
  \Omega^D \Sigma^D (Y/\partial Y)
$$

between

1. the configuration space of arbitrary points in $\mathbb{R}^d \times Y$ vanishing at the boundary (Def. \ref{ConfigurationSpacesOfnPoints})

1. the [[iterated loop space|d-fold loop space]] of the $d$-fold [[reduced suspension]] of the [[quotient space]] $Y / \partial Y$ (regarded as a [[pointed topological space]] with basepoint $[\partial Y]$).

In particular when $Y = \mathbb{D}^k$ is the [[closed ball]] of [[dimension]] $k \geq 1$ this gives a [[homotopy equivalence]]

$$
  Conf\left(
    \mathbb{R}^D, \mathbb{D}^k
  \right)
  \overset{cc}{\longrightarrow}
  \Omega^D S^{ D + k }
$$

with the [[iterated loop space|d-fold loop space]] of the [[n-sphere|(d+k)-sphere]].

=--

([May 72, Theorem 2.7](#May72), [Segal 73, Theorem 3](#Segal73), see [Bödigheimer 87, Example 13](#Boedigheimer87))

+-- {: .num_prop #StableSplittingOfMappingSpacesOutOfEuclideanSpace}
###### Proposition
**([[stable splitting of mapping spaces]] out of [[Euclidean space]]/[[n-spheres]])**

For

1. $d \in \mathbb{N}$, $d \geq 1$ a [[natural number]] with $\mathbb{R}^d$ denoting the [[Cartesian space]]/[[Euclidean space]] of that [[dimension]],

1. $Y$ a [[manifold]], with [[inhabited set|non-empty]] [[manifold with boundary|boundary]] so that  $Y / \partial Y$ is [[connected topological space|connected]],

there is a [[stable weak homotopy equivalence]]

$$
  \Sigma^\infty Conf(\mathbb{R}^d, Y)
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
$$

between

1. the [[suspension spectrum]] of the [[configuration space of points|configuration space]] of an arbitrary number of points in $\mathbb{R}^d \times Y$ vanishing at the boundary and distinct already as points of $\mathbb{R}^d$ (Def. \ref{ConfigurationSpacesOfnPoints})

1. the [[direct sum]] (hence: [[wedge sum]]) of [[suspension spectra]] of the [[configuration space of points|configuration spaces]] of a fixed number of points in $\mathbb{R}^d \times Y$, vanishing at the boundary and distinct already as points in $\mathbb{R}^d$ (also Def. \ref{ConfigurationSpacesOfnPoints}).

Combined with the [[stabilization]] of the [[Cohomotopy charge map]] [[homotopy equivalence]] from Prop. \ref{ScanningMapEquivalenceOverCartesianSpace} this yields a [[stable weak homotopy equivalence]]

$$
  Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y))
  =
  Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y))
  =
  \Omega^d \Sigma^d (Y/\partial Y)
  \underoverset{\Sigma^\infty cc}{\simeq}{\longrightarrow}
  \Sigma^\infty Conf(\mathbb{R}^d, Y)
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
$$

between the latter direct sum and the [[suspension spectrum]] of the [[mapping space]] of pointed [[continuous functions]] from the [[n-sphere|d-sphere]] to the $d$-fold [[reduced suspension]] of $Y / \partial Y$.

=--

([Snaith 74, theorem 1.1](#Snaith74), [Bödigheimer 87, Example 2](#Boedigheimer87))

In fact by [Bödigheimer 87, Example 5](#Boedigheimer87) this equivalence still holds with $Y$ treated on the same footing as $\mathbb{R}^d$, hence with $Conf_n(\mathbb{R}^d, Y)$ on the right replaced by $Conf_n(\mathbb{R}^d \times Y)$ in the well-adjusted notation of Def. \ref{ConfigurationSpacesOfnPoints}:

$$
  Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y))
  =
  Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y))
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)
$$

#### Relation to classifying space of the symmetric group

Let $X= \mathbb{R}^\infty$. Then

* the _unordered_ configuration space of $n$ points in $\mathbb{R}^\infty$ is a model for the [[classifying space]] $B \Sigma(n)$ of the [[symmetric group]] $\Sigma(n)$;

  (e.g. [Bödigheimer 87, Example 10](#Boedigheimer87))

* the _ordered_ configuration space of $n$ points, equipped with the canonical $\Sigma(n)$-[[action]], is a model for the $\Sigma(n)$-[[universal principal bundle]].

$\,$

#### Relation to James construction

The [[James construction]] of $X$ is [[homotopy equivalence|homotopy equivalent]] to the [[configuration space of points]] $C(\mathbb{R}^1, X)$ of points in the [[real line]] with labels taking values in $X$.

(e.g. [Bödigheimer 87, Example 9](#Boedigheimer87))

$\,$

#### In twisted Cohomotopy
 {#RelationToTwistedCohomotopy}

The May-Segal theorem \ref{ScanningMapEquivalenceOverCartesianSpace} generalizes from [[Euclidean space]] to [[closed manifold|closed]] [[smooth manifolds]] if at the same time one passes from plain [[Cohomotopy]] to [[twisted Cohomotopy]], twisted, via the [[J-homomorphism]], by the [[tangent bundle]]:

+-- {: .num_prop #ScanningMapEquivalenceOverClosedManifold}
###### Proposition

Let

1. $X^n$ be a [[smooth manifold|smooth]] [[closed manifold]] of [[dimension]] $n$;

1. $1 \leq k \in \mathbb{N}$ a [[positive number|positive]] [[natural number]].

Then the [[Cohomotopy charge map]] constitutes a [[weak homotopy equivalence]]

$$
  \underset{
    \color{blue}
    { \phantom{a} \atop \text{ J-twisted Cohomotopy space}}
  }{
    Maps_{{}_{/B O(n)}}
    \Big(
      X^n
      \;,\;
      S^{
        \mathbf{n}_{def} + \mathbf{k}_{\mathrm{triv}}
      }
      \!\sslash\!
      O(n)
    \Big)
  }
  \underoverset
  {\simeq}
  {
    \color{blue}
    \text{Cohomotopy charge map}
  }
  {\longleftarrow}
  \underset{
    \mathclap{
    \color{blue}
      {
        \phantom{a}
        \atop
        {
          \text{configuration space}
          \atop
          \text{of points}
        }
      }
    }
  }{
    Conf
    \big(
      X^n,
      S^k
    \big)
  }
$$

between

1. the [[J-homomorphism|J]]-[[twisted Cohomotopy|twisted (n+k)-Cohomotopy]] space of $X^n$, hence the [[space of sections]] of the $(n + k)$-[[spherical fibration]] over $X$ which is [[associated fiber bundle|associated]] via the [[tangent bundle]] by the [[O(n)]]-[[action]] on $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$

1. the [[configuration space of points]] on $X^n$ with labels in $S^k$.

=--

([Bödigheimer 87, Prop. 2](#Boedigheimer87), following [McDuff 75](#McDuff75))

+-- {: .num_prop #ScanningMapEquivalenceOverClosedFramedManifold}
###### Remark

In the special case that the [[closed manifold]] $X^n$ in Prop. \ref{ScanningMapEquivalenceOverClosedManifold} is [[parallelizable manifold|parallelizable]], hence that its [[tangent bundle]] is [[trivial bundle|trivializable]], the statement of Prop. \ref{ScanningMapEquivalenceOverClosedManifold} reduces to this:

Let

1. $X^n$ be a [[parallelizable manifold|parallelizable]] [[closed manifold]] of [[dimension]] $n$;

1. $1 \leq k \in \mathbb{N}$ a [[positive number|positive]] [[natural number]].

Then the [[Cohomotopy charge map]] constitutes a [[weak homotopy equivalence]]

$$
  \underset{
    \color{blue}
    { \phantom{a} \atop \text{ Cohomotopy space}}
  }{
    Maps
    \Big(
      X^n
      \;,\;
      S^{
       n + k
      }
    \Big)
  }
  \underoverset
  {\simeq}
  {
    \color{blue}
    \text{Cohomotopy charge map}
  }
  {\longleftarrow}
  \underset{
    \mathclap{
    \color{blue}
      {
        \phantom{a}
        \atop
        {
          \text{configuration space}
          \atop
          \text{of points}
        }
      }
    }
  }{
    Conf
    \big(
      X^n,
      S^k
    \big)
  }
$$

between

1. $(n+k)$-[[Cohomotopy]] space of $X^n$, hence the [[space of maps]] from $X$ to the [[n-sphere|(n+k)-sphere]]

1. the [[configuration space of points]] on $X^n$ with labels in $S^k$.

=--

### Rational maps to complex projective space

A similar relation holds for mapping spaces not to spheres, but to [[complex projective spaces]]:

\begin{prop}\label{RationalMapsToCPnAsConfigurationsOfPoints}
  The [[homotopy of rational maps|homotopy type of the space of rational maps]] from the [[Riemann sphere]] to [[complex projective space|complex projective $n$-space]] $\mathbb{C}P^n$ of algebraic [degree](homotopy+of+rational+maps#NotionsOfDegree) $d$ is that of the configuration space of at most $d$ points in $\mathbb{R}^2$ with labels in $S^{2n-1}$:

$$
    Maps_{
      {rat}
    }^{deg = d}
    \big(
      \Sigma
      ,\,
      \mathbb{C}P^n
    \big)
  \;\simeq_{htpy}\;
  \underset{ \leq d}{Conf}
  \big(
    \mathbb{R}^2;
    S^{2k+1}
  \big)
$$
\end{prop}

([Cohen & Shimamoto 91, Theorem 1](#CohenShimamoto91))

### Knizhnik-Zamolodchicov connection

[[!include Knizhnik-Zamolodchikov-Kontsevich construction -- definition]]

### Action by little $n$-disk operad and by Goodwillie derivatives

Under some conditions and with suitable degrees/shifts,
configuration spaces of points canonically have the [[structure]] of [[algebras over an operad]] over the [[little n-disk operad]] and the [[Goodwillie derivatives of the identitity functor]].

For more see [there](Goodwillie+derivatives+of+the+identity+functor#Properties)

\linebreak

### Homology and stabilization in homology
 {#HomologyAndStabilization}

Let $X$ be a [[topological space]] which is the [[interior]] of a [[compact topological space|compact]] [[manifold with boundary]] $\overline{X}$. We may think of the [[boundary]] $\partial \overline X$ as consisting of the "points at infinity" in $X$.

In particular, there are then inclusion maps

\[
  \label{InclusionOfUnorderedConfigurationSpaces}
  Conf_n
  \big(
    X
  \big)
  \overset{i_n}{\longrightarrow}
  Conf_{n+1}
  \big(
    X
  \big)
\]

of the unordered configuration space of $n$ points in $X$ (Def. \ref{UnorderedUnlabeledConfigurations}) into that of $n + 1$ points,
formalizing the idea of "adding a point at infinity" to a configuration.
More formally, these maps are given by pushing configuration points away from the boundary a little and then adding a new point near to a point on the boundary of $X$.

([Randal-Williams 13, Section 4](#RandalWilliams13))

The [[homotopy class]] of these maps depends (just) on the [[connected component]] of the [[boundary]] $\partial \overline{X}$ at which one chooses to bring in the new point. But for any choice, they have the following effect on [[cycles]] in [[ordinary homology]]:

+-- {: .num_prop #HomologicalStabilizationForUnorderedConfigurationSpaces}
###### Proposition
**(homological stabilization for unordered configuration spaces)**

Let $X$ be

* a [[connected topological space|connected]] [[smooth manifold]]

* which is the [[interior]] of a [[compact topological space|compact]] [[manifold with boundary]]

* of [[dimension]] $dim(X) \geq 2$.

Then for all $n \in \mathbb{N}$ the inclusion [[maps]] (eq:InclusionOfUnorderedConfigurationSpaces) are such that on [[ordinary homology]] with [[integer]] [[coefficients]] these maps induce [[split monomorphisms]] in all degrees,

$$
  H_\bullet
  \big(
    Conf_n(X)
    ,
    \mathbb{Z}
  \big)
  \overset{
    H_\bullet( i_n, \mathbb{Z} )
  }{\hookrightarrow}
  H_\bullet
  \big(
    Conf_{n+1}(X)
    ,
    \mathbb{Z}
  \big)
$$

and in degrees $\leq n/2$ these are even [[isomorphisms]]

$$
  H_p
  \big(
    Conf_n(X)
    ,
    \mathbb{Z}
  \big)
  \underoverset{\simeq}{
    H_p( i_n, \mathbb{Z} )
  }{\hookrightarrow}
  H_p
  \big(
    Conf_{n+1}(X)
    ,
    \mathbb{Z}
  \big)
  \phantom{AAAA}
  \text{for}
  \;
  p \leq n/2
  \,.
$$

Finally, for [[ordinary homology]] with [[rational number|rational]] [[coefficients]], these maps induce [[isomorphisms]] all the way up to degree $n$:

$$
  H_p
  \big(
    Conf_n(X)
    ,
    \mathbb{Q}
  \big)
  \underoverset{\simeq}{
    H_p( i_n, \mathbb{Q} )
  }{\hookrightarrow}
  H_p
  \big(
    Conf_{n+1}(X)
    ,
    \mathbb{Q}
  \big)
  \phantom{AAAA}
  \text{for}
  \;
  p \leq n
  \,.
$$

=--

([Randal-Williams 13, Theorem A and Threorem B](#RandalWilliams13))

\linebreak

### Gauss-Manin connections
 {#ReferencesGaussManinConnections}

On [[Gauss-Manin connections]] over [[configuration spaces of points]]:

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin Connections for Arrangements, I Eigenvalues*, Compositio Math. **136** (2003) 299-316 $[$[arXiiv:math/0105063](https://arxiv.org/abs/math/0105063), [doi:10.1023/A:1023262022279](https://doi.org/10.1023/A:1023262022279)$]$

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin connections for arrangements, II Nonresonant weights*, Amer. J. Math. **127** (2005) 569-594 $[$[arXiv:math/0207114](https://arxiv.org/abs/math/0207114), [jstor:40067930](https://www.jstor.org/stable/40067930)$]$

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin connections for arrangements, III Formal connections*, Trans. Amer. Math. Soc. **357** (2005) 3031-3050 $[$[arXiv:math/0307210](https://arxiv.org/abs/math/0307210), [doi:10.1090/S0002-9947-04-03621-9](https://doi.org/10.1090/S0002-9947-04-03621-9)$]$

and review in the context of [hypergeometric solutions](BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) to the [[Knizhnik-Zamolodchikov equation]]:

* {#EtingofFrenkelKirillov98} [[Pavel Etingof]], [[Igor Frenkel]], [[Alexander Kirillov]], Section 7.5 in: *Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations*, Mathematical surveys and monographs **58**, American Mathematical Society (1998) $[$[ISBN:978-1-4704-1285-2](https://bookstore.ams.org/surv-58), [review pdf](http://www.ams.org/journals/bull/2000-37-02/S0273-0979-00-00853-3/S0273-0979-00-00853-3.pdf)$]$

\linebreak

### Rational homotopy type
 {#RationalHomotopyType}

We discuss aspects of the [[rational homotopy type]] of configuration spaces of points. See also at _[[graph complex]]_.

#### Rational cohomology
 {#Cohomology}

+-- {: .num_prop #RealCohomologyOfConfigurationSpaceOfOrderedPointsInEuclideanSpace}
###### Proposition
**([[real cohomology]] of configuration spaces of ordered points in [[Euclidean space]])**

The [[real cohomology|real]] [[cohomology ring]] of the configuration spaces $\underset{{}^{\{1,\cdots,n\}}}{Conf}\big( \mathbb{R}^D\big)$
(Def. \ref{OrderedUnlabeledConfigurations})
of $n$ ordered unlabeled points  in [[Euclidean space]] $\mathbb{R}^D$

is [[generators and relations|generated]] by elements in degree $D-1$

$$
  \omega_{i j}
  \;\;
  \in
  H^{D-1}
  \Big(
    \underset{
      {}^{\{1, \cdots, n\}}
    }{
      Conf
    }
    \big(
      \mathbb{R}^D
    \big),
    \mathbb{R}
  \Big)
$$

for $i, j \in \{1, \cdots, n\}$

subject to these three [[generators and relations|relations]]:

1. **(anti-)symmetry)**

   $$\omega_{i j} = (-1)^D \omega_{j i} $$

1. **nilpotency**

   $$\omega_{i j} \wedge \omega_{i j} \;=\; 0$$

1. **3-term relation**

   $$
     \omega_{i j}
       \wedge
     \omega_{j k}
       +
     \omega_{j k}
       \wedge
     \omega_{k i}
       +
     \omega_{k i}
       \wedge
     \omega_{i j}
       =
     0
   $$

Hence:

\[
  \label{RealCohomologyOfUnorderedConfigurationSpaceInEuclideanSpace}
  H^\bullet
  \Big(
    \underset{
      {}^{\{1,\cdots,n\}}
    }{Conf}
    \big(
      \mathbb{R}^D
    \big),
    \mathbb{R}
  \Big)
  \;\simeq\;
  \mathbb{R}\Big[ \big\{\omega_{i j} \big\}_{i, j \in \{1, \cdots, n\}} \Big]
  \Big/
  \left(
    \array{
      \omega_{i j} = (-1)^D \omega_{j i}
      \\
      \omega_{i j} \wedge \omega_{i j} = 0
      \\
      \omega_{i j} \wedge \omega_{j k} + \omega_{j k} \omega_{k i} + \omega_{k i} \wedge \omega_{i j} = 0
     }
     \;\;
     \text{for}\;
     i,j \in \{1, \cdots, n\}
  \right)
\]

=--

This is due to [Cohen 1976](#Cohen76), following  [Arnold 1969](#Arnold69) and [Cohen 1073](#Cohen73). See also [Félix & Tanré 2003, Section 2](#FelixTanre03);
[Lambrechts & Tourtchine 2009, Section 3](#LambrechtsTourtchine09).

See also at _[[Fulton-MacPherson compactification]]_ the section _[de Rham cohomology](Fulton-MacPherson+operad#deRhamCohomology)_.

+-- {: .num_remark #RealCohomologyOfConfigurationSpaceInTermsOfGraphCohomology}
###### Remark
**([[real cohomology]] of the configuration space in terms of [[graph cohomology]])**

In the [[graph complex]]-model for the [[rational homotopy type]] of the ordered unlabeled [[configuration space of points]]
$\underset{{}^{\{1,\cdots,n\}}}{Conf}\big( \mathbb{R}^D\big)$ the three relations in Prop. \ref{RealCohomologyOfConfigurationSpaceOfOrderedPointsInEuclideanSpace} are incarnated as follows:

1. a graph changes sign when one of its edges is reversed ([this Def.](graph+complex#SignRulesForGraphs))

1. a graph with [[parallel edges]] is a vanishing graph ([this Def.](graph+complex#VanishingGraphs))

1. the graph coboundary of a single trivalent internal vertex ([this Example](graph+complex#ThreeTermRelation)).

=--

\linebreak

#### Rational homotopy and Whitehead products
 {#RationalHomotopyGroups}

Write again

$$
  Conf_n\big( \mathbb{R}^D \big)
  \;\coloneqq\;
  \big( \mathbb{R}^D \big)^n \setminus FatDiag
$$

for the configuration space of $n$ ordered points in [[Euclidean space]].

+-- {: .num_prop #RationalHomotopyOfConfigurationSpaceOfOrderedPointsInEuclideanSpace}
###### Proposition

The [[Whitehead product]] [[super Lie algebra]] of [[rationalization|rationalized]] [[homotopy groups]]

$$
  L^n
  \;\coloneqq\;
  \pi_{\bullet+1}\Big(  Conf_n\big( \mathbb{R}^D \big) \Big)
  \otimes_{\mathbb{Z}} \mathbb{Q}
$$

is [[generators and relations|generated]] from elements

$$
  \omega^{i j}
  \;\in\;
  \pi_2
  \Big(
    Conf_n\big( \mathbb{R}^D \big)
  \Big)
  \otimes_{\mathbb{Z}} \mathbb{Q}
  \phantom{AAA}
  i \neq j \in \{1, \cdots, n\}
  \,,
$$

subject to the following [[generators and relations|relations]]:

1. $\omega^{i j} = (-1)^D \omega^{j i}$

1. $\big[ \omega^{i j}, \omega^{k l} \big]$ $\;\;\;$ if $i,j,k,l$ are pairwise distinct;

1. $\big[ \omega^{i j}, \omega^{j k} + \omega^{k i}  \big] = 0$.

=--

This is due to [Kohno 02](#Kohno02). See also [Lambrechts-Tourtchine 09, Section 3](#LambrechtsTourtchine09).

\linebreak

### Relation to weight systems, chord diagrams and Vassiliev invariants

[[weight systems are cohomology of loop space of configuration space]]:

+-- {: .num_prop}
###### Proposition
**(integral [[horizontal weight systems]] are [[integral cohomology]] of [[based loop space]] of [[ordered configuration space of points]] in [[Euclidean space]])**

For [[ground ring]] $R = \mathbb{Z}$ the [[integers]], there is, for each [[natural number]] $n$, a canonical [[isomorphism]] of [[graded abelian groups]] between

1. the integral [[weight systems]]

   $$
     \mathcal{W}^{pb}_n
     \;\coloneqq\;
     Hom_{\mathbb{Z} Mod}
     \big(
       \underset{
         \mathcal{A}^{pb}_n
       }{
         \underbrace{
           \mathbb{Z}
           \langle
             \mathcal{D}^{pb}_n
           \rangle
           /(2T,4T)
         }
       }
       ,
       \mathbb{Z}
     \big)
   $$

   on [[horizontal chord diagrams]] of $n$ strands (elements of the set $\mathcal{D}^{pb}$)

1. the [[integral cohomology]] of the [[based loop space]] of the [[ordered configuration space of n points]] in 3d [[Euclidean space]]:

$$
  H \mathbb{Z}^\bullet
  \big(
    \Omega
    \underset{
      {}^{\{1,\cdots,n\}}
    }{Conf}
    (\mathbb{R}^3)
  \big)
  \;\simeq\;
  (\mathcal{W}^{pb}_n)^\bullet
  \;\simeq\;
  Gr^\bullet( \mathcal{V}^{pb}_n )
  \,.
$$

(the second equivalence on the right is the fact that [[weight systems are associated graded of Vassiliev invariants]]).

=--

This is stated as [Kohno 02, Theorem 4.1](#Kohno02)

+-- {: .num_prop}
###### Proposition
**([[weight systems]] are inside [[real cohomology]] of [[based loop space]] of [[ordered configuration space of points]] in [[Euclidean space]])**

For [[ground field]] $k = \mathbb{R}$ the [[real numbers]],
there is a canonical [[injection]] of the [[real vector space]] $\mathcal{W}$ of [[framed weight systems]] ([here](weight+system#eq:SpaceOfWeightSystemsAsLinearDual)) into the [[real cohomology]] of the [[based loop spaces]] of the ordered [[configuration spaces of points]] in 3-[[dimension|dimensional]] [[Euclidean space]]:

$$
  \mathcal{W}
  \;\overset{\;\;\;\;}{\hookrightarrow}\;
  H\mathbb{R}^\bullet
  \Big(
    \underset{n \in \mathbb{N}}{\sqcup}
    \Omega
    \underset{{}^{\{1,\cdots,n\}}}{Conf}
    \big(
      \mathbb{R}^3
    \big)
  \Big)
$$

=--

This is stated as [Kohno 02, Theorem 4.2](#Kohno02)

[[!include chord diagrams and weight systems -- table]]

### Berry-Robbins and Atiyah-Sutcliffe construction
 {#AtiyahSutcliffeConstruction}

In the course of providing a [[geometry|geometric]]-[[proof]] of the [[spin-statistics theorem]], [Berry & Robbins 1997](Atiyah-Sutcliffe+conjecture#BerryRobbins97) asked, at each [[natural number]] $n \in \mathbb{N}$, for a [[continuous function|continuous]] and [[symmetric group|$Sym(n)$]]-[[equivariant function]]

\[
  \label{TheBerryRobbinsMap}
  \underset{ {}^{\{1,\cdots, n\}}  }{Conf}\big(\mathbb{R}^3\big)
  \xrightarrow{\phantom{AAA}}
  \mathrm{U}(n)/\big(\mathrm{U}(1)\big)^n
\]

* from the [[configuration space of points|configuration space of $n$ points]] (ordered and unlabeled) in [[Euclidean space]] $\mathbb{R}^3$

* to the [[coset space]] of the [[unitary group]] $\mathrm{U}(n)$ by its [[maximal torus]], hence the complete [[flag manifold]] of flags in $\mathbb{C}^n$,

both equipped with the evident [[group action]] by the [[symmetric group]] $Sym(n)$.

For the first non-empty case $n = 2$ this readily reduces to asking for a [[continuous map]] of the form $\mathbb{R}^3 \setminus \{0\} \xrightarrow{\;\;} \mathbb{C}P^1 \simeq S^2$ which is [[equivariant function|equivariant]] with respect to passage to antipodal points. This is immediately seen to be given by the radial projection.
But this special case turns out not to be representative of the general case, as this simple construction idea does not generalize to $n \gt 2$.

That a continuous and $Sym(n)$-equivariant Berry-Robbins map (eq:TheBerryRobbinsMap) indeed exists for all $n$ was proven in [Atiyah 2000](Atiyah-Sutcliffe+conjecture#Atiyah00).

In this article, [[Michael Atiyah|Atiyah]] turned attention to the stronger question asking for a [[smooth function|smooth]] and $Sym(n) \times $[[SO(3)|$SO(3)$]]-[[equivariant]] function (eq:TheBerryRobbinsMap) and provided an elegant proof strategy for this stronger statement, which however hinges on some [[conjecture|conjectural]] positivity properties of a certain [[determinant]] (discussed in more detail and with first numerical evidence in [Atiyah 2001](Atiyah-Sutcliffe+conjecture#Atiyah01)), interpreted as the [[electromagnetism|electrostatic]] [[energy]] of $n$-[[particles]] in $\mathbb{R}^3$.

Extensive numerical checks of this stronger but conjectural construction was recorded, up to $n \lt 30$ , in [Atiyah & Sutcliffe 2002](Atiyah-Sutcliffe+conjecture#AtiyahSutcliffe02), together with a refined formulation of the [[conjecture]], whence it came to be known as the *[[Atiyah-Sutcliffe conjecture]]*.

The [[Atiyah-Sutcliffe conjecture]] has been [[proof|proven]] for $n = 3$ in [Atiyah 2000](Atiyah-Sutcliffe+conjecture#Atiyah00)/[01](Atiyah-Sutcliffe+conjecture#Atiyah01) and for $n = 4$ by [Eastwood & Norbury 01](Atiyah-Sutcliffe+conjecture#EastwoodNorbury01). A general proof is claimed in [Atiyah & Malkoun 18](Atiyah-Sutcliffe+conjecture#AtiyahMalkoun18).

\linebreak

## Occurrences and Applications
  {#OccurrencesAndApplications}

### Compactification

The [[Fulton-MacPherson compactification]] of configuration spaces of points in $\mathbb{R}^d$ serves to exhibit them as models for the [[little n-disk operad]].

### Stable splitting of mapping spaces

The [[stable splitting of mapping spaces]] says that [[suspension spectra]] of suitable [[mapping spaces]] are equivalently [[wedge sums]] of [[suspension spectra]] of configuration spaces of points.

### Correlators as differential forms on configuration spaces

In [[Euclidean field theory]] the [[correlators]] are often regarded as [[distributions of several variables]] with [[singularities]] on the [[fat diagonal]]. Hence they become [[non-singular distributions]] after [[restriction of distributions]] to the corresponding configuration space of points.

For more on this see at _[[correlators as differential forms on configuration spaces of points]]_.

## Related concepts

* [[Hilbert scheme]]

* [[arrangements of hyperplanes]]

* [[Knizhnik-Zamolodchikov connection]]

* [[representation stability]]

## References

### General

The [[braid group]] regarded as the [[fundamental group]] of a configuration space of points is considered (neither of them under these names, though) already in:

* {#Hurwitz1891} [[Adolf Hurwitz]], §II of: *Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten*, Mathematische Annalen **39** (1891) 1–60 [[doi:10.1007/BF01199469](https://doi.org/10.1007/BF01199469)]

there regarded as [[group action|acting]] on [[Riemann surfaces]] forming [[branched covers]], by movement of the branch points.

The concept of configuration spaces is then re-discovered/re-vived by:

* {#FadellNeuwirth62} [[Edward Fadell]], [[Lee Neuwirth]], *Configuration spaces*, Math. Scand. __10__ (1962) 111-118 $[$[doi:10.7146/math.scand.a-10517](https://doi.org/10.7146/math.scand.a-10517), [MR141126](http://www.ams.org/mathscinet-getitem?mr=141126)$]$

* {#FoxNeuwirth62} [[Ralph H. Fox]], [[Lee Neuwirth]], *The braid groups*, Math. Scand. **10** (1962) 119-126
$[$[doi:10.7146/math.scand.a-10518](https://doi.org/10.7146/math.scand.a-10518), [pdf](https://www.mscand.dk/article/view/10518/8539), [MR150755](http://www.ams.org/mathscinet-getitem?mr=150755)$]$

  > (focus on the resulting [[braid groups]])

See also early occurences in [[physics]], listed [below](#ReferencesInQuantumFieldTheory).

General accounts:

* [[Edward Fadell]], [[Sufian Husseini]], _Geometry and topology of configuration spaces_, Springer Monographs in Mathematics (2001)  [[doi:10.1007/978-3-642-56446-8](https://link.springer.com/book/10.1007/978-3-642-56446-8), [MR2002k:55038](http://www.ams.org/mathscinet-getitem?mr=2002k:55038)] xvi+313

* {#Cohen09} [[Frederick R. Cohen]], *Introduction to configuration spaces and their applications*, in *Braids* Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore **19** (2009)  183-261  [[doi:10.1142/9789814291415_0003](https://doi.org/10.1142/9789814291415_0003), [doi:10.1142/7550](https://doi.org/10.1142/7550 ), [pdf](https://www.mimuw.edu.pl/~sjack/prosem/Cohen_Singapore.final.24.december.2008.pdf)]

* [[Craig Westerland]], _Configuration spaces in geometry and topology_, 2011 ([pdf](https://www.austms.org.au/Publ/Gazette/2011/Nov11/TechPaperWesterland.pdf))

  (in [[geometry]] and [[topology]])

* {#Knudsen18} [[Ben Knudsen]], _Configuration spaces in algebraic topology_ ([arXiv:1803.11165](https://arxiv.org/abs/1803.11165))

  (in [[algebraic topology]])

* {#Williams20} [[Lucas Williams]], *Configuration Spaces for the Working Undergraduate*,  Rose-Hulman Undergraduate Mathematics Journal, **21** 1 (2020) Article 8. ([arXiv:1911.11186](https://arxiv.org/abs/1911.11186), [rhumj:vol21/iss1/8](https://scholar.rose-hulman.edu/rhumj/vol21/iss1/8))

In relation to the [[space of finite subsets]]:

* {#Handel00} David Handel, _Some Homotopy Properties of Spaces of Finite Subsets of Topological Spaces_, Houston Journal of Mathematics, Electronic Edition Vol. 26, No. 4, 2000 (pdf[hjm:Vol26-4](https://www.math.uh.edu/~hjm/Vol26-4.html))

* {#FelixTanre2010} [[Yves Félix]], [[Daniel Tanré]] _Rational homotopy of symmetric products and Spaces of finite subsets_, Contemp. Math 519 (2010): 77-92 ([pdf](http://tanre.org/Pro/Articles_files/SpnFiniteDef.pdf))

The [[algebra over an operad|algebra]]-[[structure]] of configuration spaces over [[little n-disk operads]]/[[Fulton-MacPherson operads]]:

* {#Markl99} [[Martin Markl]], _A compactification  of  the  real  configuration  space  as  an  operadic completion, J. Algebra 215 (1999), no. 1, 185–204

On topological complexity of [[configuration space]]:

* {#FarberGrant08} [[Michael Farber]], [[Mark Grant]], _Topological Complexity of the Klein bottle_ (2008), [arXiv:0806.4111](https://arxiv.org/pdf/0806.4111.pdf)

### Cohomotopy charge map
 {#ReferencesCohomotopyChargeMap}

The [[Cohomotopy charge map]] ("electric field map", "scanning map") and hence the relation of configuration spaces to [[Cohomotopy]] goes back to

* {#May72} [[Peter May]], *The geometry of iterated loop spaces*, Springer (1972) [[doi:10.1007/BFb0067491](https://link.springer.com/book/10.1007/BFb0067491), [pdf](https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf)]

* {#Segal73} [[Graeme Segal]], *Configuration-spaces and iterated loop-spaces*, Invent. Math. __21__ (1973), 213–221.  ([doi:10.1007/BF01390197](https://doi.org/10.1007/BF01390197), [pdf](http://dodo.pdmi.ras.ru/~topology/books/segal.pdf), MR 0331377)

Generalization of these constructions and results is due to

* {#McDuff75} [[Dusa McDuff]], _Configuration spaces of positive and negative particles_, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)

* {#Boedigheimer87} [[Carl-Friedrich Bödigheimer]], *Stable splittings of mapping spaces*, in: *Algebraic topology*, Lecture Notes in Mathematics **1286**, Springer (1987) 174-187 [[pdf](http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf), [[BoedigheimerStableSplittings87.pdf:file]], [doi:10.1007/BFb0078741](https://doi.org/10.1007/BFb0078741)]

* {#ManthorpeTillmann} Richard Manthorpe, [[Ulrike Tillmann]], _Tubular configurations: equivariant scanning and splitting_, Journal of the London Mathematical Society, Volume 90, Issue 3 ([arxiv:1307.5669](https://arxiv.org/abs/1307.5669), [doi:10.1112/jlms/jdu050](https://doi.org/10.1112/jlms/jdu050))

Generalization to [[equivariant homotopy theory]]:

* {#RourkeSanderson00} [[Colin Rourke]], [[Brian Sanderson]], _Equivariant Configuration Spaces_, J. London Math. Soc. 62 (2000) 544-552 ([arXiv:math/9712216](https://arxiv.org/abs/math/9712216))

and strengthening in the special case $G = \mathbb{Z}/2$:

* {#Xicotencatl06} Miguel Xicoténcatl, *On $\mathbb{Z}_2$-equivariant loop spaces*, Recent developments in algebraic topology, 183—191, Contemp. Math. 407, 2006 ([pdf](https://ncatlab.org/schreiber/files/Xicotencatl_EquivariantLoopSpaces.pdf))

The relevant construction for the [[group completion]] of the configuration space

* [Segal 73, Theorem 1](#Segal73)

* {#Salvatore01} [[Paolo Salvatore]], _Configuration spaces with summable labels_, In: Aguadé J., Broto C., [[Carles Casacuberta]]  (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics **196** Birkhäuser (2001) [[arXiv:math/9907073](https://arxiv.org/abs/math/9907073)]

and from the point of view of [[cobordism categories]]:

* [[Oscar Randal-Williams]], section 10 of: _Embedded Cobordism Categories and Spaces of Manifolds_, 	Int. Math. Res. Not. IMRN 2011, no. 3, 572-608 ([arXiv:0912.2505](https://arxiv.org/abs/0912.2505))

On the [[homotopy type]] of the [[space of maps|space of]] [[rational functions]] from the [[Riemann sphere]] to itself (related to the [[moduli space of monopoles]] in $\mathbb{R}^3$ and to the [[configuration space of points]] in $\mathbb{R}^2$):

* [[Graeme Segal]], _The topology of spaces of rational functions_, Acta Math. Volume 143 (1979), 39-72 ([euclid:1485890033](https://projecteuclid.org/euclid.acta/1485890033))

See also

* {#Kallel98} [[Sadok Kallel]], _Spaces of particles on manifolds and Generalized Poincaré Dualities_, The Quarterly Journal of Mathematics, Volume 52, Issue 1, 1 March 2001 ([doi:10.1093/qjmath/52.1.45](https://doi.org/10.1093/qjmath/52.1.45))

* Shingo Okuyama, Kazuhisa Shimakawa, _Interactions of strings and equivariant homology theories_, ([arXiv:0903.4667](https://arxiv.org/abs/0903.4667))

For relation to [[instantons]] via [[topological Yang-Mills theory]]:

* {#AtiyahJones78} [[Michael Atiyah]], [[John David Stuart Jones]], _Topological aspects of Yang-Mills theory_, Comm. Math. Phys. Volume 61, Number 2 (1978), 97-118 ([arXiv:1103904210](https://projecteuclid.org/euclid.cmp/1103904210))

An analogous statement for [[homotopy of rational maps]] related to [[Yang-Mills monopoles]]:

* {#CohenShimamoto91} [[Ralph L. Cohen]], [[Don H. Shimamoto]], *Rational Functions, Labelled Configurations, and Hilbert Schemes*, Journal of the London Mathematical Socienty **43** 2 (1991) 509-528 ([doi:10.1112/jlms/s2-43.3.509](https://doi.org/10.1112/jlms/s2-43.3.509))

In the context of speculations regarding [[Galois theory]] over the [[sphere spectrum]]:

* {#MoravaBeardsly17} [[Jack Morava]], [[Jonathan Beardsley]], _Toward a Galois theory of the integers over the sphere spectrum_,  Journal of Geometry and Physics Volume 131, September 2018, Pages 41-51 ([arXiv:1710.05992](https://arxiv.org/abs/1710.05992))

### Stable splitting of mapping spaces

The appearance of configuration spaces as summands in [[stable splittings of mapping spaces]] is originally due to

* {#Snaith74} [[Victor Snaith]],  _A stable decomposition of $\Omega^n S^n X$_, Journal of the London Mathematical Society 7 (1974), 577 - 583 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/snaiths.pdf))

An alternative proof is due to

* {#Cohen80} [[Ralph Cohen]], _Stable proof of stable splittings_, Math. Proc. Camb. Phil. Soc., 1980, 88, 149 ([doi:10.1017/S030500410005742X](https://doi.org/10.1017/S030500410005742X), [pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/247D9F951F8AB99000E4FF6CB2DB9EA2/S030500410005742Xa.pdf/div-class-title-stable-proofs-of-stable-splittings-div.pdf))

Review and generalization is in

* [Boedigheimer 87](#Boedigheimer87)

and the relation to the [[Goodwillie-Taylor tower]] of mapping spaces is pointed out in

* [Arone 99](#Arone99)

### In Goodwillie-calculus
 {#ReferencesInGoodwillieCalculus}

The configuration spaces of a space $X$ appear as the [[Goodwillie derivatives]] of its [[mapping space]]/[[nonabelian cohomology]]-[[functor]] $Maps(X,-)$:

* {#Arone99} [[Greg Arone]], _A generalization of Snaith-type filtration_, Transactions of the American Mathematical Society 351.3 (1999): 1123-1150. ([pdf](https://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02405-8/S0002-9947-99-02405-8.pdf))

* {#Ching05} [[Michael Ching]], _Calculus of Functors and Configuration Spaces_, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 ([pdf](https://www3.amherst.edu/~mching/Work/skye.pdf))

### Compactification
 {#ReferencesCompactification}

A [[compactification]] of configuration spaces of points was introduced in

* [[William Fulton]], [[Robert MacPherson]], _A compactification of configuration spaces_, Ann. of Math. (2), 139(1):183–225, 1994.

and an [[operad]]-[[structure]] defined on it ([[Fulton-MacPherson operad]]) in

* [[Ezra Getzler]], [[John Jones]], _Operads, homotopy algebra and iterated integrals for double loop spaces_ ([arXiv:hep-th/9403055](https://arxiv.org/abs/hep-th/9403055))

Review includes

* {#LambrechtsVolic14} [[Pascal Lambrechts]], [[Ismar Volić]], section 5 of _Formality of the little $N$-disks operad_, Memoirs of the American Mathematical Society ; no. 1079, 2014  ([arXiv"0808.0457](https://arxiv.org/abs/0808.0457), [doi:10.1090/memo/1079](http://dx.doi.org/10.1090/memo/1079))

This underlies the models of configuration spaces by [[graph complexes]], see there for more.

### Homology and cohomology
 {#ReferencesCohomology}

General discussion of [[ordinary homology]]/[[ordinary cohomology]] of configuration spaces of points:

* {#Arnold69} [[Vladimir Arnold]], _The cohomology ring of the colored braid group_, Mat. Zametki, 1969, Volume 5, Issue 2, Pages 227–231 ([mathnet:mz6827](http://mi.mathnet.ru/eng/mz6827))

* {#Cohen73} [[Fred Cohen]], _Cohomology of braid spaces_, Bull. Amer. Math. Soc. Volume 79, Number 4 (1973), 763-766 ([euclid:1183534761](https://projecteuclid.org/euclid.bams/1183534761))

* {#Cohen76} [[Fred Cohen]], _The homology of $C_{n+1}$-Spaces, $n \geq 0$_, In: _The Homology of Iterated Loop Spaces_, Lecture Notes in Mathematics, vol 533. Springer 1976 ([doi:10.1007/BFb0080467](https://doi.org/10.1007/BFb0080467))

* [[Carl-Friedrich Bödigheimer]], [[Fred Cohen]], L. Taylor, _On the homology of configuration spaces_, Topology Vol. 28 No. 1, p. 111-123 1989 ([pdf](https://core.ac.uk/download/pdf/82124359.pdf))

* E. Ossa, _On the cohomology of configuration spaces_, In: Broto C., [[Carles Casacuberta]], Mislin G. (eds.), _Algebraic Topology: New Trends in Localization and Periodicity_,  Progress in Mathematics, vol 136. Birkhäuser Basel (1996) ([doi:10.1007/978-3-0348-9018-2_26](https://doi.org/10.1007/978-3-0348-9018-2_26))

* [[Yves Félix]], [[Jean-Claude Thomas]], _Rational Betti numbers of configuration spaces_, Topology and its Applications, Volume 102, Issue 2, 8 April 2000, Pages 139-149 (doi:10.1016/S0166-8641(98)00148-5)

* {#RandalWilliams13} [[Oscar Randal-Williams]], _Homological stability for unordered configuration spaces_, The Quarterly Journal of Mathematics, Volume 64, Issue 1, March 2013, Pages 303–326 ([arXiv:1105.5257](https://arxiv.org/abs/1105.5257))

* {#FelixTanre03} [[Yves Félix]], [[Daniel Tanré]], _The cohomology algebra of unordered configuration spaces_, Journal of the LMS, Vol 72, Issue 2 ([arxiv:math/0311323](https://arxiv.org/abs/math/0311323), [doi:10.1112/S0024610705006794](https://doi.org/10.1112/S0024610705006794))

* {#Palmer12} [[Martin Palmer]], *Configuration spaces and homological stability*, Oxford University Research Archive (2012) [[pdf](https://mdp.ac/thesis/thesis.pdf), [web](https://mdp.ac/thesis/index.html), [[Palmer-ConfigurationSpaces.pdf:file]]]

* [[Ben Knudsen]], _Betti numbers and stability for configuration spaces via factorization homology_, Algebr. Geom. Topol. 17 (2017) 3137-3187 ([arXiv:1405.6696](https://arxiv.org/abs/1405.6696))

* Thomas Church, _Homological stability for configuration spaces of manifolds_ ([arxiv:1602.04748](https://arxiv.org/abs/1602.04748))

* Christoph Schiessl, _Betti numbers of unordered configuration spaces of the torus_ ([arxiv:1602.04748](https://arxiv.org/abs/1602.04748))

* Christoph Schiessl, _Integral cohomology of configuration spaces of the sphere_ ([arxiv:1801.04273](https://arxiv.org/abs/1801.04273))

* Weiyan Chen, *Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting* ([arXiv:1603.03931](https://arxiv.org/abs/1603.03931))

* Dan Petersen, _Cohomology of generalized configuration spaces_ ([arXiv:1807.07293](https://arxiv.org/abs/1807.07293))

* [[Victor Vassiliev]], *Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials* ([arXiv:1809.05632](https://arxiv.org/abs/1809.05632))

* [[Victor Vassiliev]], *Cohomology of spaces of Hopf equivariant maps of spheres* ([arXiv:2102.07157](https://arxiv.org/abs/2102.07157))

* Roberto Pagaria, _The cohomology rings of the unordered configuration spaces of the torus_, Algebr. Geom. Topol. 20 (2020) 2995–3012 ([doi:10.2140/agt.2020.20.2995](https://www.doi.org/10.2140/agt.2020.20.2995))

* [[Christian Blanchet]], [[Martin Palmer]], [[Awais Shaukat]], *Heisenberg homology on surface configurations* [[arXiv:2109.00515](https://arxiv.org/abs/2109.00515)]

* [[Awais Shaukat]], [[Christian Blanchet]], *Weakly framed surface configurations, Heisenberg homology and Mapping Class Group action*, Archiv der Mathematik **120** (2023) 99–109 [[arXiv:2206.11475](https://arxiv.org/abs/2206.11475), [doi:10.1007/s00013-022-01793-3](https://doi.org/10.1007/s00013-022-01793-3)]

* [[Christian Blanchet]], *Heisenberg homologies of surface configurations*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#BlanchetMar2023)]

Review with focus on the phenomenon of [[representation stability]]:

* {#RollandWilson22}  [[Rita Jimenez Rolland]], [[Jennifer C. H. Wilson]], *Stability properties of moduli spaces*, Notices of the AMS [**69** 4](https://www.ams.org/notices/202204/202204FullIssue.pdf) (April 2022) [[arXiv:2201.04096](https://arxiv.org/abs/2201.04096), [pdf](https://www.ams.org/journals/notices/202204/rnoti-p522.pdf), [web](https://www.ams.org/journals/notices/202204/noti2452/noti2452.html)]

Expressing the [[rational cohomology]] of [[ordered configuration spaces of points]] via [[factorization homology]] and [[Ran spaces]]:

* Quoc P. Ho, _Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras_ ([arXiv:2004.00252](https://arxiv.org/abs/2004.00252))

Discussion of the [[Morava E-theory]] of configuration spaces of points:

* [[Lukas Brantner]], [[Jeremy Hahn]], [[Ben Knudsen]], *The Lubin-Tate Theory of Configuration Spaces: I* ([arXiv:1908.11321](https://arxiv.org/abs/1908.11321))

### Homotopy

Discussion of [[homotopy groups]] of configuration spaces:

* [Kohno 02](#Kohno02)

* {#LambrechtsTourtchine09} [[Pascal Lambrechts]], [[Victor Tourtchine]], _Homotopy graph-complex for configuration and knot spaces_, Transactions of the AMS, Volume 361, Number 1, January 2009, Pages 207–222 ([arxiv:math/0611766](https://arxiv.org/abs/math/0611766))

* [[Sadok Kallel]], Ines Saihi, _Homotopy Groups of Diagonal Complements_, Algebr. Geom. Topol. 16 (2016) 2949-2980 ([arXiv:1306.6272](https://arxiv.org/abs/1306.6272))

### Rational homotopy type

Discussion of the [[rational homotopy type]]:

* [[Igor Kriz]], _On the Rational Homotopy Type of Configuration Spaces_, Annals of Mathematics
Second Series, Vol. 139, No. 2 (Mar., 1994), pp. 227-237 ([jstor:2946581](https://www.jstor.org/stable/2946581))

### Cohomology modeled by graph complexes

That the [[de Rham cohomology]] of (the [[Fulton-MacPherson compactification]] of) configuration spaces of points may be modeled by [[graph complexes]] (exhibiting [[formality of the little n-disk operad]]) is due to

* {#Kontsevich99b} [[Maxim Kontsevich]], around Def. 15 and Lemma 3 in _Operads and Motives in Deformation Quantization_, Lett.Math.Phys.48:35-72,1999 ([arXiv:math/9904055](https://arxiv.org/abs/math/9904055))

nicely reviewed in [Lambrechts-Volic 14](#LambrechtsVolic14)

Further discussion of the graph complex as a model for the [[de Rham cohomology]] of  [[configuration spaces of points]] is in

* Najib Idrissi, _The Lambrechts-Stanley Model of Configuration Spaces_, Invent. Math, 2018 ([arXiv:1608.08054](https://arxiv.org/abs/1608.08054), [doi:10.1007/s00222-018-0842-9](https://dx.doi.org/10.1007/s00222-018-0842-9))

* {#CamposWillwacher16} [[Ricardo Campos]], [[Thomas Willwacher]], _A model for configuration spaces of points_ ([arXiv:1604.02043](https://arxiv.org/abs/1604.02043))

* [[Ricardo Campos]], Najib Idrissi, [[Pascal Lambrechts]], [[Thomas Willwacher]], _Configuration Spaces of Manifolds with Boundary_ ([arXiv:1802.00716](https://arxiv.org/abs/1802.00716))

* [[Ricardo Campos]], Julien Ducoulombier, Najib Idrissi, [[Thomas Willwacher]], _A model for framed configuration spaces of points_ ([arXiv:1807.08319](https://arxiv.org/abs/1807.08319))

### Loop spaces of configuration spaces of points
 {#ReferencesLoopSpacesOfConfigurationSpaces}

On [[loop spaces]] of [[configuration spaces of points]]:

* [[Edward Fadell]], [[Sufian Husseini]], _The space of loops on configuration spaces and the Majer-Terracini index_, Topol. Methods Nonlinear Anal. Volume 11, Number 2 (1998), 249-271 ([euclid:tmna/1476842829](https://projecteuclid.org/euclid.tmna/1476842829))

Specifically on [[ordinary homology]]/[[ordinary cohomology]] of [[based loop spaces]] of [[configuration spaces of points]] and the relation to [[weight systems]]/[[Vassiliev invariants]]:

* {#Kohno94} [[Toshitake Kohno]], _Vassiliev invariants and de Rham complex on the space of knots_,
In: Yoshiaki Maeda, Hideki Omori, [[Alan Weinstein]] (eds.), _Symplectic Geometry and Quantization_, Contemporary Mathematics 179 (1994): 123-123 ([doi:10.1090/conm/179](http://dx.doi.org/10.1090/conm/179))

* {#CohenGitler01} [[Fred Cohen]], [[Samuel Gitler]], _Loop spaces of configuration spaces, braid-like groups, and knots_, In: Jaume Aguadé, Carles Broto, [[Carles Casacuberta]]  (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel 2001 ([doi:10.1007/978-3-0348-8312-2_7](https://doi.org/10.1007/978-3-0348-8312-2_7))

* {#Kohno02} [[Toshitake Kohno]], _Loop spaces of configuration spaces and finite type invariants_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arXiv:math/0211056](https://arxiv.org/abs/math/0211056))

* [[Fred Cohen]], [[Samuel Gitler]], _On loop spaces of configuration spaces_, Trans. Amer. Math. Soc. __354__ (2002), no. 5, 1705–1748, ([jstor:2693715](https://www.jstor.org/stable/2693715), [MR2002m:55020](http://www.ams.org/mathscinet-getitem?mr=1881013))

For more see also at *[[weight systems are cohomology of loop space of configuration space]]*.

### Graph configuration spaces

On configuration spaces of points any two of which are required to be non-coincident only if connected by an [[edge]] in a labelin [[graph]]:

* [[Sadok Kallel]], *The Homotopy Type of Graph Configuration Spaces*, [talk at](Center+for+Quantum+and+Topological+Systems#KallelOct2023) [[CQTS]] (Oct 2023) [slides:[[Kallel-GraphConfiguration.pdf:file]], video:[YT](https://youtu.be/eNdPBkjA-eQ)]

### Configurations in general position
 {#ReferencesCOnfigiurationsInGeneralPosition}

Discussion of spaces of configurations of points "in general position", where not only any pair of points is required to be non-coincident, but any $n+1$-[[tuple]] is required to [[linear span|span]] an $n$-dimensional subspace, typically considered after projective quotienting:

* [[Mikhail Kapranov]], §2.1 in: *Chow quotients of Grassmannian I*, Advances in Soviet Mathematics **16** (1993) 29–110 [[arXiv:alg-geom/9210002](https://arxiv.org/abs/alg-geom/9210002)]

* [[Nima Arkani-Hamed]], [[Thomas Lam]], [[Marcus Spradlin]], *Positive configuration space*, Commun. Math. Phys. **384** (2021) 909–954  [[arXiv:2003.03904](https://arxiv.org/abs/2003.03904), [doi:10.1007/s00220-021-04041-x](https://doi.org/10.1007/s00220-021-04041-x)]

### In quantum field theory
 {#ReferencesInQuantumFieldTheory}

In [[physics]] ([[solid state physics]]/[[particle physics]]) the configuration space of points appears early on in the discussion of [[anyon statistics]], originally in:

* [[Michael G. G. Laidlaw]], [[Cécile Morette DeWitt]], *Feynman Functional Integrals for Systems of Indistinguishable Particles*, Phys. Rev. D **3** (1971) 1375 $[$[doi:10.1103/PhysRevD.3.1375](https://doi.org/10.1103/PhysRevD.3.1375)$]$

* {#LeinaasMyrheim77} [[Jon Magne Leinaas]], [[Jan Myrheim]], pp. 22 of: _On the theory of identical particles_, _К теории тождествениых частиц_, Nuovo Cim B 37, 1–23 (1977) ([doi:10.1007/BF02727953](https://doi.org/10.1007/BF02727953))

* [[Frank Wilczek]], p. 959 of: *Quantum Mechanics of Fractional-Spin Particles*, Phys. Rev. Lett. **49** (1982) 957 (reprinted in [Wilczek 1990, p. 166-168](Frank+Wilczek#Wilczek90)) $[$[doi:10.1103/PhysRevLett.49.957](https://doi.org/10.1103/PhysRevLett.49.957)$]$

Concretely, [[anyon]]-[[wavefunctions]] are identified with [[multi-valued functions]] on a configuration space of points, see [there](braid+group+statistics#AsAFictitiousAharonovBohmEffect):

* {#ImboImboSudarshan90} Tom Imbo, Chandni Shah Imbo, E. C. G. Sudarshan, *Identical particles, exotic statistics and braid groups*, Physics Letters B **234** 1–2, (1990) 103-107 $[$doi:10.1016/0370-2693(90)92010-G, [pdf](https://lib-extopc.kek.jp/preprints/PDF/2000/0033/0033552.pdf)$]$

* [[Jürg Fröhlich]], [[Fabrizio Gabbiani]], [[Pieralberto Marchetti]], around (1.1) in: *Braid statistics in three-dimensional local quantum field theory*, in: H.C. Lee (ed.) *Physics, Geometry and Topology* NATO ASI Series, **238** Springer (1990)  $[$[doi:10.1007/978-1-4615-3802-8_2](https://doi.org/10.1007/978-1-4615-3802-8_2),  [[FroehlichGabbianiMarchetti-BraidStatistics.pdf:file]]$]$

* {#MundSchrader93} J. Mund, [[Robert Schrader]], *Hilbert Spaces for Nonrelativistic and Relativistic "Free" Plektons (Particles with Braid Group Statistics)*, in *Advances in dynamical systems and quantum physics* (Capri, 1993), World Sci. (1995) 235–259 $[$[arXiv:hep-th/9310054v2](https://arxiv.org/abs/hep-th/9310054)$]$

* {#BCMS93} Garth A. Baker, Geoff S. Canright, Shashikant B. Mulay, Carl Sundberg, *On the spectral problem for anyons*, Communications in Mathematical Physics **153** (1993) 277–295 $[$[doi:10.1007/BF02096644](https://doi.org/10.1007/BF02096644)$]$

* {#DFT97} Gianfausto Dell'Antonio, Rodolfo Figari & Alessandro Teta, *Statistics in Space Dimension Two*, Letters in Mathematical Physics **40** (1997) 235–256 $[$[doi:10.1023/A:1007361832622](https://doi.org/10.1023/A:1007361832622)$]$

* {#MurthyShankar09} [[M.V.N. Murthy]], [[Ramamurti Shankar]], p. 7, 50*Exclusion Statistics: From Pauli to Haldane* (1999, 2009) $[$[dspace:123456789/334](https://dspace.imsc.res.in/xmlui/handle/123456789/334), [pdf](https://www.imsc.res.in/xmlui/bitstream/handle/123456789/334/MR120.pdf?sequence=1), [[MurthyShankar-ExclusionStatistics.pdf:file]]$]$

Moreover, in [[quantum field theory]] one may formalize [[correlators as differential forms on configuration spaces of points]].
This perspective was originally considered specifically for [[Chern-Simons theory]] in:

* {#AxelrodSinger93} [[Scott Axelrod]], [[Isadore Singer]],  _Chern--Simons Perturbation Theory II_, J. Diff. Geom. 39 (1994) 173-213 ([arXiv:hep-th/9304087](http://arxiv.org/abs/hep-th/9304087))

which was re-amplified in:

* {#BottCattaneo97} [[Raoul Bott]], [[Alberto Cattaneo]], Remark 3.6 in _Integral invariants of 3-manifolds_, J. Diff. Geom., 48 (1998) 91-133 ([arXiv:dg-ga/9710001](https://arxiv.org/abs/dg-ga/9710001))

* {#CattaneoMnev10} [[Alberto Cattaneo]], [[Pavel Mnev]], Remark 11 in _Remarks on Chern-Simons invariants_, Commun.Math.Phys.293:803-836,2010 ([arXiv:0811.2045](https://arxiv.org/abs/0811.2045))

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

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

* {#Kontsevich93} [[Maxim Kontsevich]], _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf))

* {#Kontsevich94} [[Maxim Kontsevich]], pages 11-12 of _Feynman diagrams and low-dimensional topology_, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics __120__, Birkhäuser (1994), 97–121 ([pdf](http://www.ihes.fr/~maxim/TEXTS/Feynman%20%20diagrams%20and%20low-dimensional%20topology.pdf))

with full details and proofs in

* {#LambrechtsVolic14} [[Pascal Lambrechts]], [[Ismar Volić]], sections 6 and 7 of _Formality of the little N-disks operad_, Memoirs of the American Mathematical Society no. 1079, 2014  ([arxiv:0808.0457](https://arxiv.org/abs/0808.0457), [doi:10.1090/memo/1079](http://dx.doi.org/10.1090/memo/1079))

see also

* {#CamposIdrissiLambrechtsWillwacher18} [[Ricardo Campos]], [[Najib Idrissi]], [[Pascal Lambrechts]], [[Thomas Willwacher]], _Configuration Spaces of Manifolds with Boundary_ ([arXiv:1802.00716](https://arxiv.org/abs/1802.00716))
* {#Idrissi21} [[Najib Idrissi]], [_Real homotopy of configuration spaces_](https://idrissi.eu/research/peccot)

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

* {#BergbauerBrunettiKreimer09} [[Christoph Bergbauer]], [[Romeo Brunetti]], [[Dirk Kreimer]], _Renormalization and resolution of singularities_, ESI preprint 2010 ([arXiv:0908.0633](https://arxiv.org/abs/0908.0633), [ESI:2244](https://mat.univie.ac.at/~esiprpr/esi2244.pdf))

* {#Bergbauer09} [[Christoph Bergbauer]], _Renormalization and resolution of singularities_, talks as IHES and Boston, 2009 ([pdf](http://www.emg.uni-mainz.de/Dateien/bergbauer.pdf))

* {#Berghoff14a} [[Marko Berghoff]], _Wonderful renormalization_, 2014 ([pdf](http://www2.mathematik.hu-berlin.de/%7Ekreimer/wp-content/uploads/Berghoff-Marko.pdf), [doi:10.18452/17160](https://doi.org/10.18452/17160))

* {#Berghoff14b} [[Marko Berghoff]], _Wonderful compactifications in quantum field theory_,  Communications in Number Theory and Physics Volume 9 (2015) Number 3 ([arXiv:1411.5583](https://arxiv.org/abs/1411.5583))

Analogous discussion for [[Hilbert schemes of points]]:

* Jian Zhou, _K-Theory of Hilbert Schemes as a Formal Quantum Field Theory_ ([arXiv:1803.06080](https://arxiv.org/abs/1803.06080))

Discussion specifically in [[topological quantum field theory]] with an eye towards [[supersymmetry|supersymmetric]] field theory, in terms of the [[ordinary homology]] of [[configuration spaces of points]]:

* [[Christopher Beem]], [[David Ben-Zvi]], [[Mathew Bullimore]], [[Tudor Dimofte]], [[Andrew Neitzke]], _Secondary products in supersymmetric field theory_, Ann. Henri Poincaré 21, 1235–1310 (2020). ([arXiv:1809.00009](https://arxiv.org/abs/1809.00009), [doi:10.1007/s00023-020-00888-3](https://doi.org/10.1007/s00023-020-00888-3))

[[!include hypergeometric KZ-solutions -- references]]

### As moduli of Dp-D(p+4)-brane bound states:
 {#AsModuliOfD0D4BraneBoundStates}

Discussion of configuration spaces of _possibly coincident_ points, hence of [[symmetric products]] $X^n/Sym(n)$ as [[moduli spaces]] of [[D0-D4-brane bound states]]:

* [[Cumrun Vafa]], _Instantons on D-branes_, Nucl. Phys. B463 (1996) 435-442 ([arXiv:hep-th/9512078](https://arxiv.org/abs/hep-th/9512078))

with emphasis to the resulting [[configuration spaces of points]], as in

* [[Cumrun Vafa]], [[Edward Witten]], Section 4.1 of: _A Strong Coupling Test of S-Duality_, Nucl. Phys. B431:3-77, 1994 ([arXiv:hep-th/9408074](https://arxiv.org/abs/hep-th/9408074))

[[!redirects configuration spaces of points]]
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[[!redirects ordered configuration space of points]]
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[[!redirects graph-indexed configuration space of points]]
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confinement

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Differential cohomology
+-- {: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Confinement (e.g [Espiru 94](#Espiru94)) is the (expected) phenomenon in [[Yang-Mills theory]] generally and especially in [[quantum chromodynamics]] that the fundamental [[quarks]], which the [[Yang-Mills theory|YM]]/[[QCD]]-[[Lagrangian density]] actually describes, must form [[baryon|baryonic]] [[bound state|bound states]] which are neutral under the color charge -- the [[mesons]] and [[hadron|hadrons]] ([[proton|protons]], [[neutron|neutrons]]). Hence confinement in particular concerns the emergence and existence of [[atomic nuclei]], hence of ordinary [[matter]], which is not manifest at all in the [[quark]]-model. In order to prove, via [[QCD]] itself, that the potential energy of a colorless package of [[quarks]] tends to infinity as the distance between them grows, one must probe the long-range regime of the quark-quark interaction, in which the methods of [[perturbative quantum field theory|Feynman perturbation theory]] fail.

Part of the issue is that confinement is a [[non-perturbative effect]] (e.g [Espiru 94](#Espiru94)) outside the range of validity of [[perturbative quantum field theory]].

[[!include open problem of confinement -- references]]

## Potential solutions

### Via Skyrmions and D4-brane models
 {#ViaSkyrmions}

A good qualitative and moderate quantitative explanation of confinement in [[quantum chromodynamics]] is found in [[intersecting D-brane models]], specifically in the [[Witten-Sakai-Sugimoto model]] which [[geometric engineering of QFT|geometrically engineers]] [[QCD]] on [[D4-D8 brane bound states]].

([Witten 98](#Witten98), followed up on in [Sakai-Sugimoto 04](Ads/QCD#SakaiSugimoto04), [Sakai-Sugimoto 05](Ads/QCD#SakaiSugimoto05))

> graphics grabbed from [Erlich 09, section 1.1](Ads/QCD#Erlich09)

> graphics grabbed from [Rebhan 14](Ads/QCD#Rebhan14)

In this [[Witten-Sakai-Sugimoto model]] for [[non-perturbative effect|strongly coupled]] [[QCD]] the [[hadrons]] in [[QCD]] correspond to [[string theory|string-theoretic]]-phenomena in the [[bulk field theory]]:

1. the [[mesons]] ([[bound states]] of 2 [[quarks]]) correspond to [[open strings]] in the bulk, whose two endpoints on the [[asymptotic boundary]] correspond to the two quarks

1. [[baryons]] ([[bound states]] of $N_c$ [[quarks]]) appear in two different but equivalent ([Sugimoto 16, 15.4.1](Ads/QCD#Sugimoto16)) guises:

   1. as [[wrapped brane|wrapped]] [[D4-branes]] with $N_c$ [[open strings]] connecting them to the [[D8-brane]]

      ([Witten 98b](Ads/QCD#Witten98b), [Gross-Ooguri 98, Sec. 5](Ads/QCD#GrossOoguri98), [BISY 98](Ads/QCD#BISY98), [CGS98](Ads/QCD#CGS98))

   1. as [[skyrmions]]

      ([Sakai-Sugimoto 04, section 5.2](Ads/QCD#SakaiSugimoto04), [Sakai-Sugimoto 05, section 3.3](Ads/QCD#SakaiSugimoto05), see [Bartolini 17](Ads/QCD#Bartolini17)).

For review see [Sugimoto 16](Ads/QCD#Sugimoto16), also [Rebhan 14, around (18)](Ads/QCD#Rebhan14).

> graphics grabbed from [Sugimoto 16](Ads/QCD#Sugimoto16)

Equivalently, these baryon states are the [[Yang-Mills instantons]] on the [[D8-brane]] giving the [[D4-D8 brane bound state]] ([Sakai-Sugimoto 04, 5.7](Ads/QCD#SakaiSugimoto04)) as a special case of the general situation for [[Dp-D(p+4)-brane bound states]] (e.g. [Tong 05, 1.4](Dp-D%28p%2B4%29-brane+bound+state#Tong05)).

Strictly speaking, since the number of [[color charge|colors]] in [[quantum chromodynamics]] is not [[large N limit|large]] ($N_c = 3$), an accurate formulation of such [[holographic QCD]] requires understanding [[small N corrections]]:

For more on this see at

* _[[AdS/QCD correspondence]]_

and at

* _[[skyrmion]]_ .

### In $\mathcal{N}=2$ super Yang-Mills theory via Seiberg-Witten theory
  {#InN2SYM}

While confinement in plain [[Yang-Mills theory]] is still waiting for mathematical formalization and [[proof]] (see [Jaffe-Witten](#JaffeWitten)), there is a variant of [[Yang-Mills theory]] with more [[symmetry]], namely [[supersymmetry]], where the phenomenon has been giving a decent argument, namely in [[N=2 D=4 super Yang-Mills theory]] ([Seiberg-Witten 94](#SeibergWitten94)).

Also a strategy for a proof for [[N=1 D=4 super Yang-Mills theory]] has been proposed, see [below](#ForN1SYMViaMTheoryOnG2Manifolds).

### In $\mathcal{N} = 1$ super Yang-Mills theory via M-theory on $G_2$-manifolds
 {#ForN1SYMViaMTheoryOnG2Manifolds}

An idea for a strategy towards a proof of confinement in [[N=1 D=4 super Yang-Mills theory]] via two different but conjecturally equivalent realizations as [[M-theory on G2-manifolds]] has been given in [Atiyah-Witten 01, section 6](#AtiyahWitten01), review is in [Acharya-Gukov 04, section 5.3](#AcharyaGukov04).

The idea here is to consider a [[KK-compactification]] of [[M-theory]] on [[fibers]] which are [[G2-manifolds]] that locally around a special point
are of the form

$$
  X_{1,\Gamma}
  \;\coloneqq\;
  \big( S^3 / \Gamma \big) \times Cone\big(S^3\big)
  \phantom{AA}
  \text{or}
  \phantom{AA}
  X_{2,\Gamma}
  \;\coloneqq\;
  S^3 \times Cone\big(S^3/\Gamma\big)
$$

where

* $\Gamma$ is a [[finite subgroup of SU(2)]] that [[action|acts]] canonically by left-multiplication on $S^3 \simeq $ [[SU(2)]];

* $Cone(\cdots)$ denotes the [[metric cone]] construction.

This means that $X_{1,\Gamma}$ is a [[smooth manifold]], but $X_{2,\Gamma}$, as soon as $\Gamma$ is not the [[trivial group]], $\Gamma \neq 1$, is an [[orbifold]] with an [[ADE singularity]].

Now the lore of [[M-theory on G2-manifolds]] predicts that [[KK-compactification]]

1.  on $X_{1,\Gamma}$ yields a 4d theory without massless fields (since there are no massless modes on the [[covering space]] $S^3$ of $X_{1,\Gamma}$)

1. on the [[ADE-singularity]] $X_{2,\Gamma}$ yields [[non-abelian group|non-abelian]] [[Yang-Mills theory]] in 4d coupled to [[chiral fermions]].

So in the first case a [[mass gap]] is manifest, while non-abelian gauge theory is not visible, while in the second case it is the other way around.

But if there were an argument that [[M-theory on G2-manifolds]] is in fact equivalent for compactification both on $X_{1,\Gamma}$ and on $X_{2,\Gamma}$. To the extent that this is true, it looks like an argument that could demonstrate confinement in non-abelian 4d gauge theory.

This approach is suggested in [Atiyah-Witten 01, pages 84-85](#AtiyahWitten01). An argument that this equivalence is indeed the case is then provided in sections 6.1-6.4, based on an argument in [Atiyah-Maldacena-Vafa 00](#AtiyahMaldacenaVafa00)

### Via Calorons
 {#ViaCalorons}

It has been argued that, after [[Wick rotation]], confinement may be derived from the behaviour of [[instantons]] ([Schaefer-Shuryak 96, section III D](#SchaeferShuryak96)), or rather their [[thermal field theory|positive temperature]]-incarnations as _[[calorons]]_, [Greensite 11, section 8.5](#Greensite11):

> it is natural to wonder if [[confinement]] could be derived from some [[semiclassical approximation|semiclassical]] treatment of [[Yang–Mills theory]] based on the [[instanton]] solutions of [[nonabelian group|non-abelian]] [[gauge theories]]. The [[BPST instanton|standard]] [[instantons]], introduced by Belavin et al. ([40](BPST-instanton#BelavinPolyakovSchwartzTyupkin75)), do not seem to work; their [[field strengths]] fall off too rapidly to produce the desired  magnetic  disorder  in  the  vacuum.

> In  recent years, however, it has been realized that instanton solutions at [[thermal field theory|finite temperature]], known as _[[calorons]]_, might do the job. These caloron solutions were introduced independently by Kraan and van Baal ([41](caloron#KraanVanBaal98), [42](caloron#KraanVanBaal98b)) and Lee and Lu ([43](caloron#LeeLu98)) (KvBLL), and they have the remarkable property of containing [[monopole]] constituents which may, depending on the type of [[caloron]], be widely separated.

> $[...]$

> The [[caloron]] idea is probably the most promising current version of [[monopole]] [[confinement]] in pure non-abelian gauge theories, but it is  basically  (in certain [[gauge fixing|gauges]]) a superposition of [[monopoles]] with spherically  symmetric abelian fields, and this leads to the same questions raised in connection with monopole Coulomb gases.

See also at _[[glueball]]_.

## Related concepts

* [[asymptotic freedom]]

* [[quantization of Yang-Mills theory]]

* [[proton spin crisis]], [[proton radius problem]]

* [[QCD cosmology]]

* [[gapped Hamiltonian]]

[[!include effective field theories of nuclear physics -- contents]]

## References

### In Yang-Mills theory
 {#ReferencesInYangMillsTheory}

#### General

* {#Wilson74} [[Kenneth Wilson]],  *Confinement of quarks*, Phys. Rev. D **10** 2445 (1974) [[doi:10.1103/PhysRevD.10.2445](https://doi.org/10.1103/PhysRevD.10.2445)]

Early demonstration of confinement by [[Monte Carlo simulation]] of [[lattice field theory]] (and demonstration that $SU(2)$-[[gauge theory]] in 4d does *not* confine, as it should be):

* {#Creutz79} [[Michael Creutz]], *Confinement and the Critical Dimensionality of Space-Time*, Phys. Rev. Lett. **43**  (1979) 553 [[doi:10.1103/PhysRevLett.43.553](https://doi.org/10.1103/PhysRevLett.43.553)]

Textbook accounts:

* [[Alexander Polyakov]], Chapter V of: *Gauge Fields and Strings*, Routledge, Taylor and Francis (1987, 2021) [[doi:10.1201/9780203755082](https://doi.org/10.1201/9780203755082), [oapen:20.500.12657/50871](https://library.oapen.org/handle/20.500.12657/50871)]

* David Griffiths, chapter 8 of _Introduction to Elementary Particles_ 2nd ed., 2008. ([pdf] (https://www.physics.utah.edu/~belz/phys5110/Griffiths.pdf))

* {#Greensite11} Jeff Greensite, _An Introduction to the Confinement Problem_, Lecture Notes in Physics, Volume 821, 2011 ([doi:10.1007/978-3-642-14382-3](https://link.springer.com/book/10.1007%2F978-3-642-14382-3))

* Robert Iengo, section 9.1 of _Quantum Field Theory_ ([pdf](http://iopscience.iop.org/chapter/978-1-6432-7053-1/bk978-1-6432-7053-1ch9.pdf))

* D. Bugg (ed.), _Hadron Spectroscopy and the Confinement Problem_, Proceedings of a NATO Advanced Study Institute, Plenum Press 1996  ([doi:10.1007/978-1-4613-0375-6](https://www.springer.com/cn/book/9780306453038))

* [[Yakov Shnir]], Section 9 of: *Magnetic Monopoles*, Springer 2005 ([ISBN:978-3-540-29082-7](https://www.springer.com/gp/book/9783540252771))

  > (in view of [[Yang-Mills monopoles]])

Introductions and surveys:

* Yuri L. Dokshitzer, around section 1.2 of _QCD Phenomenology_ ([arXiv:hep-ph/0306287](https://arxiv.org/abs/hep-ph/0306287))

* {#Espiru94} D. Espriu, section 7 of _Perturbative QCD_ ([arXiv:hep-ph/9410287](http://arxiv.org/abs/hep-ph/9410287))

* Erhard Seiler, _The Confinement Problem_ ([pdf](http://physik.uni-graz.at/~dk-user/talks/Seiler20081113-14.pdf))

* Roman Pasechnik, Michal Šumbera, *Different faces of confinement*, Universe 7 (2021) 9, 330 ([arXiv:2109.07600](https://arxiv.org/abs/2109.07600), [doi:10.3390/universe7090330](https://doi.org/10.3390/universe7090330))

See also:

* Wikipedia, _[Color confinement](http://en.wikipedia.org/wiki/Color_confinement)_

* _[What is Strongly-Coupled Quantum Chromodynamics?](https://ensnaredinvacuum.wordpress.com/2015/02/11/what-is-strongly-coupled-quantum-chromodynamics/)_

* *[[Prospects in Theoretical Physics 2023 -- Understanding Confinement]]*, IAS (2023)

A formulation of confinement as an open problem of mathematical physics, together with many references, is in

* {#JaffeWitten} [[Arthur Jaffe]], [[Edward Witten]], _Quantum Yang-Mills theory_ ([pdf](http://www.claymath.org/millennium/Yang-Mills_Theory/yangmills.pdf))

Other technical reviews:

* G. M. Prosperi, _Confinement and bound states in QCD_ ([arXiv:hep-ph/0202186](http://arxiv.org/abs/hep-ph/0202186))

* {#DHMMNVW19} Christian Drischler, Wick Haxton, Kenneth McElvain, Emanuele Mereghetti, Amy Nicholson, Pavlos Vranas, André Walker-Loud, _Towards grounding nuclear physics in QCD_ ([arxiv:1910.07961](https://arxiv.org/abs/1910.07961))

See also:

* [[Yuri A. Simonov]], *The fundamental scale of QCD* ([arXiv:2103.08223](https://arxiv.org/abs/2103.08223))

* Duifje Maria van Egmond, Urko Reinosa, Julien Serreau, Matthieu Tissier,
*A novel background field approach to the confinement-deconfinement transition* ([arXiv:2104.08974](https://arxiv.org/abs/2104.08974))

* Erich Poppitz, *Notes on Confinement on $\mathbb{R}^3 \times S^1$: From Yang-Mills, super-Yang-Mills, and QCD(adj) to QCD(F)* ([arXiv:2111.10423](https://arxiv.org/abs/2111.10423))

Recognition of [[instantons]] and confinement in [[lattice gauge theory]] via [[topological data analysis]] ([[persistent homology]]):

* Daniel Spitz, Julian M. Urban, Jan M. Pawlowski, *Confinement in non-Abelian lattice gauge theory via persistent homology* [[arXiv:2208.03955](https://arxiv.org/abs/2208.03955)]

In relation to [[conformal symmetry]] and [[AdS-CFT duality]] (see also at *[[AdS-QCD]]*):

* {#KPV22} M. Kirchbach, T. Popov, J.-A. Vallejo, *The Conformal-Symmetry -- Color-Neutrality Connection in Strong Interaction* [[arXiv:2208.06484](https://arxiv.org/abs/2208.06484)]

#### Via monopole condensation
 {#ViaMonopoleCondensation}

An original suggestion that confinement in [[Yang-Mills theory]] may be understood via [[monopole]] [[condensate|condensation]] as a dual [[Meissner effect]] is due to

* {#Hooft75} [[Gerard 't Hooft]], in Proceed.of the Europ.Phys.Soc. 1975, ed.by A.Zichichi (Editrice Compositori, Bologna, 1976), p.1225.

* {#Mandelstam76} S. Mandelstam, Phys.Rep. 23C (1976) 145;

(That this is indeed the case has not yet been demonstarted for plain Yang-Mills theory, but it was later shown for [[N=2 D=4 super Yang-Mills theory]] in ([Seiberg-Witten 94](#SeibergWitten94)). What this does or does not imply for the case of [[QCD]] is discussed in ([Yung 00](#Yung00)) ).

The relation to [[QCD instantons]]/[[monopole|monopoles]] in the [[QCD vacuum]] is discussed in

* {#SchaeferShuryak96} T. Schaefer, [[Edward Shuryak]], section III D of _Instantons in QCD_, Rev. Mod. Phys.70:323-426,1998 ([arXiv:hep-ph/9610451](http://arxiv.org/abs/hep-ph/9610451))

and analogously (at [[thermal field theory|positive temperature]]) relation to _[[calorons]]_:

* [Greensite 11, section 8.5](#Greensite11)

* P. Gerhold, E.-M. Ilgenfritz, M. Müller-Preussker, _An $SU(2)$ KvBLL caloron gas model and confinement_, Nucl.Phys.B760:1-37, 2007 ([arXiv:hep-ph/0607315](https://arxiv.org/abs/hep-ph/0607315))

* {#LarsenShuryak14}  Rasmus Larsen, [[Edward Shuryak]], _Classical interactions of the instanton-dyons with antidyons_, Nucl. Phys. A **950**, 110 (2016) ([arXiv:1408.6563](https://arxiv.org/abs/1408.6563))

* {#LarsenShuryak15} Rasmus Larsen, [[Edward Shuryak]], _Interacting Ensemble of the Instanton-dyons and Deconfinement Phase Transition in the SU(2) Gauge Theory_, Phys. Rev. D 92, 094022, 2015 ([arXiv:1504.03341](https://arxiv.org/abs/1504.03341))

Further developments:

* _Dimensional Transmutation by Monopole Condensation in QCD_ ([arXiv:1206.6936](http://arxiv.org/abs/1206.6936))

Discussion of confinement via an interacting field vacuum

* {#Rafelski90} [[Johann Rafelski]], _Vacuum structure -- An Essay_, in pages 1-29 of H. Fried, Berndt Müller (eds.) _Vacuum Structure in Intense Fields_, Plenum Press 1990 ([GBooks](https://books.google.de/books?id=5uXcBwAAQBAJ&pg=PA14&lpg=PA14&dq=confinement+%22interacting+vacuum%22&source=bl&ots=xPGYJ-JOc-&sig=AoYbqWQeNRMg6hMSRZjJ3nzq8B0&hl=en&sa=X&ved=0ahUKEwjIgK68-tnYAhVCESwKHThMDykQ6AEIKTAA#v=onepage&q=confinement%20%22interacting%20vacuum%22&f=false))

See also:

* {#Simonov18} [[Yuri A. Simonov]], _Field Correlator Method for the confinement in QCD_, Phys. Rev. D 99, 056012 (2019) ([arXiv:1804.08946](https://arxiv.org/abs/1804.08946), [doi:10.1103/PhysRevD.99.056012](https://doi.org/10.1103/PhysRevD.99.056012))

### In super-Yang-Mills theory
 {#ReferencesInSuperYangMillsTheory}

Confinement in [[N=2 D=4 super Yang-Mills theory]] by a version of the monopole condensation of ([t Hooft 75](#Hooft75), [Mandelstam 76](#Mandelstam76)) was demonstrated in

* {#SeibergWitten94} [[Nathan Seiberg]], [[Edward Witten]], _Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory_,  	Nucl.Phys.B426:19-52,1994; Erratum-ibid.B430:485-486,1994 ([arXiv:hep-th/9407087](http://arxiv.org/abs/hep-th/9407087))

* {#SeibergWitten94b} [[Nathan Seiberg]], [[Edward Witten]], _Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD_,  	Nucl.Phys.B431:484-550,1994 ([arXiv:hep-th/9408099](http://arxiv.org/abs/hep-th/9408099))

Reviews with discussion of the impact on confinement in plain YM:

* {#Yung00} Alexei Yung, _What Do We Learn about Confinement from the Seiberg-Witten Theory_ [[arXiv:hep-th/0005088](http://arxiv.org/abs/hep-th/0005088)]

* [[Michael Dine]], *On the Possibility of Demonstrating Confinement in Non-Supersymmetric Theories by Deforming Confining Supersymmetric Theories* [[arXiv:2211.17134](https://arxiv.org/abs/2211.17134)]

See also:

* Michele Della Morte, Benjamin Jäger, Francesco Sannino, Justus Tobias Tsang, Felix P. G. Ziegler, *The spectrum of QCD with one flavour: A Window for Supersymmetric Dynamics* [[arXiv:2302.10514](https://arxiv.org/abs/2302.10514)]

[[!include Polyakov gauge-string duality -- references]]

### Confinement via AdS/QCD correspondence

[Polyakov's ideas](#PolyakovGaugeStringDualityReferences) were eventually picked up by discussion of confinement in the [[AdS-QCD correspondence]]; see there for many references, including:

* {#Witten98} [[Edward Witten]], _Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories_, Adv. Theor. Math. Phys.2:505-532, 1998 ([arXiv:hep-th/9803131](https://arxiv.org/abs/hep-th/9803131))

* [[David Berman]], Maulik K. Parikh, _Confinement and the AdS/CFT Correspondence_, Phys.Lett. B483 (2000) 271-276 ([arXiv:hep-th/0002031](https://arxiv.org/abs/hep-th/0002031))

* Henrique Boschi Filho, _AdS/QCD and confinement_, Seminar at the _Workshop on Strongly Coupled QCD: The confinement problem_, November 2011 ([pdf](http://www.if.ufrj.br/~boschi/pesquisa/seminarios/AdS_QCD_Confinement_UERJ_2011.pdf))

* [[Edward Witten]], *Some Milestones in the Study of Confinement*, talk at *[[Prospects in Theoretical Physics 2023 -- Understanding Confinement]]*, IAS (2023) [[web](https://www.ias.edu/video/some-milestones-study-confinement), [YT](https://youtu.be/TvIz-6YOdKs)]

> [40:13](https://youtu.be/TvIz-6YOdKs?t=2413): "Personally, I think [[AdS/QCD|this setup]] really implies that pure [[Yang-Mills theory|$SU(N)$ gauge theory]] is [[gauge-string duality|dual]] to a [[string theory]]. The 'only' problem is that to get the pure gauge theory we need to make a relevant deformation and then take the limit that the deformation parameter is large..."

### In M-theory on $G_2$-manifolds

An idea for how to demonstrate confinement in models of [[M-theory on G2-manifolds]] is given in

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]], section 6 of  _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

based on

* {#AtiyahMaldacenaVafa00} [[Michael Atiyah]], [[Juan Maldacena]], [[Cumrun Vafa]], _An M-theory Flop as a Large N Duality_, J.Math.Phys.42:3209-3220, 2001 ([arXiv:hep-th/0011256](https://arxiv.org/abs/hep-th/0011256))

See also

* [[Bobby Acharya]], _Confining Strings from $G_2$-holonomy spacetimes_ ([arXiv:hep-th/0101206](https://arxiv.org/abs/hep-th/0101206))

For review see

* {#AcharyaGukov04} [[Bobby Acharya]], [[Sergei Gukov]], section 5.3 of _M theory and Singularities of Exceptional Holonomy Manifolds_, Phys.Rept.392:121-189,2004 ([arXiv:hep-th/0409191](https://arxiv.org/abs/hep-th/0409191))

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