orbit category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
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#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
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#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _orbit category_ of a [[group]] $G$ is the category of "all kinds" of  [[orbits]] of $G$, namely of all suitable [[coset spaces]] regarded as [[G-spaces]].

## Definition

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

Given a [[topological group]] $G$ the __orbit category__ $\operatorname{Orb}_G$ (denoted also $\mathcal{O}_G$) is the [[category]] whose

* [[objects]] are the [[homogeneous spaces]]  ([[coset spaces]], $G$-[[orbit types]]) $G/H$, where $H$ is a [[closed subgroup]] of $G$,

* [[morphisms]]$;$are the $G$-[[equivariant function|equivariant]] [[continuous functions]].

=--

(e.g. [tom Dieck 1987, I (10.1)](#tomDieck87))

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

Warning: This should not be confused with the situation where a group $G$ acts on a groupoid $\Gamma$ so that one obtains the [[orbit groupoid]].

=--

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

For suitable [[continuous function|continuous]] [[actions]] of $G$ on a [[topological space]] $X$, every [[orbit]] of the action is [[isomorphism|isomorphic]] to one of the [[coset spaces]] $G/H$ (the [[stabilizer group]] of any point in the orbit is conjugate to $H$). This is the sense in which def. \ref{TheOrbitCategory} gives "the category of all $G$-orbits".

=--

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

Def. \ref{TheOrbitCategory} yields a [[small category|small]] [[enriched category|topologically enriched]] category (though of course if $G$ is a [[discrete group]], the enrichment of $\operatorname{Orb}_G$ is likewise discrete).

Of course, like any category, it has a [[skeleton]], but as usually defined it is not itself skeletal, since there can exist distinct subgroups $H$ and $K$ such that $G/H \cong G/K$.

=--

\begin{remark}\label{GActionsAndGOrbits}
**([[G-set|$G$-sets]] are the [[free coproduct completion]] of [[orbit category|$G$-orbits]])**
\linebreak
  Let $G \,\in\, Grp(Set)$ be a [[discrete group]].
Since every [[G-set]] $X$ decomposes as a [[disjoint union]] of [[transitive actions]], namely of [[orbits]] of [[elements]] of $X$, the defining inclusion of the orbit category into [[G-set|$G Set$]] exhibits the latter as its [[free coproduct completion]] (see also [this Prop.](free+coproduct+completion#CategoriesOfCoproductsOfConnectedObjects)).
\end{remark}

\begin{remark}
**(families of subgroups)**
\linebreak
More generally, given a family $\mathcal{F}$ of subgroups of $G$ which is closed under conjugation and taking subgroups one looks at the full subcategory $\mathrm{Orb}_F\,G \subset \operatorname{Orb}_G$ whose objects are those $G/H$ for which $H\in F$.
For such families, many of the considerations of results such as [[Elmendorf's theorem]] will still hold.
\end{remark}

## Equivariant Postnikov tower

Given a [[topological G-space]], there is (following [tom Dieck 1987, Sec. I.10](#tomDieck87)) a sequence of variants (enhancements) of the $G$-orbit category which mixes transitions between the [[fixed loci]] with [[n-truncated object in an (infinity,1)-category|n-truncations]] of their [[homotopy types]].

### Equivariant set of connected components
 {#EquivariantSetOfConnectedComponents}

(...)

[tom Dieck 1987, Sec. I.10](#tomDieck87)

(...)

### Equivariant fundamental groupoid
 {#FundamentalCategoryOfAGSpace}

\begin{definition}\label{EquivariantFundamentalGroupoid}
For $G \,\in\, Grp(Set)$ a [[discrete group]] and $G \curvearrowright X \,\in\, G Act(TopSp)$ a [[topological G-space|topological $G$-space]], its *fundamental category* ([tom Dieck 1987 (10.7)](#tomDieck87), [Lück 1989, 8.15](#Lueck89)) or *equivariant fundamental groupoid* ([Pronk & Scull 2021, Def. 3.1](#PronkScull21))

is the [[small category]] whose

* [[objects]] are [[pairs]] consisting of a [[subgroup]] $H \subset G$ and a  $G$-[[equivariant function]] $\; x_X \,\colon\, G/H \to X$, hence, equivalently, a point in the [[fixed locus]] $X^H$;

* [[morphisms]]$\;\;x_H \xrightarrow{ (\phi,[\gamma]) } x'_{H'}$  are [[pairs]] consisting of a $G$-[[equivariant function]] $G/H \xrightarrow{ \phi } G/H'$ and a relative $G$-homotopy class $[\gamma]$ of a continuous path $\gamma \,\colon\,  G/H \times [0,1] \xrightarrow{\;} X$ from $\gamma(0) = x_H$ to $\gamma_1 = \phi^\ast x'_X'$.

\end{definition}

\begin{example}
  For $G \curvearrowright X \,=\, G \curvearrowright \ast$ the point, its fundamental category (equivariant fundamental groupoid) according to Def. \ref{EquivariantFundamentalGroupoid} is the plain [[orbit category]] $G Orbts$ (Def. \ref{TheOrbitCategory}).
\end{example}

\begin{proposition}
\label{EquivariantFundamentalGroupoidUnderGrothendieckConstruction}
**([[equivariant fundamental groupoid]] under [[Grothendieck construction]])**
\linebreak
If we write
$$
  \tau_1 \esh \big( X^{(-)}\big)
  \;\in\;
  Sh\big( G Orbt, Grpd_1 \big)
$$
for the [[fundamental groupoid]] of $X$ in its incarnation as a [[presheaf]] of [[fixed loci]] over the [[orbit category]], hence for the [[(2,1)-functor]]

\[
  \label{FundamentalGroupoidOfAPresheafOfFixedLoci}
  \array{
    G Orbt^{op}
    &\xrightarrow{ \tau_1 \esh \big( X^{(-)}\big) }&
    Grpd_1
    \\
    G/H
    &\mapsto&
    \tau_1 \esh (X^H)
    \mathrlap{\,,}
  }
\]
which sends $G/H$ to the ordinary [[fundamental groupoid]] of the [[fixed locus]] $X^H$, then the equivariant fundamental groupoid of $G \curvearrowright X$ (Def. \ref{EquivariantFundamentalGroupoid}) is equivalently the [[Grothendieck construction]]

\[
  \label{EquivariantFundamentalGroupoidAsGrothendieckConstruction}
  \underset
    {G/H \in G Orbt}
    {\int}
  \,
  \tau_1
  \,
  \esh
  (X^{H})
  \;\;\;
  \in
  \;
  Grpd_1
\]

on (eq:FundamentalGroupoidOfAPresheafOfFixedLoci).
\end{proposition}

This follows by direct unwinding of the definitions; see also [Pronk & Scull 2021, below Def. 3.1](#PronkScull21).

### Equivariant universal cover
 {#EquivariantUniversalCover}

\begin{definition}
\label{EquivariantUniversalCovering}
**(equivariant universal covering)**
\linebreak
  For $G \in Grp(Set)$ a [[discrete group]] and $G \curvearrowright X \,\in\, G Act(TopSp)$ a well-behaved [[topological G-space|topological $G$-space]], its *equivariant [[universal cover]]* ([tom Dieck 1987 (10.13)](#tomDieck87)) "is" the [[functor]] from the fundamental equivariant groupoid (eq:EquivariantFundamentalGroupoidAsGrothendieckConstruction) of $G \curvearrowright X$ to [[TopSp]]

$$
  \array{
    \underset
      {G/H \in G Orbt}
      {\int}
    \,
    \tau_1
    \,
    \esh
    (X^{H})
    &\xrightarrow{\;}&
    TopSp
    &\xrightarrow{\esh}&
    Grpd_{\infty}
    \\
    x_H
    &\mapsto&
    \widehat{X^H_x}
  }
$$

which sends $x_H \,\in\, X^H$ to the [[universal covering]] $\widehat{X_x}$ of the connected component of $x_H \,\in\, X$ given by relative-homotopy-equivalences classes of [[paths]] based at $x_H$.
\end{definition}

Write

\[
  \label{GEquivariantHomotopyTheory}
  G Grpd_\infty \;\coloneqq\; Sh\big( G Orbt, \, Grpd_\infty \big)
\]

for the [[(infinity,1)-category of (infinity,1)-presheaves|$\infty$-category of $\infty$-presheaves]] on the $G$-[[orbit category]], hence for the proper $G$-[[equivariant homotopy theory]].

For $G \curvearrowright X \,\in\, G Act(TopSp)$ a [[topological G-space|topological $G$-space]], its equivariant homotopy type is

$$
  \esh \big(X^{(-)}\big)
  \;\;
  \in
  \;
  G Grpd_\infty
  \,.
$$

\begin{proposition}
For $G \curvearrowright X \,\in\, G Act(TopSp)$ a [[topological G-space|topological $G$-space]], its equivariant universal covering according to Def. \ref{EquivariantUniversalCovering} is, under Prop. \ref{EquivariantFundamentalGroupoidUnderGrothendieckConstruction} and the [[(infinity,1)-Grothendieck construction]] equivalent to the [[n-truncated object in an (infinity,1)-category|1-truncation unit]]

$$
  \array{
    \esh \big( X^{(-)}\big)
    \\
    \big\downarrow {}^{\mathrlap{ \eta^{\tau_1} }}
    \\
    \tau_1 \, \esh \big( X^{(-)}\big)
  }
  \;\;\;
  \in
  \;
  \big( G Grpd_\infty\big)_{/ \tau_1 \, \esh \big( X^{(-)}\big)}
  \,,
$$

in that

$$
  \array{
    \underset{G/H \in G Orbt}{\int}
    \esh \big( X^{H}\big)
    &\xrightarrow{\;}&
    \widehat{Grpd}_\infty
    \\
    \big\downarrow
    &\swArrow&
    \big\downarrow
    \\
    \underset{G/H \in G Orbt}{\int}
    \tau_1 \, \esh \big( X^{H}\big)
    &
    \xrightarrow{ \widehat{ X^{(-)}_{(-)} } }
    &
    Grpd_\infty
    \,.
  }
$$

\end{proposition}

> bottom right corner not quite right, will fix...

\linebreak

## Examples
 {#Examples}

+-- {: .num_example #OrbitCategoryOfCyclicGroupOfOrder2}
###### Example
**([[orbit category]] of [[cyclic group of order 2|Z/2Z]])**

For equivariance group the [[cyclic group of order 2]]:

$$
  G \;\coloneqq\;  \mathbb{Z}_2 \;\coloneqq\; \mathbb{Z}/2\mathbb{Z}
  \,.
$$

the [[orbit category]] looks like this:

\[
  \label{OrbitCategoryOfZMod2}
  \mathbb{Z}_2
  Orbits
  \;=\;
  \left\{
  \array{
    \mathbb{Z}_2/1
    &
      \overset{
        \phantom{AAAAA}
      }{
        \longrightarrow
      }
    &
    \mathbb{Z}_2/\mathbb{Z}_2
    \\
    Aut = \mathbb{Z}_2
    &&
    Aut = 1
  }
  \right\}
\]

i.e.:

$$
  \begin{aligned}
    \mathbb{Z}_2 Orbits
    \big(
      \mathbb{Z}_2/\mathbb{Z}_2
      \,,\,
      \mathbb{Z}_2/\mathbb{Z}_2
    \big)
    \;\simeq\;
    1
    \\
    \mathbb{Z}_2 Orbits
    \big(
      \mathbb{Z}_2/1
      \,,\,
      \mathbb{Z}_2/\mathbb{Z}_2
    \big)
    \;\simeq\;
    \ast
    \\
    \mathbb{Z}_2 Orbits
    \big(
      \mathbb{Z}_2/\mathbb{Z}_2
      \,,\,
      \mathbb{Z}_2/1
    \big)
    \;\simeq\;
    \varnothing
    \\
    \mathbb{Z}_2 Orbits
    \big(
      \mathbb{Z}_2/1
      \,,\,
      \mathbb{Z}_2/1
    \big)
    \;\simeq\;
    \mathbb{Z}_2
  \end{aligned}
$$

=--

\begin{imagefromfile}
  "file_name": "OrbitCategoriesOfSmallCyclicGroups.jpg",
  "width": 600,
  "margin": {
    "top": 0,
    "right": 10,
    "bottom": 10,
    "left": 0,
    "unit": "px"
  }
\end{imagefromfile}

## Properties

### General properties

* Orbit categories as well as fundamental categories of $G$-spaces are [[EI-categories]].

### Relation to $G$-spaces and Elmendorf's theorem

[[Elmendorf's theorem]] (see there for details) states that the [[(∞,1)-category of (∞,1)-presheaves]] on the orbit category $Orb_G$ are [[equivalence of (∞,1)-categories|equivalent]] to the [[localization of an (∞,1)-category|localization]] of [[topological spaces]] with $G$-[[action]] at the _[[weak homotopy equivalences]] on [[fixed point spaces]]_.

$$
  L_{we} G Top \simeq PSh_\infty(Orb_G)
  \,.
$$

### Relation to global equivariant homotopy theory
 {#RelationToGlobalHomotopyTheory}

The $G$-orbit category is the [[slice (∞,1)-category]] of the [[global orbit category]] $Orb$ (the version with [[faithful functors]] as morphisms) over the [[delooping]] $\mathbf{B}G$:

$$
  Orb_G \simeq Orb_{/\mathbf{B}G}
  \,.
$$

This means that in the general context of [[global equivariant homotopy theory]], the orbit category appears as follows.

For more on this see also at *[[cohesion of global- over G-equivariant homotopy theory]]* the section *[the adjunction of sites](cohesion+of+global-+over+G-equivariant+homotopy+theory#TheAdjunctionOfSites)*.

[[!include equivariant homotopy theory -- table]]

### Relation to Mackey functors

Orbit categories are used often in the treatment of [[Mackey functor]]s from the theory of [[locally compact group]]s and in the definition of [[Bredon cohomology]].

### Relation to Bredon equivariant cohomology

It appears in [[equivariant stable homotopy theory]], where the $H$-fixed [[homotopy groups]] of a space form a [[presheaf]] on the [[homotopy category]] of the orbit category (e.g. [page 8, 9 here](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf#page=8)).

### Base change of equivariance groups
 {#BaseChangeOfEquivarianceGroups}

\begin{example}\label{LeftBaseChangeAlongCoveringOfEquivarianceGroup}
**(left base change along covering of equivariance group)**
\linebreak
Let $\widehat{G} \overset{p}{\twoheadrightarrow} G $ be a surjective homomorphism of discrete groups. For $H \subset G$ a [[subgroup]], write $\widehat{H} \subset \widehat{G}$ for its [[pullback]]

\begin{tikzcd}
  \widehat{H}
  \ar[r, hook]
  \ar[d,->>]
  \ar[dr, phantom, "\mbox{\tiny(pb)}"]
  &
  \widehat{G}
  \ar[d,->>]
  \\
  H
  \ar[r, hook]
  &
  G
  \mathrlap{\,.}
\end{tikzcd}

Then there is a [[reflective subcategory]]-inclusion of orbit categories

\begin{tikzcd}
  \widehat{G}\mathrm{Orbt}
  \ar[from=r, hook', shift left=7pt, "R"]
  \ar[r, shift left=7pt, "L"]
  \ar[r, phantom, "\scalebox{.7}{$\bot$}"]
  &
  G\mathrm{Orbt}
\end{tikzcd}

where

* $R(G/H) \,=\, \widehat{G}/\widehat{H}$

* $L(\widehat{G}/K) \,=\, G/p(K)$.

A transparent way to see this is to identify (as [above](#RelationToGlobalHomotopyTheory))
$G$-orbits with the [[n-truncated object of an (infinity,1)-category|0-truncated objects]] in the $B G$-slice of $Singlrt \,\coloneqq\, Grpd_1^{cn} $:

$$
  G Orbt \;\simeq\; \left( (Grpd_1^{cn})_{/ B G} \right)_0
  \,.
$$

Under this identification, the adjunction is the canonical left [[base change]] on [[slice (infinity,1)-category|slices]] composed with the [[n-truncated object of an (infinity,1)-category|0-truncation]] reflection ([here](n-truncated+object+of+an+infinity1-category#nTruncationReflection)):

\begin{tikzcd}
  (\mathrm{Grpd}_1^{\mathrm{cn}})_{/ B \widehat{G}}
  \ar[from=r, shift left=7pt, "(B p)^\ast"]
  \ar[r, shift left=7pt, "(B p)_!"]
  \ar[r, phantom, "\scalebox{.7}{$\bot$}"]
  &
  (\mathrm{Grpd}_1^{\mathrm{cn}})_{/ B G}
  \ar[from=r, hook', shift left=7pt]
  \ar[r, shift left=7pt, "\tau_0"]
  \ar[r, phantom, "\scalebox{.7}{$\bot$}"]
  &
  \left( (\mathrm{Grpd}_1^{\mathrm{cn}})_{/ B G} \right)_0
\end{tikzcd}

That the right adjoint $R$ is fully faithful is most readily seen by direct inspection: A map between $\widehat{G}$-sets on both of which $\widehat{G}$ acts only through $G$ is evidently $\widehat{G}$-equivariant if and only if it is $G$-equivariant.
\end{example}

\linebreak

## Related concepts

* [[orbit]], [[orbit type]]

* [[equivariant homotopy theory]]

* [[global orbit category]], [[global equivariant stable homotopy theory]]

* [[Bredon cohomology]], [[equivariant cohomology]]

* [[topological G-space]]

* [[vector G-space]]

* [[equivariant dgc-algebra]]

* [[equivariant rational homotopy theory]]

## References

The notion of the orbit category (for use in [[equivariant cohomology]]/[[Bredon cohomology]]) is due to

* {#Bredon67b} [[Glen Bredon]], Section I.3. of _[[Equivariant cohomology theories]]_, Springer Lecture Notes in Mathematics Vol. 34. 1967 ([doi:10.1007/BFb0082690](https://link.springer.com/book/10.1007/BFb0082690))

announced in

* {#Bredon67a} [[Glen Bredon]], _Equivariant cohomology theories_, Bull. Amer. Math. Soc. Volume 73, Number 2 (1967), 266-268. ([educlid:1183528794](https://projecteuclid.org/euclid.bams/1183528794))

and there considered specifically for [[finite groups]].

Discussion for any [[topological group]] (and further generalization)  is considered (together with the [[model category|model category theoretic]] proof of [[Elmendorf's theorem]]) in:

* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]], Sec. 2.1 and Thm. 3.1 of: _Singular functors and realization functors_, Indagationes Mathematicae (Proceedings) Volume 87, Issue 2, 1984, Pages 147-153 (doi:10.1016/1385-7258(84)90016-7)

Textbook accounts:

* {#tomDieck87} [[Tammo tom Dieck]], Section I.10 of: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

* {#Lueck89} [[Wolfgang Lück]], (8.16) in _Transformation Groups and Algebraic K-Theory_, Lecture Notes in Mathematics **1408** (Springer 1989) ([doi:10.1007/BFb0083681](https://doi.org/10.1007/BFb0083681))

* {#May96} [[Peter May]], Section I.4 of: _Equivariant homotopy and cohomology theory_ CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996 ([ISBN:978-0-8218-0319-6](https://bookstore.ams.org/cbms-91), [pdf](http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf), [pdf](https://ncatlab.org/nlab/files/MayEtAlEquivariant96.pdf))

Lecture notes:

* {#tomDieck09} [[Tammo tom Dieck]], Section 1.3 of: _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf))

For more in relation to [[global equivariant homotopy theory]] see

* [[Charles Rezk]], _[[Global Homotopy Theory and Cohesion]]_

The notion of fundamental category of a $G$-space is due to

* [tom Dieck 1987](#tomDieck87)

* [Lück 1989, 8.15](#Lueck89)

with further discussion in

* {#PronkScull21} [[Dorette Pronk]], [[Laura Scull]], *The Equivariant Fundamental Groupoid as an Orbifold Invariant*, Homology, Homotopy and Applications **23** 1 (2021) ([arXiv:1908.01201](https://arxiv.org/abs/1908.01201), [doi:10.4310/HHA.2021.v23.n1.a3](https://dx.doi.org/10.4310/HHA.2021.v23.n1.a3))

[[!redirects orbit categories]]

[[!redirects equivariant set of connected components]]
[[!redirects equivariant sets of connected components]]

[[!redirects equivariant fundamental groupoid]]
[[!redirects equivariant fundamental groupoids]]

[[!redirects equivariant universal cover]]
[[!redirects equivariant universal covers]]

[[!redirects equivariant universal covering]]
[[!redirects equivariant universal coverings]]


orbit groupoid

If a [[group]] $G$ [[action|acts]] on a [[groupoid]] $\Gamma$, then there is an _orbit groupoid_ $\Gamma /\! / G$ which is a groupoid with trivial $G$-action and with a $G$-morphism $\Gamma \to \Gamma /\! / G$ universal for $G$-morphisms to groupoids with trivial $G$-action. So in principle   $\Gamma /\! / G$ is obtained from $\Gamma$ by identifying  $g.\gamma$ and $\gamma$  for all $g \in G$, $\gamma \in \Gamma$.

Note that if a group $G$ acts on a space $X$ then it has an induced action on the [[fundamental groupoid]] $\Pi_1 X$, and so there is an induced morphism

$$\alpha: \Pi_1 X /\! / G \to \Pi_1 (X/G). $$

So there is interest in when this morphism is an isomorphism.

+--{: .un_theorem}
$\alpha$ is an isomorphism if $X$ is Hausdorff, has a universal cover, and the action of $G$ on $X$ is discontinuous.
=--

This allows some calculation of the [[fundamental group]]s of [[orbit space]]s.

# References #

* R. Brown, _Topology and groupoids_, Booksurge, 2006, Chapter 11.

* J. Taylor, "Quotients of groupoids by the action of a group, Math. Proc. Camb. Phil. Soc.,  103 (1988) 239--249.


orbit method

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Geometric quantization
+-- {: .hide}
[[!include geometric quantization - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Orbit method_ (or _Kirillov's method_, or _method of coadjoint orbits_) is a method in [[geometric representation theory]] concerned with identifying [[unitary representations]] of [[Lie groups]] with the canonical $G$-[[actions]] on spaces of [[sections]] of certain [[line bundles]] over [[coadjoint orbits]] of the Lie group. In terms of [[quantum physics]] this realizes $G$-[[representations]] as actions of [[global gauge groups]] of [[quantum operators]] on [[spaces of quantum states]] under _[[geometric quantization]]_.

More in detail, the [[linear dual space]] $\mathfrak{g}^*$ of a (finite-dimensional real) [[Lie algebra]] $\mathfrak{g}$ has a canonical structure of a [[Poisson manifold]] -- its _[[Lie-Poisson structure]]_ --, namely for any $a\in \mathfrak{g}^*$,

$$
  \{ f, g\}(a)
  \;\coloneqq\;
  \big\langle
    [\mathrm{d} f_a, \mathrm{d} g_a]
    ,\,
    a
  \big\rangle
  \,.
$$

This Poisson manifold is [[foliation|foliated]] by [[symplectic leaves]] which are the [[coadjoint orbits]]. The [[line bundles]] in question are the _[[prequantum line bundles]]_ of these [[symplectic manifolds]].

Hence, in the language of [[quantum physics]], the orbit methods identifies unitary representations of Lie groups $G$ with the $G$-action on [[spaces of quantum states]] of the [[geometric quantization]] of a [[classical mechanical system]] with a global $G$-[[symmetry]].

Many important classes of unitary representations are obtained by that method.

Notably in the case of [[compact Lie groups]], co-adjoint orbits are [[flag manifolds]] and the _[[Borel-Weil theorem]]_ says that under certain further conditions the expected unitary representations are obtained.

The case of non-compact Lie groups is much less well understood, see for instance [Graham & Vogan 1999](#GrahamVogan), [Vogan 1999](#Vogan99).

## Definitions and constructions
 {#DefinitionsAndConstructions}

We list and discuss the basic notions, definitions and constructions in the context of the orbit method. A useful review is also in [Beasley 2011, section 4](#Beasley11).

### The group and its Lie algebra

Throughout, let $G$ be a [[semisimple Lie group|semisimple]] [[compact Lie group]]. For some considerations below we furthermore assume it to be [[simply connected topological space|simply connected]].

Write $\mathfrak{g}$ for its [[Lie algebra]]. Its canonical (up to scale) binary [[invariant polynomial]] we write

$$
  \langle -,-\rangle
  \;\colon\;
  \mathfrak{g} \otimes \mathfrak{g}
  \longrightarrow
  \mathbb{R}
  \,.
$$

Since this is non-degenerate, we may equivalently think of this as an [[isomorphism]]

$$
  \mathfrak{g} \simeq \mathfrak{g}^*
$$

that identifies the [[vector space]] underlying the Lie algebra with its [[dual vector space]] $\mathfrak{g}^*$.

### The coadjoint orbit and the coset space/ flag manifold
 {#TheCoadjointOrbitAndTheCosetSpaceAndFlagManifold}

We discuss the [[coadjoint orbits]] of $G$ and their relation to the [[coset space]]/[[flag manifolds]] of $G$.

Write

1. $T \hookrightarrow G$ inclusion of the [[maximal torus]] of $G$.

1. $\mathfrak{t} \hookrightarrow \mathfrak{g}$ the corresponding [[Cartan subalgebra]]

In all of the following we consider an element $\langle\lambda,-\rangle \in \mathfrak{g}^*$.

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

For $\langle\lambda,-\rangle \in \mathfrak{g}^*$ write

$$
  \mathcal{O}_\lambda
  \xhookrightarrow{\phantom{--}}
  \mathfrak{g}^*
$$

for its [[coadjoint orbit]]

$$
  \mathcal{O}_{\lambda}
  \;=\;
  \Big\{
    Ad_g^*\big(
      \langle\lambda,-\rangle
    \big)
    \in
    \mathfrak{g}^*
  \big\vert
    g \in G
  \Big\}
$$

and

$$
  G_\lambda \xhookrightarrow{\phantom{--}} G
$$

for the [[stabilizer subgroup]] of $\langle \lambda,-\rangle$ under the coadjoint action.

=--

\begin{proposition}\label{CoadjointOrbitsAsCosetSpaces}
There is an [[isomorphism]]

$$
  \array{
    G/G_\lambda
    &
    \xrightarrow{\;\; \sim \;\;}
    &
    \mathcal{O}_\lambda
    \\
    g G_\lambda
    &\mapsto&
    Ad_g^* \langle\lambda,-\rangle
   \mathrlap{\,.}
  }
$$
\end{proposition}

\begin{definition}\label{RegularElement}

An element $\langle\lambda,-\rangle \in \mathfrak{g}^*$ is called *regular* if its [[coadjoint action]] [[stabilizer subgroup]] coincides with the [[maximal torus]]: $G_\lambda \simeq T$.

\end{definition}

+-- {: .num_example }
###### Example

For generic values of $\lambda$, the dual $\langle \lambda, - \rangle$ it is regular (Def. \ref{RegularElement}).
The element in $\mathfrak{g}^*$ farthest from regularity is $\lambda = 0$ for which $G_\lambda = G$ instead.

=--

### The symplectic form

We describe a canonical [[symplectic form]] on the [[coadjoint orbit]]/[[coset space]] $\mathcal{O}_\lambda \simeq G/G_\lambda$ (from Prop. \ref{CoadjointOrbitsAsCosetSpaces}).

Write $\theta \in \Omega^1(G, \mathfrak{g})$ for the [[Maurer-Cartan form]] on $G$.

+-- {: .num_defn #The2FormOnG}
###### Definition

Write

$$
  \Theta_\lambda \coloneqq \langle \lambda, \theta \rangle \in \Omega^1(G)
$$

for the [[differential 1-form]] obtained by pairing the value of the [[Maurer-Cartan form]] at each point with the fixed element $\lambda \in \mathfrak{g}^*$.

Write

$$
  \nu_\lambda
  \;\coloneqq\;
  \mathrm{d}_{dR} \Theta_\lambda
$$

for its [[de Rham differential]].

=--

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

The 2-form $\nu_\lambda$ from def. \ref{The2FormOnG}

1. satisfies

   $$
     \nu_\lambda
     \;=\;
    \frac{1}{2}
    \big\langle
      \lambda, [\theta\wedge \theta]
    \big\rangle
    \,,
   $$

1. descends to a closed $G$-invariant 2-form on the [[coset space]], to be denoted by the same symbol

   $$
     \nu_\lambda \;\in\; \Omega^2_{cl}(G/G_\lambda)^G
     \,.
   $$

1. which is non-degenerate and hence defines a [[symplectic form]] on $G/G_\lambda$.

=--

### The prequantum bundle

We discuss the [[geometric prequantization]] of the [[symplectic manifold]] given by the [[coadjoint orbit]] $\mathcal{O}_\lambda$ equipped with its [[symplectic form]] $\nu_\lambda$ of def. \ref{TheSymplecticFormOnTheCoset}.

Assume now that $G$ is [[simply connected topological space|simply connected]].

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

The [[weight lattice]] $\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t}$ of the Lie group $G$ is [[isomorphism|isomorphic]] to the group of [[group characters]]

$$
  \Gamma_{wt}
  \xrightarrow{\;\; \sim \;\;}
  Hom_{LieGrp}\big(T,U(1)\big)
  \,,
$$

where the identification takes $\langle \alpha , -\rangle \in \mathfrak{t}^*$ to $\rho_\alpha \colon T \to U(1)$ given on $t = \exp(\xi)$ for $\xi \in \mathfrak{t}$ by

$$
  \rho_\alpha
    \;\colon\;
  \exp(\xi)
    \;\mapsto\;
  \exp(i \langle \alpha, \xi\rangle)
  \,.
$$

=--

\begin{proposition}\label{IntegralityOfSymplecticForm}
The [[symplectic form]] $\nu_\lambda \in \Omega^2_{cl}(G/T)$ of prop. \ref{TheSymplecticFormOnTheCoset} is [[integral form|integral]] precisely if $\langle \lambda, - \rangle$ is in the [[weight lattice]].
\end{proposition}

\begin{remark}
Integrality of the symplectic form (via Prop. \ref{IntegralityOfSymplecticForm}) is known as the [[geometric quantization|*Bohr--Sommerfeld condition*]], which is equivalent to the existence of a [[prequantum line bundle]].
\end{remark}

### The Hamiltonian $G$-action / coadjoint moment map

The group $G$ canonically [[group action|acts]] on the [[coset space]] $G/G_{\lambda}$ (by multiplication from the left). We discuss a lift of this action to a [[Hamiltonian action]] with respect to the [[symplectic manifold]] structure $(G/T, \nu_\lambda)$ of Prop. \ref{TheSymplecticFormOnTheCoset}, equivalently a [[momentum map]] exhibiting this Hamiltonian action.

### Wilson loops and 1d Chern-Simons $\sigma$-models with target the coadjoint orbit
 {#WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit}

[Above](#DefinitionsAndConstructions)  we discussed how an [[irreducible representation|irreducible]] [[unitary representation]] of $G$ is encoded by the [[prequantization]] of a  [[coadjoint orbit]] $(\mathcal{O}_\lambda, \nu_\lambda)$. Here we discuss how to express [[Wilson loops]]/[[holonomy]] of $G$-[[principal connections]] in this representation as the [[path integral]] of a topological particle charged under this background field, whose [[action functional]] is that of a [[1-dimensional Chern-Simons theory]].

This was hinted at in [Witten 89, p. 22, 23](#Witten89), details are in [Beasley 2011, section 4](#Beasley11).

Let $A|_{S^1} \in \Omega^1(S^1, \mathfrak{g})$ be a [[Lie algebra valued 1-form]] on the circle, equivalently a $G$-[[principal connection]] on the circle.

For

$$
  R \;\colon\; G \to Aut(V)
$$

a [[linear representation]] of $G$, write

$$
  W_{S^1}^R(A)
  \;\coloneqq\;
  hol^R_{S^1}(A)
  \;\coloneqq\;
  Tr_R\big( tra_{S^1}(A) \big)
$$

for the [[holonomy]] of $A$ around the circle in this representation, which is the [[trace]] of its [[parallel transport]] around the circle (for any basepoint). If one thinks of $A$ as a [[background gauge field]] then this is alse called a *[[Wilson loop]]*.

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

Let the [[action functional]]

$$
  \exp\big(
    \mathrm{i}
    CS_\lambda(-)^A
  \big)
  \;\colon\;
  [S^1, G/T] \longrightarrow U(1)
$$

be given by sending $g T \,\colon\, S^1 \to G/T$ represented by $g \,\colon\, S^1 \to G$ to

$$
  \exp\Big(
    \mathrm{i}
    \textstyle{\int}_{S^1}
    \langle \lambda, A^g\rangle
   \Big)
   \;\in\;
   U(1)
  \,,
$$

where

$$
  A^g \,\coloneqq\, Ad_g(A) + g^* \theta
$$

is the [[gauge transformation]] of $A$ under $g$.

=--

+-- {: .num_prop #WilsonLoopIsPartitionFunctionOf1dCSSigmaModel}
###### Proposition

The [[Wilson loop]] of $A$ over $S^1$ in the unitary irreducible representation $R$ is proportional to the [[path integral]] of the 1-dimensional [[sigma-model]] with

1. [[target space]] the [[coadjoint orbit]] $\mathcal{O}_\lambda \simeq G/T$ for $\langle \lambda, - \rangle$ the [[weight (in representation theory)|weight]] corresponding to $R$ under the [[Borel-Weil-Bott theorem]],

1. [[action functional]] the functional of def. \ref{ActionFunctionalForTopologicalChargedParticle}:

   $$
     W_{S^1}^R(A)
     \;\propto\;
     \underset{g T \in [S^1, \mathcal{O}_\lambda]}{\int}
     \exp\Big(
       \mathrm{i}
       \textstyle{\int}_{S^1}
       \langle \lambda, A^g\rangle
     \Big)
     \;
     D(g T)
     \,.
   $$

=--

See for instance [Beasley 2011 (4.55)](#Beasley11).

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

Notice that since $\mathcal{O}_\lambda$ is a [[manifold]] of [[finite number|finite]] [[dimension]], the [[path integral]] for a point particle with this target space can be and has been defined rigorously, see at _[[path integral]]_.

=--

## Formulation in higher geometry
 {#FormulationInHigherGeometry}

We discuss here a natural equivalent reformulation of the [above](#DefinitionsAndConstructions) ingredients of the orbit method in terms of the [[higher differential geometry]] of [[smooth ∞-groupoids]], and specifically in terms of the [[extended prequantum field theory]] of [[Chern-Simons theory]] with [[Wilson line]] [[QFT with defects|defects]] ([FSS](#FSS)).

1. [Survey](#FormulationInHigherGeometrySurvey)

1. [Definitions and constructions](#FormulationInHigherGeometryDefinitions)

1. [Nonabelian charged particle trajectories – Wilson loops](#GaugeAndGravityWilsonLoops)

1. [3d Chern-Simons theory with Wilson loops](#ExtendedChern-SimonsTheoryAndWilsonLoops).

### Survey
 {#FormulationInHigherGeometrySurvey}

We discuss how for $\lambda \in \mathfrak{g}$ a regular element, there is a canonical diagram of [[smooth infinity-groupoid|smooth]] [[moduli stacks]] of the form

$$
  \array{
     \mathcal{O}_\lambda &\stackrel{\simeq}{\to}& G/T &\stackrel{\mathbf{\theta}}{\to}& \Omega^1(-,\mathfrak{g})//T &\stackrel{\langle \lambda, - \rangle}{\to}& \mathbf{B} U(1)_{conn}
   \\
    && \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\mathbf{J}}}
    \\
    && * &\stackrel{}{\to}& \mathbf{B}G_{conn} &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^3 U(1)_{conn}
  }
  \,,
$$

where

1. $\mathbf{J}$ is the canonical [[2-monomorphism]];

1. the left square is a [[homotopy pullback]] square, hence $\mathbf{\theta}$ is the [[homotopy fiber]] of $\mathbf{J}$;

1. the bottom map is the [[extended Lagrangian]] for $G$-[[Chern-Simons theory]], equivalently the universal [[Chern-Simons circle 3-bundle with connection]];

1. the top map denoted $\langle \lambda,- \rangle$ is an [[extended Lagrangian]] for a [[1-dimensional Chern-Simons theory]];

1. the total top composite modulates a [[prequantum circle bundle]] which is a [[prequantization]] of the canonical [[symplectic manifold]] structure on the [[coadjoint orbit]] $\Omega_\lambda \simeq G/T$.

### Definitions and constructions
 {#FormulationInHigherGeometryDefinitions}

Write $\mathbf{H} = $ [[Smooth∞Grpd]] for the [[cohesive (∞,1)-topos]] of smooth $\infty$-groupoids.

For the following, let $\langle \lambda, - \rangle \in \mathfrak{g}^*$ be a _regular_ element, def. \ref{RegularElement}, so that the [[stabilizer subgroup]] is identified with a [[maximal torus]]: $G_\lambda \simeq T$.

As usual, write

$$
  \mathbf{B}G_{conn} \simeq \Omega^1(-,\mathfrak{g})//G
  \in
  \mathbf{H}
$$

for the [[moduli stack]] of $G$-[[principal connections]].

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

Write

$$
  \mathbf{J}
  \coloneqq
  \left( \;
   \Omega^1(-,\mathfrak{g})//T
     \;\to\;
     \Omega^1(-,\mathfrak{g})//G \simeq \mathbf{B}G_{conn}
    \;
  \right)
  \in
  \mathbf{H}^{(\Delta^1)}
$$

for the canonical map, as indicated.

=--

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

The map $\mathbf{J}$ is the differential refinement of the [[delooping]] $\mathbf{B}T \to \mathbf{B}G$ of the defining inclusion. By the general discussion at [[coset space]] we have a [[homotopy fiber sequence]]

$$
  \array{
    \mathcal{O}_\lambda \simeq G/T &\to& \mathbf{B}T & \simeq *//T
    \\
    && \downarrow
    \\
    && \mathbf{B}G & \simeq *//G
  }
  \,.
$$

By the discussion at _[[∞-action]]_ this exhibits the canonical [[action]] $\rho$ of $G$ on its [[coset space]]: it is the [[universal associated ∞-bundle|universal rho-associated bundle]].

=--

The following proposition says what happens to this statement under differential refinement

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

The [[homotopy fiber]] of $\mathbf{J}$ in def. \ref{InclusionOfModuliStacks} is

$$
  \mathbf{\theta} \colon G/T \stackrel{}{\to} \Omega^1(-,\mathfrak{g})//T
$$

given over a test manifold $U \in $ [[CartSp]] by the map

$$
  \mathbf{\theta}_U \colon C^\infty(U,G/T) \to \Omega^1(U,\mathfrak{g})
$$

which sends $g \mapsto g^* \theta$, where $\theta$ is the [[Maurer-Cartan form]] on $G$.

=--

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

This is a general phenomenon in the context of [[Cartan connections]]. See there at _[Definition -- In terms of smooth moduli stacks](Cartan+connection#InTermsOfSmoothModuliStacks)_.

=--

+-- {: .proof}
###### Proof
**of prop. \ref{ThetaAsHomotopyFiberOfJ}**.

We compute the [[homotopy pullback]] of $\mathbf{J}$ along the point inclusion by the [[factorization lemma]] as discussed at _[homotopy pullback -- Constructions](homotopy%20pullback#ConstructionsGeneral)_.

This says that with $\mathbf{J}$ presented canonically as a map of presheaves of groupoids via the above definitions, its homotopy fiber is presented by the presheaf of groupids $hofib(\mathbf{J})$ which is the [[limit]] [[cone]] in

$$
  \array{
    hofib(\mathbf{J}) &\to& &\to& \Omega^1(-, \mathfrak{g})//T
    \\
    \downarrow && \downarrow && \downarrow
    \\
    && (\mathbf{B}G_{conn})^I &\to& \mathbf{B}G_{conn}
    \\
    \downarrow && \downarrow
    \\
    * &\stackrel{}{\to}& \mathbf{B}G_{conn}
  }
  \,.
$$

Unwinding the definitions shows that $hofib(\mathbf{J})$ has

1. [[objects]] over a $U \in $ [[CartSp]] are equivalently morphisms $0 \stackrel{g}{\to} g^* \theta$ in $\Omega^1(U,\mathfrak{g})//C^\infty(U,G)$, hence equivalently elements $g \in C^\infty(U,G)$;

1. [[morphisms]] are over $U$ [[commuting diagram|commuting triangles]]

   $$
     \array{
       g_1^* \theta &&\stackrel{t}{\to}&& g_2^* \theta
       \\
       & {}_{\mathllap{g_1}}\nwarrow && \nearrow_{\mathrlap{g_2}}
       \\
       && 0
     }
   $$

   in $\Omega^1(U,\mathfrak{g})//C^\infty(U,G)$ with $t \in C^\infty(U,T)$, hence equivalently morphisms

   $$
     g_1 \stackrel{t}{\to} g_2
   $$

   in $C^\infty(U,G)//C^\infty(U,T)$. This shows that $hofib(\mathbf{J}) \simeq G/T$.

1. The canonical map $hofib(\mathbf{J}) \to \Omega^1(-,\mathfrak{g})//T$ picks the top horizontal part of these commuting triangles hence equivalently sends $g$ to $g^* \theta$.

=--

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

There is yet one more fiber sequence of similar structure. If we let $L G \coloneqq [S^1, G]$ denote the free [[loop group]], then there is a fiber sequence

$$
  \array{
    G/T &\to& L G / T
    \\
    && \downarrow
    \\
    && L G / G & \simeq \Omega G
  }
  \,.
$$

The [[geometric quantization]] of $L G / T$ yields the [[positive energy representations]] of the [[loop group]] $L G_\mathcal{C}$. See at _[loop group -- Properties -- Representations](loop+group#Representations)_ for more on this.

=--

+-- {: .num_prop #Extended1dCSLagrangianFromLambda}
###### Proposition

If $\langle \lambda ,- \rangle \in \Gamma_{wt} \hookrightarrow \mathfrak{g}^*$ is in the [[weight lattice]], then there is a morphism of [[moduli stacks]]

$$
  \langle \lambda, - \rangle
  \;\colon\;
  \Omega^1(-,\mathfrak{g})//T
  \to
  \mathbf{B}U(1)_{conn}
$$

in $\mathbf{H}$ given over a test manifold $U \in $ [[CartSp]] by the [[functor]]

$$
  \langle \lambda, - \rangle_U
  \;:\;
  \Omega^1(U,\mathfrak{g})//C^\infty(U,T) \to \Omega^1(U)//C^\infty(U,U(1))
$$

which is given on objects by

$$
  A \mapsto \langle \lambda, A\rangle
$$

and which maps morphisms labeled by $\exp(\xi) \in T$, $\xi \in C^\infty(-,\mathfrak{t})$ as

$$
  \exp(\xi) \mapsto \exp( i \langle \lambda, \xi \rangle )
  \,.
$$

=--

+-- {: .proof}
###### Proof

That this construction defines a map $*//T \to *//U(1)$ is
the statement of prop. \ref{WeightsAndCharacters}. It remains to check that the differential 1-forms gauge-transform accordingly.

For this the key point is that since $T \simeq G_\lambda$ stabilizes $\langle \lambda , - \rangle$ under the [[coadjoint action]], the [[gauge transformation]] law for points $A \colon U \to \mathbf{B}G_{conn}$, which for $g \in C^\infty(U,G)$ is

$$
  A \mapsto Ad_g A + g^* \theta
  \,,
$$

maps for $g = exp( \xi ) \in C^\infty(U,T) \hookrightarrow C^\infty(U,G)$ to the gauge transformation law in $\mathbf{B}U(1)_{conn}$:

$$
  \begin{aligned}
    \langle \lambda, A \rangle
    & \mapsto
    \langle \lambda, Ad_g A\rangle + \langle \lambda, g^* \theta\rangle
    \\
     & = \langle \lambda, A \rangle + d  \langle\lambda, \xi \rangle
  \end{aligned}
$$

=--

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

The composite of the canonical maps of prop. \ref{ThetaAsHomotopyFiberOfJ} and prop. \ref{Extended1dCSLagrangianFromLambda} modulates a canonical [[circle bundle with connection]] on the [[coset space]]/[[coadjoint orbit]]:

$$
  \langle \lambda, \mathbf{\theta}\rangle
  \colon
  G/T
  \stackrel{\mathbf{\theta}}{\to}
  \Omega^1(-,\mathfrak{g})//T
  \stackrel{\langle \lambda, - \rangle}{\to}
  \mathbf{B}U(1)_{conn}
  \,.
$$

=--

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

The [[curvature]] 2-form of the circle bundle $\langle \lambda, \mathbf{\theta}\rangle$ from remark \ref{ThePrequantumBundleFromCanonicalMaps} is the [[symplectic form]] of prop. \ref{TheSymplecticFormOnTheCoset}. Therefore $\langle \lambda, \mathbf{\theta}\rangle$ is a [[prequantization]] of the [[coadjoint orbit]] $(\mathcal{O}_\lambda \simeq G/T, \nu_\lambda)$.

=--

+-- {: .proof}
###### Proof

The curvature 2-form is modulated by the composite

$$
  \omega
  \colon
  G/T
  \stackrel{\mathbf{\theta}}{\to}
  \Omega^1(-,\mathfrak{g})//T
  \stackrel{\langle \lambda, - \rangle}{\to}
  \mathbf{B}U(1)_{conn}
  \stackrel{F_{(-)}}{\to}
  \Omega^2_{cl}
  \,.
$$

Unwinding the above definitions and propositions, one finds that this is given over a test manifold $U \in $ [[CartSp]] by the map

$$
  \omega_U \colon C^\infty(G/T) \to \Omega^2_{cl}(U)
$$

which sends

$$
  [g] \mapsto d \langle \lambda,  g^* \theta \rangle
  \,.
$$

=--

### Nonabelian charged particle trajectories – Wilson loops
{#GaugeAndGravityWilsonLoops}

Let $\Sigma$ be an [[orientation|oriented]] [[closed manifold|closed]] [[smooth manifold]] of [[dimension]] 3 and let

$$
  C
  \;\colon\;
  S^1 \hookrightarrow \Sigma
$$

be a [[submanifold]] inclusion of the [[circle]]: a [[knot]] in $\Sigma$.

Let $R$ be an [[irreducible representation|irreducible]]
[[unitary representation]] of $G$ and let $\langle \lambda,-\rangle$ be a [[weight (in representation theory)|weight]] corresponding to it by the [[Borel-Weil-Bott theorem]].

Regarding the inclusion $C$ as an object in the [[arrow category]] $\mathbf{H}^{\Delta^1}$, say that a [[gauge field]] configuration for $G$-[[Chern-Simons theory]] on $\Sigma$ with
[[Wilson loop]] $C$ and labeled by the [[representation]] $R$ is a map

$$
  \phi
  \;\colon\;
  C
  \to
  \mathbf{J}
$$

in the [[arrow category]] $\mathbf{H}^{(\Delta^1)}$ of the ambient [[cohesive (∞,1)-topos]]. Such a map is equivalently by a square

$$
  \array{
    S^1 &\stackrel{(A|_{S^1})^g}{\to}& \Omega^1(-,\mathfrak{g})//T
    \\
    \downarrow^{\mathrlap{C}}
    &\swArrow_{g}&
    \downarrow^{\mathrlap{\mathbf{J}}}
    \\
    \Sigma
    &\stackrel{A}{\to}&
    \mathbf{B}G_{conn}
  }
$$

in $\mathbf{H}$.
In components this is

* a $G$-[[principal connection]] $A$ on $\Sigma$;

* a $G$-valued function $g$ on $S^1$

which fixes the field on the circle defect to be $(A|_{S^1})^g$, as indicated.

Moreover, a _[[gauge transformation]]_ between two such fields $\kappa \colon \phi \Rightarrow \phi'$ is a $G$-gauge transformation of $A$ and  a $T$-gauge transformation of $A|_{S^1}$ such that these intertwine the component maps $g$ and $g'$. If we keep the bulk gauge field $A$ fixed, then his means that two fields $\phi$ and $\phi'$ as above are gauge equivalent precisely if there is a function $t \;\colon\; S^1 \to T$ such that $g = g' t$, hence gauge [[equivalence classes]] of fields for fixed bulk gauge field $A$ are parameterized by their components $[g] = [g'] \in [S^1, G/T]$ with values in the coset space, hence in the coadjoint orbit.

For every such field configuration we can evaluate two [[action functionals]]:

1. that of 3d [[Chern-Simons theory]], whose [[extended Lagrangian]] is $\mathbf{c} \colon \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$;

1. that of the [[1-dimensional Chern-Simons theory]] discussed [above](#WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit) whose [[extended Lagrangian]] is $\langle \lambda, -\rangle \colon \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn}$, by prop. \ref{Extended1dCSLagrangianFromLambda}.

These are obtained by postcomposing the above square on the right by these [[extended Lagrangians]]

$$
  \array{
    S^1 &\stackrel{(A|_{S^1})^g}{\to}& \Omega^1(-,\mathfrak{g})//T
    &\stackrel{\langle \lambda, -\rangle}{\to}&
    \mathbf{B}U(1)_{conn}
    \\
    \downarrow^{\mathrlap{C}}
    &\swArrow_{g}&
    \downarrow^{\mathrlap{\mathbf{J}}}
    \\
    \Sigma
    &\stackrel{A}{\to}&
    \mathbf{B}G_{conn}
    &\stackrel{\mathbf{c}}{\to}&
    \mathbf{B}U(1)_{conn}
  }
$$

and then preforming the [[fiber integration in ordinary differential cohomology]] over $S^1$ and over $\Sigma$, respectively.

For the bottom map this gives the ordinary action functional of [[Chern-Simons theory]]. For the top map inspection of the proof of prop. \ref{Extended1dCSLagrangianFromLambda} shows that this gives the 1d Chern-Simons action whose [[partition function]] is the [[Wilson loop]] observable by prop. \ref{WilsonLoopIsPartitionFunctionOf1dCSSigmaModel} above.

### 3d Chern-Simons theory with Wilson loops
 {#ExtendedChern-SimonsTheoryAndWilsonLoops}

We discuss how an [[extended Lagrangian]]
for $G$-[[Chern-Simons theory]] with [[Wilson loop]] [[QFT with defects|defects]] is naturally obtained from the [above](#FormulationInHigherGeometryDefinitions) [[higher geometry|higher geometric]] formulation of the orbit method. In particular we discuss how the relation between Wilson loops and [[1-dimensional Chern-Simons theory]] [[sigma-models]] with [[target space]] the [[coadjoint orbit]], as discussed  [above](#WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit) is naturally obtained this way.

More formally, we have an extended Chern-Simons theory as follows.

The [[moduli stack]] of fields $\phi \colon C \to \mathbf{J}$ in $\mathbf{H}^{(\Delta^1)}$ as above is the [[homotopy pullback]]

$$
  \array{
    \mathbf{Fields}(S^1 \hookrightarrow \Sigma)
    &\stackrel{}{\to}&
    [S^1, \Omega^1(-,\mathfrak{g})//T]
    \\
    \downarrow &\swArrow_\simeq& \downarrow
    \\
    [\Sigma, \mathbf{B}G_{conn}]
    &\to&
    [S^1, \mathbf{B}G_{conn}]
  }
$$

in $\mathbf{H}$, where square brackets indicate the [[internal hom]] in $\mathbf{H}$.

Postcomposing the two projections with the two [[transgressions]] of the [[extended Lagrangians]]

$$
  \exp(2 \pi i \int_\Sigma[\Sigma, \mathbf{c}])
  \;\colon\;
  [\Sigma, \mathbf{B}G_{conn}]
  \stackrel{[\Sigma, \mathbf{c}]}{\to}
  [\Sigma, \mathbf{B}^3 U(1)_{conn}]
  \stackrel{\exp(2 \pi i \int_\Sigma (-))}{\to}
  U(1)
$$

and

$$
  \exp(2 \pi i \int_\Sigma[S^1, \langle \lambda, -\rangle])
  \;\colon\;
  [S^1, \Omega^1(-,\mathfrak{g})//T]
  \stackrel{[\Sigma, \langle \lambda , -\rangle]}{\to}
  [S^1, \mathbf{B} U(1)_{conn}]
  \stackrel{\exp(2 \pi i \int_{S^1} (-))}{\to}
  U(1)
$$

to yield

$$
  \array{
    \mathbf{Fields}(S^1 \hookrightarrow \Sigma)
    &\stackrel{}{\to}&
    [S^1, \Omega^1(-,\mathfrak{g})//T]
    &\stackrel{\exp(2 \pi i \int_{S^1} [S^1, \langle \lambda, -\rangle] ) }{\to}&
    U(1)
    \\
    \downarrow &\swArrow_\simeq& \downarrow
    \\
    [\Sigma, \mathbf{B}G_{conn}]
    &\to&
    [S^1, \mathbf{B}G_{conn}]
    \\
    \downarrow^{\mathrlap{\exp(2\pi i \int_{\Sigma_2} [\Sigma_3, \mathbf{c}])}}
    \\
    U(1)
  }
$$

and then forming the product yields the action functional

$$
  \exp(2 \pi i \int_{S^1}[S^1, \langle -\rangle])
   \cdot
  \exp(2 \pi i \int_{\Sigma}[\Sigma, \mathbf{c}])
  \;:\;
  \mathbf{Fields}(S^1 \hookrightarrow \Sigma)
  \to U(1)
  \,.
$$

This is the action functional of 3d $G$-[[Chern-Simons theory]] on $\Sigma$ with Wilson loop $C$ in the representation determined by $\lambda$.

Similarly, in [[codimension]] 1 let $\Sigma_2$ now be a 2-dimensional closed manifold, thought of as a slice of $\Sigma$ above, and let $\coprod_i {*} \to \Sigma_2$ be the inclusion of points, thought of as the punctures of the Wilson line above through this slice. Then we have [[prequantum bundles]] given by [[transgression]] of the extended Lagrangians to codimension 1

$$
  \exp\left(2 \pi i \int_{\Sigma_2}\left[\Sigma, \mathbf{c}\right]\right)
  \;\colon\;
  \left[\Sigma_2, \mathbf{B}G_{conn}\right]
  \stackrel{\left[\Sigma_2, \mathbf{c}\right]}{\to}
  \left[\Sigma_2, \mathbf{B}^3 U(1)_{conn}\right]
  \stackrel{\exp\left(2 \pi i \int_{\Sigma_2} \left(-\right)\right)}{\to}
  \mathbf{B}U\left(1\right)_{conn}
$$

and

$$
  \exp\left(2 \pi i \int_{\coprod_i {*}}\left[\coprod_i {*}, \left\langle \lambda, -\right\rangle\right]\right)
  \;\colon\;
  \left[\coprod_i {*}, \Omega^1\left(-,\mathfrak{g}\right)//T\right]
  \stackrel{[\coprod_i {*}, \langle \lambda , -\rangle]}{\to}
  \left[\coprod_i {*}, \mathbf{B} U(1)_{conn}\right]
  \stackrel{\exp\left(2 \pi i \int_{\coprod_i {*}} \left(-\right)\right)}{\to}
  \mathbf{B}U(1)_{conn}
$$

and hence a total prequantum bundle

$$
  \exp\left(2 \pi i \int_{\coprod_i {*}}\left[\coprod_i {*}, \langle \beta, -\rangle\right]\right)
   \otimes
  \exp\left(2 \pi i \int_{\Sigma_2}\left[\Sigma_2, \mathbf{c}\right]\right)
  \;:\;
  \mathbf{Fields}\left(\coprod_i {*} \hookrightarrow \Sigma\right)
  \to \mathbf{B}U\left(1\right)_{conn}
  \,.
$$

One checks that this is indeed the correct prequantization as considered in ([Witten 89, p. 22](#Witten89)).

## Formulation in equivariant K-theory (Dirac induction)
 {#DiracInduction}

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

For $G$ a [[compact Lie group]] with [[Lie algebra]] $\mathfrak{g}^\ast$, the [[push-forward in generalized cohomology|push-forward]] in compactly supported [[twisted K-theory|twisted]] $G$-[[equivariant K-theory]] to the point (the $G$-equivariant [[index]]/[[Dirac induction]]) produces the [[Thom isomorphism]]

$$
  ind_{\mathfrak{g}^\ast}
    \;\colon\;
  K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt}
   \stackrel{\simeq}{\to}
  K_G^0(\ast) \simeq Rep(G)
  \,.
$$

Moreover, for $i \colon \mathcal{O} \hookrightarrow \mathfrak{g}^\ast$ a regular [[coadjoint orbit]], [[push-forward in generalized cohomology|push-forward]] involves a [[twisted K-theory|twist]] $\sigma$ of the form

$$
  Rep(G) \simeq K_G^0(\ast)
  \stackrel{ind_{\mathcal{O}}}{\leftarrow}
  K_G^{\sigma(\mathcal{O}) + dim(\mathcal{O})}(\mathcal{O})
  \stackrel{i_!}{\to}
  K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt}
$$

and

1. $i_!$ is surjective

1. $ind_{\mathcal{O}} = ind_{\mathfrak{g}^\ast} \circ i_!$.

=--

This is ([FHT II, (1.27), theorem 1.28](#FHT)).

## Formulation in equivariant elliptic cohomology

The [above](#DiracInduction) formulation of the orbit method in
[[equivariant K-theory]] has a higher order generalization where one replaces [[equivariant K-theory]] with [[equivariant cohomology|equivariant]] [[elliptic cohomology]]. Here the "elliptic" orbit method directly knows about the [[representation theory]] of the [[loop group]]. ([Ganter 12](#Ganter12)).

## Theorems

* [[Borel-Weil-Bott theorem]]

## Examples

* [[geometric quantization of the 2-sphere]]

## References
 {#References}

Introductions and surveys:

* [[Alexandre Kirillov]], _[[Lectures on the Orbit Method]]_, Graduate Studies in Mathematics, 64, American Mathematical Society, (2004)

  [[David Vogan]], _Review of: Lectures on the orbit method_ ([pdf](http://www.ams.org/journals/bull/2005-42-04/S0273-0979-05-01065-7/S0273-0979-05-01065-7.pdf))

* [[David Vogan]], _Geometry and representations of reductive groups_ (2007) ([pdf](http://www-math.mit.edu/~dav/rittC.pdf))

* J. Maes, _A introduction to the orbit method_, Master thesis (2011) ([pdf](http://testweb.science.uu.nl/ITF/teaching/2011/Jeroen%20Maes.pdf), [pdf slides](http://www.imus.us.es/FSMYT12/Talk_Jeroen_Maes.pdf), [web](http://igitur-archive.library.uu.nl/student-theses/2011-0622-200341/UUindex.html))

* Craig Jackson, _Symplectic manifolds, geometric quantization, and unitary representations of Lie groups_ ([pdf](http://go.owu.edu/~chjackso/Papers/topic.pdf))

* Reyer Sjamaar, _Notes on the orbit method and quantization_ (1997) ([[SjamaarOrbitMethod.pdf:file]])

See also:

* Wikipedia, _[orbit method](http://en.wikipedia.org/wiki/Orbit_method)_

Original references:

* В. А. Гинзбург, _Метод орбит в теории представлений комплексных групп Ли_, Функц. анализ и его прил., 1981, том 15, в. 1, стр. 23–37,  [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1690&what=fullt&option_lang=rus); transl. V. A. Ginzburg, _Method of orbits in the representation theory of complex Lie groups_, Funct. Analysis and Its Appl. __1981__, 15:1, 18–28, [doi](http://dx.doi.org/10.1007/BF01082375)

* [[Bertram Kostant]], _Orbits and quantization theory_, Proc. ICM Nice 1970, 395-406, [djvu:597 K](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0395.0406.ocr.djvu), [pdf:1.1 M](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0395.0406.ocr.pdf)

* [[Bertram Kostant]], _Quantization and unitary representations. I. Prequantization_, in: Lectures in Modern Analysis and Applications III, Lec. Notes in Math. __170__, 87–208, [MR294568](http://www.ams.org/mathscinet-getitem?mr=294568); Russ. transl. by A. Kirillov: Uspehi Mat. Nauk __28__ (1973), no. 1(169), 163–225, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=4837&volume=28&year=1973&issue=1&fpage=163&what=fullt&option_lang=eng)

* [[Alexandre Kirillov]], _Унитарные представления нильпотентных групп Ли_, , Uspehi. Mat. Nauk. 17 (1962), 57-110, [Rus. pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=6513&what=fullt&option_lang=rus); transl. _Unitary representations of nilpotent Lie groups_, Russian Math. Surveys, 1962, 17:4, 53–104, [doi](http://dx.doi.org/10.1070%2FRM1962v017n04ABEH004118), [MR142001](http://www.ams.org/mathscinet-getitem?mr=142001)

* L. Auslander, [[Bertram Kostant]], _Quantization and representations of solvable Lie groups_, Bull. Amer. Math. Soc. __73__, 1967, 692–695, [pdf](http://www.ams.org/journals/bull/1967-73-05/S0002-9904-1967-11829-9/S0002-9904-1967-11829-9.pdf); _Polarization and unitary representations of solvable Lie groups_, Invent. Math. __14__ (1971), 255–354, [MR293012](http://www.ams.org/mathscinet-getitem?mr=293012), [doi](http://dx.doi.org/10.1007/BF01389744)

* {#GrahamVogan} W. Graham, [[David Vogan]], _Geometric quantization for nilpotent coadjoint orbits_, in Geometry and Representation Theory of real and p-adic groups.
Birkhäuser, Boston-Basel-Berlin (1998)

* {#Vogan99} [[David Vogan]], _The method of coadjoint orbits for real reductive groups_, in Representation Theory of Lie Groups. IAS/Park City Mathematics Series 8 (1999), 179–238

Discussion with an eye towards application in [[gauge theory]] and in particular for [[Wilson loop]] observables in [[Chern-Simons theory]], hinted at on

* {#Witten89} [[Edward Witten]], p. 22, 23 of _Quantum Field Theory and the Jones Polynomial_ Commun. Math. Phys. 121 (3) (1989) 351–399. MR0990772 ([project EUCLID](http://projecteuclid.org/euclid.cmp/1104178138))

is in section 4 of

* {#Beasley11} [[Chris Beasley]], _Localization for Wilson Loops in Chern-Simons Theory_, in J. Andersen, H. Boden, A. Hahn, and B. Himpel (eds.) _Chern-Simons Gauge Theory: 20 Years After_, AMS/IP Studies in Adv. Math. **50**, AMS (2011) [[arXiv:0911.2687](http://arxiv.org/abs/0911.2687), [ISBN:9780821853535](https://www.intlpress.com/site/pub/pages/books/items/00000258/index.php)]

referring to

* [[Shmuel Elitzur]], [[Greg Moore]], A. Schwimmer, [[Nathan Seiberg]], _Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory_, Nucl. Phys. B 326 (1989) 108–134.

A program of applying the orbit method to real nilpotent orbits of real semisimple Lie groups (closely related to [[quantization via the A-model]]) is in

* Ranee Brylinski, _Geometric Quantization of Real Minimal Nilpotent Orbits_, DGA, vol. 9 (1998), 5-58 ([arXiv:math/9811033](http://arxiv.org/abs/math/9811033))

  * _Quantization of the 4-dimensional nilpotent orbit of $SL(3,\mathbb{R})$_, Canad. J. Math. 49(1997), 916-943 ([web](http://cms.math.ca/cjm/a148867))

  * _Instantons and Kähler Geometry of Nilpotent Orbits_ ([arXiv:math/9811032](http://arxiv.org/abs/math/9811032))

Discussion of the orbit method in terms of [[equivariant K-theory]] and [[Dirac induction]] is in

* {#HFT} [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], part II, section 1 of _[[Loop Groups and Twisted K-Theory]]_

* {#Hochs12} [[Peter Hochs]], section 2.2 of _Quantisation of presymplectic manifolds, K-theory and group representations_ ([arXiv:1211.0107](http://arxiv.org/abs/1211.0107))

The generalization of this to [[elliptic cohomology]] is discussed in

* {#Ganter12} [[Nora Ganter]], _The elliptic Weyl character formula_ ([arXiv:1206.0528](http://arxiv.org/abs/1206.0528))

Generalization to [[supergeometry]] is discussed in:

* Gijs M. Tuynman, _Geometric Quantization of Superorbits: a Case Study_ ([arXiv:0901.1811](http://arxiv.org/abs/0901.1811))

* Alexander Alldridge, Joachim Hilgert, Tilmann Wurzbacher, _Superorbits_ ([arXiv:1502.04375](https://arxiv.org/abs/1502.04375))

A generalization to [[higher geometry]] and [[2-group]] [[2-representations]] is proposed in

* [[Bruce Bartlett]], _On unitary 2-representations of finite groups and topological quantum field theory_ ([arXiv:0901.3975](http://arxiv.org/abs/0901.3975))

The above discussion of the interpretation of the orbit method in terms of higher [[moduli stacks]] for [[differential cohomology]] appears in

* {#FSS} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], section 3.4.5 of _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_, in Damien Calaque et al. (eds.) _Mathematical Aspects of Quantum Field Theories_, Mathematical Physics Studies, Springer 2014 ([arXiv:1301.2580](http://arxiv.org/abs/1301.2580))

[[!redirects method of coadjoint orbits]]


orbit type

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{:toc}

## Idea

In discussion of [[topological G-spaces]] (such as in [[equivariant homotopy theory]] and/or [[equivariant differential topology]]) by an _orbit type_ one means the [[isomorphism class]] of [[orbits]] of the [[equivariance group]].

Since every [[orbit]] of a [[group]] $G$ (typically taken to be a [[topological group]]) is [[isomorphism|isomorphic]], as a [[topological G-space]], to a [[coset space]] $G/H$, and since these coset spaces, in turn, are isomorphic to each other precisely if the [[subgroups]] $H$ are [[conjugate subgroup|conjugate]] to each other (see also at _[[orbit category]]_), the $G$-orbit types are often identified with the [[conjugacy classes]] $(H)$ of the [[subgroups]] of $G$.

## Related concepts

* [[orbit]]

* [[orbit category]]

## References

* {#Bredon72} [[Glen Bredon]], Section I.4 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

[[!redirects orbit types]]


order

> This entry is about the concept in [[order theory]]. See at [[group order]] for the concept of the same name in [[group theory]].

***

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{:toc}

## In the sense of order theory

An **order** on a set $S$ is (usually) a binary [[relation]] that is, at the very least, [[transitive relation|transitive]].

Actually, there are several different notions of order that are each useful in their own ways:

* [[direction]]
* [[strict total order]]
* [[partial order]]
* [[preorder]]
* [[strict preorder]]
* [[total order]]
* [[well-order]]

The closely related notion of a [[cyclic order]] is not actually a binary relation but a ternary relation.

The study of orders is [[order theory]].

## In the sense of group theory

A mostly unrelated notion from [[group theory]] is _[[order of a group]]_, meaning the [[cardinality]] $|G|$ of the [[underlying set]] of a [[group]] $G$, especially when this is [[finite set|finite]].

By extension, one speaks of the [[order of an element]] $x \in G$, as the order of the [[cyclic group|cyclic]] [[subgroup]] $\langle x\rangle$ generated by the element.  For example, the order of a [[permutation]] $\pi \in S_n$ is the least integer $1 \le k\le n$ such that $\pi^k = id$.

Sometimes one thinks of an [[infinite set|infinite]] group as having order [[zero]].  The orders then have the natural order relation of [[divisibility]].

## Other meanings

The term 'order' can also be used fairly generically as a synonym of 'degree' or 'rank', as in [[first-order logic]], the order of a [[differential equation]], etc.  Of course, these various orders form a [[well-order]], so this is not entirely unrelated either.

[[!redirects order]]
[[!redirects orders]]
[[!redirects order relation]]
[[!redirects order relations]]


order category > history

< [[preorder]] and [[partial order]]


order of a distribution

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## Definition

Let $X \subset \mathbb{R}^n$ be an [[open subset]] of [[Euclidean space]]. One way to state the [[continuous function|continuity]] condition on a [[distribution]] $u$ on $X$, being a [[continuous linear functional]] on the space $C^\infty_c(X)$ of [[bump functions]] is to require that for all [[compact topological space|compact]] [[subspaces]] $K$ there exist constants $C$ and $k$ such that the [[absolute value]] of $u(b)$ is bounded by $C$ times the [[suprema]] of the [[sums]] of the [[absolute values]] of all [[derivatives]] of $b$ of order bounded by $k$:

$$
  \underset{ {K \subset X} \atop {\text{compact}}}{\forall}
  \left(
    \underset{ {C_K \in \mathbb{R}} \atop {k_K \in \mathbb{N}} }{\exists}
    \left(
      \underset{b \in C^\infty_c(X)}{\forall}
      \left(
        {\vert u(b) \vert}
        \leq
        C_K \underset{\vert \alpha\vert \leq k_K}{\sum}sup(\partial^\alpha b)
      \right)
    \right)
  \right)
  \,.
$$

+-- {: .num_defn #OrderOfADistribution}
###### Definition
**(order of a distribution)**

The distribution $u \in \mathcal{D}'(X)$ has _order_ $\leq k \in \mathbb{N}$ if $k$ serves as a global choice (valid for all compact $K \subset X$) in the above formula, hence if:

$$
  \underset{ {K \subset X} \atop {\text{compact}}}{\forall}
  \left(
    \underset{ {C_K \in \mathbb{R}}  }{\exists}
    \left(
      \underset{b \in C^\infty_c(X)}{\forall}
      \left(
        {\vert u(b) \vert}
        \leq
        C_K \underset{\vert \alpha\vert \leq k}{\sum}sup(\partial^\alpha b)
      \right)
    \right)
  \right)
  \,.
$$

If $u$ is of order $\leq k$ for some $k \in \mathbb{N}$, it is said to have _finite order_.

=--

(e.g. [Hoermander 90, def. 2.1.1, below (2.1.2)](#Hoermander90))

## Examples

+-- {: .num_example #CompactLySupportedDistributionsHaveFiniteOrder}
###### Example

Every [[compactly supported distribution]] has finite order.

=--

As a special case of example \ref{CompactLySupportedDistributionsHaveFiniteOrder}:

+-- {: .num_example}
###### Example

A [[point-supported distribution]] is a sum of [[derivatives]] of the [[delta distribution]] at that point. This has order $k$ if the order of the derivatives is bounded by $k$.

=--

([Hörmander 90, theorem 2.3.4](#Hoermander90))

## Related concepts

* [[support of a distribution]]

* [[scaling degree of a distribution]]

## References

* {#Hoermander90} [[Lars Hörmander]], _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

[[!redirects order of distributions]]
[[!redirects orders of distributions]]


order of a group

> This entry is about the concept in [[group theory]]. See at _[[order]]_ for the concept in [[order theory]].

***

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## Definition

### Order of a group
 {#OrderOfAGroup}

For $G$ a [[discrete group]], its **order** is the [[cardinality]] of the underlying [[set]].

Hence for $G$ a [[finite group]], its order $\vert G\vert \in \mathbb{N}$ is a [[natural number]], the number of its group elements.

### Order of an element of a group
 {#OrderOfAnElementOfAGroup}

For $G$ a [[group]] and $g \in G$ an [[element]], the **order** of $g$ is the smallest [[natural number]] $n$ such that the $n$-fold group product of $g$ with itself is the [[neutral element]]:

$$
  order(g) \coloneqq min \{ n \in \mathbb{N} | g^n = e\}
  \,.
$$

The _[[exponent of a group]]_ is the [[least common multiple]] of the order of all elements of the group.

### Order of a group scheme

Sometimes the term ''order'' refers to the [[height of a group scheme|height]] of a ([[group scheme|group]]) [[scheme]] $X$ over a [[field]] (of [[characteristic]] $p$) which is defined to be the dimension of the associated ring of functions $O(X)$ as a $k$-[[vector space]]. Another term for this notion is ''rank''. If this group scheme is moreover [[p-divisible group|p-divisible]] - which means that is is in fact a codirected diagram of [[group scheme|group schemes]] of order $p^{v h}$; in this case $h$ is called the order or height of $X$.

## Properties

For $G$ a [[finite group]] and $g \in G$ an element, there is a close relation between

1. the order $\vert G\vert$ of $G$ in the sense [above](#OrderOfAGroup);

1. the order $order(g)$ of $g$ the sense [above](#OrderOfAnElementOfAGroup)

### Orders of cyclic subgroups

The element $g$ [[generators and relations|generates]] a [[cyclic group|cyclic]] [[subgroup]] $C_g \subset G$. Evidently, the order of the element $g$ equals the order $\vert C_g \vert$ of this cyclic subgroup that it generates:

\[
  \label{OrderOfElementIsOrderOfGeneratedCyclicGroup}
  order(g) =\vert C_g\vert
  \,.
\]

### Lagrange's theorem

By [[Lagrange's theorem]], if $G$ a [[finite group]] and $H \subset G$ a [[subgroup]], then the order $\vert G\vert$ of $G$ is divisible by the order $\vert H\vert$ of $H$. The multiple is called the [[index of a subgroup|subgroup index]] $[G : H] \in \mathbb{N}$:

$$
  {\vert G\vert}
  \;=\;
  [G : H] \, {\vert H\vert}
  \,.
$$

Hence with (eq:OrderOfElementIsOrderOfGeneratedCyclicGroup) it follows that with $g \in G$ any element, the order of $g$ divides the order of $G$:

$$
  {\vert G\vert}/ order(g)
  \;=\;
  [G : C_g]
  \,.
$$

## Related concepts

* [[exponent of a group]]

* [[p-group]]

## References

See also

* Wikipedia, _Order (group theory)_

[[!redirects order of an element]]

[[!redirects order of a group element]]

[[!redirects group order]]


order polynomial

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## Idea

The **order polynomial** $\Omega_P(n)$ of a finite [[partially ordered set]] $P$ counts the number of ways that $P$ can be extended to a [[linear order]] of size $n$.

## Definition

By a _linear extension_ of $P$ of size $n$, we mean an [[order-preserving function]] from $P$ to $(n) = \{ 1 \lt \cdots \lt n \}$.
To see that
$$\Omega_P(n) = |Hom(P,(n))|$$
defines a [[polynomial]] in $n$, first observe that any function $P \to (n)$ factors as a [[surjection]] from $P$ onto some $(k) = \{ 1 \lt \cdots \lt k \}$ (where $k \le n$), followed by an [[injection]] from $(k)$ to $(n)$. The total number of order-preserving functions from $P$ to $(n)$ can therefore be calculated explicitly as
$$\Omega_P(n) = \sum_{k=1}^{|P|} e_k \binom{n}{k}$$
where $e_k$ is the number of surjective order-preserving functions from $P$ to $(k)$. Hence $\Omega_P(n)$ is a polynomial of degree equal to the cardinality of $P$.

## Examples

The order polynomial of $(3) = \{ 1 \lt 2 \lt 3 \}$ is
$$n + 2 \binom{n}{2} + \binom{n}{3} = \frac{n^3 + 3n^2 + 2n}{6}$$
For example, evaluating the polynomial at $n=3$ confirms that there are 10 order-preserving functions from $(3)$ to itself.

The order polynomial of the 3-element poset
$$P = \array{&&c&& \\ &\nearrow& &\nwarrow& \\ a &&&& b}$$
is $n + 3 \binom{n}{2} + 2\binom{n}{3} = \frac{2n^3 + 3n^2 + n}{6}$.
Evaluating at $n=2$, we compute that there are 5 order-preserving functions from $P$ onto $(2)$.

## Properties

### Relation to zeta polynomial

Let $P^\downarrow \cong (2)^P$ denote the lattice of [[lower sets]] in $P$. The order polynomial of a finite poset $P$ is related to the [[zeta polynomial]] of its lattice of lower sets $P^\downarrow$ by the equation[^Note]
$$\Omega_P(n+2) = Z_{P^\downarrow}(n)$$
This can be seen as a consequence of the [[currying]] isomorphisms
$$Hom([n], P^\downarrow) \cong Hom([n] \times P, (2)) \cong Hom(P, [n]^\downarrow)$$
together with the isomorphisms $[n]^\downarrow \cong [n+1] \cong (n+2)$.

[^Note]: Note that the definition we use for $Z_P(n)$ has an index shift from the definition that seems to be more standard in combinatorics (see the footnote at [[zeta polynomial]]), so this equation is sometimes expressed as $\Omega_P(n) = Z_{P^\downarrow}(n)$.

### Relation to the strict order polynomial

Let $\bar{\Omega}_P(n)$ denote the number of [[strict order-preserving functions]] from $P$ to $(n)$, that is, functions $f : P \to (n)$ such that $x \lt y$ implies $f(x) \lt f(y)$. This again defines a polynomial in $n$, called the **strict order polynomial** of $P$. The strict order polynomial of a finite poset $P$ is related to its order polynomial by the equation
$$\bar{\Omega}_P(n) = (-1)^p \Omega_P(-n)$$
where $p = |P|$ is the cardinality of $P$. This is an example of a _combinatorial reciprocity theorem_ in the sense of [Stanley (1974)](#Stanley74).

(Notice that we are evaluating the order polynomial at a negative integer, even though the putative definition $\Omega_P(n) = |Hom(P,(n))|$ only makes sense for natural numbers $n$. This is justified because the value of a polynomial on the natural numbers determines its value on arbitrary integers.)

As an example, the strict order polynomial of the 3-element poset $P$ defined above is $\bar{\Omega}_P(n) = \frac{2n^3 - 3n^2 + n}{6}$. Evaluation at $n=2$ confirms that there is exactly one strict monotone map from $P$ to a 2-element linear order (sending both $a$ and $b$ to the first element and $c$ to the second element).

Through [[Möbius inversion]], this equation relating the strict order polynomial to the order polynomial can be connected to the previous one relating the order polynomial to the zeta polynomial. Since the lattice of lower sets $P^\downarrow$ has [[bottom]] and [[top]] elements, we know that
$$
\Omega_P(n) = Z_{P^\downarrow}(n-2) = \zeta^n(\emptyset,P)
$$
where $\zeta^n$ is the $n$-fold convolution of the zeta function of $P^\downarrow$. One can similarly work out that
$$
\bar{\Omega}_P(n) = (-1)^p\mu^n(\emptyset,P)
$$
where $\mu^n$ is the $n$-fold convolution of the [[Möbius function]] of $P^\downarrow$. The equation $\bar{\Omega}_P(n) = (-1)^p \Omega_P(-n)$ then follows formally from the fact that $\mu = \zeta^{-1}$.

### Computational complexity

Counting the number of total linear extensions of a poset (i.e., number of linear extensions of an $n$-element poset to a linear order of size $n$) is a #P-complete problem [(Brightwell and Winkler 1991)](#BW1991), and thus so is the problem of computing the leading coefficient of the order polynomial (cf. [Browning, Hopkins, and Kelley 2017](#BHK2017)).

## Related concepts

* [[zeta polynomial]]
* [[incidence algebra]]
* [[Möbius inversion]]

## References

* Richard P. Stanley. Ordered structures and partitions. Memoirs of the AMS 119, 1972. ([doi](http://www.ams.org/books/memo/0119/)) (Free copy available from the [author's website](http://www-math.mit.edu/~rstan/pubs/), see #9.)
 {#Stanley72}

* Richard P. Stanley. Combinatorial Reciprocity Theorems. Advances in Mathematics 14:194-253, 1974. (Free copy available from the [author's website](http://www-math.mit.edu/~rstan/pubs/), see #22.)
 {#Stanley74}

* Joseph P. S. Kung, [[Gian-Carlo Rota]], Catherine H. Yan. Combinatorics: The Rota Way. Cambridge, 2009.

* {#BW1991} Graham Brightwell and Peter Winkler. Counting Linear Extensions. _Order_ 8:225-242, 1991. [doi](https://doi.org/10.1007/BF00383444)

* {#BHK2017} Thomas Browning, Max Hopkins, and Zander Kelley. Doppelgangers: the Ur-Operation and Posets of Bounded Height. [arXiv:1710.140407](https://arxiv.org/abs/1710.10407)

category: combinatorics


order profile

Two [[finite group]]s have the same **order profile** precisely if for each [[natural number]] $m$, they have the same number of elements of [[order]] $m$.


order theory

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## Idea ##

Order theory, the study of [[orders]], is one of the main branches of [[mathematics]], but it often gets short shrift.  Many (but not all) of the ideas of [[category theory]] that don\'t apply to [[groupoids]] or [[sets]] already apply to [[posets]].  (Of course, order theory also has features of interest for its own sake.)

## As low-dimensional higher category theory

In the context of [[(n,r)-category]] theory, ordinary [[category theory]] is the study of $(1,1)$-[[1-category|categories]]. Following the spirit of [[negative thinking]], we find that one dimension below this is not just [[set theory]] as the study of $(0,0)$-[[0-category|categories]], but a richer theory, that of $(0,1)$-categories. These are [[poset]]s: sets with order. Thus order theory is the study of $(0,1)$-categories.

A comparative discussion of order theory as compared to category theory is at [[category theory vs order theory]].

[[!redirects order theories]]


order topology

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## Definition

Given a [[linearly ordered set]] $(S, \lt)$, its __order topology__ is the [[topological structure|topology]] on $S$ generated from the [[subbase of a topology|sub-basis]] $\beta \subset P(S)$

$$
  \beta
    \coloneqq
  \left\{
    (a,\infty),
    (-\infty,a)
  \right\}_{a \in S}
  \,.
$$

whose elements are the "open half rays"

$$
  (a, \infty)
  \coloneqq
  \left\{
    s \in S \,\vert\, a \lt s
  \right\}
    \phantom{AAAA}
  (-\infty, a)
  \coloneqq
  \left\{
    s \in S \,\vert\, s \lt a
  \right\}
  \,.
$$

## Examples

* The usual [[Euclidean space]] [[metric topology]] on the [[real numbers]] is equal to the order topology with respect to the canonical ordering of the real numbers. The analogous statement then holds for the [[rational numbers]] $\mathbb{Q} \subset \mathbb{R}$ equipped with their [[subspace topology]].

## Related concepts

* [[specialization topology]],

* [[Scott topology]]

## References

* Wikipedia, _[Order topology](https://en.wikipedia.org/wiki/Order_topology)_

[[!redirects order topology]]
[[!redirects order topologies]]


order-theoretic structure in quantum mechanics

The following surveys how basic [[theorems]] about the standard foundation of [[quantum mechanics]] imply an accurate [[geometry|geometric]] incarnation of the "[[phase space]] in [[quantum mechanics]]" by an [[order theory|order-theoretic structure]] that combines with an [[algebra|algebraic]] [[structure]] to a [[ringed topos]], the "[[Bohr topos]]". While the notion of [[Bohr topos]] has been _motivated_ by the [[Kochen-Specker theorem]], the point here is to highlight that taking into account further theorems about the standard foundations of [[quantum mechanics]], the notion effectively follows automatically and provides an accurate and useful description of the geometry of "quantum phase space" also in [[quantum field theory]] formulated in the style of [[AQFT]].

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## The quantum phase space as a Jordan-geometry given by a ringed topos
  {#QuantumPhaseSpaceAsJordanGeometry}

Back when [[quantum mechanics]] was discovered in the first half of the 20th century, it was eventually formalized as [[mathematical physics]] and the traditional modern formulation emerged, where, in [[AQFT]] perspective, [[quantum observables]] are represented by suitable [[linear operators]] on a [[Hilbert space]] and more generally by elements of a [[C*-algebra]] [[algebra of observables|of observables]], and where [[quantum states]] are certain (namely positive and normalized) linear functionals on these [[C*-algebra|C*]]-[[algebras of observables]] ([[states on star-algebras]]).

But one may still ask if the [[axioms]] in the definition of _[[C*-algebra]]_ accurately capture the intended [[physics]]. This or similar questions were discussed back in the middle of the 20th century, when these notions were still in flux. Specifically in the 1930s [[Pascual Jordan]] argued ([Jordan 32](#Jordan32)) that the [[associative algebra]] [[structure]] on the observables is more [[structure]] than supported by the physics of states and observation, that instead only the underlying structure of what is now called the _[[Jordan algebra]]_ should matter.

Much later in 1978 this idea was formally validated by the [[Alfsen-Shultz theorem]] ([Alfsen-Shultz 78](#AlfsenShultz78)). This states that the [[space of quantum states]] for given [[quantum observables]] depends indeed only on the underlying [[Jordan algebra]] structure. This is not too surprising: the definition of a [[state on an operator algebra]] does not even mention the [[associative algebra]] structure but mentions only the [[positive operator|positivity]] structure, which is what the [[Jordan algebra]] captures.

Despite these insights, [[Jordan algebras]] found and find only marginal attention in [[mathematical physics]]. In a [review from 2004](#ASReview04) of the book of Alfsen and Shultz it says that back then Jordan algebras were hoped to shed light on conceptual problems of genuine [[quantum field theory]], but that these hopes never materialized. However, more recent developments change this picture a bit, we come to that below.

Much more recently in 2010, the [[Harding-Döring-Hamhalter theorem]] sheds a new light on the role of [[Jordan algebra]] structure. This theorem states mild conditions under which a [[Jordan algebra]] structure on [[quantum observables]] is equivalently encoded in the [[poset of commutative subalgebras]] of the full [[C*-algebra]].

These commutative subalgebras themselves are of course of old fame in quantum mechanics, they are the "[[classical contexts]]" given by tuples of [[quantum observables]] that all commute with each other and hence which can all be [[measurement|measured]] simultaneously without the [[uncertainty principle]] interfering. These [[classical contexts]] played a crucial role in the discussion of the foundation of quantum mechanics in the first half of the 20th century: back then people argued that for $A$ and $B$ two [[quantum observables]] which do not commute with each other it is unclear what it means physically to form their sum $A + B$ or their product $A B$, hence that a [[quantum state]] should be demanded to be a linear (and positive normed) [[linear functional|functional]] on all [[poset of commutative subalgebras|commutative subalgebras]], but not necessarily on the whole non-commutative algebra. Such a notion of "quantum state on all [[classical contexts]]" was called a _[[quasi-state]]_.

The issue of whether quasi-states are a more accurate description of quantum [[measurement]] was settled in 1957 by [[Gleason's theorem]] ([Gleason 57](#Gleason57)), which says that given a  [[Hilbert space]] of [[dimension]] greater than 2, then the [[quasi-states]] are automatically [[quantum states]] also on the full non-commutative [[algebra of observables]]. Typically this is viewed as making the notion of [[quasi-state]] obsolete, but since that is a formally weaker notion the opposite attitude makes sense: what is traditionally taken as the definition of [[quantum state]] is more accurately thought of as being a [[quasi-state]], hence something that is intrinsically related not to a non-commutative algebra of observables, but to the "[[classical contexts]]" of its [[poset of commutative subalgebras]].

Indeed, by combining the [[Alfsen-Shultz theorem]] with the [[Harding-Döring-Hamhalter theorem]], we have (under the pertinent mild assumptions) that two algebras of observables have the same [[space of quantum states]] already when they have the same [[poset of commutative subalgebras]]. Notice that where [[Gleason's theorem]] only involves the commutative subalgebras themselves, this Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem crucially involves their [[order]] of inclusions, hence the actual [[poset]] structure of their inclusions.

Therefore there is an intrinsic [[order theory|order theoretic]] aspect in the standard foundations of [[quantum mechanics]]. But it is not _just_ order theory, for it is not the [[poset]] structure of inclusions alone that matters in the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem, but that poset structure together with the actual [[commutative C*-algebra]] structure of each [[classical context]].

There is an elegant way to combine these two aspects: a system of [[commutative ring|commutative algebras]] together with the [[order]] of their inclusions is equivalently a single [[algebra object]] [[internalization|internal to]] the [[sheaf topos]] over the [[poset]]. With some basics of [[topos theory]] in hand this is a trivial statement, but in view of the above it is worthwhile to make explicit: the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem ([Harding-Döring 10](#HardingDoering10), [Hamhalter 11](#Hamhalter11)) says that the collection of [[quantum observables]] in [[quantum mechanics]] is accurately formalized by a single [[commutative C*-algebra]] [[internalization|internal]] to a [[sheaf topos]] over a [[poset]].

It has been argued that this serves as a formalization of the views on [[quantum mechanics]] that in the middle of the 20th century [[Niels Bohr]] expressed in extensive but informal writing (see [Scheibe 73](#Scheibe73)): he said roughly that whatever [[quantum mechanics]] is, it must be expressible and must be expressed through classical contexts. In honor of this intuition, the above [[toposes]] deserve to be called _[[Bohr toposes]]_, following the term "Bohrification" in ([Heunen-Landsman-Spitters 09](#HeunenLandsmanSpitters09)).

In fact, a [[Bohr topos]] is fairly trivial as far as [[toposes]] go, since, by the above, it is just a reflection -- precisely: the "[[localic reflection]]" -- of a purely [[order theory|order theoretic structure]]. But the key is that passing to the topos over the [[poset]] provides a home for the context-wise [[commutative C*-algebra]] structure which makes the [[Bohr topos]] have the additional structure of a _[[ringed topos]]_. This is the additional [[algebra|algebraic]] datum on top of the purely  [[order theory|order theoretic]] datum in the [[Jordan algebra]] structure of [[quantum observables]].

Ringed toposes have of course a long tradition in [[geometry]] (most famously in [[algebraic geometry]]). By [[Grothendieck]]'s foundational work, laid out in the [thesis](#Hakim72) of [[Monique Hakim]], ringed toposes form a general foundation for structured [[geometry]]. More recently this was further strengthened and refined by [[Jacob Lurie]] by the notion of [[structured (infinity,1)-topos|higher ringed toposes]], which we will see appear in [[quantum field theory]] below. In as far as the [[quantum observables]] in [[quantum mechanics]] are supposed to be the [[Isbell duality|dual]] of the _[[phase space]]_, it is therefore natural to have this "quantum phase space" be realized as a [[ringed topos]].

Notice that this is a different geometric interpretation of "quantum phase space" than the traditional idea that quantum phase space is an object in [[non-commutative geometry]]! Here by the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem we see that it is not actually accurate to say that quantum phase space is dually given by a non-commutative [[C*-algebra]], as in fact it is given dually just by a [[Jordan algebra]]. The [[ringed topos|ringed]] [[Bohr topos]] provides a natural [[geometry|geometric]] interpretation of this, one might call it "Jordan geometry".

This geometric nature of the Bohr topos becomes more manifest as we consider its [[opposite category]]. By [[Gelfand duality]] this carries not an internal [[commutative C*-algebra]] but its [[Gelfand spectrum]]: an actual internal [[topological space]]. This internal topological space has been called the _[[spectral presheaf]]_ ([Butterfield-Hamilton-Isham 98](#ButterfieldHamiltonIsham98)).

While here we find this [[spectral presheaf]] as an accurate dual description of the space of [[quantum observables]] in [[quantum mechanics]] based on the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem, interest in the spectral presheaf originally came from the observation that it provides a clear geometric formulation of yet another theorem about the foundations of [[quantum mechanics]], namely of the _[[Kochen-Specker theorem]]_ ([Kochen-Specker 67](#KochenSpecker67)).

This again amplifies the role of the "classical contexts" of commutative subalgebras: one may ask if there is a [[hidden variable]] description of [[quantum mechanics]] that allows to assign actual values to all [[quantum observables]] such that in all [[classical contexts]] this assignment behaves as an actual [[classical observable]] in that it provides a (star-)[[homomorphism]] of [[associative algebras]] from the [[commutative C*-algebra]] of the classical context to the real numbers. The [[Kochen-Specker theorem]] rules out such a [[hidden variable theory]] by stating that when the [[algebra of observables]] is that of [[bounded operators]] on a [[Hilbert space]] of [[dimension]] greater than 2, then such a "hidden variable" assignment cannot exist.

In ([Butterfield-Isham 98](#IshamButterfield98)) it was observed, that this statement has a natural geometric interpretation in the [[Bohr topos]]: it simply says that the [[spectral presheaf]], hence the "Jordan-algebraic geometry" incarnation of quantum phase space, has no [[global element]]. This statement in turn is a characterization of how the quantum phase space is "exotic" as far as [[spaces]] go. It behaves like a non-trivial space, and yet there is no way to map a point into it as a whole, maps of points into it exist only locally.

In summary, the [[Bohr topos]] incarnation of the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter characterization of quantum observables not only accurately captures the nature of quantum observables, but also makes other subtle nature of [[quantum mechanics]] becomes more explicitly evident than in other formulations.

To see how one can get more out of the [[Bohr topos]] incarnation of the quantum phase space, it serves to pass from plain [[quantum mechanics]] to the more general context of [[quantum field theory]]. Here the original [[Haag-Kastler axioms]] of [[AQFT]] demand that to a region of [[spacetime]] is to be assigned the [[quantum observables]] as a [[C*-algebra]]/[[von Neumann algebra]]. But by the above discussion it is rather only the underlying [[Jordan algebra]] structure that matters, hence the [[Bohr topos]]. In light of this a [[local net of observables]] in [[AQFT]] is naturally regarded as a [[presheaf]] of [[ringed toposes]] on [[spacetime]], assigning the respective [[Bohr topos]] of local observables to each local region of spacetime.

Such "Bohrification of local nets of observables" were analyzed in ([Nuiten 12](#Nuiten12)). There it was found that the natural structure of the transition functions of local nets of Bohr toposes of observables by [[geometric morphisms]] automatically capture the [[causal locality]] condition of [[local quantum field theory]]. This is now called "Nuiten's lemma" in ([Wolters-Halvorson 13](#WoltersHalvorson13)). Moreover, ([Nuiten 12](#Nuiten12)) shows that a natural [[descent]] condition on spacetime nets of [[Bohr toposes]] is equivalent to _[strong locality](local+net#StrongLocality)_ of the [[quantum field theory]], something slightly weaker than _[Einstein causality](local+net#EinsteinLocality)_, which implies it. Since, by the above, the [[Bohr topos]] is the geometric incarnation of the [[Jordan algebra]] structure on [[quantum observables]], one might see this as a reply to the alleged lack of implications (in [the AS review, 04](#ASReview04)) of [[Pascual Jordan|Pascual Jordan's]] ideas from the 1930s to quantum field theory.

Be that as it may, notice that generally local systems of [[ringed toposes]] are to be expected to naturally encode [[quantum field theory]] on general grounds: the modern [[AQFT]]-style formulation of [[classical field theory]]/[[prequantum field theory]] via [[factorization algebras]] or similar assigns to subsets of [[spacetime]] the [[derived critical locus]] of local [[field (physics)|fields]] extremizing the given [[local action functional]], hence the [[derived geometry|derived space]] of solutions to the [[Euler-Lagrange equations]] of motion: the [[covariant phase space]] (pre-quantum). In [[physics]] this [[derived critical locus]] is modeled explicitly as a [[BV-BRST formalism|BV-complex]], but when realized in the full technical beauty of [[derived geometry]] it is in fact a [[structured (infinity,1)-topos|higher ringed topos]], as explained by [[Jacob Lurie]]. In view of this incarnation of [[classical field theory]] in [[AQFT]]-perspective as a net of [[structured (infinity,1)-topos|higher ringed topos]], it seems rather natural that under [[quantization]] it remains a net of ringed toposes, sending ringed toposes incarnating classical [[covariant phase spaces]] to ringed toposes incarnating their quantized version as quantum phase spaces.

## Relation to the traditional non-commutative geometry
 {#RelationToTheNonCommutativePhaseSpace}

In the above we highlighted that by [the fundamental theorems](#ListOfTheoremsInvoked) on the foundations of [[quantum mechanics]] -- going back to insights of [[Pascual Jordan]] way back in 1930 and formally affirmed by more recent results by Alfsen-Shultz and Harding-Döring -- it follows that, accurately speaking, quantum phase space is not really an object in [[noncommutative geometry]], but rather in a kind of _non-associative_ "Jordan geometry" which is naturally captured by the [[ringed topos|ringed]] [[Bohr topos]] over the [[poset of commutative subalgebras]].

This observation indeed puts doubt on the long and widely held believe that the quantum phase space is an object in [[noncommutative geometry]], a belief that in fact motivated much of the development of [[noncommutative geometry]] in the first place.

But that this is not really true was "well known" all along: it is pointed out for instance in the foundational text ([Bates-Weinstein 97](#BatesWeinstein97)) on [[geometric quantization]]. On page 80 there is highlighted how given a classical [[phase space]] represented by a [[Poisson manifold]] $(X, \{-,-\})$, hence of a [[manifold]] that carries

1. a commutative algebra of classical observables

1. equipped with a compatible non-commutative [[Lie bracket]] $\{-,-\}$ (the [[Poisson bracket]])

the [[quantization]] of this data is to be thought of as applying to these two items separately:

1. a non-associative but commutative [[Jordan algebra]] structure of quantum observables is the [[deformation quantization]] of the commutative algebra structure of classical observables;

1. a non-commutative [[Lie bracket]] structure is the deformation quantization of the Poisson bracket

and that if the quantization of both items is given by a single [[C*-algebra]] structure, then the non-associative commutative Jordan algebra structure is the one induced by the anticommutator

$$
  A \circ B = \tfrac{1}{2}(A B + B A)
$$

while the non-commutative algebra structure is that given by the commutator

$$
  [A,B] = \tfrac{1}{2}(A B - B A)
$$

See also at _[Jordan algebra -- Origin in quantum physics](Jordan+algebra#OriginInQuantumPhysics)_. This splitting of the notion of quantization into a Lie-algebraic and a Jordan algebraic aspect is formalized in the notion of _[[Jordan-Lie-Banach algebra]]_.

To understand that this makes good sense notice that this decomposition is that into [[kinematics]] and [[dynamics]] of [[quantum mechanics]]:

1. [[kinematics]] --- the construction of the [[quantum observables]] and of the [[space of quantum states]] alone indeed does _not_ involve the associative product of the [[C*-algebra]];

1. [[dynamics]]  ---  the commutator [[Lie bracket]] structure is used to impose quantum [[Hamiltonian flows]] $\exp([A,-])$, hence (time) propagation along the [[trajectories]] generated by the observables $A$.

But then observe in addition that when we pass from [[quantum mechanics]] to [[quantum field theory]] axiomatized as [[AQFT]], then in fact time propagation is no longer implemented by [[inner automorphisms]] of the form $\exp([A,-])$. Indeed it is impossible for any realistic [[physical system]] with infinitely many degrees of freedom (such as a [[field (physics)|field]]) to have time evolution given by an [[inner automorphism]]. That this does work for [[quantum mechanics]] is really an artifact of the degenerate case of finitely many degrees of freedom considered there.

Instead, in [[AQFT]] the [[spacetime]]-evolution of the quantum fields is all encoded in the transition functions of the [[local net of observables]]. By the [above](#QuantumPhaseSpaceAsJordanGeometry) discussion, this is however, accurately speaking, not actually to be thought of as a net of [[C*-algebras]], but rather as a net of [[Jordan algebras]], hence as a net of [[Bohr toposes]]. This way the need for the non-commutative algebraic structure disappears and only the non-associative commutative [[Jordan algebra]] structure remains.

In this context it may be noteworthy to recall what is a well-kept secret: despite much work on [[AQFT]] with the traditional [[Haag-Kastler axioms]] that demand to assign [[C*-algebras]] to regions of spacetime, there is to date not a single non-[[free field theory|free field]] example in [[spacetime]] [[dimension]] greater than 3 of these axioms. (There are plenty of interesting example in dimension 2, though, see at _[[conformal net]]_ and pointers given there.)

On the other hand, more recently the variant of the [[AQFT]] axioms known as _[[factorization algebras]]_ has been shown to admit plenty of interesting examples of [[quantum field theory]]. Comparison of the axioms is not straightforward and should be taken with a grain of salt, but it is maybe noteworthy that a [[factorization algebra]] is indeed a net that assigns to a region of spacetime _not_ an algebra structure. The algebra structure there is instead all encoded into the way that spacetime regions are included into each other.

## List of theorems invoked
 {#ListOfTheoremsInvoked}

For reference, the following lists that theorems about the standard foundations of quantum mechanics that are being referred to [above](QuantumPhaseSpaceAsJordanGeometry):

* [[Gleason's theorem]]

* [[Alfsen-Shultz theorem]]

* [[Harding-Döring-Hamhalter theorem]]

* [[Kochen-Specker theorem]]

* [[Nuiten's lemma]]

## Related entries

* [[interpretation of quantum mechanics]]

[[!include states and observables -- content]]

## References

*  [[Pascual Jordan]], Über eine Klasse nichtassociativer
hyperkomplexer Algebren, _Nachr. Ges. Wiss. Göttingen
(1932), 569--575.
 {#Jordan32}

* [[Pascual Jordan]], [[John von Neumann]] and [[Eugene Wigner]], On an algebraic generalization of the quantum mechanical formalism, _Ann. Math._ **35** (1934), 29--64.
 {#JordanvNeumannWigner34}

* A.M. Gleason, _Measures on the closed subspaces of a Hilbert space_, Journal of Mathematics and Mechanics,  Indiana Univ. Math. J. 6 No. 4 (1957), 885–893 ([web](http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050))
  {#Gleason57}

* [[Simon Kochen]], [[Ernst Specker]], _The problem of hidden variables in quantum mechanics_ , Journal of Mathematics and Mechanics 17, 59–87 (1967), ([pdf](http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1968/17/17004/pdf))
  {#KochenSpecker67}

* [[Monique Hakim]], _Topos annelés et schémas relatifs_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).
  {#Hakim72}

* Erhard Scheibe, _The logical analysis of quantum mechanics_ . Oxford: Pergamon Press, 1973.

 {#Scheibe73}

* [[Erik Alfsen]], [[Frederic Shultz]], _A Gelfand Neumark theorem for Jordan algebras_, Advances in Math., 28 (1978), 11-56.
  {#AlfsenShultz78}

* [[Erik Alfsen]], H. Hanche-Olsen, [[Frederic Shultz]], _State spaces of $C^\ast$-algebras_, Acta Math., 144 (1980), 267-305.
  {#AlfsenHOShultz80}

* Sean Bates, [[Alan Weinstein]], _Lectures on the geometry of quantization_  American Mathematical Society in the Berkeley Mathematics Lecture Notes series, 1997 ([pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf))
  {#BatesWeinstein97}

* [[Jeremy Butterfield]], [[Chris Isham]], _A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations_ ([arXiv:quant-ph/9803055](http://arxiv.org/abs/quant-ph/9803055))
  {#IshamButterfield98}

* [[Jeremy Butterfield]], John Hamilton, [[Chris Isham]], _A topos perspective on the Kochen-Specker theorem_, _I. quantum states as generalized valuations_, Internat. J. Theoret. Phys. 37(11):2669--2733, 1998, [MR2000c:81027](http://www.ams.org/mathscinet-getitem?mr=1669557), [doi](http://dx.doi.org/10.1023/A:1026680806775); _II. conceptual aspects and classical analogues_ Int. J. of Theor. Phys. 38(3):827--859, 1999, [MR2000f:81012](http://www.ams.org/mathscinet-getitem?mr=1697983), [doi](http://dx.doi.org/10.1023/A:1026652817988); _III. Von Neumann algebras as the base category_, Int. J. of Theor. Phys. 39(6):1413--1436, 2000, [arXiv:quant-ph/9911020](http://arxiv.org/abs/quant-ph/9911020), [MR2001k:81016](http://www.ams.org/mathscinet-getitem?mr=1788498),[doi](http://dx.doi.org/10.1023/A:1003667607842); _IV. Interval valuations_, Internat. J. Theoret. Phys. __41__ (2002), no. 4, 613–639, [MR2003g:81009](http://www.ams.org/mathscinet-getitem?mr=1902067), [doi](http://dx.doi.org/10.1023/A:1015276209768)
  {#ButterfieldHamiltonIsham98}

* review of Alfsen-Shultz, 2004 ([pdf](http://www.ams.org/journals/bull/2004-41-04/S0273-0979-04-01019-5/S0273-0979-04-01019-5.pdf))
 {#ASReview04}

* [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], _Bohrification of operator algebras and quantum logic_, in _Deep Beauty_ Cambridge University Press (2009) ([arXiv:0909.3468](http://arxiv.org/abs/0909.3468), [arXiv:0905.2275](http://arxiv.org/abs/0905.2275))
  {#HeunenLandsmanSpitters09}

* [[John Harding]], [[Andreas Döring]], _Abelian subalgebras and the Jordan structure of a von Neumann algebra_ ([arXiv:1009.4945](http://arxiv.org/abs/1009.4945))
 {#HardingDoering10}

* [[Jan Hamhalter]], _Isomorphisms of ordered structures of abelian $C^\ast$-subalgebras of $C^\ast$-algebras_, J. Math. Anal. Appl. 383 (2011) 391–399 ([journal](dx.doi.org/10.1016/j.jmaa.2011.05.035))
 {#Hamhalter11}

* [[Joost Nuiten]], _[[schreiber:bachelor thesis Nuiten|Bohrification of local nets of observables]]_, [Proceedings of QPL 2011](http://qpl.science.ru.nl/) [EPTCS 95](http://rvg.web.cse.unsw.edu.au/eptcs/content.cgi?QPL2011), 2012, pp. 211-218 ([arXiv:1109.1397](http://arxiv.org/abs/1109.1397))
  {#Nuiten12}

* Sander Wolters, [[Hans Halvorson]], _Independence Conditions for Nets of Local Algebras as Sheaf Conditions_ ([arXiv.1309.5639](http://arxiv.org/abs/1309.5639))
  {#WoltersHalvorson13}