symplectic geometry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

__Symplectic geometry__ is a branch of [[differential geometry]] studying [[symplectic manifold]]s and some generalizations; it originated as a formalization of the mathematical apparatus of [[classical mechanics]] and geometric optics (and the related [[semiclassical approximation|WKB-method]] in [[quantum mechanics]] and, more generally, the method of stationary phase in [[harmonic analysis]]). A wider branch including symplectic geometry is [[Poisson geometry]] and a sister branch in odd dimensions is [[contact geometry]]. A special and central role in the subject belongs to certain real-like half-dimensional submanifolds, called [[Lagrangian submanifold|lagrangian (or Lagrangean) submanifolds]], which are in some sense classical points. Symplectic geometry radically changed after the 1985 article of [[Mikhail Gromov]] on pseudoholomorphic curves and the subsequent work of [[Andreas Floer]] giving birth to symplectic topology or "hard methods" of symplectic geometry.

## Applications

* In its application to [[physics]], symplectic geometry is the fundamental mathematical language for [[Hamiltonian mechanics]], [[geometric quantization]], geometrical optics.

* A tremendous amount of insight into higher [[Lie theory]] ([[Lie groupoids]], [[Lie ∞-groupoids]], [[Lie ∞-algebroids]]) has derived from [[Alan Weinstein]]'s long-term project of understanding the role of symplectic geometry in [[geometric quantization]]. See there for more details.

+--{.query}
[[Zoran Skoda]]: it is true that large influence of Weinstein's program can not be overestimated, but it is not the origin of these considerations; it rather builds up on earlier fundamental works of Kirillov, Kostant, Souriau who invented geometric quantization, all of them originally in symplectic context; and the florishing of the subject from mid 1960s till mid 1980-s is related to their work; and other related tracks of Guillemin, Sternberg, Kashiwara, Karasev, Arnold and so on; and vast developments in harmonic analysis and representation theory (Kostant, Auslander, Vogan, Wallach, Stein...), [[microlocal analysis]] ([[Kashiwara]], Saito, Hormander, Maslov, Karasev, Duistermaat...),  [[integrable system]]s/[[quantum group]]s (this is more into more general Poisson geometry: Lie-Poisson groups, classical r-matrices, bihamiltonian systems...), and  related approaches to quantization (Berezin method, [[coherent state]]s...).
=--

* [[T-duality]] and (related) [[mirror symmetry]] interchange the symplectic data and complex algebraic geometry data. Some cases of both the symplectic and complex algebraic picture can be unified and studied in a broader concept of [[generalized complex geometry]].

## Related concepts

* There is a [[vertical categorification]] of symplectic geometry to [[higher symplectic geometry]]. This involves  [[multisymplectic geometry]] and the geometry of [[symplectic Lie n-algebroid]]s. And their combination.

  See [[higher symplectic geometry]]

* [[holomorphic symplectic geometry]]

* [[Arnold conjecture]]

* [[phase space]], [[Hamiltonian flow]], [[prequantum line bundle]]

* [[symplectic formulation of quantum mechanics]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

Textbook accounts:

* [[Rolf Berndt]], _An introduction to symplectic geometry_, Graduate Studies in Mathematics **26** (2001) [[ams:gsm-26](https://bookstore.ams.org/gsm-26), [pdf](http://www.ams.org/bookstore/pspdf/gsm-26-prev.pdf)]

* [[Ana Cannas da Silva]], *Lectures on Symplectic Geometry*, Lecture Notes in Mathematics **1764**, Springer (2008) [[doi:10.1007/978-3-540-45330-7](https://doi.org/10.1007/978-3-540-45330-7), [pdf](https://www.math.tecnico.ulisboa.pt/~acannas/Books/lsg.pdf)]

Lecture notes:

* Konstantin Athanassopoulos, *Notes on Symplectic Geometry*, lecture notes (2015) [[pdf](http://users.math.uoc.gr/~athanako/symplectic.pdf), [[Athanassopoulos-Symplectic.pdf:file]]]

Discussion from the point of view of [[homological algebra]] of [[abelian sheaves]] is in

* C. Viterbo, _An introduction to symplectic topology through sheaf theory_ (2010) ([pdf](http://www.math.polytechnique.fr/cmat/viterbo/Eilenberg/Eilenberg.pdf))

The notion of symplectic geometry may be understood as the mathematical structure that underlies the physics of [[Hamiltonian mechanics]]. A classical monograph that emphasizes this point of view:

* [[Vladimir Arnol'd]], _[[Mathematical methods of classical mechanics]]_, Graduate Texts in Mathematics **60**, Springer (1978) [[doi:10.1007/978-1-4757-1693-1](https://doi.org/10.1007/978-1-4757-1693-1)]

For more on this see [[Hamiltonian mechanics]].

Symplectic geometry is also involved in [[geometric optics]], [[geometric quantization]] and the study of oscillatory integrals and [[microlocal analysis]]. Books concentrating on these topics include

* [[Victor Guillemin]], [[Shlomo Sternberg]], *Geometric asymptotics*, Mathematical Surveys and Monographs **14**, AMS (1977) [[ams:surv-14](https://bookstore.ams.org/surv-14)]

* Sean Bates, [[Alan Weinstein]], _[[Lectures on the geometry of quantization]]_, [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)

* J.J. Duistermaat, _Fourier integral operators_, Progress in Mathematics, Birkhäuser 1995 (and many other references at [[microlocal analysis]]).

* [[Alan Weinstein]], _Symplectic geometry_, (survey)  Bull. Amer. Math. Soc. 5 (1981), 1-13, [doi](http://dx.doi.org/10.1090/S0273-0979-1981-14911-9)
* N. R. Wallach, _Symplectic geometry and Fourier analysis_, Math. Sci. Press, Brookline, Mass., 1977.
* wikipedia [symplectic geometry](http://en.wikipedia.org/wiki/Symplectic_geometry)
Application: "Jacobi's elimination of nodes" is moved to [[equivariant localization and elimination of nodes]].

A proof of the [[Arnold conjecture]] in symplectic geometry is in

* [[Mohammed Abouzaid]], [[Andrew J. Blumberg]], _Arnold Conjecture and Morava K-theory_, ([arXiv:2103.01507](https://arxiv.org/abs/2103.01507))


symplectic geometry - contents

**[[symplectic geometry]]**

[[higher symplectic geometry]]

## Background

* [[geometry]]

* [[differential geometry]]

## Basic concepts

* [[almost symplectic structure]], [[metaplectic structure]], [[metalinear structure]]

* [[symplectic form]], [[n-plectic form]]

* [[symplectic Lie n-algebroid]]

  * [[symplectic manifold]]

  * [[Poisson manifold]], [[Poisson Lie algebroid]]

    [[Poisson n-algebra]]

  * [[Courant Lie 2-algebroid]]

* [[symplectic infinity-groupoid]]

  * [[symplectic groupoid]]

* [[symplectomorphism]], [[symplectomorphism group]]

  * [[symplectic vector field]], [[Hamiltonian vector field]]

    * [[Hamiltonian]], [[Hamiltonian form]]

  * [[symplectic gradient]]

* [[Hamiltonian action]], [[moment map]]

* [[symplectic reduction]], [[BRST-BV formalism]]

* [[isotropic submanifold]], [[Lagrangian submanifold]], [[polarization]]

## Classical mechanics and quantization

* [[Hamiltonian mechanics]]

* [[quantization]]

  [[deformation quantization]],

  **[[geometric quantization]]**, [[higher geometric quantization]]

  * [[geometric quantization of symplectic groupoids]]

  * [[prequantum line bundle]], [[prequantum circle n-bundle]]

    * [[contact manifold]], [[contactomorphism]]

    * [[contact form]], [[Reeb vector field]]

  * [[quantomorphism group]], [[quantomorphism n-group]]

    * [[Poisson bracket]], [[Poisson algebra]]

    * [[Poisson bracket Lie n-algebra]]

    * [[Heisenberg Lie algebra]], [[Heisenberg Lie n-algebra]]

    * [[Heisenberg group]]

[Edit this sidebar](/nlab/edit/symplectic+geometry+-+contents)


symplectic gradient

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _symplectic gradient_ is the analog in [[symplectic geometry]] of the [[gradient]] in [[Riemannian geometry]].

## Definition

Let $(X,\omega)$ be a [[symplectic manifold]] and $H \in C^\infty(X)$ a [[function]].

The **symplectic gradient** of $H$ is the [[vector field]]

$$
  X_H := \omega^{-1} d_{dR} H \in \Gamma(T X)
  \,,
$$

where $d_{dR} : C^\infty(X) \to \Omega^1(X)$ is the [[de Rham differential]].

This is the unique vector field $X_H$ such that

$$
  d_{dR} H = \omega(-,X_H)
$$

The function $H$ in this context is called an [[Hamiltonian]] and the vector field $H_X$ an [[Hamiltonian vector field]].

Equivalently, the vector field $X_H$ is defined by the condition

$$
X_H(f)=\{H,f\}
$$
for any $f \in C^\infty(X)$, where $\{\,,\,\}$ is the [[Poisson bracket]] on $(M,\omega)$.

## Examples

If $(M,g)$ is $\mathbb{R}^{2n}$ endowed with the standard [[symplectic form]] $\omega=dp_i\wedge dq^i$, then

$$
X_f= \sum_{i=1}^n\frac{\partial f}{\partial p_i}\frac{\partial}{\partial q^i}-\frac{\partial f}{\partial q^i}\frac{\partial}{\partial p_i}.
$$

[[!redirects symplectic gradients]]


symplectic group

> See also _[[quaternionic unitary group]]_.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $n \in \mathbb{N}$, the **symplectic group** $Sp(2n, \mathbb{R})$ is one of the [[classical Lie groups]].

It is the subgroup of the [[general linear group]] $GL(2n, \mathbb{R})$ of elements preserving the canonical [[symplectic form]] $\Omega$ on the [[Cartesian space]] $\mathbb{R}^{2n}$, that is: the group consisting of those [[matrices]] $A$ such that

$$
  A^T \Omega A = \Omega
  \,.
$$

The symplectic group should not be confused with the [[compact symplectic group]] $Sp(n)$, which is the [[maximal compact subgroup]] of the _complex_ symplectic group $Sp(2n,\mathbb{C})$.

## Properties

### Maximal compact subgroup
 {#MaximalCompactSubgroup}

The [[maximal compact subgroup]] of the symplectic group $Sp(2n, \mathbb{R})$ is the [[unitary group]] $U(n)$.

### Homotopy groups
 {#HomotopyGroups}

By the [above](#MaximalCompactSubgroup) the [[homotopy groups]] of the symplectic group are those of the corresponding [[unitary group]].

In particular the first [[homotopy group]] of the symplectic group is the [[integers]]

$$
  \pi_1(Sp(2n,\mathbb{R})) \simeq \mathbb{Z}
  \,.
$$

The unique connected [[double cover]] obtained from this is the [[metaplectic group]] [[group extension|extension]] $Mp(2n) \to Sp(2n, \mathbb{R})$.

## Related concepts

* [[affine symplectic group]]

* [[metaplectic group]]

* [[extended affine symplectic group]]

* [[orthosymplectic supergroup]]

* A higher analog of the symplectic group in [[2-plectic geometry]] is the [[exceptional Lie group]] [[G2]] (see there for more details).

* [[quaternionic unitary group]]

## References

The term "symplectic group" was suggested in

* [[Hermann Weyl]], _The Classical Groups: their invariants and representations_ (1939, p. 165)

by

> The name "complex group" formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word "complex" in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective "symplectic." Dickson calls the group the "Abelian linear group" in homage to Abel who first studied it.

On [[homotopy groups]]:

* {#MimuraToda63} Mamoru Mimura, Hiroshi Toda, _Homotopy groups of symplectic groups_, J. Math. Kyoto Univ. Volume 3, Number 2 (1963), 251-273 ([Euclid:1250524819](https://projecteuclid.org/euclid.kjm/1250524819))

O. Meara's book studies symplectic group of a finite dimensional symplectic or even alternating space (space with an alternating form, not necessarily nondegenerate) over an arbitrary field

* O. T. Meara, _Symplectic groups_, Math. Surveys __16__, Amer. Math. Soc. 1978

A generalization, a symplectic group over a noncommutative algebra with involution is studied in

* D. Alessandrini, A. Berenstein, [[V. Retakh]], E. Rogozinnikov,  A. Wienhard, _Symplectic groups over noncommutative algebras_ Sel. Math. New Ser. __28__, 82 (2022) [doi](https://doi.org/10.1007/s00029-022-00787-x)

[[!redirects symplectic group]]
[[!redirects symplectic groups]]


symplectic groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[Poisson manifold]] may be thought of as a [[Poisson Lie algebroid]], a [[Lie algebroid]] with extra structure: called an [[n-symplectic manifold]] for $n = 1$.

By [[Lie integration]] this Lie algebroid should integrate to a [[Lie groupoid]] with extra structure. Symplectic groupoids are supposed to be those objects that integrate [[n-symplectic manifold]] aka [[Poisson manifold]]s in this sense.

The [[category algebra|groupoid algebra]] of these symplectic groupoids are [[C-star algebra]]s that may be regarded as the [[quantization]] of the original [[Poisson manifold]]. This is described in the references below.

## Definition
 {#Definition}

The original definition of [Weinstein 1987](#Weinstein87) is this:

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

A **symplectic Lie groupoid** is a [[Lie groupoid]] $\mathbf{X}_\bullet$ whose manifold of [[morphisms]] $\mathbf{X}_1$ is equipped with a [[symplectic manifold|symplectic structure]] whose symplectic form $\omega \in \Omega^2_{closed}(\mathbf{X}_1)$ is _multiplicative_ in that the alternating sum of its canonical [[pullback of a differential form|pullbacks]] to the space $\mathbf{X}_2$ of composable morphisms vanishes:

$$
 0 =
  \delta \omega = pr_1^* \omega - compose^* \omega
  + pr_2^* \omega
  \,.
$$

=--

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

The manifold of [[objects]] $\mathbf{X}_0$ of a symplectic Lie groupoid $\mathbf{X}_\bullet$, def. \ref{WeinsteinsDefintion}, carries the structure of a [[Poisson manifold]] which is unique, up to [[isomorphism]], with the property that the target map $t \colon \mathbf{X}_1 \to \mathbf{X}_0$ is a [[homomorphism]] of [[Poisson manifolds]] (canonically regarding the [[symplectic manifold]] $(\mathbf{X}_1, \omega)$ as a Poisson manifold).

The [[Poisson manifolds]] that arise this way as $\mathbf{X}_0$ of a symplectic Lie groupoid are called **integrable Poisson manifolds**.

=--

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

Reformulated more abstractly, def. \ref{WeinsteinsDefintion} says that the differential form $\omega$, when extended to a triple

$$
  (0, \omega, 0)
  \in
  \oplus_{k = 0,1,2} \Omega^{3-k}(\mathbf{X}_{k})
$$

is a [[cocycle]] of degree 3 in the [[de Rham complex]] of $\mathbf{X}$, identified with the [[simplicial de Rham complex]] of the [[nerve]] $X_\bullet$ of $X$.

=--

This observation leads to the following generalization

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

A **pre-quasi symplectic groupoid** is a [[Lie groupoid]] $\mathbf{X}$ equipped with a [[differential 2-form]] $\omega_2 \in \Omega^2(\mathbf{X}_1)$ and a differential 3-form $\omega_3 \in \Omega^3(\mathbf{X}_0)$ such that

$$
  (0, \omega_2, \omega_3) \in \oplus_{k = 0,1,2} \Omega^{3-k}(\mathbf{X}_{k})
$$

is a [[cocycle]] in the [[simplicial de Rham complex]] of $\mathbf{X}_\bullet$, hence such that

$$
  \delta \omega_2 = 0
$$

$$
  d\omega_2 + \delta \omega_3 = 0
  \,,
$$

where $\delta = \sum_{k} (-1)^k \partial_k^*$ is the alternating sum of the [[pullback of a differential form|pullbacks]] along the face maps of the [[nerve]] $\mathbf{X}_\bullet$.

=--

This appears as ([Xu, def. 2.1](#Xu), [LG-Xu, def. 2.1](#LGX)). This structure is called a **twisted presymplectic groupoid** in ([BCWZ, def. 2.1](#BCWZ)).

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

Since therefore a (pre-)symplectic groupoid is really a Lie groupoid equipped with a cocycle in degree-3 [[de Rham cohomology]] (instead of degree 2 as for a symplectic manifold),  it is really rather an object in [[2-plectic geometry]].

=--

## Properties

### Lie integration and Poisson manifolds

Every [[Lie groupoid]] [[Lie integration|integrating]] a [[Poisson Lie algebroid]] is naturally a symplectic Lie groupoid. Picking always the unique source-simply connected integrating [[Lie groupoid]] produces a [[functor]]

$$
  \Sigma : PoissonManifolds \to SymplecticGroupoids
  \,.
$$

When the [[Poisson manifold]] we start with happens to be a [[symplectic manifold]], then its symplectic Lie groupoid is always the [[fundamental groupoid]] of $X$:

$$
  (X,\pi)\;\; symplectic
  \;\;\Rightarrow\;\;
  \Sigma(X,\pi) \simeq_{iso} \Pi(X)
  \,.
$$

When $X$ is [[simply connected topological space|simply connected]] such that $\Pi(X)$ is the [[codiscrete category|codiscrete groupoid]] $Pair(X)$ we have that the symplectic form on $Mor(\Pi(X)) = X \times X$ is $\omega \otimes (-\omega)$, for $\omega$ the [[symplectic manifold|symplectic form]] on $X$.

Conversely, for every symplectic groupoid $\mathbf{X}$ there is a unique [[Poisson manifold]] structure on its manifold $\mathbf{X}_0$ of objects such that the [[codomain]] map $t \colon \mathbf{X}_1 \to \mathbf{X}_0$ is a [[homomorphism]] of [[Poisson manifolds]]. (For instance [Racaniere, theorem 6.3](#Racaniere)) One says also that $\mathbf{X}$ **integrates the Poisson manifold** $\mathbf{X}_0$.

### Symplectic realization

The source map of a symplectic groupoid over a Poisson manifold constitutes a _[[symplectic realization]]_ of this Poisson manifold, hence its canonical desingularization via [[Lie integration]]. See at _[[symplectic realization]]_ for more.

### In geometric quantization of Poisson manifolds

In the _groupoid approach to quantization_  symplectic groupoids are used to discuss [[geometric quantization]] not just of [[symplectic manifold]]s but more generally of [[Poisson manifold]]s.

See [[geometric quantization of symplectic groupoids]].

### As Lie integration of Poisson Lie algebroid

(...)

### A reduced phase space of open Poisson sigma-model

The symplectic groupoid of a Poisson manifold is also the
[[reduced phase space]] of the open sector of the
corresponding [[Poisson sigma-model]]. ([Cattaneo-Felder 01](#CattaneoFelder01))

## Examples

### Of Lie-Poisson structure
 {#OfLiePoissonStructure}

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

Let $G$ be a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$ and consider the [[dual vector space]] $\mathfrak{g}^*$ equipped with its
[[Lie-Poisson structure]]. Then the [[action groupoid]] $\mathfrak{g}^* \sslash G$ of the [[coadjoint action]] carries a multiplicative symplectic form $\omega$ induced by the identification of the manifold of [[morphisms]] with the [[cotangent bundle]] of the group, $G \times \mathfrak{g}^* \simeq T^* G$, induced by right translation from the [[Poincare form]] on the [[cotangent bundle]]. This makes $(\mathfrak{g}^* //G, \omega)$ a symplectic groupoid which [[Lie integration|Lie integrates]] the [[Lie-Poisson structure]] on $\mathfrak{g}^*$.

=--

This seems to be due to [Weinstein 1991, ex. 3.2](#Weinstein91), see [Bursztyn & Crainic 2005, ex. 4.3](#BursztynCrainic), [Nuiten 2013, p. 111](#Nuiten13).

## Related concepts

* [[symplectic manifold]]

* [[symplectic Lie n-algebroid]]

* [[symplectic infinity-groupoid]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

The notion of symplectic groupoids was apparently proposed independently by Karasëv, [[Alan Weinstein|Weinstein]], and Zakrzewski, all motiviated from the problem of [[quantization]].

* {#Weinstein87} [[Alan Weinstein]], _Symplectic groupoids and Poisson manifolds_, Bull. Amer. Math. Soc. (N.S.) **16** (1987) 101-104 [[euclid:bams/1183553676](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-16/issue-1/Symplectic-groupoids-and-Poisson-manifolds/bams/1183553676.full)]

* [[Alan Weinstein]], _Symplectic groupoids, geometric quantization, and irrational rotation algebras_, in:
_Symplectic geometry, groupoids, and integrable systems_ (Berkeley, CA, 1989), Springer (1991) 281-290 [[doi:10.1007/978-1-4613-9719-9_19](https://doi.org/10.1007/978-1-4613-9719-9_19), MR1104934]

* [[Alan Weinstein]], _Tangential deformation quantization and polarized symplectic groupoids_, in: _Deformation theory and symplectic geometry_ (Ascona, 1996), 301-314, Kluwer (1997) [[ISBN:9780792345251](https://link.springer.com/book/9780792345251), [MR1480730](http://www.ams.org/mathscinet-getitem?mr=1480730)]

* M. V. Karas\"ev, _The Maslov quantization conditions in higher cohomology and analogs of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds II_, Selecta Mathematica Sovietica 8, pp. 235–257, 1989.

* S. Zakrzewski, Quantum and classical pseudogroups I, II, Commun. Math. Phys. 134 (1990)

See also the references at _[[geometric quantization of symplectic groupoids]]_ .

Lecture notes include

* {#Racaniere} Sébastien Racanière, _Lie algebroids, Lie groupoids and Poisson geometry_ ([pdf](http://empg.maths.ed.ac.uk/Activities/GCY/Racaniere.pdf))

The notion of pre-quasi-symplectic groupoids is introduced and the intepretation of symplectic groupoids in [[higher geometry]] is made fairly explicit in

* {#Xu} [[Ping Xu]], _Momentum Maps and Morita Equivalence_, J. Diff. Geom ([arXiv:math/0307319](http://arxiv.org/abs/math/0307319))

* {#LGX} [[Camille Laurent-Gengoux]], [[Ping Xu]], _Quantization of pre-quasi-symplectic groupoids and their Hamiltonian spaces_ in _The Breadth of Symplectic and Poisson Geometry_, Progress in Mathematics **232** (2005)  423-454 [[arXiv:math/0311154](http://arxiv.org/abs/math/0311154), [doi:10.1007/0-8176-4419-9_14](https://doi.org/10.1007/0-8176-4419-9_14)]

These "pre-quasi-symplectic groupoids" had been called "twisted presymplectic groupoids" in

* {#BCWZ} [[Henrique Bursztyn]], [[Marius Crainic]], [[Alan Weinstein]], [[Chenchang Zhu]], _Integration of twisted Dirac brackets_ ([arXiv:math/0303180](http://arxiv.org/abs/math/0303180))

The identification with reduced phase spaces of the open Poisson sigma-model is in

* {#CattaneoFelder01} [[Alberto Cattaneo]], [[Giovanni Felder]], _Poisson sigma models and symplectic groupoids_, in _Quantization of Singular Symplectic Quotients_, (ed. [[Klaas Landsman]], M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198 (Birkhäuser,
2001), 61–93. ([arXiv:math/0003023](http://arxiv.org/abs/math/0003023))

Further developments:

* {#Weinstein91} [[Alan Weinstein]], *Noncommutative geometry and geometric quantization*, in P. Donato et al. (eds.) _Symplectic geometry and Mathematical physics_, Progr. Math **99** Birkhäuser (1991) 446-461.  [doi:10.1007/978-1-4757-2140-9_23](https://doi.org/10.1007/978-1-4757-2140-9_23).

* [[Ping Xu]], _Morita equivalence and symplectic realizations of Poisson manifolds_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 25 no. 3 (1992) ([NUMDAM](http://www.numdam.org/item?id=ASENS_1992_4_25_3_307_0))

* {#BursztynCrainic} [[Henrique Bursztyn]], [[Marius Crainic]], *Dirac structures, momentum maps and quasi-Poisson manifolds*, in: *The Breadth of Symplectic and Poisson Geometry*, Progress in Mathematics **232**,  Birkhäuser (2005) 1-40 [[doi:10.1007/0-8176-4419-9_1](https://doi.org/10.1007/0-8176-4419-9_1)]

* F. Bonechi, N. Ciccoli, N. Staffolani, M. Tarlini, _The quantization of the symplectic groupoid of the standard Podles sphere_ ([arXiv:1004.3163](http://arxiv.org/abs/1004.3163))

* {#Nuiten13} [[Joost Nuiten]], pp. 108 in: *[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]*, MSc thesis, Utrecht (August 2013) [[pdf](/schreiber/files/thesisNuiten.pdf), [dspace:1874/282756](https://dspace.library.uu.nl/handle/1874/282756), [talk slides](/schreiber/files/thesisNuitenTalk.pdf)]

The [[formal groupoid]] version of symplectic groupoids is discussed in

* [[Alberto S. Cattaneo|Alberto Cattaneo]], Benoit Dherin, [[Giovanni Felder]], _Formal symplectic groupoid_, Comm. Math. Phys. __253__ (2005), no. 3, 645–674 [math.SG/0312380](http://arxiv.org/abs/math/0312380) [doi](HTTP://dx.doi.org/10.1007/s00220-004-1199-z){#CattaneoDherinFelder}; _Formal Lagrangian operad_, [math.SG/0505051](http://arxiv.org/abs/math/0505051)

[[!redirects symplectic Lie groupoid]]
[[!redirects symplectic groupoids]]
[[!redirects symplectic Lie groupoids]]


symplectic infinity-groupoid

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _symplectic $\infty$-groupoid_ is a [[smooth ∞-groupoid]] equipped with a [[symplectic form]], or, more generally, with an [[n-plectic form]].

This is the generalization of the notion of [[symplectic manifold]] to [[higher symplectic geometry]]. It is also the image under [[Lie integration]] of the notion of [[symplectic Lie n-algebroid|symplectic L-∞ algebroid]], which is also a higher analog of symplectic manifolds, but in an [[infinitesimal object|infinitesimal]] way.

Notice that every symplectic manifold is in particular a [[Poisson manifold]] and that the structure of a Poisson manifold is equivalently encoded in the corresponding [[Poisson Lie algebroid]]. A _[[symplectic groupoid]]_ is the [[Lie integration]] of such a Poisson Lie algebroid. Therefore, strictly speaking, already "ordinary" symplectic geometry secretly involves [[Lie groupoid]]s. This insight is exploited in the refinement of [[geometric quantization of symplectic groupoids]].

## Definition

For any $n \in \mathbb{N}$, a _[[symplectic Lie n-algebroid]]_ $(\mathfrak{P}, \omega)$ is an [[L-∞ algebroid]] $\mathfrak{P}$ that is equipped with a quadratic and non-degenerate $L_\infty$-[[invariant polynomial]].

Under [[Lie integration]] $\mathfrak{P}$ integrates to a [[smooth infinity-groupoid|smooth n-groupoid]] $\tau_n \exp(\mathfrak{P})$. Under the [[∞-Chern-Weil homomorphism]] the invariant polynomial induces an [[smooth infinity-groupoid -- structures|differential form on an ∞-groupoid]]

$$
  \omega : \tau_n \exp(\mathfrak{P}) \to \flat_{dR} \mathbf{B}^{n+2} \mathbb{R}
$$

representing a class $[\omega] \in H^{n+2}_{dR}(\tau_n \exp(\mathfrak{P}))$.

Let

$$
  SymplSmooth\infty Grpd
  \hookrightarrow
  Smooth\infty Grpd/(\coprod_{n}\mathbf{\flat}_{dR}\mathbf{B}^{n+2}\mathbb{R})
$$

be the full [[sub-(∞,1)-category]] of the [[over-(∞,1)-topos]] of [[Smooth∞Grpd]] over the de Rham coefficient objects on those objects in the image of this construction.

We say an object on $SymplSmooth \infty Grpd$ is a **symplectic smooth $\infty$-groupoid**.

(There are evident variations of this for the ambient [[Smooth∞Grpd]] replaced by some variant, such as [[SynthDiff∞Grpd]] or [[SmoothSuper∞Grpd]].)

## Examples
  {#ByLieIntegration}

The [[symplectic form]] $\omega$ on a [[symplectic Lie n-algebroid]] $\mathfrak{a}$ is Lie theoretically an [[invariant polynomial]]. Therefore by [[infinity-Chern-Weil theory]] it induces a moprhism

$$
  \exp(\omega)
    :
  \tau_n\exp(\mathfrak{a})
    \to
  \mathbf{\flat}_{dR} \mathbf{B}^{n+2} \mathbb{R}
$$

from the [[Lie integration]] of $\mathfrak{a}$ to the de Rham coefficient object: this is an $(n+2)$-form on a [[smooth ∞-groupoid]] (as discussed at [smooth ∞-groupoid -- structures -- de Rham cohomology](http://ncatlab.org/nlab/show/smooth+infinity-groupoid+--+structures#StrucDeRham)) and hence equips $\exp(\mathfrak{a})$ with the structure of a symplectic $\infty$-groupoid.

We spell this out in some special cases.

### $n = 0$ -- Symplectic manifolds

A [[symplectic Lie n-algebroid|symplectic Lie 0-algebroid]] is simply a [[symplectic manifold]], and so is its [[Lie integration]].

### $n = 1$ -- Symplectic groupoids from Poisson Lie algebroids
 {#SymplecticGroupoidsFromPoissonLieAlgebroids}

We discuss the [[Lie integration]] of [[Poisson Lie algebroids]] to [[symplectic groupoids]]. For more details and applications of this see at _[[extended geometric quantization of 2d Chern-Simons theory]]_.

Let $\mathfrak{P}$ be the [[Poisson Lie algebroid]] corresponding to a [[Poisson manifold]] that comes from a [[symplectic manifold]] $(X,\omega)$.

The [[symplectic groupoid]] associated to this is (by the discussion there) supposed to be the [[fundamental groupoid]]
$\Pi_1(X)$ of $X$ equipped on its space of morphisms with the differential form $p_1^* \omega - p_2^* \omega$, where $p_1,p_2$ are the two endpoint projections from paths in $X$ to $X$.

We demonstrate in the following how this is indeed the result of applying the [[∞-Chern-Weil homomorphism]] to this situation.

For simplicity we shall start with the simple situation where $(X,\omega)$ has a global [[Darboux coordinate chart]] $\{x^i\}$. Write $\{\omega_{i j}\}$ for the components of the [[symplectic form]] in these coordinates, and $\{\omega^{i j}\}$ for the components of the [[inverse]].

Then the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{P})$ is generated from $\{x^i\}$ in degree 0 and $\{\partial_i\}$ in degree 1, with differential given by

$$
  d_{CE} x^i = - \omega^{i j} \partial_j
$$

$$
  d_{CE} \partial_i = \frac{\partial \pi^{j k}}{\partial x^i} \partial_j \wedge \partial_k = 0
  \,.
$$

The differential in the corresponding [[Weil algebra]] is hence

$$
  d_{W} x^i = - \omega^{i j} \partial_j + \mathbf{d}x^i
$$

$$
  d_{W} \partial_i = \mathbf{d} \partial_i
  \,.
$$

By the discussion at [[Poisson Lie algebroid]], the symplectic [[invariant polynomial]] is

$$
  \mathbf{\omega} = \mathbf{d} x^i \wedge \mathbf{d} \partial_i \in W(\mathfrak{P})
  \,.
$$

Clearly it is useful to introduce a new basis of generators with

$$
  \partial^i := -\omega^{i j} \partial_j
  \,.
$$

In this new basis we have a manifest isomorphism

$$
  CE(\mathfrak{P}) = CE(\mathfrak{T}X)
$$

with the [[Chevalley-Eilenberg algebra]] of the [[tangent Lie algebroid]] of $X$.

Therefore the [[Lie integration]] of $\mathfrak{P}$ is the [[fundamental groupoid]] of $X$, which, since we have assumed global Darboux oordinates and hence [[contractible]] $X$, is just the [[pair groupoid]]:

$$
  \tau_1 \exp(\mathfrak{P}) = \Pi_1(X) = (X \times X \stackrel{\overset{p_2}{\to}}{\underset{p_1}{\to}} X)
  \,.
$$

It remains to show that the symplectic form on $\mathfrak{P}$ makes this a [[symplectic groupoid]].

Notice that in the new basis the invariant polynomial reads

$$
  \begin{aligned}
    \mathbf{\omega}
      &=
     - \omega_{i j} \mathbf{d}x^i \wedge \mathbf{d} \partial^j
     \\
      & = \mathbf{d}( \omega_{i j} \partial^i \wedge \mathbf{d}x^j)
   \end{aligned}
$$

and that we may regard this as a morphism of $L_\infty$-algebroids

$$
  \mathbf{\omega}
   :
  \mathfrak{T}\mathfrak{P} \to \mathfrak{T}b^3 \mathbb{R}
$$

The corresponding [[infinity-Chern-Weil theory|infinity-Chern-Weil homomorphism]] that we need to compute is given by the [[∞-anafunctor]]

$$
  \array{
    \exp(\mathfrak{P})_{diff}
     &\stackrel{\exp(\mathbf{\omega})}{\to}&
    \exp(b \mathbb{R})_{dR}
    &\stackrel{\int_{\Delta^\bullet}}{\to}&
    \mathbf{\flat}_{dR}\mathbf{B}^3 \mathbb{R}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \exp(\mathfrak{P})
  }
  \,.
$$

Over a test space $U$ in degree 1 an element in $\exp(\mathfrak{P})_{diff}$ is a pair
$(X^i, \eta^i)$

$$
  X^i \in C^\infty(U \times \Delta^1)
$$

$$
  \eta^i \in \Omega^1_{vert}(U \times \Delta^1)
$$

subject to the verticality constraint, which says that along $\Delta^1$ we have

$$
  d_{\Delta^1} X^i + \eta^i_{\Delta^1} = 0
  \,.
$$

The vertical morphism $\exp(\mathfrak{P})_{diff} \to \exp(\mathfrak{P})$ has in fact a [[section]] whose image is given by those pairs for which $\eta^i$ has no leg along $U$. We therefore find the desired form on $\exp(\mathfrak{P})$ by evaluating the top morphism on pairs of this form.

Such a pair is taken by the top morphism to

$$
  \begin{aligned}
    (X^i, \eta^j)
     & \mapsto
   \int_{\Delta^1}
    \omega_{i j} F_{X^i} \wedge F_{\partial^j}
   \\
    & =
  \int_{\Delta^1}
    \omega_{i j} (d_{dR} X^i + \eta^i) \wedge d_{dR} \eta^j
  \in
  \Omega^3(U)
  \end{aligned}
  \,.
$$

Using the above verticality constraint and the condition that $\eta^i$ has no leg along $U$, this becomes

$$
  \cdots =
  \int_{\Delta^1}
  \omega_{i j}
    d_U X^i
  \wedge
    d_U d_{\Delta^1} X^j
  \,.
$$

By the [[Stokes theorem]] the integration over $\Delta^1$ yields

$$
  \cdots
    =
  \omega_{i j} d_{dR} x^i \wedge \eta^j|_{0}
   -
  \omega_{i j} d_{dR} x^i \wedge \eta^j|_{1}
  \,.
$$

This completes the proof.

### $n = 2$ -- Symplectic 2-groupoids from Courant Lie 2-algebroids

## Geometric quantization of symplectic $\infty$-groupoids
  {#HigherGeometricQuantization}

### Idea
 {#GeometricQuantizationIdea}

The notion of _[[symplectic manifold]]_ formalizes in [[physics]] the concept of a _[[classical mechanical system]]_ . The notion of _[[geometric quantization]]_ of a symplectic manifold is one formalization of the general concept in physics of _[[quantization]]_ of such a system to a _[[quantum mechanical system]]_ .

Or rather, the notion of [[symplectic manifold]] does not quite capture the most general systems of [[classical mechanics]]. One generalization requires passage to _[[Poisson manifolds]]_ . The original methods of [[geometric quantization]] become meaningless on a Poisson manifold that is not symplectic.

However, a Poisson structure on a manifold $X$ is equivalent to the structure of a [[Poisson Lie algebroid]] $\mathfrak{P}$ over $X$. This is noteworthy, because the latter _is_ again symplectic, as a [[Lie algebroid]], even if the underlying Poisson manifold is not symplectic: it is a _[[symplectic Lie n-algebroid|symplectic Lie algebroid]]_ .

Based on related observations it was suggested that the notion of _[[symplectic groupoid]]_ (see the references there) should naturally replace that of _symplectic manifold_ for the purposes of geometric quantization to yield a notion of _[[geometric quantization of symplectic groupoids]]_ .

Since a symplectic manifold can be regarded as a [[symplectic Lie n-algebroid|symplectic Lie 0-algebroid]] and also as a symplectic [[smooth infinity-groupoid|smooth 0-groupoid]], this step amounts to a kind of [[categorification]] of [[symplectic geometry]].

More or less implicitly, there has been strong evidence that this shift in perspective is substantial: the _[[deformation quantization]]_ (see there for references) of a [[Poisson manifold]] turns out to be constructible in terms of [[correlator]]s of the 2-dimensional [[TQFT]] called the _[[Poisson sigma-model]]_ associated with the corresponding [[Poisson Lie algebroid]]. The fact that this is 2-dimensional and not 1-dimensional, as the [[quantum mechanical system]] that it thus encodes, is a direct reflection of this [[categorification]] shift of degree -- see _[[holographic principle]]_ for more on this.

On general abstract grounds this already suggests that it makes sense to pass via [[higher category theory|higher categorification]] further to
[[Courant Lie 2-algebroid|symplectic Lie 2-algebroid]]s, and generally [[symplectic Lie n-algebroid]]s, as well as to symplectic [[2-groupoid]]s, symplectic [[3-groupoids]], etc. up to symplectic $\infty$-groupoids.

Formal hints for such a generalization had been noted in ([Ševera](#Severa)), in particular in its concluding table. More indirect -- but all the more noteworthy -- hints came from [[quantum field theory]], where it was observed that a generalization of symplectic geometry to _[[multisymplectic geometry]]_ of degree $n$ more naturally captures the description of $n$-dimensional [[QFT]] (notice that [[quantum mechanics]] may be understood as $(0+1)$-dimensional QFT). For, observe that the symplectic form on a [[symplectic Lie n-algebroid]] is, while always "binary", nevertheless a representative of [[de Rham cohomology]] in degree $(n+2)$.

There is a natural formalization of these higher symplectic structures in the context of any [[cohesive (∞,1)-topos]]. Moreover, with ([FRS](#FRS)) we may observe that symplectic forms on [[L-∞ algebroid]]s have a natural interpretation in [[∞-Lie theory]]: they are $L_\infty$-[[invariant polynomial]]s. This means that the [[∞-Chern-Weil homomorphism]] applies to them.

We shall show below that all notions of geometric quantization of symplectic $\infty$-groupoids have a natural interpretation in terms of these canonical structures. For instance the higher "prequantum line bundle" is nothing but the [[circle n-bundle with connection]] that the [[∞-Chern-Weil homomorphism]] assigns to the symplectic form, regarded as an $L_\infty$-[[invariant polynomial]], and the corresponding "[[holographic principle|holographic]]" [[TQFT]] -- the _[[AKSZ sigma-model]]_ -- is that given by the induced [[schreiber:∞-Chern-Simons theory|∞-Chern-Simons functional]].

### Prequantum circle $(n+1)$-bundle
 {#PrequantumBundles}

#### Idea

What is called _[[geometric quantization|(geometric) prequantization]]_ is a refinement of symplectic 2-forms to [[curvature]] 2-forms on a [[line bundle]] with [[connection on a bundle|connection]]. This is called a choice of _prequantum line bundle_ for the given symplectic form.

This has an evident generalization to closed forms of degree $(n+2)$. If integral, these may be refined to a [[curvature]] $(n+2)$-form on a _[[circle n-bundle with connection]]_ . Since in the context of [[smooth ∞-groupoid]]s we can have circle $n$-bundles over other smooth $\infty$-groupoids, this means that we canonically have the notion of **prequantum circle $(n+1)$-bundles** on a symplectic $n$-groupoid.

Moreover, since, as discussed above, the symplectic form on a symplectic $n$-groupoid may be regarded as the image of an [[invariant polynomial]] under the unrefined [[∞-Chern-Weil homomorphism]]

$$
  \omega : X \to \mathbf{\flat}_{dR} \mathbf{B}^{n+2}\mathbb{R}
  \,,
$$

the passage to the prequantum $(n+1)$-bundle with connection corresponds to passing to the _refined_ [[∞-Chern-Weil homomorphism]]

$$
  \hat \omega : X \to \mathbf{B}^{n+1}U(1)_{conn}
$$

(as discussed there).

#### Definition

Let $(X, \omega)$ be a symplectic $\infty$-groupoid. Then $\omega$ represents a class

$$
  [\omega] \in H^{n+2}_{dR}(X)
  \,.
$$

We say this form is _integral_ if it is in the image of the [[curvature]]-projection

$$
  curv : H^{n+1}_{diff}(X,U(1)) \to H^{n+2}_{dR}(X)
$$

from the [[ordinary differential cohomology]] of $X$.

In this case we say a **prequantum [[circle n-bundle with connection|circle (n+1)-bundle with connection]]** for $(X,\omega)$ is a lift of $\omega$ to $\mathbf{H}_{diff}(X, \mathbf{B}^{n+1}U(1))$.

Write $\hat X \to X$ for the underlying [[circle n-group|circle (n+1)-group]]-[[principal ∞-bundle]].

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

If $(X, \omega)$ indeed comes from the [[Lie integration]] of a [[symplectic Lie n-algebroid]] $(\mathfrak{P}, \omega)$ such that the [[period]]s of the [[L-∞ cocycle]] $\pi$ that $\omega$ [[Chern-Simons element|transgresses to]] are integral, then $\hat X$ is the [[Lie integration]] of the [[L-∞ extension]]

$$
  b^{n}\mathbb{R} \to  \hat \mathfrak{P} \to \mathfrak{P}
$$

classified by $\pi$:

$$
  \hat X \simeq \tau_{n+1} \exp(\hat \mathfrak{P})
  \,.
$$

=--

#### Examples

##### $n= 1$ -- Ordinary prequantum line bundle

See [[geometric quantization of symplectic groupoids]].

##### $n = 2$ -- String Lie 2-algebra

For $\mathfrak{g}$ a [[semisimple Lie algebra]] with  quadratic [[invariant polynomial]] $\omega$, the pair $(b \mathfrak{g}, \omega)$ is a [[symplectic Lie n-algebroid|symplectic Lie 2-algebroid]] ([[Courant Lie 2-algebroid]]) over the point.

In this case the infinitesimal prequantum line 2-bundle is the [[delooping]] of the [[string Lie 2-algebra]]

$$
  \widehat b \mathfrak{g} \simeq b \mathfrak{string}
$$

and the prequantum [[circle n-group|circle 2-group]] [[principal 2-bundle]] is the [[delooping]] of the [[smooth string 2-group]]

$$
  (\hat X \to X) =
  (\mathbf{B}String \to \mathbf{B}G)
  \,.
$$

## Poisson $L_\infty$-algebras
 {#HamiltonianVectorFields}

### Idea

A [[Hamiltonian vector field]] on an ordinary [[symplectic manifold]] is a [[vector field]] $v$ whose contraction with the [[symplectic form]] yields an exact form

$$
  \iota_v \omega = d \alpha
  \,.
$$

This definition generalizes verbatim to [[n-plectic geometry]].

We observe [below](#OrdinaryHamiltonianVectorFields) that this condition is equivalent to the fact that the [[flow]] $\exp(v) : X \to X$ of $v$ preserves the connection on any [[prequantum line bundle]], up to homotopy (up to [[gauge transformation]]). In this form the definition has an immediate generalization to symplectic $n$-groupoids.

### Definition

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

Let $\omega : X \to \mathbf{\flat}_{dR} \mathbf{B}^{n+2} U(1)$ be a symplectic $(n-1)$-groupoid and let

$$
  \hat \omega
  :
  X \to  \mathbf{B}^{n+2} U(1)_{conn}
$$

be a [prequantization](#PrequantumBundles) [[circle n-bundle with connection]].

Regard it as an object in the [[over-(∞,1)-topos]] $\mathbf{H}/\mathbf{B}^{n+2}U(1)_{conn}$.

Consider the internal [[automorphism ∞-group]]

$$
  \underline{Aut}_{\mathbf{H}/\mathbf{B}^{n+1}U(1)_{conn}}(X)
  \in \mathbf{H}
$$

of auto-[[equivalence in an (infinity,1)-category|equivalences]] that respect the [[connection on an infinity-bundle|∞-connection]] that refines $\omega$.

* Its image under $p_! : \mathbf{H}_{/\mathbf{B}^n U(1)_{conn}} \to \mathbf{H}$ we call the **[[Hamiltonian symplectomorphism]] $\infty$-group.

* Its [[∞-Lie algebra]] we call the **[[Poisson algebra|Poisson ∞-Lie algebra]]** of $(X, \omega)$.

=--

### Examples

#### Ordinary Hamiltonian vector fields
 {#OrdinaryHamiltonianVectorFields}

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

For $\omega : X \to \mathbf{\flat}_{dR} \mathbf{B}^2 U(1)$ an ordinary [[symplectic manifold]], regarded as a symplectic 0-groupoid, the general definition \ref{HamiltonianVectorFieldsOnGrpd} reproduces the standard notion of [[Hamiltonian vector fields]].

=--

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

An Hamiltonian diffeomorphism is given by a diagram

$$
  \array{
     X &&\stackrel{\phi}{\to} && X
     \\
     & {}_{\mathllap{\hat \omega}}\searrow
     &\swArrow_{\alpha}&
     \swarrow_{\mathrlap{\hat \omega}}
     \\
     && \mathbf{B} U(1)_{conn}
  }
  \,,
$$

where $\phi$ is an ordinary [[diffeomorphism]]. To compute the Lie algebra of this, we need to consider smooth 1-parameter families of such and differentiate them.

Assume first that the connection 1-form in $\hat \omega$ is globally defined $A \in \Omega^1(X)$ with $d A = \omega$. Then the above diagram is equivalent to

$$
  (\phi(t)^* A - A) = d \alpha(t)
  \,,
$$

where $\alpha(t) \in C^\infty(X)$. Differentiating this at 0 yields the [[Lie derivative]]

$$
  \mathcal{L}_v A = d \alpha'
  \,,
$$

where $v$ is the [[vector field]] of which $t \mapsto \phi(t)$ is the [[flow]].

By [[Cartan calculus]] this is

$$
  d \iota_v A + \iota_v d_{dR} A  = d \alpha'
$$

hence

$$
  \iota_v \omega = d (\alpha' - \iota_v A)
  \,.
$$

This says that for $v$ to be Hamiltonian, its contraction with $\omega$ must be exact. This is precisely the definition of [[Hamiltonian vector field]]s. The corresponding [[Hamiltonian]] here is $\alpha'-\iota_v A$.

In the general case that the prequantum [[circle n-bundle with connection]] is not trivial, we can present it by a [[Cech cohomology|Cech cocycle]] on the [[Cech nerve]] $C(P_* X \to X)$ of the based [[path space]] [[surjective submersion]] (regarding $P_* X$ as a [[diffeological space]] and choosing one base point per connected component, or else assuming without restriction that $X$ is connected).

Any [[diffeomorphism]] $\phi = \exp(v) : X \to X$ lifts to a diffeomorphism $ P_*\phi : P_* X \to P_* X$ by setting $P_* \phi(\gamma) : (t \in [0,1]) \mapsto \exp(t v)(\gamma(t))$.

So we get a diagram

$$
  \array{
     C(P_* \to X) &&\stackrel{P_*\phi}{\to} && C(P_* \to X)
     \\
     & {}_{\mathllap{\hat \omega}}\searrow
     &\swArrow_{\alpha}&
     \swarrow_{\mathrlap{\hat \omega}}
     \\
     && \mathbf{B} U(1)_{conn}
  }
$$

of [[simplicial presheaves]]. Now the same argument as above applies on $P_* X$.

=--

## Related concepts

* [[higher symplectic geometry]]

  * [[symplectic Lie n-algebroid]]

  * [[symplectic groupoid]]

* [[higher geometric quantization]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

Some ideas pointing to higher symplectic groupoids were indicated in

* [[Pavol ?evera]], _Some title containing the words "homotopy" and "symplectic", e.g. this one_ ([arXiv:math/0105080](http://arxiv.org/abs/math/0105080)).
 {#Severa}

Aspects of the relation to [[multisymplectic geometry]] are in

* [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))
 {#Rogers}

A discussion of higher symplectic geometry in a general context is in

* [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantization]]_

See also section 4.3 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ .

Some ingredients for the geometric quantization of symplectic Lie $n$-algebroids are constructed in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher Chern-Weil Derivation of AKSZ Sigma-Models]]_
  {#FRS}

[[!redirects symplectic ∞-groupoid]]
[[!redirects symplectic ∞-groupoids]]
[[!redirects symplectic infinity-groupoids]]

[[!redirects geometric quantization of symplectic ∞-groupoids]]
[[!redirects geometric quantization of symplectic infinity-groupoids]]

[[!redirects n-plectic ∞-groupoid]]
[[!redirects n-plectic infinity-groupoid]]
[[!redirects n-plectic ∞-groupoids]]
[[!redirects n-plectic infinity-groupoids]]

[[!redirects n-plectic smooth ∞-groupoid]]
[[!redirects n-plectic smooth infinity-groupoid]]
[[!redirects n-plectic smooth ∞-groupoids]]
[[!redirects n-plectic smooth infinity-groupoids]]


symplectic integrator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _symplectic integrator_ is a numerical discretization scheme for solving [[Hamilton's equations]] which takes into account the [[symplectic structure]] and, in particular, the [[conservation laws]], at the discretization level, thus resulting in better long-time behaviour of numerical solutions than that of generic discretization schemes. There are analogues for [[classical field theory]], which take into account the resulting [[multisymplectic geometry|multisymplectic structure]].

## References

* wikipedia [symplectic integrator](http://en.wikipedia.org/wiki/Symplectic_integrator), [multisymplectic integrator](http://en.wikipedia.org/wiki/Multisymplectic_integrator)

* Denis Donnelly, Edwin Rogers, _Symplectic integrators: An introduction_,  Am. J. Phys. __73__, 938 (2005) [doi](https://doi.org/10.1119/1.2034523)
* [symplectic numerical integration](http://www.maths.cam.ac.uk/undergrad/course/na/ib/symplectic_integrators/symplectic_integrators.php) at DAMTP
* Daniel W. Markiewicz, _Survey on symplectic integrators_,
[pdf](http://math.berkeley.edu/~alanw/242papers99/markiewicz.pdf)
* Robert McLachlan, Klas Modin, Olivier Verdier, _Collective Lie--Poisson integrators on $\mathbb{R}^{3}$_, [arXiv:1307.2387](https://arxiv.org/abs/1307.2387)
* A. Lew, J. E. Marsden, M. Ortiz, M. West, _An overview of variational integrators_, In L. P. Franca (ed.), Finite Element Methods: 70's and Beyond. Barcelona (2003).
* Jerrold E. Marsden, George W. Patrick, Steve Shkoller, _Multisymplectic geometry, variational integrators, and nonlinear PDEs_, Commun. Math. Physics __199__:2 (1998) 351-395 [math.DG/9807080](https://arxiv.org/abs/math/9807080), [doi](https://doi.org/10.1007/s002200050505)
* François Demoures, François Gay-Balmaz, [[Tudor S. Ratiu]], _Multisymplectic variational integrators and space/time symplecticity_, [arXiv:1310.4772](https://arxiv.org/abs/1310.4772)
* Ernst Hairer, _Backward analysis of numerical integrators and symplectic methods_, [ps](www.unige.ch/~hairer/preprints/symplect.ps)
* Sebastian Reich, _Backward error analysis for numerical integrators_, SIAM J. Numer. Anal. 36, 1549--1570, 1996 [citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.6972)
* Y. B. Suris, _Hamiltonian Runge-Kutta type methods and their variational formulation_ (1990)

The idea can be adapted to dissipative systems as well:

* Guilherme França, Michael I. Jordan, René Vidal, _On dissipative symplectic integration with applications to gradient-based optimization_, [arxiv/2004.06840](https://arxiv.org/abs/2004.06840)

category: applications
[[!redirects multisymplectic integrator]]


symplectic leaf

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $(X, \{-,-\})$ a [[Poisson manifold]], a _symplectic leaf_ is a maximal connected [[submanifold]] $Y \hookrightarrow X$ on which the [[Poisson bracket]] restricts to a [[symplectic manifold]] structure.

$X$ is [[foliation|foliated]] by its symplectic leaves.

## References

Regular foliations by symplectic leafs have originally been found and studied in

* F. Bayen, M. Flato, C. Fronsdal, A. Lichnerovicz & D. Sternheimer, _Deformation theory and quantization_, Ann. Phys. I l l (1978) 61-151.

A detailed technical review is in the notes

* Jordan Watts, _An introduction to Poisson manifolds_ (2007) ([pdf](http://www.math.toronto.edu/karshon/grad/2006-07/poisson.pdf))

[[!redirects symplectic leafs]]
[[!redirects symplectic leaves]]


symplectic Lie algebra

[[!redirects symplectic Lie algebras]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[Lie algebra]] of a [[symplectic group]] or [[quaternion unitary group]].

## Related concepts

* [[general linear Lie algebra]]

* [[orthogonal Lie algebra]]


symplectic Lie n-algebroid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[Lie n-algebroid]] is _symplectic_ if it is equipped with a non-degenerate binary [[invariant polynomial]]. This generalizes the notion of a [[symplectic form]] on a [[symplectic manifold]], to which it reduces for $n = 0$.

## Definition

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

A **symplectic Lie $n$-algebroid** is a pair

$$
  (\mathfrak{a}, \langle -,- \rangle)
$$

consisting of

* a [[Lie n-algebroid]] $\mathfrak{a}$;

* a binary [[invariant polynomial]] $\langle- , - \rangle$ of degree $(n+2)$

  (a closed element in the shifted elements of the [[Weil algebra]] $W(\mathfrak{a})$)

  which is non-degenerate.

=--

## Properties

The [[Chern-Simons element]] that witnesses this transgression is the Lagrangian of the corresponding [[AKSZ theory]] [[sigma-model]] with $\mathfrak{a}$ as its target space and the invariant polynomial $\langle -,- \rangle$ as the ([[curvature]] of) its background [[gauge field]].

## Examples

### $n=0$: symplectic manifold

A 0-Lie algebroid is just a [[smooth manifold]] $X$.

Its [[Chevalley-Eilenberg algebra]] is the algebra of smooth functions on $X$

$$
  CE(X) = C^\infty(X)
  \,.
$$

The [[Weil algebra]] of $X$ is

$$
  W(X) = \Omega^\bullet(X)
$$

the [[de Rham complex|de Rham algebra]] of $X$. A degree 2-[[invariant polynomial]] on $X$ is therefore a non-degenerate closed [[differential form|2-form]] $\omega \in \Omega^2(X)$, a [[symplectic manifold|symplectic 2-form]].

+-- {: .standout}

A [[symplectic manifold]], being a pair

$$
  (X,\;\; \omega)
$$

consisting of a  [[smooth manifold]] $X$ and a symplectic 2-form $\omega$, is a symplectic Lie 0-algebroid.

=--

### $n=1$: Poisson manifold

For a [[Poisson manifold]] $X$ with Poisson bivector $\pi \in \Gamma(T X) \wedge \Gamma(T X)$ the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{a})$ of the corresponding [[Poisson Lie algebroid]]

$$
  \mathfrak{a} := \mathfrak{P}(X,\pi)
$$

is that of multi-vector fields on $X$, equipped with the differential $d_{CE(\mathfrak{a})} = [\pi, -]_{Sch}$ given by the [[Schouten bracket]].

If we work locally in coordinates  then $CE(\mathfrak{a})$ is generated from degree 0 elements $x^i$ and degree 1 elements $\partial_i$. The differential is

$$
  d_{CE(\mathfrak{a})} = [\pi, -]_{Sch}
  \,.
$$

The Poisson tensor is $\nu := \pi = \pi^{i j} \partial_i \wedge \partial_j$ and that this is a [[∞-Lie algebra cohomology|Lie algebroid cocycle]] is the fact that

$$
  d_{CE(\mathfrak{a})} \pi = [\pi,\pi]_{Sch} = 0
  \,.
$$

By definition the [[Weil algebra]] $W(\mathfrak{a})$ is generated from the $x^i$, the $\partial_i$ and their shifted partners $\mathbf{d}x^i$ and $\mathbf{d}\partial_i$. The differential here is

$$
  d_{W(\mathfrak{a})} = [\pi , - ] + \mathbf{d}
  \,.
$$

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

The [[invariant polynomial]] $\omega$ that is in transgression with the cocycle  $\nu = \pi$ is

$$
  \omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i)
  \;\;\;
  \in inv(\mathfrak{a})
  \,.
$$

=--

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

One checks directly that the element

$$
  cs_\omega =
  \pi^{i j} \partial_i \wedge \partial_j
  +
  x^i \wedge \mathbf{d} \partial_i
$$

is a Chern-Simons transgression element for $\nu$ and $\omega$,

i.e. $d_{W(\mathfrak{a})} cs(\omega) = \omega$. The restriction of $cs_\omega$ to $CE(\mathfrak{a})$ is evidently the Poisson tensor $\pi$.

=--

More details on this at [[Chern-Simons element]].

+-- {: .standout}

For a [[Poisson manifold]] $X$ with Poisson tensor $\pi = \pi^{i j} \partial_i \wedge \partial_j$, the pair

$$
  (\mathfrak{P}(X,\pi), \;\;\;
\omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i))
$$

consisting of the [[Poisson Lie algebroid]] $\mathfrak{P}(X,\pi)$ and of the [[invariant polynomial]] $\omega$ that is in transgression with its canonical 2-cocycle $\nu = \pi$ (the Poisson tensor) is a symplectic [[Lie algebroid]].

=--

### $n=2$: Courant algebroid

A $2$-symplectic manifold encodes and is encoded by the structure of a [[Courant algebroid]].

A Courant 2algebroid over the point if given by a [[semisimple Lie algebra]] with the symplectic form being the [[Killing form]]. The coresponding Poisson tensor is the canonical 3-cocycle $\langle -, [-,-] \rangle$ on a semisimple Lie algebra. The extension classified by this is the [[string Lie 2-algebra]].

## Relation to other concepts

### To $\infty$-Chern-Simons theory

Since the symplectic form on a symplectic Lie $n$-Algebroid may be understood [[Lie theory|Lie theoretically]] as an [[invariant polynomial]] on an [[L-∞ algebroid]], every symplectic Lie $n$-algebroid serves as a [[target space]] for an [[schreiber:∞-Chern-Simons theory]]: this is [[AKSZ theory]].

We have

* for $n = 1$ the [[Poisson sigma-model]];
* for $n = 2$ the [[Courant sigma model]]
* and so on.

### To multisymplectic geometry

There is also the closely related notion  of [[multisymplectic geometry]]. See

* [[John Baez]], [[Chris Rogers]], _Categorified Symplectic Geometry and the String Lie 2-Algebra_, ([arXiv](http://arxiv.org/abs/0901.4721))

for some relations of this to the above situation for $n = 2$. Essentially multisymplectic geometry studies the higher $n$-ary brackets induced from the binary graded symplectic form discussed here. The relation between these two pictures is the same as that between as studied in the context of [[hemistrict Lie 2-algebra]]s.

An article with more details on this:

* [[Chris Rogers]], _Courant algebroids from categorified symplectic geometry_ ([pdf](http://math.ucr.edu/~chris/2plectic-algebroid_DRAFT.pdf)).

## Related concepts

* [[L-∞ algebroid]]

* [[symplectic groupoid]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

The notion originates somewhere in the school of [[Alan Weinstein]]'s school of [[higher category theory|higher categorial]] [[symplectic geometry]]. The first published appearance of the notion at least for $0 \leq n \leq 3$ is

* [[Dmitry Roytenberg]], _Courant algebroids, derived brackets and even symplectic supermanifolds_, PhD thesis [[arXiv:9910078](http://arxiv.org/abs/math/9910078)]

A good writeup of this material is in

* [[Dmitry Roytenberg]], _On the structure of graded symplectic supermanifolds and Courant algebroids_, in: [[Theodore Voronov]] (ed.),  _Quantization, Poisson Brackets and Beyond_, Contemp. Math. **315**, Amer. Math. Soc. (2002) [[arXiv:0203110](http://arxiv.org/abs/math/0203110), [ISBN:978-0-8218-7905-4](https://bookstore.ams.org/conm-315)]

The idea for all $n$ was then sketched, together with many other ideas about [[L-infinity algebroid]]s in the article with the nice title

* [[Pavol ?evera]], _Some title containing the words "homotopy" and "symplectic", e.g. this one_ ([arXiv](http://arxiv.org/abs/math/0105080))

What we call $n$-symplectic manifold here is called $\Sigma_n$-manifold there.

**Warning** This article here uses the term "$n$-symplectic" in a related but not identical sense to the one used here:

* M. de Leon, D. Martin de Diego, M. Salgado, S. Vilariño, _K-symplectic formalism on Lie algebroids_ ([arXiv](http://lanl.arxiv.org/abs/0905.4585))

A discussion of aspects of how [[multisymplectic geometry]] related to $n$-symplectic manifolds is in

* [[Chris Rogers]], _Courant algebroids from categorified symplectic geometry_ ([pdf](http://math.ucr.edu/~chris/2plectic-algebroid_DRAFT.pdf))
arXiv:1001.0040v1 [math-ph]

* [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))

A discussion of symplectic Lie n-algebroids from an [[infinity-Lie theory]] perspective as discussed here is in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]_, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 ([arXiv:1108.4378](http://arxiv.org/abs/1108.4378))

The [[H-cohomology]] of graded symplectic forms is considered in

* {#Severa05} [[Pavol Severa]], p. 1 of _On the origin of the BV operator on odd symplectic supermanifolds_, Lett Math Phys (2006) 78: 55. ([arXiv:0506331](https://arxiv.org/abs/math/0506331))

[[!redirects symplectic Lie n-algebroids]]

[[!redirects n-symplectic manifold]]


symplectic manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

A **symplectic manifold** is

* a [[smooth manifold]] $X$ of even [[dimension]] $dim X = 2 n$;

* equipped with a **symplectic form**:

  * a closed smooth [[differential form|2-form]] $\omega \in \Omega^2_{cl}(X)$;

  * such that $\omega$ is **non-degenerate**, which means equivalently that

    * $\omega^{\wedge n}=\omega\wedge\omega\wedge\cdots\wedge\omega$ has the maximal [[rank]] at every point $p\in X$;

    * $(\wedge^2 T^*_p X,\omega_p)$ is a [[symplectic vector space]] for every point $p\in X$.

=--

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

A $2n$-dimensional [[topological manifold]] $X$ is

* a real __symplectic manifold__

* equipped with  a **symplectic atlas**:

  * an [[atlas]] consisting of smooth [[chart]]s $\phi_i:U_i\to X$ as usual,

  * such that the transition functions $\phi_j^{-1}\circ\phi_i:\phi_i^{-1}(\phi_i(U_i)\cap\phi_j(U_j))\to \phi_j^{-1}(\phi_i(U_i)\cap\phi_j(U_j))$ preserve the standard symplectic form $\omega_0=\sum_{i=1}^n dx_i\wedge dp_i$ on $\mathbb{R}^{2n}$ with the basis $(x_1,\ldots,x_n,p_1,\ldots,p_n)$.

=--

+-- {: .num_note #TangentCotangentIsomorphism}
###### Remark

The non-degenracy of the symplectic form implies that it defines an [[isomorphism]]

$$
  \omega(-,-) : \Gamma(T X) \to \Gamma(T^* X)
$$

between [[section]]s of the [[tangent bundle]] -- [[vector field]]s --  and [[section]]s of the [[cotangent bundle]] -- [[differential form|differential 1-form]]s -- on $X$ by the map

$$
  (v \in T_x X) \mapsto (\omega(v,-) \in T^*_x X)
  \,.
$$

=--

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

The vector fields in the image of the exact 1-forms under the isomorphism, remark \ref{TangentCotangentIsomorphism}, are called **[[Hamiltonian vector fields]]**.

=--

This means that for $H \in C^\infty(X)$ a [[smooth function]] and $d H$ its differential 1-form, the corresponding [[Hamiltonian vector field]] $v_H \in \Gamma(T X)$ is the unique vector field such that

$$
  d H = \omega(v_H, -)
  \,
$$

Equivalently, for $\phi \mathbb{R}^{2n} \to X$ a [[coordinate chart]] of $X$ and $\phi^*\omega = \omega_{i j} d x^i \wedge d x^j$ the symplectic form on this patch, the [[Hamiltonian vector field]] $v_H$ is

$$
  v_H =  \frac{\partial H}{\partial x^i} (\omega^{-1})^{i j} \partial_j
  \,.
$$

## Properties

### Darboux coordinates

By [[Darboux's theorem]] every symplectic manifold has an [[atlas]] by [[coordinate charts]] $\mathbb{R}^{2n} \simeq U \hookrightarrow X$ on which the [[symplectic form]] takes the canonical form $\omega|_U = \sum_{k = 1}^n d x^{2k} \wedge d x^{2 k+1}$.

### Relation to almost symplectic structure
 {#SymplecticStructure}

The existence of a 2-form $\omega \in \Omega^2(X)$ which is non-degenerate (but not necessarily closed) is equivalent to the existence of a [[G-structure|Sp-structure]] on $X$, a [[reduction of the structure group]] of the [[tangent bundle]] along the inclusion of the [[symplectic group]] into the [[general linear group]]

$$
  Sp(2n) \hookrightarrow GL(2n)
  \,.
$$

Such an _Sp(2n)-structure_ is also called an _almost symplectic structure_ on $X$. Adding the extra condition that $d \omega = 0$ -- the condition for [[integrability of G-structures]] -- makes it a genuine symplectic structure. See at _[integrability of G-structures -- Examples -- Symplectic structure](integrability+of+G-structures#ExampleSymplecticStructure)_.

A _[[metaplectic structure]]_ on a symplectic or almost symplectic manifold is in turn [[lift of the structure group]] to the [[metaplectic group]].

### Relation to almost Hermitian and Kähler structure

By the [above](#SymplecticStructure), a symplectic manifold structure is an
[[integrable G-structure|integrable]] $Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})$-structure. Further [[reduction of the structure group]] along the [[maximal compact subgroup]] inclusion of the [[unitary group]] $U(n) \hookrightarrow Sp(2n,\mathbb{R})$ yields is an [[almost Hermitian structure]]. If that is again [[integrable G-structure|first order integrable]] then it is _[[Kähler structure]]_.

Such  a refinement from symplectic to Kähler structure is also called a choice of _[[Kähler polarization]]_.

### Symplectomorphisms

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

For $(X, \omega)$ a symplectic manifold, the [[vector field]]s
$v \in \Gamma(T X)$ that generate [[diffeomorphism]]s that preserve the symplectic structure are precisely the locally [[Hamiltonian vector fields]].

=--

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

The condition in question is that the [[Lie derivative]]

$$
  L_v \omega = 0
$$

vanishes. By [[Cartan's magic formula]] and using that $d \omega = 0$ this is equivalently

$$
  d \iota_v \omega = 0
  \,.
$$

By the [[Poincare lemma]] it follows that there is locally a function $H$ with $d H = \iota_v \omega$.

=--

### Poisson structure

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

For $(X,\omega)$ a symplectic manifold,
define a bilinear skew-symmetric map

$$
  \{-,-\} : C^\infty(X) \otimes C^\infty(X) \to C^\infty(X)
$$

by

$$
  \{F,H\} := \iota_{v_F} \iota_{v_H} \omega
  \,.
$$

=--

In a [[coordinate chart]] this says that

$$
  \{F,H\} = (\frac{\partial F}{\partial x^i}) (\omega^{-1})^{i j}
    (\frac{\partial H}{\partial x^j})
  \,.
$$

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

The bracket $\{-,-\}$ makes $C^\infty(X)$ a [[Poisson algebra]].

=--

## Examples

* every [[Kähler manifold]] is canonically also a symplectic manifold;

* the [[critical locus]] over any [[local action functional]] becomes a symplectic manifold after dividing out symmetries: the reduced [[phase space]].

## Related concepts

* [[bilinear form]], [[quadratic form]], [[sesquilinear form]]

* [[Kähler form]], [[Hermitian form]]

* [[MSp]]

* [[symplectic orbifold]]

The notion of symplectic manifold is equivalent to that of [[symplectic Lie n-algebroid]] for $n = 0$. (See there.)

* [[isotropic submanifold]], [[coisotropic submanifold]]

* [[symplectic singularity]], [[symplectic resolution]]

* [[symplectic reduction]]

* [[symplectic connection]]

* [[geometric quantization]], [[canonical momentum]]

* [[holomorphic symplectic manifold]]

* [[contact manifold]]

* [[G2-manifold]]

[[!include (co)isotropic subspaces - table]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

See the references at [[symplectic geometry]].

Discussion of the [[torsion of a G-structure|torsion-invariants]] of almost symplectic structures includes

* {#AlbuquerquePicken11} Rui Albuquerque, Roger Picken, _On invariants of almost symplectic connections_ ([arXiv:1107.1860](http://arxiv.org/abs/1107.1860))

The generalization of the notion of symplectic manifolds to [[dg-manifolds]] is sometimes known as _[[NQ-supermanifolds|PQ-supermanifolds]]_ , due to

* M. Alexandrov, [[Maxim Kontsevich|M. Kontsevich]], [[Albert Schwarz|A. Schwarz]], O. Zaboronsky, _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997 ([arXiv:hep-th/9502010](http://arxiv.org/abs/hep-th/9502010))

On [[symplectic orbifolds]]:

* [[Misha Verbitsky]], _Holomorphic symplectic geometry and orbifold singularities_, Asian J. Math. 4 (2000), no. 3, 553-563 ([arXiv:math/9903175](https://arxiv.org/abs/math/9903175))

On a proposal for [[homotopy theory]] of symplectic manifolds:

* [[Vardan Oganesyan]], *The first step towards symplectic homotopy theory* [[arXiv:2304.10529] (https://arxiv.org/abs/2304.10529)]

[[!redirects symplectic manifolds]]
[[!redirects symplectic form]]
[[!redirects symplectic forms]]

[[!redirects symplectic structure]]
[[!redirects almost symplectic structure]]

[[!redirects symplectic orbifold]]
[[!redirects symplectic orbifolds]]

[[!redirects symplectic structures]]
[[!redirects almost symplectic structures]]


symplectic orbifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _symplectic orbifold_  is an [[orbifold]] with [[symplectic structure]], hence like a [[symplectic manifold]] but possibly with [[singularities]] modeled on [[fixed points]] of local [[actions]] of [[finite groups]] of [[symplectomorphisms]].

## Related concepts

* [[Riemannian orbifold]]

* [[complex orbifold]]

* [[Lorentzian orbifold]]

## References

* Leonor Godinho, _Blowing Up Symplectic Orbifolds_, Annals of Global Analysis and Geometry 20, 117–162 (2001) ([doi:10.1023/A:1011628628835](https://doi.org/10.1023/A:1011628628835))

* [[Weimin Chen]], _Resolving symplectic orbifolds with applications to finite group actions_, Journal of Gökova Geometry Topology **12** (2018), 1-39 ([arXiv:1708.09428](https://arxiv.org/abs/1708.09428))

* Juan Ángel Rojo Carulli,   _Orbifolds and geometric structures_ (2019) ([eprints.ucm.es/56824/](https://eprints.ucm.es/56824/))

On [[orbifold cohomology]] of symplectic orbifolds:

* [[Vasily Dolgushev]], [[Pavel Etingof]], _Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology_, Int. Math. Res. Not. 2005, no. 27, 1657-1688 ([arXiv:math/0410562](https://arxiv.org/abs/math/0410562))

* Tara Holm, Tomoo Matsumura, _Equivariant cohomology for Hamiltonian torus actions on symplectic orbifolds_, Transformation Groups 17, 717–746 (2012) ([arXiv:1008.3315](https://arxiv.org/abs/1008.3315), [doi:10.1007/s00031-012-9192-7](https://doi.org/10.1007/s00031-012-9192-7))

* Cheol-Hyun Cho, Mainak Poddar, _Holomorphic orbi-discs and Lagrangian Floer cohomology of symplectic toric orbifolds_, J. Differential Geom. Volume 98, Number 1 (2014), 21-116 ([euclid:jdg/1406137695](https://projecteuclid.org/euclid.jdg/1406137695))

[[!redirects symplectic orbifolds]]


symplectic realization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $(X, \pi)$ a [[Poisson manifold]], a **symplectic realization** of it is a [[symplectic manifold]] $(Y, \omega)$ and a Poisson map $(Y, \omega) \to (X, \pi)$ such that $Y \to X$ is a   [[surjective submersion]] $Y \to X$.

## Properties

### Solution in terms of symplectic groupoids

For any [[symplectic groupoid]] $\Sigma$ with base a [[Poisson manifold]] $P$ the target map is a symplectic realization of $P$ and  the source map is a symplectic realization  of the opposite structure. Thus $\Sigma$ with its symplectic structure may be regarded as a desingularization of  $P$ with its Poisson structure. Since the [[symplectic groupoid]] is the [[Lie integration]] of the [[Poisson Lie algebroid]] of the Poisson manifold, symplectic realization has been reduced to a problem in [[Lie theory]].


symplectic reduction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

> This entry is being splitt-off from _[[BRST-BV formalism]]_. For the moment see the discussion there at _[Hamiltonian BFV](BV-BRST+formalism#HamiltonianBV)_.

## Properties

* [[Guillemin-Sternberg geometric quantization conjecture]]

## Related concepts

* [[Hamiltonian action]]

The analog for [[presymplectic manifolds]] is [[presymplectic reduction]].

The analog for [[Poisson manifolds]] is [[Poisson reduction]].

[[!include symplectic reduction - table]]

* [[quantization commutes with reduction]]

## References

The original influential article is

* [[Jerrold Marsden]], [[Alan Weinstein]], _Reduction of symplectic manifolds with symmetry_, Reports on Math. Phys., 5, (1974), 121-130 ([web](http://www.cds.caltech.edu/~marsden/bib/1974/01-MaWe1974/))

Reviews include

* [[José Figueroa-O'Farrill]], chapter 2+3 of PhD thesis  ([pdf](http://empg.maths.ed.ac.uk/Activities/BRST/Ch2+3PhD.pdf))
 {#FOF}

Textbook accounts include

* [[Bertram Kostant]], [[Shlomo Sternberg]], _Symplectic reduction, BRS cohomology and infinite-dimensional Clifford algebras_, Annals of Physics __176__:1 (15 May 1987) pp. 49–113 [MR893479](http://www.ams.org/mathscinet-getitem?mr=893479) doi

[[!redirects symplectic reductions]]

[[!redirects symplectic quotient]]
[[!redirects symplectic quotients]]

[[!redirects Marsden-Weinstein reduction]]
[[!redirects Marsden-Weinstein reductions]]


symplectic reduction - table

Reductions of ([[presymplectic manifold|pre]]-)[[symplectic manifolds]]:

| [[symplectic geometry]] | [[physics]] |
|--|--|
| [[presymplectic manifold]] | [[covariant phase space]] |
| $\downarrow$ [[gauge reduction]] | $\downarrow$ quotient by [[gauge symmetry]] |
| [[symplectic manifold]] |  [[reduced phase space]] |
| $\downarrow$ [[symplectic reduction]] | $\downarrow$ quotient by [[global symmetry]] |
| [[symplectic manifold]] | [[reduced phase space]] |


symplectic reflection algebra

Symplectic reflection algebra is certain multiparametric deformation of smash product of a finite group of automorphisms of a symplectic vector space and the polynomial algebra of that vector space.  It is related to the coordinate ring of a universal Poisson deformation of the quotient singularity of that action. Rational Cherednik algebras are a special case.

* [[Pavel Etingof]], [[Victor Ginzburg]], _Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism_, Invent. math. __147__, 243--348 (2002) [doi](https://doi.org/10.1007/s002220100171) arXiv:[math.AG/0011114](https://arxiv.org/abs/math/0011114)
* [[Pavel Etingof]], Wee Liang Gan, [[Victor Ginzburg]], Alexei Oblomkov, _Harish-Chandra homomorphisms and [[symplectic reflection algebra]]s for wreath-products_, arXiv:[math.RT/0511489](https://arxiv.org/abs/math/0511489)
* Pavel Etingof, _Symplectic reflection algebras and affine Lie algebras_, [arXiv:1011.4584](https://arxiv.org/abs/1011.4584)
* [[I. G. Gordon]], _Symplectic reflection algebras_, in: Trends in Representation Theory of Algebras and Related Topics (Andrzej Skowroński, editor)
* K. Brown, I. Gordon, _Poisson orders, symplectic reflection algebras and representation theory_, J. Reine Angew. Math. __559__ (2003) 193--216
* [[Ivan Losev]], _Procesi bundles and symplectic reflection algebras_, In: M. Hitrik, D. Tamarkin, B. Tsygan, S. Zelditch (eds), Algebraic and Analytic Microlocal Analysis. AAMA 2013. Springer Proc. in Math. & Stat. __269__; [arXiv:1501.00643](https://arxiv.org/abs/1501.00643)

category: algebra


symplectic representation

#Contents#
* table of contents
{:toc}

## Idea

Given a [[group]] $G$, a *symplectic representation* of $G$ is a [[representation]] of $G$ on a [[symplectic vector space]], that is a $F$-vector space (where $F$ is a [[field]]) with a [[symplectic form]]
(a nondegenerate skew-symmetric [[bilinear form]]) $\omega$, preserving that form: $\omega(g.x,g.y) = \omega(x,y)$ for all $x,y\in G$.

## References

* Wikipedia:  [Symplectic representation](https://en.wikipedia.org/wiki/Symplectic_representation)

category: algebra


symplectic singularities.

A **symplectic singularity** is a normal algebraic variety $X$ such that some (equivalently any) resolution of singularities of $X$ carries an algebraic 2-form $\Omega$ which is closed, and non-degenerate on the regular locus of the resolution.

A **symplectic resolution** is a resolution $\tilde X\to X$ where the 2-form is closed and non-degenerate on all of $\tilde X$.  A symplectic resolution is necessarily [[Calabi-Yau]].

A **conical symplectic resolution** is an affine symplectic singularity which carries a conical $\mathbb{G}_m$-action (one with non-negative weights on the coordinate ring of $X$, and zero-weight space spanned by $1$) that acts on the symplectic form on the regular locus with weight $n\gt 0$.

[[!redirects symplectic resolution]]

[[!redirects conical symplectic resolution]]


symplectic singularity

#Contents#
* table of contents
{:toc}

## Idea

A [[normal variety]] is called a **[[symplectic variety]]** if its [[smooth variety|smooth part]] admits a holomorphic [[symplectic form]], whose pull-back to any resolution extends to a holomorphic 2-form. If this 2-form is itself symplectic, the given resolution is called a **symplectic resolution**.  More generally, an algebraic variety is said to have **symplectic singularities** if every point in the variety has a neighborhood which carries a symplectic structure.

Symplectic resolutions are analogous to [[hyper Kähler manifolds]].

## Related concepts

* [[symplectic duality]]

## References

The term was introduced in

* [[Arnaud Beauville]], _Symplectic singularities_ (1999) ([arXiv:math/9903070](https://arxiv.org/abs/math/9903070))

Useful surveys include:

* [[Dmitry Kaledin]], _Geometry and topology of symplectic resolutions_, Algebraic Geometry: 2005 Summer Research Institute, edited by Dan Abramovich ([arXiv:math.AG/0608143](https://arxiv.org/abs/math/0608143))
* Baohua Fu, _A survey on symplectic singularities and symplectic resolution_, Annales Mathématiques Blaise Pascal 13 (2006) ([web](http://ambp.cedram.org/item?id=AMBP_2006__13_2_209_0))

from which some of the above text is taken.

[[!redirects symplectic singularities]]


symplectic spinor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

By [[analogy]] of [[spin]] and [[spin^c]] with [[Mp]] and [[Mp^c]] and in view of [[spin representations]], the standard Segal-Shale-Weil [[metaplectic representation]] is also called the _symplectic spinor representation_.

Accordingly, an [[associated bundle]] for this representation is also called a _symplectic spinor bundle_ (in [[analogy]] with [[spinor bundles]]). This plays a role in [[metaplectic quantization]].

## References

The original article is

* {#Kostant74} [[Bertram Kostant]], _Symplectic Spinors_ Symposia Mathematica, Vol. XIV, pp. 139–152. Academic Press, London, 1974.

See also

* Wikipedia, _[Symplectic spinor bundle](SegalShaleWeilRepresentation)_

[[!redirects symplectic spinors]]

[[!redirects symplectic spin representation]]
[[!redirects symplectic spin representations]]

[[!redirects symplectic spinor bundle]]
[[!redirects symplectic spinor bundles]]


symplectic topology

Every (paracompact Hausdorff) [[differentiable manifold]] can be equipped with a [[Riemannian structure]]. Therefore the existence of a metric does not impose constraints on a toplogy of a manifold. The local Riemannian structure of a Riemannian manifold on the other hand can be very different with the same global topology: there are even locally many nonisometric Riemannian structures on the same manifold/neighborhood. All even dimensional manifolds allow *locally* a (unique up to isomorphism) symplectic structure. Symplectic manifolds, due Darboux theorem, are all locally isomorphic, but according to the spectacular findings of Gromov in 1985, there are still global invariants of symplectic structure as well as topological constraints on even-dimensional manifolds permitting symplectic structure.

## Related $n$Lab entries

* [[Floer homology]]

* [[Arnold conjecture]]

## Literature

* [[Dusa McDuff]], [[Dietmar Salamon]], _Introduction to symplectic topology_, 2 ed. Oxford Mathematical Monographs 1998. x+486 pp.
* [[Yong-Geun Oh]], _Symplectic topology and Floer homology_, [pdf](http://www.math.wisc.edu/~oh/all.pdf)
 {#Oh}
* _Pseudo holomorphic curves in symplectic manifolds_, Inventiones Mathematicae __82__, n. 2 (1985) 307–347.


symplectic vector field

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### For symplectic manifolds

For $(X, \omega)$ a [[symplectic manifold]], a [[vector field]] $v \in \Gamma(T X)$ on $X$ is called a **symplectic vector field** if its [[Lie derivative]] preserves the [[symplectic form]]:

$$
  \mathcal{L}_v \omega = 0
  \,.
$$

### For $n$-plectic manifolds

The analogous definition applies to [[n-plectic geometry]].

## Properties

### Relation to symplectomorphisms

The [[flow]] generated by a symplectic vector field is an auto-[[symplectomorphism]].

### Relation to Hamiltonian vector fields

By [[Cartan's magic formula]] and using that $\omega$ is by definition a closed form, the equation $\mathcal{L}_v \omega = 0$ is equivalent to

$$
  d_{dR} \iota_v \omega = 0
  \,,
$$

hence equivalent to the condition that the contraction of $v$ in $\omega$ is a [[closed differential form]]. If this contraction even is an [[exact differential form]] in that there is a function $h \in C^\infty(X)$ such that

$$
  \iota_v \omega = d_{dR} h
  \,,
$$

then the symplectic vector field $v$ is called a [[Hamiltonian vector field]] and $h$ is called its [[Hamiltonian]] function.

[[!redirects symplectic vector fields]]


symplectic vector space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[vector space]] $V$ over a field $k$ is __symplectic__ if it is equipped with an [[exterior form|exterior 2-form]] $\omega \in \Lambda^2_k V$ such that $\omega^{\wedge n}=\omega\wedge\omega\wedge\cdots\wedge\omega$ has the maximal rank.

A subspace $W\subset V$ in a symplectic vector space is **isotropic** if $\omega(v,v) = 0$ for all $v\in W$ and Lagrangean (or **lagrangian**) if it is maximal isotropic (not proper subspace in any isotropic subspace). See [wikipedia](http://en.wikipedia.org/wiki/Symplectic_vector_space).

## Related concepts

* [[symplectic manifold]]

* [[quadratic Hamiltonian]]

* [[symplectic group]], [[affine symplectic group]], [[metaplectic group]], [[extended affine symplectic group]], [[Heisenberg group]]

[[!include (co)isotropic subspaces - table]]

## References

* O. T. O'Meara, _Symplectic groups_, Math. Surveys __16__, Amer. Math. Soc. 1978. xi+122 pp.

[[!redirects symplectic vector spaces]]


symplectomorphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Symplectomorphisms are the [[homomorphisms]] of [[symplectic manifolds]].

In the context of [[mechanics]] where symplectic manifolds model [[phase spaces]], symplectomorphisms are essentially what are called _[[canonical transformations]]_.

## Definition

### Symplectomorphisms

A **symplectomorphism** or **symplectic diffeomorphism** from a [[symplectic manifold]] $(X_1,\omega_1)$ to a symplectic manifold $(X_2,\omega_2)$ is a [[diffeomorphism]] $\phi : X_1 \to X_2$ preserving the [[symplectic form]], i.e. such that

$$
  \phi^* \omega_2 = \omega_1
  \,.
$$

### Auto-symplectomorphisms
 {#AutoSymplectomorphisms}

The symplectomorphisms from a [[symplectic manifold]] $(X, \omega)$ to itself form an infinite-dimensional [[Lie group]] that is a [[subgroup]] of the [[diffeomorphism group]] of $X$, the _[[symplectomorphism group]]_:

$$
  Sympl(X, \omega)
  \hookrightarrow
  Diff(X)
  \,.
$$

Its [[Lie algebra]]

$$
  \mathfrak{SymplVect}(X, \omega) \hookrightarrow \mathfrak{Vect}(X)
$$

is that of [[symplectic vector fields]]: those [[vector fields]]
$v \in  \mathfrak{Vect}(X)$ such that their [[Lie derivative]] annihilates the [[symplectic form]]

$$
  \mathcal{L}_v \omega = 0
  \,.
$$

The further [[subgroup]] corresponding to those symplectic vector fields which are [[flows]] of [[Hamiltonian vector fields]] coming from a smooth family of [[Hamiltonians]]

$$
  \mathfrak{HamVect}(X, \omega) \hookrightarrow \mathfrak{SymplVect}(X, \omega) \hookrightarrow \mathfrak{Vect}(X)
$$

is the group of **Hamiltonian symplectomorphisms** or **Hamiltonian diffeomorphisms**.

$$
  HamSympl(X,\omega) \hookrightarrow Sympl(X, \omega) \hookrightarrow Diff(X)
  \,.
$$

### $n$-Plectomorphisms

In the generalization to [[n-plectic geometry]] there are accordingly _$n$-plectomorphisms_. See at _[[higher symplectic geometry]]_.

## Properties

### Preservation of volume

Inasmuch as a symplectic manifold $(M, \omega)$ carries a canonical [[volume form]] $\omega^{\wedge n}$, it is clear that a symplectomorphism is locally volume-preserving.

### Relation to Poisson brackets

The [[Lie algebra]] given by the [[Poisson bracket]] of a [[symplectic manifold]] $(X, \omega)$ is that of a [[central extension]] of the group of Hamiltonian symplectomorphisms. (It [[Lie integration|integrates]] to the [[quantomorphism group]].)

The central extension results form the fact that the Hamiltonian associated with every [[Hamiltonian vector field]] is well defined only up to the addition of a constant function.

If $(X, \omega)$ is a [[symplectic vector space]] then there is corresponding to it a [[Heisenberg Lie algebra]]. This sits inside the Poisson bracket algebra, and accordingly the [[Heisenberg group]] is a subgroup of the group of (necessarily Hamiltonian) symplectomorphisms of the symplectic vector space, regarded as a symplectic manifold.

### Relation to Lagrangian correspondences

A symplectomorphisms $\phi \;\colon\; (X_1, \omega_1) \longrightarrow (X_2, \omega_2)$ canonically induces a [[Lagrangian correspondence]] between $(X_1, \omega_1)$ and $(X_2,\omega_2)$, given by its [[graph]].

### Extensions under geometric quantization

[[!include geometric quantization extensions - table]]

## Examples

### Linear symplectomorphisms

Given a [[symplectic vector space]] $(V,\omega)$ regarded as a [[symplectic manifold]], then those symplectomorphisms which are [[linear maps]] on $V$ form, under composition, the [[symplectic group]] $Sp(V,\omega)$.

The linear Hamiltonian symplectomorphisms are also known as the _[[Hamiltonian matrices]]_.

### A curious example: volumes of balls

The following example, due to Andreas Blass and Stephen Schanuel, is a [[categorification|categorified]] way to calculate [[volumes]] of even-dimensional [[balls]].

In any dimension $n$, the volume of the unit ball in $\mathbb{R}^n$ (with respect to the [[Lebesgue measure]]) is

$$
  vol(B_n) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}
$$

where $\Gamma$ is the Euler [[Gamma function]]. In dimension $2 n$, this gives

$$
  vol(B_{2 n}) = \frac{\pi^n}{n!}
$$

Meanwhile, we may regard $\pi^n$ as the volume of the $n$-dimensional complex [[polydisc]], viz. the $n^{th}$ cartesian power of the complex 1-disc $B_{2} = \{z: {|z|} \leq 1\}$, on which the [[symmetric group]] $S_n$ acts by [[permutation|permuting]] [[coordinates]]. The volume of the orbit space $B_2^n/S_n$ is clearly $\pi^n/n!$.

+-- {: .un_thm}
###### Theorem (Blass-Schanuel)
Given $(z_1, \ldots, z_n) \in \mathbb{C}^n$, write coordinates $z_j$ in polar coordinate form $z_j = r_j e^{i \theta_j}$, and define an $S_n$-invariant map $\phi \colon B_2^n \to B_{2 n}$ by first permuting the $z_j$ so that $r_1 \geq r_2 \geq \ldots \geq r_n$ and then mapping $(z_1, \ldots, z_n)$ to
$$(\sqrt{r_1^2 - r_2^2}e^{i\theta_1}, \sqrt{r_2^2 - r_3^2}e^{i(\theta_1 + \theta_2)}, \ldots, \sqrt{r_{n-1}^2-r_n^2}e^{i(\theta_1 + \theta_2 + \ldots + \theta_{n-1})}, r_n e^{i(\theta_1 + \theta_2 + \ldots + \theta_n)})$$
Then $\phi$ induces a continuous well-defined map $B_2^n/S_n \to B_{2 n}$. Furthermore, when restricted to the set $P_n$ of $(z_1, \ldots, z_n)$ for which the $r_j$ are all distinct, $\phi$ induces a smooth symplectic isomorphism mapping $P_n/S_n$ onto the set $Q_n$ of $(w_1, \ldots, w_n) \in B_{2 n}$ for which $w_j \neq 0$ for $1 \leq j \leq n-1$.
=--

In other words, writing $z_j = x_j + i y_j$ the symplectic 2-form

$$\sum_{j=1}^n d x_j \wedge d y_j = \sum_{j=1}^n r_j d r_j \wedge d\theta_j$$

is preserved by pulling back along $\phi \colon P_n/S_n \to Q_n$. Since symplectic maps are locally volume-preserving, and since $P_n$ and $Q_n$ are almost all of $B_2^n$ and $B_{2 n}$ respectively, this gives a proof that the volume of $B_{2 n}$ is $\pi^n/n!$ (alternate to standard purely computational proofs).

## Related concepts

* [[canonical transformation]]

* [[Hamilton's equations]]

* [[symplectic integrator]]

## References

Lecture notes include

* Augustin Banyaga, _Introduction to the geometry of hamiltonian diffeomorphisms_ ([pdf](https://www.personal.psu.edu/auw4/dakar.pdf))

The example of volumes of balls is discussed in

* Andreas Blass, Stephen Schanuel, On the volumes of balls ([ps](http://www.math.lsa.umich.edu/~ablass/vol.ps)).

[[!redirects symplectomorphisms]]

[[!redirects Hamiltonian symplectomorphism]]
[[!redirects Hamiltonian symplectomorphisms]]
[[!redirects hamiltonian symplectomorphism]]
[[!redirects hamiltonian symplectomorphisms]]

[[!redirects n-plectomorphism]]
[[!redirects n-plectomorphisms]]
[[!redirects Hamiltonian n-plectomorphism]]
[[!redirects Hamiltonian n-plectomorphisms]]

[[!redirects symplectic diffeomorphism]]
[[!redirects symplectic diffeomorphisms]]
[[!redirects Hamiltonian diffeomorphism]]
[[!redirects Hamiltonian diffeomorphisms]]


symplectomorphism group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### Traditional

For $(X,\omega)$ a [[symplectic manifold]], the _symplectomorphism group_

$$
  Diff(X,\omega) \hookrightarrow Diff(X)
$$

is the [[subgroup]] of the [[diffeomorphism group]] of $X$ on the [[diffeomorphisms]].

### In higher symplectic geometry

Analogous constructions apply when symplectic manifolds are generalized to [[n-plectic infinity-groupoids]]: for $(X, \omega)$ an [[n-plectic manifold]], and **$n$-plectomorphism** is a [[diffeomorphism]] $\phi : X \to X$ that preserves the $n$-plectic form $\phi^* X \simeq X$.

## Examples

* The linear part of the 2-plectomorphism group/3-plectomorphism group of the [[Cartesian space]] $\mathbb{R}^7$ equipped with its associative 3-form $\omega = \lanlge (-), (-) \times (-)\rangle$ is the [[exceptional Lie group]] [[G2]]. See there for more details.

## Related concepts

A further subgroup is that of [[Hamiltonian symplectomorphisms]]. The [[group extension]] of that whose elements are pairs consisting of a Hamiltonian diffeomorphism and a choice of [[Hamiltonian]] for this is the [[quantomorphism group]].

The [[Lie algebra]] of the symplectomorphism group is that of [[symplectic vector fields]].

[[!include geometric quantization extensions - table]]

[[!redirects symplectomorphism groups]]

[[!redirects Hamiltonian symplectomorphism group]]
[[!redirects Hamiltonian symplectomorphism groups]]

[[!redirects n-plectomorphism ∞-group]]
[[!redirects n-plectomorphism infinity-group]]

[[!redirects Hamiltonian symplectomorphism n-group]]
[[!redirects Hamiltonian symplectomorphism n-groups]]

[[!redirects 2-plectomorphism group]]
[[!redirects 2-plectomorphism groups]]

[[!redirects 3-plectomorphism group]]
[[!redirects 3-plectomorphism groups]]


Syméon Bozapalides

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=98641)

## Selected writings

On [[total category]]-objects:

* [[Syméon Bozapalides]], *Quelques Remarques sur les Cosmos Elementaires*, Δελτίο της Ελληνικής Μαθηματικής Εταιρίας 15.15 (1974) 144-151 [[pdf](https://hdml.di.ionio.gr/pdfs/journals/1.pdf), [[Bozapalides-CosmosElementaires.pdf:file]]]

Early discussion of [[fibrations in a 2-category]]:

* [[Syméon Bozapalides]]. _Fibrations et faisceaux dans une 2-catégorie_, Archivum Mathematicum 12.4 (1976) 199-208. [[10338.dmlcz/106944](https://dml.cz/handle/10338.dmlcz/106944), [pdf](https://dml.cz/bitstream/handle/10338.dmlcz/106944/ArchMath_012-1976-4_4.pdf)]

On [[formal category theory|formal]] [[2-category theory]] (2-categorical [[coends]], [[enriched bicategories]] etc.):

* [[Syméon Bozapalides]], *Théorie formelle des bicatégories*, Ph.D. thesis, Paris (1976) [scan:pdf]
  > (supervised by [[Charles Ehresmann]])

On generalised [[ends]], which were a first step toward [[homotopy coherent diagram|homotopy coherent]] ends:

* [[Syméon Bozapalides]], *Les fins cartésiennes
généralisées*, Archivum Mathematicum* **13** (1977) 75–87 [[eudml:17939](https://eudml.org/doc/17939)]

and on lax presheaves:

* [[Syméon Bozapalides]], *Some remarks on Lax presheafs* Illinois J. Math. *24* (1980) 676 – 680 [[doi:10.1215/ijm/1256047482](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-24/issue-4/Some-remarks-on-lax-presheafs/10.1215/ijm/1256047482.full)]

category: people

[[!redirects Syméon Bozapalides]]