symmetric space

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#### Manifolds and cobordisms
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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> For a [[topological space]] satisfying the $R_0$ regularity condition (which states that the [[specialisation preorder]] is [[symmetric relation|symmetric]], hence an [[equivalence relation]]), see [[symmetric topological space]].

***

# Contents
* table of contents
{: toc}

## Idea

A _symmetric space_ is a specially nice [[homogeneous space]], characterized by the property that for each [[point]] there is a symmetry fixing that point and acting as $-1$ on its [[tangent space]].  An example would be the [[sphere]], the [[Euclidean plane]], or the [[hyperbolic plane]].

## Definitions

A **symmetric space** is classically defined to be a [[quotient]] [[manifold]] of the form $G/H$, where $G$ is a [[Lie group]] and the [[subgroup]] $H$ is the set of fixed points of some [[involution]] $\sigma : G \to G$, that is, a smooth [[homomorphism]] with $\sigma^2 = 1_G$.   Using the involution, every point $a \in G/H$ gives rise to a [[smooth function]]

$$   a \triangleright -  : G/H \to G/H $$

fixing the point $a$ and acting as $-1$ on the tangent space of $a$.  This operations satisfies the laws of an involutory [[quandle]].

More precisely, a **symmetric pair** is a pair $(G,H)$ where $G$ is a [[Lie group]] and the subgroup $H$ is the set of fixed points of some [[involution]] $\sigma : G \to G$.  Different pairs $(G,H)$, $(G',H')$ can give what is normally considered the same symmetric space $G/H \cong G'/H'$.   In other words, not every morphism of symmetric spaces arises from a morphism of symmetric pairs.

To avoid this problem, we can define a symmetric space as a smooth manifold $M$ with a smooth map $\triangleright : M\times M\to M$ such that for all $x,y,z\in M$

1. $x \triangleright x = x$ (idempotence)
2. $x \triangleright (x\triangleright y) = y$
3. $x \triangleright (y \triangleright z) = (x \triangleright y)\triangleright (x \triangleright z)$ ([[left self-distributivity]])
4. for every $x$ there is a neighborhood $U\subset M$ such that $x \triangleright y = y$ implies $x = y$ for all $z\in U$.

This amounts to an involutory [[quandle]] object $Q$ in the category of smooth manifolds, with the property that each point $a \in Q$ is an _isolated_ fixed point of the map $a \triangleright - : Q \to Q$.

## Related entries

* [[homogeneous space]]

* [[quandle]]

* [[Lie triple system]]

* [[transvection]]

## References

The definition in terms of quandles coincides with the classical definition in the case of connected symmetric spaces.  For details, including a comparison of other definitions of symmetric space, see:

* Wolgang Bertram, _The geometry of Jordan and Lie structures_, Lecture Notes in Mathematics **1754**, Springer, Berlin, 2000.

The relation to quandles is given in Theorem I.4.3.  Bertram attributes this result to part I, chapter II of

* Ottmar Loos, _Symmetric Spaces I, II_, Chapter II, Benjamin, New York, 1969.

See also:

* Sigurdur Helgason, _Differential geometry, Lie groups and symmetric spaces_,

* S. Helgason, _Group representations and symmetric spaces_, Proc. ICM. Nice 1970, vol. 2, 313-320, [pdf](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0313.0320.ocr.pdf), [djvu](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0313.0320.ocr.djvu)

[[!redirects symmetric spaces]]


Symmetric spectra

This page collects material related to the lecture notes

* [[Stefan Schwede]],

  _Symmetric spectra_,

  2012

  ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

on [[stable homotopy theory]] in terms of [[symmetric spectra]].

See also the _[[Lectures on Equivariant Stable Homotopy Theory]]_ and on _[[Global homotopy theory]]_ for the model on [[orthogonal spectra]] and [[equivariant stable homotopy theory]].

#Contents#
* table of contents
{:toc}

## Chapter I. Basics

### 1. Symmetric spectra

* [[symmetric spectrum]]

* [[ring spectrum]]

* [[module spectrum]]

Basic Examples:

* [[sphere spectrum]]

  [[stable homotopy groups of spheres]]

  [[Hopf fibration]]

* [[suspension spectrum]]

* [[Eilenberg-MacLane spectrum]] [[HA]]

  (the claim about [[Eilenberg-MacLane spaces]] here is proven in ([Aguilar-Gitler-Prieto 02, corollary 6.4.23](Eilenberg-MacLane+space#AguilarGitlerPrieto02))
)

* [[Thom spectrum]]

  [[MO]], [[MU]]

* [[topological K-theory]], [[KU]]

### 2. Properties of naive homotopy groups

* [[stable homotopy groups]]

* [suspension isomorphism](homotopy+group+of+a+spectrum#SuspensionIsomorphismOfStableHomotopyGroups)

### 3. Basic constructions

#### 3.1 Symmetric spectra of simplicial sets

* [[powering]] and [[copowering]] over [[sSet]]

#### 3.2 Constructions

* [[limits]] and [[colimits]]

* [[shift spectrum]]

* [[semistable symmetric spectrum]]

* [[mapping spectrum]]

* [[free symmetric spectrum]]

#### 3.3 Constructions involving ring spectra

#### 3.4 Symmetric Spectra and $I$-spaces

* [[I-space]]

### 4. Stable equivalences

* [[spectrification]]

### 5. Smash product

* [[smash product of spectra]]

### 6. Homotopy groups

### 7. Relation to other kinds of spectra

#### 7.1 Orthogonal spectra

* [[orthogonal spectrum]]

#### 7.2 Unitary spectra

#### 7.3 Continuous and simplicial functors

* [[excisive functor]], [[model structure for excisive functors]]

#### 7.4 $\Gamma$-spaces

* [[Gamma-space]]

#### 7.5 Permutative categories

* [[permutative category]]

#### 7.6 Spectra as enriched functors

* [[enriched functor]]

### 8. Naive versus true homotopy groups

### History, credits, further reading

* [[highly structured spectra]]

* [[symmetric monoidal smash product of spectra]]

## Chapter II. The stable homotopy category

### 1. The stable homotopy category

* [[stable homotopy category]]

### 2. Triangulated structure

* [[triangulated category]]

### 3. Derived smash product

* [[smash product of spectra]]

### 4. Grading

* [[ring spectrum]]

### 5. Generator

### 6. Homology and cohomology

#### 6.1 Generalized homology and cohomology

* [[generalized (Eilenberg-Steenrod) cohomology]]

* [[generalized homology]]

* [[Milnor exact sequence]]

* [[Kronecker pairing]]

* [[universal coefficient theorem]]

#### 6.2 Ordinary homology and cohomology

* [[ordinary cohomology]]

* [[Künneth theorem]]

* [[Hurewicz theorem]]

#### 6.3 Moore spectra

* [[Moore spectrum]]

### 7. Finite spectra

* [[Spanier-Whitehead category]]

* [[finite spectra]]

### 8. Connective covers and Postnikov sections

* [[connective cover]]

* [[Postnikov tower]]

### 9. Localization and completion

* [[localization]], [[formal completion]]

* [[fracture theorem]]

### 10. The Steenrod algebra

* [[cohomology operation]]

* [[Steenrod algebra]]

#### 10.1 Examples and applications

#### 10.2 Hopf algebra structure

* [[Hopf algebra]]

#### 10.3 The Adams spectral sequence

* [[Adams spectral sequence]]

## Chapter III. Model structures

* [[model structure on symmetric spectra]]

### 1. Symmetric spectra in a simplicial category

### 2. Flat cofibrations

### 3. Level model structures

### 4. Stable model structures

* [[model structure on symmetric spectra]]

### 5. Operads and their algebras

* [[operad]]

* [[algebra over an operad]]

### 6. Model structures for algebras over an operad

* [[model structure on algebras over an operad]]

### 7. Connective covers of structured spectra

* [[connective cover]]

## Chapter IV. Module spectra

## Appendix A. Miscellaneous tools

### Model category theory

* [[model category]]

### Compactly generated spaces

* [[compactly generated topological spaces]]

* [[classical model structure on topological spaces]]

### Simplicial sets

* [[simplicial sets]]

* [[classical model structure on simplicial sets]]

### Equivariant homotopy theory

* [[equivariant homotopy theory]]

## Errata

* p. 273. the definition of [[Kronecker pairing]] (construction 6.13) is missign the application of the multiplication map

* p. 331. the second "vertically" should be "horizontally"

* p. 331. "with coordinated"

* p. 333. "stable stemes"

* p. 344. uncompiled source code "eqrefeq-general" etc.

category: reference


symmetric spectrum

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#Contents#
* table of contents
{:toc}

## Idea

Symmetric spectra are one version of [[highly structured spectra]] that support a [[symmetric monoidal smash product of spectra]]. A _symmetric spectrum_ is a sequence of [[topological spaces]]/[[simplicial sets]] which are compatibly equipped with an [[action]] of the [[symmetric group]].

The [[category]] of _symmetric spectra_ is a [[presentable (∞,1)-category|presentation]] of the [[symmetric monoidal (∞,1)-category]] [[stable (infinity,1)-category of spectra|of spectra]], with the special property that it implements the [[smash product of spectra]] such as to yield itself a [[symmetric monoidal category|symmetric]] [[monoidal model category|monoidal]] [[model category of spectra]]: the _[[model structure on symmetric spectra]]_. This implies in particular that with respect to this [[symmetric smash product of spectra]] an [[E-∞ ring]] is presented simply as a plain [[commutative monoid]] [[internalization|in]] symmetric spectra. This is of course such that truncating down to the [[homotopy category]] yields equivalently the [[stable homotopy category]] with its usual [[smash product of spectra]].

The main technical issue with symmetric spectra is that the naive definition of [[homotopy groups]] for them is not general homotopy correct, one needs to replace by a "[[semistable symmetric spectrum]]" first, see [below](#HomotopyGroups). This problem goes away for [[orthogonal spectra]] (these however need to be based on [[topological spaces]] instead of [[simplicial sets]]).

## Definition

### In components
 {#InComponents}

A symmetric spectrum is a [[sequential spectrum]] equipped with an [[action]] of the [[symmetric group]] on each component space, such that the structure maps [[intertwiner|intertwine]] these actions combined with the canonical permutation action on the [[n-spheres]]:

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

A **symmetric spectrum** $X$ in [[sSet]] is

1. a sequence $\{X_n| n \in \mathbb{N}\}$ of [[pointed object|pointed]] [[simplicial sets]];

1. a basepoint preserving left [[action]] of the [[symmetric group]] $\Sigma_n$ on $X_n$;

1. a sequence of morphisms of pointed simplicial sets $\sigma_n \colon X_n \wedge S^1 \longrightarrow X_{n+1}$

such that

* for all $n, k \in \mathbb{N}$ the [[composition|composite]]

  $$
    X_n \wedge S^{k} \stackrel{\sigma_n \wedge id}{\longrightarrow} X_{n+1} \wedge S^{k-1} \stackrel{\sigma_{n+1}\wedge id}{\longrightarrow} \cdots \stackrel{\sigma_{n+k-1}}{\longrightarrow} X_{n+k}
  $$

  [[intertwiner|intertwines]] the $\Sigma_{n} \times \Sigma_k$-[[action]].

A [[morphism]] of symmetric spectra $f\colon X \longrightarrow Y$ is

* a sequence of morphisms of pointed simplicial sets $f_n \colon X_n \longrightarrow Y_n$

such that

1. each $f_n$ [[intertwiner|intetwines]] the $\Sigma_n$-[[action]];

1. the following [[commuting diagram|diagrams commute]]

   $$
     \array{
        X_n \wedge S^1 &\stackrel{f_n \wedge id}{\longrightarrow}& Y_n \wedge S^1
        \\
        \downarrow^{\mathrlap{\sigma^X_n}} && \downarrow^{\mathrlap{\sigma^Y_n}}
        \\
        X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1}
     }
     \,.
   $$

=--

([Hovey-Shipley-Smith 00, def. 1.2.1](#HoveyShipleySmith00), [Schwede 12, def. 1.1](#Schwede12))

### As $\mathbb{S}_{Sym}$-modules
 {#AsSModules}

A more concise equivalent formulation is the following:

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

Write $Sym$ for the standard [[skeletal category|skeleton]] of the [[core]] of [[FinSet]], with [[objects]] labeled by [[natural numbers]], all morphisms [[automorphisms]] and the automorphisms of $n$ forming the [[symmetric group]] $\Sigma_n$. The [[disjoint union]] monoidal structure on [[FinSet]] gives $Sym$ the structure of a [[monoidal category]] with tensor product given by addition of natural numbers on objects.

Write $[Sym,sSet^{\ast/}]$ for the [[functor category]] equipped with the induced [[Day convolution]] product. Write

$$
  \mathbb{S}_{Sym} \in [Sym,sSet^{\ast/}]
$$

for the functor which sends $n$ to the minimal simplicial [[n-sphere]] $S^n = (S^1)^{\wedge^n}$ and which sends elements of the symmetric group to the corresponding endomorphisms of $S^n$ exchanging [[smash product]] factors.

Regard $\mathbb{S}_{Sym}$ as a [[monoid object]] with respect to [[Day convolution]], whose monoidal structure is given via the [discussion there](Day+convolution#Monoids) by the system of canonical natural morphisms

$$
  S^{n_1}\wedge S^{n_2} \longrightarrow S^{n_1 + n_2}
  \,.
$$

=--

+-- {: .num_defn #SymmetricSpectrumAsSModule}
###### Proposition

Symmetric spectra in the sense of def. \ref{SymmetricSpectrum} are equivalently right [[module objects]] in $([Sym,sSet^{\ast/}], \otimes_{Day})$ over the [[monoid object]] $\mathbb{S}_{Sym}$, according to def. \ref{SphereSpectrumAsDayConvolutionMonoidOverSym}:

$$
  SymSpec(sSet)
  \simeq
  \mathbb{S}_{Sym}Mod_r
  \,.
$$

=--

This point of view was highlighted and almost made explicit in ([Mandell-May-Schwede-Shipley 01](#MandellMaySchwedeShipley01)).

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

By the discussion [there](Day+convolution#Monoids), right $\mathbb{S}_{Sym}$-modules with respect to [[Day convolution]] are equivalently right [[modules over monoidal functors]] over the monoidal functor corresponding to $\mathbb{S}_{Sym}$ as in def. \ref{SphereSpectrumAsDayConvolutionMonoidOverSym}. This means that they are functors $X \colon Sym \longrightarrow sSet^{\ast/}$ equipped with [[natural transformations]]

$$
  X_p \wedge S^{q}  \longrightarrow X_{p+q}
$$

satisfying the evident [[categorification|categorified]] [[action]] property. In the present case this action property says that these morphisms are determined by

$$
  X_p \wedge S^1 \longrightarrow X_{p+1}
$$

under the isomorphisms $S^p \simeq S^1 \wedge S^{p-1}$. Naturality of all these morphisms as functors on $Sym$ is the equivariance under the symmetric group actions in def. \ref{SymmetricSpectrum}.

=--

For more on this see at _[[Model categories of diagram spectra]] -- [part I](Model%20categories%20of%20diagram%20spectra#PartI)_.

## Properties

### Smash product
 {#SmashProduct}

[[smash product of spectra]] modeled on symmetric spectra

e.g. ([Schwede 12, I.3](#Schwede12))

### Homotopy groups
 {#HomotopyGroups}

The main technical issue with symmetric spectra is that the evident definition of [[stable homotopy groups]] of symmetric spectra $X$ as

$$
  \pi_n(X) = \underset{\longrightarrow}{\lim}_k \pi_{k+n}(X_k)
$$

does _not_ in general come out in the homotopy correct way; instead one needs to replace by a "[[semistable symmetric spectrum]]" first, which however is hard to control ([Hovey-Shipley-Smith 00, section 3.1](#HoveyShipleySmith00), [Schwede 12, chapter I, sections 2 and 6 and 8](#Schwede12)), survey includes ([Malkiewich 14, section 2.3](#Malkiewich14)).

Among the various (now) common models for [[spectra]], this issue is unique to symmetric spectra, see also the comment in ([Mandell-May-Schwede-Shipley 01, p. 3](#MandellMaySchwedeShipley01)).

In particular, this problem goes away for the concept of [[orthogonal spectra]], which otherwise is very similar to that of symmetric spectra and shares most of its advantages (except that it does not admit a version based on [[simplicial sets]]).

Technically, the issue comes down to the fact that the [[quotients]] of [[symmetric groups]] $\Sigma_q/\Sigma_{q-n}$ do not become highly connected for large $q$ the way the quotients $O(q)/O(n-q)$ of [[orthogonal groups]] do ([Mandell-May-Schwede-Shipley 01, proof of lemma 8.6, top of p. 26](#MandellMaySchwedeShipley01)). See [this proposition](Model+categories+of+diagram+spectra#AdjunctsOfFreeSpectrumInclusionsAreOrAreNotStableWeakHomotopyEquivalences) at _[[Model categories of diagram spectra]]_.

## Related concepts

* [[symmetric sequence]]

* [[free symmetric spectrum]]

* [[symmetric ring spectrum]]

[[model structure on spectra]], [[symmetric monoidal smash product of spectra]]

* **symmetric spectrum**, [[model structure on symmetric spectra]]

* [[orthogonal spectrum]], [[model structure on orthogonal spectra]]

* [[S-module]], [[model structure on S-modules]]

## References

The orginal article is

* {#HoveyShipleySmith00} [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], *Symmetric spectra*, J. Amer. Math. Soc. **13** (2000) 149-208 [[arXiv:math/9801077](http://arxiv.org/abs/math/9801077), [doi:10.1090/S0894-0347-99-00320-3](https://doi.org/10.1090/S0894-0347-99-00320-3)]

A comprehensive textbook account is in

* {#Schwede12} [[Stefan Schwede]], _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

Further discussion of the [[model structure on symmetric spectra]] beyond ([Hovey-Shipley-Smith 00](#HoveyShipleySmith00)) and [Schwede 12, part III](#Schwede12) includes

* {#MandellMaySchwedeShipley01} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], _[[Model categories of diagram spectra]]_, Proceedings of the London Mathematical Society, 82 (2001), 441-512 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf))

More or less brief reviews include

* {#Malkiewich14} [[Cary Malkiewich]], section 2.3 of _The stable homotopy category_, 2014 ([pdf](http://math.stanford.edu/~carym/stable.pdf))

* [[Alexander Kupers]], _Symmetric spectra_ ([pdf](http://ncatlab.org/nlab/files/SymmetricSpectra.pdf))

* [[Mitya Boyarchenko]], _Introduction to symmetric spectra_ (prepared for a [Geometric Langlands Seminar](http://www.math.uchicago.edu/~mitya/langlands.html)) Part I ([pdf](http://www.math.uchicago.edu/~mitya/langlands/spectra/iss1.pdf)) and Part II ([pdf](http://www.math.uchicago.edu/~mitya/langlands/spectra/iss2.pdf))

[[!redirects symmetric spectra]]


symmetric topological space

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## Definition

A [[topological space]] $(X,\tau)$ is called a **symmetric topological space** or an **$R_0$-topological space** if for all elements $x \in X$ and $y \in X$, if for all [[open sets]] $U \in \tau$, $x \in U$ implies that $y \in U$, then for all open sets $U \in \tau$, $y \in U$ implies that $x \in U$. Or equivalently, if its [[specialization preorder]] is an [[equivalence relation]].

## Properties

A $T_1$-topological space is a symmetric topological space which is also a [[Kolmogorov topological space]]. The [[quotient topological space|quotient]] of a symmetric topological space $(X, \tau)$ by its [[specialization order]] equivalence relation is a $T_1$-topological space, the [[Kolmogorov quotient]] of $X$.

## Examples

Every [[set]] $X$ with its [[power set]] $\mathcal{P}(X)$ is a symmetric topological space with respect to the specialization order equivalence relation defined as
$$x \equiv y \coloneqq \forall P \in \mathcal{P}(X).P(x) \iff P(y)$$
for all $x \in X$ and $y \in X$.

## Related concepts

* [[specialization order]]
* [[separation axiom]]

| [[preorder]] | [[partial order]] | [[equivalence relation]] | [[equality]] |
| - | - | - | - |
| [[topological space]] | [[ Kolmogorov topological space]] | [[symmetric topological space]] | [[accessible topological space]] |

[[!redirects R0]]
[[!redirects R_0]]

[[!redirects R0 space]]
[[!redirects R0 spaces]]
[[!redirects R0-space]]
[[!redirects R0-spaces]]
[[!redirects R_0 space]]
[[!redirects R_0 spaces]]
[[!redirects R_0-space]]
[[!redirects R_0-spaces]]

[[!redirects symmetric topological space]]
[[!redirects symmetric topological spaces]]
[[!redirects R0 topological space]]
[[!redirects R0 topological spaces]]
[[!redirects R_0 topological space]]
[[!redirects R_0 topological spaces]]
[[!redirects R0-topological space]]
[[!redirects R0-topological spaces]]
[[!redirects R_0-topological space]]
[[!redirects R_0-topological spaces]]


symmetric topos

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## Idea

The **symmetric topos** $M(\mathcal{E})$ of a Grothendieck topos $\mathcal{E}$ is the topos analogue of the [[symmetric algebra]] construction. Its importance stems from the property that for a topos $\mathcal{E}$ the symmetric topos $M(\mathcal{E})$ classifies [[Lawvere distribution|topos distributions]] on $\mathcal{E}$.

## Definition

## Properties

## Example

## Related entries

* [[Lawvere distribution]]

* [[bagdomain topos]]

* [[cosheaf]]

* [[locally connected geometric morphism]]

* [[totally connected geometric morphism]]

* [[lax-idempotent 2-monad]]

## References

* [[Marta Bunge]], _Cosheaves and Distributions on Toposes_ , Alg. Univ. **34** (1995) pp.469-484.

* [[Marta Bunge]], [[Aurelio Carboni]], _The Symmetric Topos_  ,  JPAA **105** (1995) pp.233-249.

* [[Marta Bunge]], [[Jonathon Funk]], _Spreads and the Symmetric Topos_ , JPAA **113** (1996) pp.1-38.

* [[Marta Bunge]], [[Jonathon Funk]], _Spreads and the Symmetric Topos II_ , JPAA **130** (1998) pp.49-84.

* {#BF99} [[Marta Bunge]], [[Jonathon Funk]], _On a bicomma object condition for KZ-doctrines_ , JPAA **143** (1999) pp.69-105.

* [[Marta Bunge]], [[Jonathon Funk]], _Singular Coverings of Toposes_ , Springer LNM **1890** Heidelberg 2006.

* [[Peter Johnstone]], _Sketches of an Elephant I_ , Oxford UP 2002.  (sec. B4.5, pp.456-467) {#elephant}

* [[Andrew Pitts|Andrew M. Pitts]], _On product and change of base for toposes_ , Cah. Top. Géom. Diff. Cat. **XXVI** no.1 (1985) pp.43-61.

* [[Steve Vickers]], _Cosheaves and Connectedness in Formal Topology_ , Ann. Pure Appl. Logic **163** no.2 (2012) pp.157-174. ([preprint](www.cs.bham.ac.uk/~sjv/ConnFT.pdf))


Symmetry

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This page compiles pointers related to:

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]:

  \linebreak

  **Symmetry**

  \linebreak

  (2021)

  [pdf](https://unimath.github.io/SymmetryBook/book.pdf)

which is a gentle exposition to [[group theory]] ("[[symmetry]]") in the language of [[univalent foundations of mathematics]] ([[homotopy type theory]] with the [[univalence axiom]]).

Formalization in [[Agda]]:

* [[UniMath project]]: [agda-unimath/src/group-theory](https://github.com/UniMath/agda-unimath/tree/master/src/group-theory)

category: reference


symmetry

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#Contents#
* table of contents
{:toc}

## Idea

A [[symmetry]] is (in most classical situations) invariance under a [[group]] [[action]], or [[infinitesimal space|infinitesimally]], invariance under a [[Lie algebra]] action. Indeed, historically the mathematical term "[[group]]" is a contraction of _group of symmetries_ (e.g. [Klein 1872](#Klein1872)). More recently,the concept of symmetry has been a popular topic of study in the Physics literature under the name of [[generalized symmetry]].

In [[physics]], see _[[local symmetry]]_, _[[global symmetry]]_, _[[spontaneous symmetry breaking]]_.

## Examples

* [[Euclidean group]], [[space group]]/[[crystallographic group]]

* [[Lorentz group]]

* ...

## Related concepts

* [[automorphism group]]

* [[invariant]], [[invariant theory]]

* [[stabilizer subgroup]], [[Erlangen program]]

* [[symmetry of a Lagrangian]]

* [[quantum symmetry]]

* [[supersymmetry]]

## References

Historical articles

* {#Klein1872} [[Felix Klein]], _Vergleichende Betrachtungen über neuere geometrische Forschungen_ (1872)

  translation by M. W. Haskell, _A comparative review of recent researches in geometry_ , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. ([retyped pdf](http://math.ucr.edu/home/baez/erlangen/erlangen_tex.pdf), [[KleinRetyped.pdf:file]], [scan of original](http://math.ucr.edu/home/baez/erlangen/erlangen.pdf))

  > ([[Erlanger program]])

* [[Hermann Weyl]], *Symmetry*, Journal of the Washington Academy of Sciences **28** 6 (1938) 253-271 [[jstor:24530200](https://www.jstor.org/stable/24530200)]

On [[transformation groups]]:

* [[Willard Miller]], *Symmetry Groups and Their Applications*, Pure and Applied Mathematics **50** (1972) 16-60 [[ISBN:9780080873657](https://www.elsevier.com/books/symmetry-groups-and-their-applications/miller/978-0-12-497460-9)]

On symmetry and introducing the language of [[homotopy type theory]] for [[univalent foundations of mathematics]]:

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$

[[!redirects symmetries]]
[[!redirects symmetry group]]
[[!redirects symmetry groups]]

[[!redirects infinitesimal symmetry]]
[[!redirects infinitesimal symmetries]]


symmetry of a Lagrangian density

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#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Throughout, let

1. $\Sigma$ be a [[smooth manifold]] of [[dimension]] $p+1$, thought of as [[spacetime]];

1. $E \overset{fb}{\longrightarrow} \Sigma$ a [[fiber bundle]] thought of as a _[[field bundle]]_

1. $\mathbf{L} \in \Omega^{p+1,0}_\Sigma(E)$ a [[Lagrangian density]].

+-- {: .num_defn #EvolutionaryVectorField}
###### Definition
**([[evolutionary vector fields]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]]. Then an _[[evolutionary vector field]]_ $v$ on $E$ is  "variational vertical vector field" on $E$, hence a smooth [[bundle]] [[homomorphism]]
out of the [[jet bundle]] (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})

$$
  \array{
    J^\infty_\Sigma E
    && \overset{v}{\longrightarrow} &&
    T_\Sigma E
    \\
    & {}_{\mathllap{jb_{\infty,0}}}\searrow && \swarrow_{\mathllap{}}
    \\
    && E
  }
$$

to the [[vertical tangent bundle]] $T_\Sigma E \overset{}{\to} \Sigma$ of $E \overset{fb}{\to} \Sigma$.

In the special case that the [[field bundle]] is a [[trivial vector bundle]] over [[Minkowski spacetime]] as in example \ref{TrivialVectorBundleAsAFieldBundle}, this means that an evolutionary vector field is
a [[tangent vector field]] (example \ref{TangentVectorFields}) on $J^\infty_\Sigma(E)$ of the special form

$$
  \begin{aligned}
    v
     & =
    v^a \partial_{\phi^a}
    \\
    & =
    v^a\left(
      (x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots
    \right) \partial_{\phi^a}
  \end{aligned}
  \,,
$$

where the [[coefficients]] $v^a \in C^\infty(J^\infty_\Sigma(E))$ are general [[smooth functions]]
on the [[jet bundle]] (while the cmponents are [[tangent vectors]] along the field coordinates $(\phi^a)$,
but not along the spacetime coordinates $(x^\mu)$ and not along the jet coordinates $\phi^a_{,\mu_1 \cdots \mu_k}$).

We write

$$
  \Gamma_E^{ev}\left( T_\Sigma E \right)
  \;\in\;
  \Omega^{0,0}_\Sigma(E) Mod
$$

for the space of evolutionary vector fields,
regarded as a [[module]] over the $\mathbb{R}$-[[associative algebra|algebra]]

$$
  \Omega^{0,0}_\Sigma(E)
  \;=\;
  C^\infty\left( J^\infty_\Sigma(E) \right)
$$

of [[smooth functions]] on the [[jet bundle]].

=--

An [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField}) describes
an infinitesimal change of field values _depending_ on, possibly, the point in spacetime and the values of the field
and all its derivatives (locally to finite order, by prop. \ref{JetBundleIsLocallyProManifold}).

This induces a corresponding infinitesimal change of the derivatives of the fields, called the _prolongation_
of the evolutionary vector field:

+-- {: .num_prop #EvolutionaryVectorFieldProlongation}
###### Proposition
**(prolongation of [[evolutionary vector field]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]].

Given an [[evolutionary vector field]] $v$ on $E$ (def. \ref{EvolutionaryVectorField})
there is a unique [[tangent vector field]] $\hat v$ on the [[jet bundle]] $J^\infty_\Sigma(E)$ such that

1. $\hat v$ agrees on field coordinates (as opposed to jet coordinates) with $v$:

   $$
     (jb_{\infty,0})_\ast(\hat v) = v
     \,,
   $$

   which means in the special case that $E \overset{fb}{\to} \Sigma$ is a [[trivial vector bundle]] over [[Minkowski spacetime]] that $\hat v$ is of the form

   $$
     \label{GenericComponentsForProlongationOfEvolutionaryVectorField}
     \hat v
     \;=\;
     \underset{
       = v
     }{
     \underbrace{
     v^a \partial_{\phi^a}
     }}
     \,+\,
     \hat v^a_{\mu} \partial_{\phi^a_{,\mu}}
     +
     \hat v^a_{\mu_1 \mu_2} \partial_{\phi^a_{,\mu_1 \mu_2}}
     +
     \cdots
   $$

1. contraction with $\hat v$ (def. \ref{ContractionOfFormsWithVectorFields}) anti-commutes with the [[total derivative|total spacetime derivative]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}):

   $$
     \label{ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative}
     \iota_{\hat v} \circ d + d \circ \iota_{\hat v} = 0
     \,.
   $$

In particular [[Cartan's homotopy formula]] (prop. \ref{CartanHomotopyFormula})
for the [[Lie derivative]] $\mathcal{L}_{\hat v}$ holds with respect to the [[variational derivative]] $\delta$:

$$
 \label{HomotopyFormulaForLieDerivativeAlongProlongationOfEvolutionaryVectorField}
  \mathcal{L}_{\hat v} = \delta \circ \iota_{\hat v} + \iota_{\hat v} \circ \delta
$$

Explicitly, in the special case that the [[field bundle]] is a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}) $\hat v$ is given by

$$
  \label{ProlongationOfEvolutionaryVectorFieldExplicit}
  \hat v = \underoverset{n = 0}{\infty}{\sum} \frac{d^n v^a}{ d x^{\mu_1} \cdots d x^{\mu_n} } \partial_{\phi^a_{\mu_1 \cdots \mu_n}}
  \,.
$$

=--

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

It is sufficient to prove the coordinate version of the statement.
We prove this by [[induction]] over the maximal jet order $k$. Notice that the coefficient of $\partial_{\phi^a_{\mu_1 \cdots \mu_k}}$
in $\hat v$ is given by the contraction $\iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k}$ (def. \ref{ContractionOfFormsWithVectorFields}).

Similarly (at "$k = -1$") the component of $\partial_{\mu_1}$ is given by $\iota_{\hat v} d x^{\mu}$. But by the second condition
above this vanishes:

$$
  \begin{aligned}
    \iota_{\hat v} d x^\mu
    & =
    d \iota_{\hat v} x^\mu
    \\
    & =
    0
  \end{aligned}
$$

Moreover, the coefficient of $\partial_{\phi^a}$ in $\hat v$ is fixed by the first condition above to be

$$
  \iota_{\hat v} \delta \phi^a
  =
  v^a
  \,.
$$

This shows the statement for $k = 0$. Now assume that the statement is true up to some $k \in \mathbb{N}$.
Observe that the coefficients of all $\partial_{\phi^a_{\mu_1 \cdots \mu_{k+1}}}$ are fixed by the
contractions with $\delta \phi^a_{\mu_1 \cdots \mu_{k} \mu_{k+1}} \wedge d x^{\mu_{k+1}}$. For this we find again from the second
condition and using $\delta \circ d + d \circ \delta = 0$ as well as the induction assumption that

$$
  \begin{aligned}
    \iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_{k+1}} \wedge d x^{\mu_{k+1}}
    & =
    \iota_{\hat v}  \delta d \phi^a_{\mu_1 \cdots \mu_k}
    \\
    & =
    d \iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k}
    \\
    &
    = d \frac{d^k v^a}{d x^{\mu_1} \cdots d x^{\mu_k}}
    \\
    & =
    \frac{d^{k+1}v^a }{d x^{\mu_1} \cdots d x^{\mu_{k+1}}} d x^{\mu_{k+1}}
    \,.
  \end{aligned}
$$

This shows that $\hat v$ satisfying the two conditions given exists uniquely.

Finally formula (eq:HomotopyFormulaForLieDerivativeAlongProlongationOfEvolutionaryVectorField) for the [[Lie derivative]] follows from the second of the two conditions with [[Cartan's homotopy formula]]
$\mathcal{L}_{\hat v} = \mathbf{d} \circ \iota_{\hat v} + \iota_{\hat v} \circ \mathbf{d}$ (prop. \ref{CartanHomotopyFormula})
together with $\mathbf{d} = \delta + d$.

=--

+-- {: .num_prop #EvolutionaryVectorFieldLieAlgebra}
###### Proposition
**([[evolutionary vector fields]] form a [[Lie algebra]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]]. For any two [[evolutionary vector fields]]
$v_1$, $v_2$ on $E$ (def. \ref{EvolutionaryVectorField}) the [[Lie bracket]] of [[tangent vector fields]]
of their prolongations $\hat v_1$, $\hat v_2$ (def. \ref{EvolutionaryVectorFieldProlongation})
is itself the prolongation $\widehat{[v_1, v_2]}$ of a unique evolutionary vector field $[v_1,v_2]$.

This defines the structure of a [[Lie algebra]] on evolutionary vector fields.

=--

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

It is clear that $[\hat v_1, \hat v_2]$ is still [[vertical vector field|vertical]], therefore, by
prop. \ref{EvolutionaryVectorFieldProlongation}, it is sufficient to show that contraction $\iota_{[v_1, v_2]}$ with this
vector field (def. \ref{ContractionOfFormsWithVectorFields}) anti-commutes with the [[horizontal derivative]] $d$,
hence that $[d, \iota_{[\hat v_1, \hat v_2]}] = 0$.

Now $[d, \iota_{[\hat v_1, \hat v_2]}]$ is an
operator that sends vertical 1-forms to horizontal 1-forms and vanishes on
horizontal 1-forms. Therefore it is sufficient to see that this operator
in fact also vanishes on all vertical 1-forms. But for this it is
sufficient that it commutes with the vertical derivative.
This we check by [[Cartan calculus]], using $[d,\delta] = 0$ and $[d, \iota_{\hat v_i}]$, by assumption:

$$
  \begin{aligned}
    {[ \delta, [ d,\iota_{[\hat v_1, \hat v_2]}] ]}
    & =
    - [d, [\delta, \iota_{[\hat v_1, \hat v_2]}]]
    \\
    & =
    - [d,  \mathcal{L}_{[\hat v_1, \hat v_2]}]
    \\
    & =
    -[d, [\mathcal{L}_{\hat v_1}, \iota_{\hat v_2}]  ]
    \\
    & =
    - [d, [ [\delta, \iota_{\hat v_1}], \iota_{\hat v_2}  ]]
    \\
    & =
    0
    \,.
  \end{aligned}
$$

=--

Now given an evolutionary vector field, we want to consider the [[flow]] that it
induces on the [[space of field histories]]:

+-- {: .num_defn #FlowOfFieldHistoriesAlongEvolutionaryVectorField}
###### Definition
**([[flow]] of [[field histories]] along [[evolutionary vector field]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]]
and let $v$ be an [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField})
such that the ordinary [[flow]] of its prolongation $\hat v$ (prop. \ref{EvolutionaryVectorFieldProlongation})

$$
  \exp(t \hat v)
  \;\colon\; J^\infty_\Sigma(E) \longrightarrow J^\infty_\Sigma(E)
$$

exists on the [[jet bundle]] (e.g. if the order of derivatives of field coordinates that it depends
on is bounded).

For $\Phi_{(-)} \colon U_1 \to  \Gamma_\Sigma(E)$ a collection of  [[field histories]]
(hence a plot of the [[space of field histories]]) the _[[flow]]_ of $v$
through $\Phi_{(-)}$ is the [[smooth function]]

$$
  U_1 \times \mathbb{R}^1
   \overset{\exp(v)(\Phi_{(-)})}{\longrightarrow}
  \Gamma_\Sigma(E)
$$

whose unique factorization $\widehat{\exp(v)}(\Phi_{(-)})$ through the space of jets of field histories
(i.e. the [[image]] $im(j^\infty_\Sigma)$ of [[jet prolongation]])

$$
  \array{
    && im(j^\infty_\Sigma) &\hookrightarrow& \Gamma_\Sigma(J^\infty_\Sigma(E))
    \\
    &
    {}^{\mathllap{\widehat{\exp(v)}(\Phi_{(-)})}}
    \nearrow
    &
    \Big\downarrow^{\mathrlap{\simeq}}
    \\
    U_1 \times \mathbb{R}^1
      &\underset{ \exp(v)(\Phi) }{\longrightarrow}&
    \Gamma_{\Sigma}(E)_{}
  }
$$

takes a plot $t_{(-)} \;\colon\; U_2 \to \mathbb{R}^1$ of the [[real line]]
(regarded as a [[super formal smooth set|super smooth set]]), to the plot

$$
  \label{LocalDataForFlowOfImplicitInfinitesimalGaugeSymmetry}
  (\exp(t(-) \hat v) \circ j^\infty_\Sigma(\Phi_{(-)})
    \;\colon\:
  U_1 \times U_2   \longrightarrow \Gamma_\Sigma\left( J^\infty_\Sigma(E) \right)
$$

of the [[smooth space|smooth]] [[space of sections]] of the [[jet bundle]].

(That $\exp(t(-) \hat v)$ indeed flows jet prolongations $j^\infty_\Sigma(\Phi(-))$ again to jet prolongations
is due to its defining relation to the [[evolutionary vector field]] $v$ from prop. \ref{EvolutionaryVectorFieldProlongation}.)

=--

+-- {: .num_defn #SymmetriesAndConservedCurrents}
###### Definition
**([[infinitesimal symmetries of the Lagrangian]] and [[conserved currents]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).

Then

1. an _[[infinitesimal symmetry of the Lagrangian]]_ is a variation $v$ (def. \ref{Variation})
   which arises as the prolongation $\hat v$ (prop. \ref{EvolutionaryVectorFieldProlongation})
   of  an [[evolutionary vector field]] $v$ (def. \ref{EvolutionaryVectorField}) such that the [[Lie derivative]] $\mathcal{L}_v$ of the Lagrangian density along $\hat v$ is a [[total derivative|total spacetime derivative]]

   $$
     \mathcal{L}_v \mathbf{L} = d \tilde J_v
   $$

1. an _[[on-shell]] [[conserved current]]_ is a horizontal $p$-form $J \in \Omega^{p,0}_\Sigma(E)$
   whose [[total derivative|total spacetime derivative]] vanishes on the prolonged [[shell]] (eq:ShellInJetBundle)

   $$
     d J\vert_{\mathcal{E}^\infty} = 0
     \,.
   $$

=--

+-- {: .num_prop #NoethersFirstTheorem}
###### Proposition
**([[Noether's theorem]] I)**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).

If $v$ is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents})
with $\mathcal{L}_v \mathbf{L} = d \tilde J_v$, then

$$
  J_v \coloneqq \tilde J_v - \iota_v \Theta_{BFV}
$$

is an [[on-shell]] [[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}), for $\Theta_{BFV}$
a presymplectic potential (eq:PresymplecticPotential) from def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}.

=--

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

By [[Cartan's homotopy formula]] for the [[Lie derivative]] (prop. \ref{CartanHomotopyFormula})
and the decomposition of the variational derivative $\delta \mathbf{L}$ (eq:dLDecomposition)
and the fact that contraction $\iota_{\hat v}$ with the prolongtion of an evolutionary vector field vanishes on horizontal differential forms
(eq:GenericComponentsForProlongationOfEvolutionaryVectorField) and anti-commutes with the horizontal differential (eq:ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative), by def. \ref{EvolutionaryVectorField},
we may re-express the defining equation for the symmetry as follows:

$$
  \begin{aligned}
    d \tilde J_v
    & =
    \mathcal{L}_v \mathbf{L}
    \\
    & =
    \iota_v \underset{= \delta_{EL}\mathbf{L} - d \Theta_{BFV}}{\underbrace{\mathbf{d} \mathbf{L}}}
    +
    \mathbf{d} \underset{= 0}{\underbrace{\iota_v \mathbf{L}}}
    \\
    & =
    \iota_v \delta_{EL} \mathbf{L}
    + d \iota_v \Theta_{BFV}
  \end{aligned}
$$

which is equivalent to

$$
  d(\underset{= J_v}{\underbrace{\tilde J_v - \iota_v \Theta_{BFV}}}) = \iota_v \delta_{EL}\mathbf{L}
$$

Since, by definition of the [[shell]] $\mathcal{E}$, the form $\frac{\delta_{EL} \mathbf{L}}{\delta v}$ vanishes on $\mathcal{E}$
this yields the claim.

=--

+-- {: .num_prop #FlowAlongImplicitInfinitesimalGaugeSymmetryPreservesOnShellSpaceofFieldHistories}
###### Proposition
**([[flow]] along [[infinitesimal symmetry of the Lagrangian]] preserves [[on-shell]] [[space of field histories]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).

For $v$ an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents})
the [[flow]] on the [[space of field histories]] (example \ref{DiffeologicalSpaceOfFieldHistories}) that it induces by def. \ref{FlowOfFieldHistoriesAlongEvolutionaryVectorField}
preserves the space of [[on-shell]] field histories (from prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}):

$$
  \array{
    \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
      &\hookrightarrow&
    \Gamma_\Sigma(E)
    \\
    {\mathllap{\exp(\hat v)\vert_{\delta_{EL}\mathbf{L} = 0}  }}
    \uparrow
      &&
    \uparrow {\mathrlap{\exp(\hat v)}}
    \\
    \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
      &\hookrightarrow&
    \Gamma_\Sigma(E)
  }
$$

=--

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

By def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime} a field history
$\Phi \in \Gamma_\Sigma(E)$ is [[on-shell]] precisely if its [[jet prolongation]] $j^\infty_\Sigma(E)$ (def. \ref{JetProlongation})
factors through the [[shell]] $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ (eq:ShellInJetBundle).
Hence by def. \ref{FlowOfFieldHistoriesAlongEvolutionaryVectorField} the statement is equivalently that
the ordinary flow (prop. \ref{CartanHomotopyFormula}) of $\hat v$ (def. \ref{EvolutionaryVectorFieldProlongation})
on the [[jet bundle]] $J^\infty_\Sigma(E)$ preserves the [[shell]]. This in turn means that it preserves the
vanishing locus of the [[Euler-Lagrange form]] $\delta_{EL} \mathbf{L}$.

For this it is sufficient to show that the [[derivative]] of the components of the [[Euler-Lagrange form]] along $\hat v$ vanish on the [[prolonged shell]]

$$
  \hat v \left( \frac{\delta_{EL} L}{\delta \phi^a} \right)
  \;=\;
  0
  \phantom{AA}
  on \mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)
  \,.
$$

This is the statement of [Olver 95, theorem 5.53](#Olver95).

=--

## References

* {#Olver95} [[Peter Olver]], _Applications of Lie groups to differential equations_, Springer; _Equivalence, invariants, and symmetry_, Cambridge Univ. Press 1995.

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symmetry protected topological phase

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

In [[solid state physics]], by a *[[phase of matter]]* which is

* *symmetry protected topological* ("SPT", [Gu & Wen 09](#GuWen09), [PBTO 09](#PBTO09))

or more generally

* *symmetry enriched topological* ("SET", [CGLW 11, p. 3](#CGLW11), [CLM 12, p. 2](#CLM12))

one means a [[topological phase of matter]] which is $G$-*[[equivariance|equivariantly]]* non-trivial, in that it cannot be [[quantum adiabatic theorem|adiabatically]] deformed to a trivial phase *while respecting some* $G$-[[symmetry]].
In case of SPT one in addition requires that the underlying topological phase (forgetting the symmetry) *is* trivial, while in case of SET this constraint is not implied.

Since, if one forgets (theoretically) the $G$-[[equivariance]], a $G$-equivariantly non-trivial SPT may be trivial as a plain topological phase, and two distinct SETs may be equivalent as plain topological phases,
one may regard the $G$-symmetry as "protecting" an SPT phase from decaying and as  "enriching" one plain phase to several distinct SET phases -- whence the terminology.

In other words:

1. distinct SPT/SET phases with a given symmetry cannot be [[quantum adiabatic theorem|adiabatically]] deformed into each other, in particular not without going through a [[phase transition]], *if* the whole deformation preserves the symmetry;

1. but they may possibly be transformed into each other this way if the symmetry is [[broken symmetry|broken]] during the deformation.

{#TypesOfSymmetries} There are the following types of symmetry groups $G$ to which the concept of crystalline SPT/SET phases applies (as well as to any of their [[semidirect product group|semidirect]] combinations):

\begin{imagefromfile}
    "file_name": "TypesOfSymsOfCrystTopInsulators-220515.jpg",
    "float": "right",
    "width": 550,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 0,
        "right": -10,
        "left": 10
    },
    "caption": "(table from [SS 22](#SS22))"
\end{imagefromfile}

1. $G$ may be an "external" **spatial** symmetry, namely a [[subgroup]] of the [[crystallographic group|crystallographic point group]] of the underlying [[crystal|crystalline]] material, canonically [[group action|acting]] on the [[Brillouin torus]] (BT) $\mathbb{T}^d$;

   for example *inversion* $I$ is the sign reversal [[involution]] on $\mathbb{R}^d \twoheadrightarrow \mathbb{T}^d$.

   If a [[topological insulator]]-phase is "protected" by a [[crystallographic group|crystallographic point group symmetry]] this way (possibly including [[time-reversal symmetry]]), then one speaks of a *[[topological crystalline insulator]]*.

1. $G$ may be [[time reversal symmetry]] or [[charge]] reversal, which acts as $I$ on the Brillouin torus, but in addition acts on the [[Hamiltonian]] by [[complex conjugation]] and, respectively, sign reversal;

1. $G$ may be an **[[internal symmetry]]** which acts not on the position but on the "internal degrees of freedom" of (electrons in the) substance, which are, typically, located at each atomic site (whence also: "on-site symmetry").

   The prime example of an internal symmetry are ([[finite subgroup of SU(2)|finite]]) [[subgroups]] of the [[Spin(3)]]$\simeq$[[SU(2)]]-group which [[spin representation|acts]] on the [[electron]] [[spins]]. This will be a symmetry to the extent that spin [[interactions]] (such as the [[spin-orbit coupling]] or the intrinsic [[Lorentz force]] due to an external [[magnetic field]]) are negligible.

\begin{remark}\label{OnTerminology}
**(terminology)**

1. Beware that some authors insist on using the specific term "SPT" only for items further down in this list.
For instance the claim below that SPT's are "classified by [[group cohomology]]" applies really to the last item (see [below](#ClassificationInternalSymmetries)).

   The first two items instead, for the case of free fermion systems at least, are expected to have a classification in [[twisted equivariant K-theory|twisted equivariant]] [[topological K-theory]] (see at [[K-theory classification of topological phases of matter]]).

   For the conceptual relation between these cases see [further below](#ClassificationInTermsOfKTheory).

1. Beware that the first articles on the topic ([Gu & Wen 09](#GuWen09), [PBTO 09](#PBTO09)) actually used the term "symmetry protected *topological order*". This could be perceived as somewhat of a misnomer, since in examples the (symmetry protected) "[[topological order]]" is often *trivial* (in that the [[ground state]] is non-degenerate and/or the [[Berry connection]] is [[abelian group|abelian]]) even though the underlying (symmetry protected) [[topological phase of matter|topological phase]] is non-trivial (ie. the [[valence bundle]] of a [[topological insulator]] has a non-trivial [[K-theory classification of topological phases of matter|K-class]]).

   Due to this problem, the original authors argued that "SPT" could also stand for "symmetric protected *trivial order*" ([X.-G. Wen,  Sep 18, 2014](https://physics.stackexchange.com/a/136096/5603)). But then it seems more descriptive (and now fairly widely accepted) to speak of "symmetry protected topological phase". (See also at *[[red herring principle]]*.)

\end{remark}

## Examples

### Spatial symmetry

* The [[topological insulator]]/[[semi-metal]]-phase of [[graphene]] is protected  by [[time-reversal symmetry]] $T$ (and inversion symmetry $I$),  and for this historical reason this $T$- and/or $T I$-symmetry protection is taken by some authors to be the default meaning of [[topological insulator]].

* *Removing* the $T$/$I$-protection of the phase of graphene leads to *un-protected* [[Chern insulators]] such as (modeled by) the [[Haldane model]].

For more on spatial-SPT *[[topological crystalline insulator]]*-phases, see [there](topological+crystalline+insulator#ReferencesExperimentalRealization) for concrete examples.

### Internal symmetry

* The first example of SPT order is the [[Haldane phase]] of the [[Heisenberg model]] [[spin chain|spin-1 chain]] ([Haldane 83](#Haldane83)). It is a SPT phase protected by the [[internal symmetry]] [[Spin(3)|$Spin(3)$]]-group acting on the [[electron]] [[spin]] (often referred to in this context as a [[projective representation]] of [[SO(3)|$SO(3)$]]).

More examples of internal-SPT: [Ye & Wen 13](#YeWen13), [CLV 14](#CLV14), [Yang-Liu 18](#YangLiu18) ...

## Properties

### Classification
 {#Classification}

#### Spatial symmetries

Symmetry protected [[crystal|crystalline]] phases where the [[dynamics]] of the [[electrons]] may approximately be regarded [[free field|free]] (but subject to the the atomic lattice Coulomb [[background field]], see [here](Dirac+field#FreeDiracFieldInCoulombBackground)) are thought to be classified by [[twisted equivariant K-theory|twisted equivariant]] [[topological K-theory]] of the [[Brillouin torus]]. This is the statement of the [[K-theory classification of topological phases of matter]].

Beware that this case has mostly been discussed for CPT-symmetries such as [[time-reversal symmetry]] (see at *[[topological insulator]]*) and for [[crystallographic group|crystallographic symmetries]] (see at *[[topological crystalline insulator]]*), not so much for [[internal symmetry|internal]] ("on site") symmetries (but see [Wen 12](#Wen12)).

#### Internal symmetries
 {#ClassificationInternalSymmetries}

In contrast, a widely cited claim [CGLW 11](#CGLW11) asserts (motivation is offered in [CLW 11, Sec. V](#CLW11)) that "*bosonic*" SPT orders for an *[[internal symmetry|internal]]* [[symmetry group]] $G_{int}$ are given by [[group cohomology]] $H^{d+1}\big(G;\, U(1)\big)$ of $G_{int}$ with [[coefficients]] in the [[circle group]] and in degree $d + 1$, for $d$ the effective [[dimension]] of the given material (in practice: $d \in \{0,1,2,3\}$).
This claim was generalized to fermionic SPT orders via a kind of group-supercohomology ([Gu & Wen14](#GuWen14)).

From X.-G. Wen ([2013, in rev 1](https://ncatlab.org/nlab/revision/symmetry+protected+topological+phase/1)):

> So the group (super-)cohomology theory may allow us to classify all SPT orders even for interacting systems, which include interacting topological insulator/superconductor.

But conceptual problems have remained with this proposal:

From [Wang & Senthil 14, p. 1](#WangSenthil14)

> this classification is now known not to be complete

{#XiongQuoteOpenProblem} From [Xiong 18](#Xiong18):

> Despite tremendous progress, a complete classification of SPT phases for arbitrary symmetries in arbitrary dimensions remains elusive. A number of classification proposals have been made in the general case: the Borel group cohomology proposal $[$[CGLW11](#CGLW11)$]$, the oriented cobordism proposal, the Freed-Hopkins proposal, and Kitaev’s proposal in the bosonic case; and the group supercohomology proposal $[$[GW14](#GuWen14)$]$, the spin cobordism proposal, the Freed-Hopkins proposal, and Kitaev’s proposal in the fermionic case. These proposals give differing predictions in certain dimensions for certain symmetry groups that cannot be attributed to differences in definitions or conventions. While more careful analysis has uncovered previously overlooked phases and input from topological field theories has brought us closer than ever to our destination, we believe that we can do much more.

From [BBCW 19, p. 3](#BBCW19):

> Although a remarkable amount of progress has been made on these deeply interrelated topics, a completely general understanding is lacking, and many questions remain. For example, although there are many partial results, the current understanding of fractionalization of quantum numbers, along with the classification and characterization of SETs is incomplete.

> Moreover, while there have been many results towards understanding the properties of extrinsic defects in topological phases, there has been no general systematic understanding and, in particular, no concrete method of computing all the rich topological properties of the defects for an arbitrary topological phase. The study of topological phase transitions between different topological phases is also missing a general theory. In this paper, we develop a general systematic framework to understand these problems.

\linebreak

The proposal of [BBCW 19](#BBCW19) (may not yet have a physics "proof" either, but) is conceptually transparent: The authors assume, as often done, that a [[topological order]] with [[anyon|anyonic]] [[defects]] is characterized by a [[unitary fusion category]] $\mathcal{C}$, and then propose that a $G_{int}$-SPT-phase $\Phi$ with this underlying topological order is what we may equivalently recognize as:

1. the [[equivalence class]] $[\Phi]$ of an [[∞-action]] of $G_{int}$ on $\mathcal{C}$, namely

1. the [[cohomology]]-class of a [[2-group]]-[[homomorphism]] from $G_{int}$ to the [[automorphism 2-group]] of $\mathcal{C}$:

   \[
     \label{GIntActionOnUnitaryFusionCategory}
     \Phi
     \;\in\;
     Hom
     \big(
       G_{int}
       ,\,
       Aut(\mathcal{C})
     \big)
     \,,
   \]

   > (this is essentially [BBCW 19, (1)](#BBCW19) -- where it says "group action", but later from (81) on ([p. 14](https://arxiv.org/pdf/1410.4540.pdf#page=14)) it transpires that the correct [[2-group]]-action is indeed meant),

   namely:

1. the [[pseudonatural transformation]]-class of a [[weak 2-functor]]

   $$
     \Phi
     \;\in\;
     Maps
     \big(
       \mathbf{B}G_{int}
       ,\,
       \mathbf{B}Aut(\mathcal{C})
     \big)
     \,,
   $$

   from the [[delooping groupoid]] of $G_{int}$ to that delooping [[2-groupoid]] of the [[automorphism 2-group]] of $\mathcal{C}$.

This is plausible (relative to the assumption that $\mathcal{C}$ characterizes the un-proteced topological order) since under an "[[internal symmetry]]" one wants to mean a global group action under which all other constructions have [[equivariant]]-[[structure]], and (eq:GIntActionOnUnitaryFusionCategory) is exactly the data that equips [[anyon]]-species (the [[simple objects]] of $\mathcal{C}$) and their [[fusion]] (the [[tensor product]]) and [[braid representation|braiding]] (the [[braiding]]) with such $G_{int}$-equivariant structure.

\linebreak

{#ButNeitherOfThese} But neither of these proposals connects recognizably to the [[twisted equivariant K-theory|twisted equivariant]] [[K-theory classification of topological phases]] (TE-K). While the latter may not apply to all cases of symmetry symmetry protected topological phases, it certainly applies to some of them, and it currently stands out among all other proposals on the classification of [[topological phases of matter]] as being the one with the most detailed support by theory and experiment. Therefore it would be reassuring to see how any other classification proposal plausibly connects to the TE-K proposal in appropriate special cases (notably for phases well-approximated by free fermion dynamics).

{#ClassificationInTermsOfKTheory} Conversely, inspection of the mathematical construction of [[twisted equivariant K-theory]] (as made explicit in [[schreiber:Equivariant principal infinity-bundles|SS 21]], [(4.1.28)](https://ncatlab.org/schreiber/files/EquivariantInfinityBundles_220324.pdf#page=201)) readily reveals the evident way in which it subsumes [[internal symmetries]] and how the resulting structure has an equivalent expression in terms of [[∞-group cohomology]]. This is an immediate consequence of the [[mapping stack]]-[[adjoint (infinity,1)-functor|$\infty$-adjunction]], as shown the following diagram

> (graphics from [[schreiber:Topological Quantum Computation in TED-K|SS 22b]])

Here the mapping stack adjunction...

* ...on the internal symmetry group factor (shown on the right) manifestly identifies the [[TE-K cohomology]]-group at the top with an [[∞-group cohomology|∞-group cohomology group]] of $G_{int}$ which plausibly matches, as before, the intuition one would have about internal symmetry groups acting on K-theoretically classified phases of matter;

* ..on the external symmetry group factor (shonw on the left) manifestly identifies the [[TE-K cohomology]]-group at the top with an [[inner local system]]-twisted sector of [[geometric fixed point spectrum|$G$-fixed]] TE-K-theory. Just this sector has been shown (in [[schreiber:Anyonic defect branes in TED-K-theory|SS 22a]], by recourse to the [[hypergeometric construction of KZ solutions]]) to host [[su(2)-anyon|SU(2)-]][[anyon]] [[wavefunctions]] (namely [[affine Lie algebra]]-[[conformal blocks]]) constituting [[braid representations]], and hence exactly the structure expected in a symmetry protected [[topological order]].

\linebreak

## Related concepts

* [[protection from quantum corrections]]

* [[TQFT]], [[topological order]],

* [[group cohomology]]

* [[entanglement]]

## References

#### Original articles

* {#GuWen09} [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], *Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order*, Phys. Rev. B **80** 155131 (2009) $[$[arXiv:0903.1069](https://arxiv.org/abs/0903.1069), [doi:10.1103/PhysRevB.80.155131](https://doi.org/10.1103/PhysRevB.80.155131)$]$

* {#PBTO09} [[Frank Pollmann]], [[Erez Berg]], [[Ari M. Turner]], [[Masaki Oshikawa]], *Symmetry protection of topological order in one-dimensional quantum spin systems*, Phys. Rev. B **85** 075125 (2012) $[$[arXiv:0909.4059](https://arxiv.org/abs/0909.4059), [doi:10.1103/PhysRevB.85.075125](https://doi.org/10.1103/PhysRevB.85.075125)$]$

* [[Xie Chen]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], *2D symmetry protected topological orders and their protected gapless edge excitations*, Phys. Rev. B 84, 235141 (2011);

See also:

* {#YeWen13} [[Peng Ye]], [[Xiao-Gang Wen]], *Projective construction of two-dimensional symmetry-protected topological phases with $\mathrm{U}(1)$, $SO(3)$, or $SU(2)$ symmetries*, Phys. Rev. B **87** 195128 (2013) $[$[doi:10.1103/PhysRevB.87.195128](https://doi.org/10.1103/PhysRevB.87.195128), [arXiv:1212.2121](https://arxiv.org/abs/1212.2121)$]$

* {#CLM12} Gil Young Cho, Yuan-Ming Lu, and [[Joel E. Moore]], *Gapless edge states of background field theory and translation-symmetric $\mathbb{Z}_2$ spin liquids*, Phys. Rev. B **86** 125101 (2012) $[$[arXiv:10.1103/PhysRevB.86.125101](https://doi.org/10.1103/PhysRevB.86.125101)$]$

* {#CLV14} [[Xie Chen]], Yuan-Ming Lu, Ashvin Vishwanath, *Symmetry-protected topological phases from decorated domain walls*, Nature Communications **5** 3507  (2014) $[$[doi:10.1038/ncomms4507](https://doi.org/10.1038/ncomms4507)$]$

* [[Juven Wang]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], _Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond_, Phys. Rev. Lett. **114** (2015) 031601 [[arxiv:1405.7689](https://arxiv.org/abs/1405.7689), [doi:10.1103/PhysRevLett.114.031601](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.031601)]

* {#YangLiu18} Jian Yang, [[Zheng-Xin Liu]], *Irreducible Projective Representations and Their Physical Applications*, J. Phys. A: Math. Theor. **51** 025207 (2018) $[$[doi:10.1088/1751-8121/aa971a](https://doi.org/10.1088/1751-8121/aa971a), [arXiv:1605.05805](https://arxiv.org/abs/1605.05805)$]$

* Yoshiko Ogata, *An invariant of symmetry protected topological phases with on-site finite group symmetry for two-dimensional Fermion systems* $[$[arXiv:2110.04672](https://arxiv.org/abs/2110.04672)$]$

#### Reviews

* [[Ching-Kai Chiu]], [[Jeffrey C.Y. Teo]], [[Andreas P. Schnyder]] [[Shinsei Ryu]], *Classification of topological quantum matter with symmetries*, Rev. Mod. Phys. **88** 035005 (2016) $[$[arXiv:1505.03535](https://arxiv.org/abs/1505.03535), [doi:10.1103/RevModPhys.88.035005](https://doi.org/10.1103/RevModPhys.88.035005)$]$

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  Sec. 10 of: *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

* [[Tudor D. Stanescu]], Section 6.3 of: *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))

See also:

* Wikipedia, *[Symmetry protected topological order](http://en.wikipedia.org/wiki/Symmetry_protected_topological_order)*

Review with focus on the case of [[topological insulators]] [[symmetry protected topological phase|protected by]] [[crystallographic group]]-[[symmetry]]:

* [[Yoichi Ando]], [[Liang Fu]], *Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials*, Annual Review of Condensed Matter Physics **6** (2015) 361-381 $[$[arXiv:1501.00531](https://arxiv.org/abs/1501.00531), [doi:10.1146/annurev-conmatphys-031214-014501](https://doi.org/10.1146/annurev-conmatphys-031214-014501)$]$

A proposal for classifying symmetry-protected [[topological order]] in terms of [[2-group|2-groupoidal]]-actions on [[modular tensor categories]]:

* {#BBCW19} [[Maissam Barkeshli]], [[Parsa Bonderson]], [[Meng Cheng]], [[Zhenghan Wang]], *Symmetry Fractionalization, Defects, and Gauging of Topological Phases*, Phys. Rev. B **100** 115147 (2019) $[$[arXiv:1410.4540](https://arxiv.org/abs/1410.4540), [doi:10.1103/PhysRevB.100.115147](https://doi.org/10.1103/PhysRevB.100.115147), [talk pdf](http://helper.ipam.ucla.edu/publications/stq2015/stq2015_12401.pdf)$]$

* [[Zhenghan Wang]], §2.2 of: *Beyond Anyons*,  Modern Physics Letters A **33** 28 (2018) 1830011  [[arXiv:1710.00464](https://arxiv.org/abs/1710.00464), [doi:10.1142/S0217732318300112](https://doi.org/10.1142/S0217732318300112)]

See also:

* {#WangSenthil14} Chong Wang, T. Senthil, *Interacting fermionic topological insulators/superconductors in three dimensions*, Phys. Rev. B **89** 195124 (2014) $[$[arXiv:1401.1142](https://arxiv.org/abs/1401.1142), [doi:10.1103/PhysRevB.89.195124](https://doi.org/10.1103/PhysRevB.89.195124)$]$

* Hao Song, Sheng-Jie Huang, [[Liang Fu]], Michael Hermele, *Topological phases protected by point group symmetry*, Phys. Rev. X **7** (2017) 011020 $[$[arXiv:1604.08151](https://arxiv.org/abs/1604.08151), [doi:10.1103/PhysRevX.7.011020](https://doi.org/10.1103/PhysRevX.7.011020)$]$

* Zhen Bi, *Physical Properties and Experimental Platform of Symmetry Protected Topological Phases* (2017) $[$[escholarship:37q1k2rj](https://escholarship.org/uc/item/37q1k2rj)$]$

Some of the above material is taken from:

* {#SS22} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Topological Quantum Computation in TED-K]]*

#### Classification

##### Of free fermionic SPT phases

* {#Wen12} [[Xiao-Gang Wen]], *Symmetry-protected topological phases in noninteracting fermion systems*, Phys. Rev. B **85** (2012) 085103 $[$[doi:10.1103/PhysRevB.85.085103](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.85.085103)$]$

##### Of bosonic and interacting SPT phases

Claim of classification of SPT phases via [[group cohomology]]:

* {#CLW11} [[Xie Chen]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], *Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations*, Phys. Rev. B **84** (2011) 235141 $[$[doi:10.1103/PhysRevB.84.235141](https://doi.org/10.1103/PhysRevB.84.235141), [arXiv:1106.4752](https://arxiv.org/abs/1106.4752)$]$

* {#CGLW11} [[Xie Chen]], [[Zheng-Cheng Gu]], Zheng-Xin Liu, [[Xiao-Gang Wen]], _Symmetry protected topological orders and the group cohomology of their symmetry group_, Phys. Rev. B **87** (2013) 155114  $[$[arXiv:1106.4772](http://arxiv.org/abs/1106.4772), [doi:10.1103/PhysRevB.87.155114](https://doi.org/10.1103/PhysRevB.87.155114)$]$

* {#CGLW12} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], *Symmetry protected topological orders and the group cohomology of their symmetry group*,  Science **338** (2012) 1604-1606 ([doi:10.1103/PhysRevB.87.155114](https://doi.org/10.1103/PhysRevB.87.155114))

* {#GuWen14} [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], *Symmetry-protected topological orders for interacting fermions -- fermionic topological non-linear sigma-models and a group super-cohomology theory*, Phys. Rev. B **90** 115141 (2014) $[$[arXiv:1201.2648](http://arxiv.org/abs/1201.2648), [doi:10.1103/PhysRevB.90.115141](https://doi.org/10.1103/PhysRevB.90.115141)$]$

Classification for free fermion SPT phases in [[twisted equivariant K-theory]]:

* [[Alexei Kitaev]], _Periodic table for topological insulators and superconductors_, Proc. L.D.Landau Memorial Conf. "Advances in Theor. Physics", June 22-26, 2008, Chernogolovka, Russia, [arxiv/0901.2686](http://arxiv.org/abs/0901.2686) (uses [[K-homology]], [[Bott periodicity]] etc.)

* {#FreedMoore12} [[Daniel S. Freed]], [[Gregory Moore|Gregory W. Moore]], _Twisted equivariant matter_, Annales Henri Poincaré December 2013, Volume 14, Issue 8, pp 1927–2023 [arxiv/1208.5055](http://arxiv.org/abs/1208.5055) (uses [[equivariant K-theory]])

For more on this see at *[[K-theory classification of topological phases of matter]]*

Proposal that the general classification involves [[2-group]]-symmetries:

* [[Anton Kapustin]], [[Ryan Thorngren]], *Higher symmetry and gapped phases of gauge theories*, in *Algebra, Geometry, and Physics in the 21st Century*, Progress in Mathematics, **324** (2017) 177-202  [[arXiv:1309.4721](https://arxiv.org/abs/1309.4721), [doi:10.1007/978-3-319-59939-7_5](https://doi.org/10.1007/978-3-319-59939-7_5)]

Proposal that the general classification involves [[equivariant cohomology|equivariant]] [[Whitehead-generalized cohomology theory]]:

* {#Xiong18} [[Charles Zhaoxi Xiong]], *Minimalist approach to the classification of symmetry protected topological phases*, J. Phys. A: Math. Theor. **51** 445001 (2018) $[$[doi:1701.00004](https://arxiv.org/abs/1701.00004), [doi:10.1088/1751-8121/aae0b1](https://doi.org/10.1088/1751-8121/aae0b1)$]$

#### Examples

The 1d Haldane phase:

* {#Haldane83} [[Duncan Haldane]], *Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the $O(3)$ nonlinear sigma model*, Physics Letters A **93** 9 (1983) 464-468 $[$doi:10.1016/0375-9601(83)90631-X $]$

Examples with [[braid group]] effects:

* [[Michael Levin]], [[Zheng-Cheng Gu]], *Braiding statistics approach to symmetry-protected topological phases*, Phys. Rev. B **86** (2012) 115109 ([doi:10.1103/PhysRevB.86.115109](https://doi.org/10.1103/PhysRevB.86.115109))

and with [[loop braid group]]-effects:

* Chao-Ming Jian, Xiao-Liang Qi, *Layer Construction of 3D Topological States and String Braiding Statistics*, Phys. Rev. X **4** (2014) 041043 $[$[doi:10.1103/PhysRevX.4.041043](https://doi.org/10.1103/PhysRevX.4.041043)$]$

and with both:

* Zhen Bi, Yi-Zhuang You, Cenke Xu, *Anyon and loop braiding statistics in field theories with a topological $\Theta$ term*, Phys. Rev. B **90** (2014) 081110(R)  ([doi:10.1103/PhysRevB.90.081110](https://doi.org/10.1103/PhysRevB.90.081110))

#### Other articles

* [[Michael Levin]], Zheng-Cheng Gu, Braiding statistics approach to symmetry-protected topological phases, Phys. Rev. B 86, 115109 (2012), arXiv:1202.3120.

* Yuan-Ming Lu, Ashvin Vishwanath, Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach, Phys. Rev. B 86, 125119 (2012), arXiv:1205.3156.

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], _Symmetry protected topological phases and generalized cohomology_ $[$[arxiv/1712.07950](https://arxiv.org/abs/1712.07950)$]$

* Yizhi You, Trithep Devakul, F. J. Burnell, Titus Neupert, *Higher order symmetry-protected topological states for interacting bosons and fermions*, Phys. Rev. B **98** (2018) 235102  ([arXiv:1807.09788v2](https://arxiv.org/abs/1807.09788), [doi:10.1103/PhysRevB.98.235102](https://doi.org/10.1103/PhysRevB.98.235102))

* Rongge Xu, Zhi-Hao Zhang, *Categorical descriptions of 1-dimensional gapped phases with abelian onsite symmetries* $[$[arXiv:2205.09656](https://arxiv.org/abs/2205.09656)$]$

Observation in [[experiment]]:

* Zhihuang Luo, Wenzhao Zhang, Xinfang Nie, Dawei Lu, *Observation of a symmetry-protected topological phase in external magnetic fields* [[arXiv2208.05357](https://arxiv.org/abs/2208.05357)]

#### Conference and seminar cycles

* seminar in Koeln [Topological states of matter](http://www.thp.uni-koeln.de/trebst/Lectures/2012-TopoSeminar.html)

* Topological Phases of Matter: Simons Center, June 10-14, 2013, videos [available](http://scgp.stonybrook.edu/archives/3464)

* [[Alexei Kitaev]], _On the classification of short-range entangled states_, [video](http://scgp.stonybrook.edu/archives/7874)

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symplectic 2-groupoid

#Contents#
* table of contents
{:toc}

## Idea

the [[Lie integration]] of a [[Courant Lie 2-algebroid]] to a [[smooth infinity-groupoid|smooth 2-groupoid]] with extra structure. The higher analog to a [[symplectic groupoid]].

## Related concepts

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


symplectic connection

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

#Contents#
* table of contents
{:toc}

## Definition

An [[affine connection]] $\nabla$ on a [[symplectic manifold]] $(X,\omega)$ is a **symplectic connection** if the [[symplectic form]] is [[covariant derivative|covariantly constant]] with respect to the connection, $\nabla \omega = 0$.

## Applications

### In deformation quantization

The second order term in the [[formal deformation quantization]] (see there) of a symplectic manifold is closely related to symplectic connections.

## References

* [[eom]], _[Symplectic connection](http://www.encyclopediaofmath.org/index.php/Symplectic_connection)_

[[!redirects symplectic connections]]


symplectic derived stack

Symplectic derived stacks and shifted symplectic derived stacks
are studied in

* T. Pantev, [[Bertrand Toen|B. Toen]], M. Vaquie, G. Vezzosi, _Shifted symplectic structures_, Publ. Math. IHES [arxiv/1111.3209](http://arxiv.org/abs/1111.3209)
* Oren Ben-Bassat, _Multiple derived Lagrangian intersections_, [arxiv/1309.0596](http://arxiv.org/abs/1309.0596)

Some of the structures are implicit in

* [[Maxim Kontsevich]], _Renormalization via OPE_, conference talk, Institut Camille Jordan, Lyon Oct 17, 2010 ([conference page](http://math.univ-lyon1.fr/~calaque/Colloque-schedule.htm))

Related entries: [[higher symplectic geometry]], [[derived critical locus]], [[Batalin-Vilkovisky formalism]], [[derived stack]]


symplectic dual pair

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

#Contents#
* table of contents
{:toc}

## Idea

A kind of [[correspondence]] space between [[Poisson manifolds]].

## Definition

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

For $(X_1, \pi_1)$ and $(X_2, \pi_2)$ two [[Poisson manifolds]], a _symplectic dual pair_ between them is a [[correspondence]] [[symplectic manifold]] $(Z,\pi_Z = \omega^{-1})$

$$
  (X_1, \pi_1) \stackrel{i_1}{\leftarrow} (Z,\omega) \stackrel{i_2}{\to} (X_2, \pi_2)
$$

such that for all $f_1 \in C^\infty(X_1)$ and $f_2 \in C^\infty(X_2)$ the [[Poisson bracket]] on $Z$ vanishes:

$$
  \{i_1^\ast f_1, i_2^\ast f_2\}_{\omega} = 0
  \,.
$$

=--

## Examples

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

A [[Lagrangian correspondence]] between [[symplectic manifolds]], regarded as [[Poisson manifolds]], is a symplectic dual pair.

=--

## Morita Equivalence

One may also impose additional conditions on $i_1$ and $i_2$. Suppose $i_1$ and $i_2$ are Poisson and anti-Poisson maps that are complete, constant rank with connected, simply-connected fibers satisfying the symplectic orthogonality of $i_1^\ast C^\infty (X_1)$ and $i_2^\ast C^\infty (X_2)$. Then this gives the notion of Morita equivalence between $X_1$ and $X_2$ as Poisson manifolds.

## Related concepts

* [[symplectic category]]

## References

The notion was introduced in

* M.V. Karasev, (1989), _The Maslov quantization conditions in higher cohomology and analogs of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds I, II_ Selecta Math. Soviet. 8, 213–234, 235–258.

and

* [[Alan Weinstein]], _The local structure of Poisson manifolds_, J. Diff. Geom. 18, 523–557 (1983)

A textbook accounts are in

* Ch. 4 _Dual pairs_, in: A. Cannas da Silva, [[Alan Weinstein]], _Geometric models for noncommutative algebras_, Berkeley Math. Lec. Notes Series, AMS 1999, [pdf](http://math.berkeley.edu/%7Ealanw/Models.pdf)
*  J.-P. Ortega, T.S. Ratiu, _Momentum maps and Hamiltonian reduction_, Progress in Math. 222, Birkhauser 2004

Other

* Paul Skerritt, Cornelia Vizman, _Dual pairs for matrix Lie groups_, [arxiv/1805.01519](https://arxiv.org/abs/1805.01519)

A review with an eye towards [[geometric quantization]] with codomain [[KK-theory]] ([[geometric quantization by push-forward]]) is in section 3 of

* [[Klaas Landsman]], _Functorial quantization and the Guillemin-Sternberg conjecture_,  Proc. Bialowieza 2002 ([arXiv:math-ph/0307059](http://arxiv.org/abs/math-ph/0307059))

[[!redirects symplectic dual pairs]]


symplectic duality

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###Context###
#### Symplectic geometry
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[[!include symplectic geometry - contents]]
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#Contents#
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##Overview##

**Symplectic duality** is the name of a hypothetical duality operation $\mathfrak{M}_0\mapsto \mathfrak{M}^!_0$ on conical [[symplectic singularities]].  At the moment, no rigorous definition exists, though there are a number of conjectured examples and a great number of connections that should tie together dual singularities.

The reader should be warned that this page and this circle of ideas are works in progress.  At the moment, two papers by [[Tom Braden|Braden]], [[Anthony Licata|Licata]], [[Nicholas Proudfoot|Proudfoot]] and [[Ben Webster|Webster]] that will lay out lots of the ideas and details behind the claims made below are still in preparation;  links will be added when they are ready to be publicly shared.

##Correspondence between geometric data##

For $\mathfrak{M}_0$ a conical symplectic singularity, there is an associated [[Harrison homology]] $HP^2( \mathfrak{M}_0)$ which is the base of a universal [[Poisson deformation]] of $\mathfrak{M}_0$; this group actually coincides with the quotient by a [[Coxeter group]] $W$ (called the **[[Weyl group]]** of $\mathfrak{M}_0$) of the $\mathbb{C}$-[[Picard group]] $H$ of a $\mathfrak{Q}$-[[Q-factorial|factorial]] [[terminal singularity|terminalization]] of $\mathfrak{M}_0$ (a [[symplectic resolution]] is an example of such a terminalization, but not all symplectic singularities have resolutions).  Let $H_{\mathbb{Z}}$ denote the Picard group of this variety.

The variety $\mathfrak{M}_0$ also has a finite-dimensional group of [[Hamiltonian vector field|Hamiltonian]] [[automorphisms]] commuting with the conical structure.  Let $G$ be a [[Levi complement]] of this group, $T$ a [[maximal torus]] of $G$ and $\mathbb{W}=N_G(T)/T$ its [[Weyl group]].  For all these pieces of data, let the corresponding object for the (hypothetical) dual be denoted by $(-)^!$

Among the basic properties we expect of $\mathfrak{M}_0$ is
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###### Conjecture

There are canonical isomorphisms of Coxeter groups $\mathbb{W}\cong W^!$ and $\mathbb{W}^!\cong W$ and
canonical equivariant  isomorphisms
$\mathfrak{t}\cong H^!$ and $\mathfrak{t}^!\cong H$, inducing isomorphisms $HP^2( \mathfrak{M}_0)\cong\mathfrak{t}^!/\mathbb{W}^!\cong \mathfrak{g}^!/G^!$.

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##Quantizations##

Every conical symplectic singularity has a canonical [[deformation quantization]]: a family $A$ of filtered non-commutative algebras over $HP^2( \mathfrak{M}_0)$, each of whose [[associated graded]] is isomorphic to the [[coordinate ring]] $\mathbb{C}[\mathfrak{M}_0]$.  One of the most interesting manifestations of the (hypothetical) duality is the effect it seems to have on the representation theory of these algebras.

Using the isomorphisms above, the choice of a quantization parameter $\lambda$ for $\mathfrak{M}_0$ thus corresponds to a $G^!$-conjugacy class of an element of $\mathfrak{g}^!$, which we can think of as a Hamiltonian vector field on $\mathfrak{M}_0^!$.  There is an element $\xi\in A_\lambda$ which corresponds to the [[Hamiltonian]] of this vector field under associated graded, and thus whose inner action on $A_\lambda$ is a lift of that of the vector field on functions on $\mathfrak{M}_0$.

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###### Definition

Category $\mathcal{O}_{\lambda;\xi}$ for $\xi$ is the category of finitely generated $A_\lambda$-modules where $\xi$ acts locally finitely with generalized eigenvalues whose real parts bounded above.

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Note that the definition of category $\mathcal{O}$ involved the choice of parameters $\lambda$ and $\xi$, and that under duality, the spaces where these parameters live switch places.

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###### Conjecture

There is a choice of "[[Kalb-Ramond field|b-fields]]" $b\in H, b^!\in H^!$ such that for $\lambda\in H_{\mathbb{Z}}, \lambda^!\in H_{\mathbb{Z}}^!$ generic, with $\xi,\xi^!$ the elements of $A^!$ and $A$ associated to these elements as described above,
the categories $\mathcal{O}_{\lambda+b;\xi^!}$ and $\mathcal{O}_{\lambda^!+b^!,\xi}^!$
generate equivalent triangulated subcategories of the [[triangulated category|triangulated categories]] of $A_{\lambda+b}$-modules and $A_{\lambda^!+b^!}$-modules.
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[This mathoverflow question](http://mathoverflow.net/questions/51289/) was an attempt to get a rigorous definition of what these b-fields are.

##The smooth case##

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###### Conjecture

The variety $\mathfrak{M}_0$ possesses a symplectic resolution of singularities $\mathfrak{M}$ if and only if $T^!$ acts on $\mathfrak{M}_0^!$ with a unique fixed point.

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Thus, if both $\mathfrak{M}_0$ and $\mathfrak{M}_0^!$ possess symplectic resolutions, they also both carry Hamiltonian torus actions with isolated fixed points.  In this case, the simple modules in category $\mathcal{O}_{\lambda;\xi}$ (for both $\lambda$ and $\xi$ generic)  are in bijection with $T$-fixed points on $\mathfrak{M}$.

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###### Conjecture

The categories $\mathcal{O}_{\lambda+b;\xi^!}$ and $\mathcal{O}_{\lambda^!+b^!,\xi}^!$ are both
[[Koszul algebra|Koszul]] and [[Koszul dual]] to each other.

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Underlying this Koszul duality should be a bijection between fixed points.  This bijection should also appear on the level of equivariant cohomology.

The map of equivariant homology $\scqup{x\in \mathfrak{M}^T} H_2^T(\{x\})\to H_2^T(\mathfrak{M})$ produces an arrangement of subspaces in $ H_2^T(\mathfrak{M})$.

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###### Conjecture
The vector spaces $H_2^T(\mathfrak{M})$ and $H_2^{T^!}(\mathfrak{M}^!)$ are canonically dual, with the spaces $H_2^T(\{x\})$ and $H_2^{T^!}(\{x^!\})$ mutual annihilators.
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In the paper

*  [[Tom Braden]], [[Anthony Licata]], [[Nicholas Proudfoot]] and [[Ben Webster]], _Localization algebras and deformations of Koszul algebras_, Selecta Mathematica  **17**, Issue 3 (2011), Page 533-572. [arXiv:arXiv:0905.1335](http://front.math.ucdavis.edu/arXiv:0905.1335)

it's shown that Koszul algebras have a similar duality property when equivariant cohomology rings are replaced with the centers of certain deformations.

##Twisting and shuffling##

The categories $\mathcal{O}_{\lambda;\xi}$ (for $\lambda$ generic) carry actions of topologically defined groups.  Let $V\subset \mathfrak{t}_{\mathbb{R}}$ be the subset of the Lie algebra of the compact real form of $T$ consisting of vector fields with isolated fixed points, and let $U\subset H_{\mathbb{R}}$ be the complement of the walls of all the Mori chambers of the $\mathbb{R}$-Picard group.  Then we have an action of $\pi_1(U/W,\lambda)$ by functors we call **twisting functors** and on $\pi_1(V/\mathbb{W},\xi)$ by  functors we call **shuffling functors**.

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###### Conjecture

The Koszul duality between $\mathcal{O}_{\lambda+b;\xi^!}$ and $\mathcal{O}_{\lambda^!+b^!,\xi}^!$ interchanges twisting and shuffling functors.

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##Examples##

All conjectured properties are proven for the following examples:

###Hypertoric varieties###
For a [[hypertoric variety]], the quantization is the torus invariant differential operators on a vector space (called the _hypertoric enveloping algebra_) which is constructed from a linear
    algebraic object called a polarized arrangement.   The symplectic dual is the
    hypertoric variety associated to the Gale dual arrangement.  [[Tom Braden|Braden]], [[Anthony Licata|Licata]], [[Nicholas Proudfoot|Proudfoot]] and [[Ben Webster|Webster]] constructed Koszul dual categories conjectured to arise this way in

*  [[Tom Braden]], [[Anthony Licata]], [[Nicholas Proudfoot]] and [[Ben Webster]], _Gale duality and Koszul duality_, Adv. Math. **225**
  (2010), no. 4, 2002--2049. [arXiv:0806.3256](http://front.math.ucdavis.edu/0806.3256)

and then proved that these categories arise in geometry, compatibly with all conjectures discussed in

*   [[Tom Braden]], [[Anthony Licata]], [[Nicholas Proudfoot]] and [[Ben Webster]], _Hypertoric category $\mathcal{O}$_, [arXiv:1010.2001](http://front.math.ucdavis.edu/1010.2001)

###Cotangent bundles and S3-varieties###
For the cotangent bundle $T^*(G/B)$, where $G$ is a
    reductive algebraic group and $B\subset G$ a Borel, the associated quantization is the universal enveloping algebra of $\mathfrak{g}$.  However, our definition of category $\mathcal{O}$ is not actually the same as the BGG category $\mathcal{O}$.  We require semi-simple action of the center, and allow non-semi-simple action  of the Cartan; in usual category $\mathcal{O}$ it is vice versa.  However, one can use a functor constructed by Soergel to construct an equivalence between these categories.

The symplectic dual to $T^*(G^\vee/B^\vee)$, $G^\vee$ is the Langlands dual of $G$.

Modulo the issues discussed above, the Koszul duality portion of duality is shown in

* [[Alexander Beilinson]], [[Victor Ginzburg]], and [[Wolfgang Soergel]], _Koszul
  duality patterns in representation theory_, J. Amer. Math. Soc. **9**
  (1996), no. 2, 473-527.

and the duality between twisting and shuffling in

* [[Volodymyr Mazorchuk]], [[Serge Ovsienko]], and [[Catharina Stroppel]], _Quadratic duals, Koszul dual functors, and applications_, Trans. Amer. Math. Soc.
  **361** (2009), no. 3, 1129-1172.

More generally, given partitions $\nu$ and $\mu$, there is a symplectic variety called an **S3-variety** $\mathfrak{X}^\nu_\mu$ which is the preimage in the cotangent bundle of the space of flags of type $\nu$ of the Slodowy slice to the nilpotent orbit with Jordan type $\mu$.  The algebra quantizing this variety is a primitive quotient of the W-algebra for the orbit of Jordan type $\mu$.  In

* [[Ben Webster]], _Singular blocks of parabolic category $\mathcal{O}$ and finite W-algebras_, Journal of Pure and Applied Algebra __215__:12 (2011) 2797--2804 [arXiv:0909.1860](https://arxiv.org/abs/0909.1860)

it's shown that the category $\mathcal{O}$'s attached to generic integral quantization parameters are in fact singular blocks of parabolic category $\mathcal{O}$'s (in the BGG sense) attached to the same Lie algebra.

The dual is again an S3 variety $\mathfrak{X}^{\mu}_{\nu}$ where $\mu$ and $\nu$ have switched roles.  The Koszul duality is that proven by BGS (referenced above), after using the equivalence to singular blocks of parabolic category $\mathcal{O}$ mentioned before.  One particularly interesting special case of this is when $\mu=0$, in which case we obtain the duality between cotangent bundle of the space of flags of type $\nu$ and the Slodowy slice to $O_{\mu}$ in the full nilcone.

###Quiver varieties and affine Grassmannian slices###

Perhaps the richest examples of symplectic singularities come from [[quiver varieties]]; these are varieties associated to a pair of weights in the [[weight lattice]] of a [[Kac-Moody algebra]].  The category $\cO$'s of these varieties are derived equivalent to algebras studied in

* [[Ben Webster]], _Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products_, [arXiv:1001.2020](http://arxiv.org/abs/1001.2020).
* [[Ben Webster]], _Knot invariants and higher representation theory II: the categorification of quantum knot invariants_, [arXiv:1005.4559](http://arxiv.org/abs/1005.4559).

with the shuffling functors coinciding with the R-matrix action from the second paper above, and the twisting functors coinciding with the Chuang-Rouquier braid group action on the category (these statements will be proved in a forthcoming paper of [[Ben Webster]]).

Heuristic considerations suggest that the dual variety to a quiver variety in finite type should be a slice between Schubert cells in the affine Grassmannian of the Langlands dual group.  Ongoing work of [[Joel Kamnitzer]], [[Ben Webster]] and [[Oded Yacobi]] is aimed at understanding the behavior of the quantization referred to above for an affine Grassmannian slice.

##Connections to physics##

This topicis particularly speculative: the relationship of
these constructions to [[physics]].

The physical theories which concern us are certain
[[gauge theories]] in 3
dimensions which carry $N=4$ [[supersymmetry]] (see _[[N=4 D=3 super Yang-Mills theory]]_); the most important examples
of these are associated to a [[compact topological space|compact]] [[simple Lie group]] $G$ and a
complex [[representation]] $V$ of $G$ and are obtained by [[Kaluza-Klein mechanism|dimensional reduction]] from 6 dimensional [[super Yang-Mills theory|super Yang-Mills]], considered for example in

* [[Nathan Seiberg]] and [[Edward Witten]], _Gauge dynamics and compactification to three dimensions_, The mathematical beauty of physics (Saclay, 1996), Adv. Ser.
  Math. Phys., vol. 24, World Sci. Publ., River Edge, NJ, 1997, pp. 333--366. ([arXiv:hep-th/9607163](http://arxiv.org/abs/hep-th/9607163))

The fields of this theory include scalar fermions
valued in the [[adjoint representation]] $\mg$ and hypermultiplets
corresponding to the representation $V$ (called "the matter content").

In

* [[Ken Intriligator]], [[Nathan Seiberg]], _Mirror symmetry in three-dimensional gauge theories_, Phys. Lett. B **387** (1996), no. 3, 513--519 ([arXiv:hep-th/9607207](http://arxiv.org/abs/hep-th/9607207))

Seiberg and Intrilligator propose a notion of a "[[mirror symmetry|mirror duality]]" between such [[gauge theories]] ([[3d mirror symmetry]]).  The most important feature of such a duality is that it interchanges two spaces attached to them, the [[Higgs branch]] and [[Coulomb branch]] of the [[moduli space]] of [[vacua]].  These are a pair of hyperkähler cones defined in the [[moduli space]] of [[vacua]] by the vanishing of some of the fields (the adjoint scalars for Higgs, the hypermultiplets for Coulomb).  After arriving at a preliminary version of the conjectures discussed above, it was pointed out (initially by [[Sergei Gukov]]) that there is a very large overlap between the list of examples above and the known list of Higgs/Coulomb pairs in physics.  For example,  Intriligator and Seiberg have shown that an [[ALE space]] forms a Higgs/Coulomb pair with the moduli space of an [[instanton]] of the corresponding simply-laced Lie group on $\mathbb{R}^2$.

In

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], and [[Yaron Oz]], _Mirror symmetry in three-dimensional gauge theories, quivers and D-branes_, Nuclear Phys. B **493** (1997), no. 1-2, 101--147.

it is shown that

* Gale dual hypertoric varieties are a Higgs/Coulomb pair.

* if an S3-variety is presented as a type A quiver variety, its dual is the dual S3-variety, also presented as a type $A$ quiver variety.

* all affine type quiver varieties are in a Higgs/Coulomb pair with another affine quiver variety, with the weight data transforming as in the rank/level duality.

These examples strongly suggest that

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###### Conjecture
  If $\mathfrak{M}$ is the [[Higgs branch]] of a [[N=4 D=3 super Yang-Mills theory]], then the dual $\mathfrak{M}^!$ exists and is the [[Coulomb branch]] of the same theory (equivalently, the Higgs branch of its [[mirror symmetry|mirror]]).

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Unfortunately, this conjecture is not as useful as one might imagine, as the construction of the Coulomb branch is rather subtle, and we have found no description of it at the level of precision which would satisfy a mathematician.  The issue is that while the "classical"
and "quantum" Higgs branches are the same, the Coulomb branch acquires "quantum corrections."  The classical Coulomb branch is easy to describe; it is $T^*H/W$ where $H$ is the abstract Cartan
subalgebra of $G$. However, to arrive at the correct conjecture for
$\mathfrak{M}^!$ one must change the hyperkähler metric on this space, and
thus also its topology, to account for the quantized nature of the
situation.  At the moment, we do not understand how to make these
changes of metric and topology precise, but we are optimistic about
the future.

## References

A recent seminar with a list of resources is

* [GRT learning seminar winter 2020: Symplectic Duality](http://www.math.toronto.edu/dbutson/ls20.html)

A survey is in

* [[Nick Proudfoot]], _[research statement](http://pages.uoregon.edu/njp/research.pdf)_

A workshop dealing with related topics is

* [Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory](http://www.fields.utoronto.ca/programs/scientific/11-12/reptheory2012/), May 7-9, 2012
Hosted by the Fields Institute, at the Univerity of Toronto


symplectic field theory

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###Context###
#### Symplectic geometry
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[[!include symplectic geometry - contents]]
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#### Quantum field theory
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[[!include functorial quantum field theory - contents]]
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#Contents#
* table of contents
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## Idea

__Symplectic field theory__ is a mathematical approach to
[[Gromov-Witten invariants]] (and invariants in [[contact geometry]] and the relative invariants of [[Legendrean submanifolds]] (relative contact homology)) in the spirit of [[topological field theory]]. It has inspired to some extent Hofer's generalized Fredholm theory on [[polyfolds]].

## References

* [[Yakov Eliashberg]], [[Alexander Givental]], [[Helmut Hofer]], _Introduction to symplectic field theory_, [math.SG/0010059](http://arxiv.org/abs/math/0010059)
* wikipedia: Floer homology:[symplectic field theory](http://en.wikipedia.org/wiki/Floer_homology#Symplectic_field_theory)
* Oliver Fabert, Paolo Rossi, _Topological recursion relations in the symplectic field theory of mapping tori and local symplectic field theory_, [arxiv/1109.0972](http://arxiv.org/abs/1109.0972)