dihedral group

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## Idea

### Dihedral groups
 {#DihedralGroups}

The dihedral group, $D_{2n}$, is a [[finite group]] of [[order of a group|order]] $2n$. It may be defined as the [[symmetry group]] of a regular $n$-gon.

For instance $D_6$ is the symmetry group of the equilateral triangle and is [[isomorphism|isomorphic]] to the [[symmetric group]], $S_3$.

For $n \in \mathbb{N}$, $n \geq 1$, the dihedral group $D_{2n}$ is thus the [[subgroup]] of the [[orthogonal group]] [[O(2)|$O(2)$]] which is [[generators and relations|generated]] from the [[finite group|finite]] [[cyclic group|cyclic]] [[subgroup]] $C_n \,\coloneqq\, \mathbb{Z}/n$ of [[SO(2)|$SO(2)$]] and the [[reflection]] at the $x$-axis (say).  It is a [[semidirect product group|semi-direct product]] of $C_n$ and a [[finite group of order 2|$C_2 \,\coloneqq\, \mathbb{Z}/2$]] corresponding to that reflection, hence fitting into a {#ShortExactSequence} [[short exact sequence]] as follows:

\begin{tikzcd}
  \mathbb{Z}/n
  \ar[r, hook]
  \ar[d, hook]
  &
  \mathrm{SO}(2)
  \ar[d, hook]
  \\
  D_{2n}
  \ar[d, ->>]
  \ar[r, hook]
  &
  \mathrm{O}(2)
  \ar[d, ->>]
  \\
  \mathbb{Z}/2
  \ar[r,-,shift left=1pt]
  \ar[r,-,shift right=1pt]
  &
  \mathbb{Z}/2
\end{tikzcd}

Under the further embedding $O(2)\hookrightarrow SO(3)$ the cyclic and dihedral groups are precisely those [[finite subgroups of SO(3)]] that, among their [[ADE classification]], are not in the exceptional series.

+-- {: .num_remark #NotationConvention}
###### Remark
**Warning on notation**

There are two different conventions for numbering the dihedral groups.

1. The above is the _algebraic convention_ in which the suffix gives the [[order]] of the group: ${\vert D_{2 n}\vert} = 2 n$.

1. In the _geometric convention_ one writes "$D_n$" instead of "$D_{2n}$", recording rather the geometric nature of the object of which it is the symmetry group.

   Also beware that there is yet another group denoted $D_n$ mentioned at _[[Coxeter group]]_.

=--

### Binary dihedral/dicyclic groups
  {#BinaryDihedralGroup}

Under the further lift through the [[spin group]]-[[double cover]] map $SU(2) \simeq Spin(3) \to SO(3)$ of the [[special orthogonal group]], the dihedral group $D_{2n}$ is covered by the _binary dihedral group_, also known as the _dicyclic group_ and denoted

$$
  2 D_{2n} = Dic_n
$$

Equivalently, this is the [[lift]] of the dihedral group $D_{2n}$ ([above](#DihedralGroups)) through the [[pin group]] [[double cover]] of the [[orthogonal group]] [[O(2)]] to [[Pin(2)]]

$$
  \array{
    2 D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Pin_-(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    \\
    D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    O(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)
  }
$$

Explicity, let $\mathbb{H} \simeq \mathbb{C} \oplus \mathrm{j} \mathbb{C}$ be the [[quaternions]] realized as the [[Cayley-Dickson double]] of the [[complex numbers]], and identify the [[circle group]]

$$
  SO(2) \simeq S\big( \mathbb{C}\big) \hookrightarrow \mathbb{H}
$$

with the [[unit circle]] in $\mathbb{C} \hookrightarrow \mathbb{H}$ this way, with group structure given by [[multiplication]] of [[quaternions]]. Then the [[Pin group]] [[Pin(2)]] is [[isomorphism|isomorphic]] to the [[subgroup]] of the [[group of units]] $\mathbb{H}^\times$ of the [[quaternions]] which consists of this copy of [[SO(2)]] together with the multiples of the imaginary quaternion $\mathrm{j}$ with this copy:

$$
  Pin_-(2)
  \;\simeq\;
  S\big( \mathbb{C}\big)
   \;\cup\;
  \mathrm{j} \cdot S\big( \mathbb{C}\big)
  \;\subset\;
  S(\mathbb{H})
  \;\simeq\;
  Spin(3)
  \,.
$$

The binary dihedral group $2 D_{2n}$ is the [[subgroup]] of that generated from

1. $ a \coloneqq \exp\left( \pi \mathrm{i} \tfrac{1}{n} \right) \in S(\mathbb{C}) \subset Pin_-(2) \subset Spin(3)$

1. $x \coloneqq  \mathrm{j} \in Pin_-(2) \subset Spin(3)$.

It is manifest that these two generators satisfy the relations

$$
  a^{2n} = 1
  \,,
  \phantom{AA}
  x^2 = a^n \; (= -1)
  \,,
  \phantom{AA}
  x^{-1} a x = a^{-1}
$$

and in fact these [[generators and relations]] fully determine $2 D_{2n}$, up to [[isomorphism]].

## Properties

### Group cohomology

The [[group cohomology]] of the dihedral group is discussed for instance at [Groupprops](#Groupprops).

### As part of the ADE pattern

[[!include ADE -- table]]

### Group presentation

The dihedral group $D_{2n}$ has a group presentation

$$\langle x,y : x^n=y^2=(xy)^2=1\rangle.$$

From this it is easy to see that it is a [[semi-direct product]] of the $C_n$ generated by $x$ and the $C_2$ generated by $y$.  The action of $y$ on $x$ is given by $\,^y x= x^{-1}$.

It is a standard example considered in elementary [[combinatorial group theory]].

## Examples

### Quaternion group $Q_8$ and triality

The first binary dihedral group $2 D_4$ is [[isomorphism|isomorphic]] to the [[quaternion group]] of order 8:

$$
  2 D_4 (= Dic_2) \simeq Q_8
  \,.
$$

In the [[ADE-classification]] this is the entry [[D4]].

[[!include character table of 2D4=Dic2=Q8]]

## Related concepts

* [[anti-cyclotomic extension]]

* [[dihedral homology]]

* [[group presentation]]

* [[Coxeter group]]

## References

Discussion in the context of the [[classification of finite rotation groups]] goes back to

* {#Klein1884} [[Felix Klein]], chapter I.4 of _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

See also

* Wikipedia, _[Dihedral group](https://en.wikipedia.org/wiki/Dihedral_group)_

* Wikipedia, _[Binary dihedral group](https://en.wikipedia.org/wiki/Dicyclic_group#Binary_dihedral_group)_

* Wikipedia, _[Dicyclic group](https://en.wikipedia.org/wiki/Dicyclic_group)_

* [[Groupprops]], _[Dicyclic group](https://groupprops.subwiki.org/wiki/Dicyclic_group)_

* [[Groupprops]], _[Linear representation theory of dicyclic groups](https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dicyclic_groups)_

* {#Groupprops} [[Groupprops]], _[Group cohomology of dihedral group:D8](http://groupprops.subwiki.org/wiki/Group_cohomology_of_dihedral_group:D8)_

* [[GroupNames]], _[Dicyclic groups $Dic_n$](https://people.maths.bris.ac.uk/~matyd/GroupNames/dicyclic.html)_

Discussion as the [[equivariance group]] in [[equivariant cohomology theory]]:

* {#Greenless01} [[John Greenlees]], Section 2 of: _Rational $SO(3)$-Equivariant Cohomology Theories_, in _Homotopy methods in algebraic topology_ (Boulder, CO, 1999), Contemp. Math. **271**,  Amer. Math. Soc. (2001) 99 ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.5444), [GBooks](https://books.google.de/books?id=X2YbCAAAQBAJ&lpg=PR7&ots=tdiPe0oWYn&dq=%22Homotopy%20methods%20in%20algebraic%20topology%22&lr&pg=PR7#v=onepage&q=%22Homotopy%20methods%20in%20algebraic%20topology%22&f=false))

and specifically in [[equivariant K-theory]] and [[KR-theory]]:

* {#BrunerGreenlees10} [[Robert Bruner]], [[John Greenlees]], Chapter 8 of: *Connective Real K-Theory of Finite Groups*, Mathematical Surveys and Monographs **169** AMS 2010 ([ISBN:978-0-8218-5189-0](https://bookstore.ams.org/surv-169))

Discussion of [[equivariant ordinary cohomology]] ([[Bredon cohomology]]) over the point but in arbitrary [[RO(G)-degree]], for [[equivariance group]] a [[dihedral group]] of [[order of a group|order]] $2p$:

* [[Igor Kriz]], [[Yunze Lu]], *On the $RO(G)$-graded coefficients of dihedral equivariant cohomology*, Mathematical Research Letters **27** 4 (2020) ([arXiv:2005.01225](https://arxiv.org/abs/2005.01225), [doi:10.4310/MRL.2020.v27.n4.a7](https://dx.doi.org/10.4310/MRL.2020.v27.n4.a7))

see also

* Yajit Jain, *Bredon Equivariant Homology of Representation Spheres*, 2014 ([pdf](https://math.mit.edu/research/undergraduate/urop-plus/documents/Jain.pdf), [[JainBredonEquivariant.pdf:file]])

[[!redirects dihedral groups]]

[[!redirects binary dihedral group]]
[[!redirects binary dihedral groups]]

[[!redirects dicyclic group]]
[[!redirects dicyclic groups]]


dihedral group of order 6

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## Idea

The [[dihedral group]] of [[group order|order]] 6 -- $D_6$

and

the [[binary dihedral group]] of [[order of a group|order]] 12 -- $2 D_{12}$

correspond to the Dynkin label [[D5]] in the [[ADE-classification]].

## Properties

$D_6$ is [[isomorphism|isomorphic]] to the [[symmetric group]] on 3 elements

$$
  D_6 \simeq S_3
  \,.
$$

These are the smallest [[non-abelian groups]].

## Related concepts

[[!include ADE -- table]]

## References

For $D_6$:

* group names, _[S3](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/S3.html)_

For $2 D_6$

* group names, _[Dic3](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/Dic3.html)_

See also

* Wikipedia, _[Dihedral group of order 6](https://en.wikipedia.org/wiki/Dihedral_group_of_order_6)_

[[!redirects binary dihedral group of order 12]]


dihedral group of order 8

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## Idea

The [[dihedral group]] of [[group order|order]] 8 -- $D_8$

and

the [[binary dihedral group]] of [[group order|order]] 16 -- $2 D_8$

at Dynkin label [[D6]] in the [[ADE-classification]]

## Related concepts

[[!include ADE -- table]]

## References

For the dihedral group $D_8$:

* group names, _[D4](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/D4.html)_

* Groupprops, _[Dihedral group:D8](https://groupprops.subwiki.org/wiki/Dihedral_group:D8)_

For the binary dihedral group $2 D_8$:

* group names, _[Dic4](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/Q16.html)_

[[!redirects binary dihedral group of order 16]]


dihedral homology

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## Idea

The idea of cyclic homology can be generalized to other [[skew-simplicial group]]s as shown by Krasauskas and independently by Fiodorowicz. The simplest analogue is the dihedral homology.

There is also a geometric analogy where

* _[[Hochschild homology]]_ is essentially the cohomology of [[free loop spaces]];

*  _[[cyclic homology]]_ is essentially the cohomology of [[free loop spaces]] modulo the [[circle group]] $SO(2)$ action by rotating on the loops;

so _dihedral homology_ is the cohomology of free loop spaces modulo the $O(2)$-action on loops (including reflections, as in the [[dihedral group]]).

## References

* Massimiliano Ungheretti, _Free loop spaces and dihedral homology_ ([arXiv:1608.08140](https://arxiv.org/abs/1608.08140))
* T. Popelensky, _On codihedral module for $\ast$-Hopf algebras_, Journal of K-Theory, 2(2) (2008) 353--360 [doi:10.1017/is008001018jkt033](https://doi.org/10.1017/is008001018jkt033)

[[!redirects dihedral set]]
[[!redirects dihedral cohomology]]


dihedron

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## Idea

A _dihedron_ is a degenerate [[Platonic solid]] with only two (identical) faces, which may be any [[regular polygon]] (including possibly the degenerate [[bigon]], which would make the dihedron very degenerate!)

[[!include ADE -- table]]

## References

* {#Klein1884} [[Felix Klein]], chapter I.4 of _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

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

[[!redirects dehedra]]


Dijkgraaf-Witten theory

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## Idea

_Dijkgraaf-Witten theory_ in dimension $n$ is the [[TFT|topological]] [[sigma-model]] [[quantum field theory]] whose [[target space]] is the [[classifying space]] of a [[discrete group]] and whose
[[background gauge field]] is a [[circle n-bundle with connection]] on $\mathbf{B}G$ (necessarily flat) which is equivalently a [[cocycle]] in the [[group cohomology]] of $G$ with coefficients in the [[circle group]].

Viewed in a broader context and generalizing: Dijkgraaf-Witten theory is the [[schreiber:∞-Chern-Simons theory]] induced from a [[characteristic class]] $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ on a [[discrete ∞-groupoid]] $\mathbf{B}G := Disc B G$. If $G$ here is an ordinary [[discrete group]] this is traditional Dijkgraaf-Witten theory, if $G$ is a discrete [[2-group]] and the background field is a circle 4-bundle, then this is called the [[Yetter model]].

This are the first two steps in filtering of target spaces by [[homotopy type]] [[truncated|truncation]] of [[schreiber:∞-Chern-Simons theory]] with discrete target spaces.

### Concise survey of the ingredients

We may think of this as  describing the [[quantum mechanics]] of an $(n-1)$-[[brane]] with $n$-dimensional [[worldvolume]] $\Sigma$ propagating on $B G$ and being [[charge]]d under a [[gauge field|higher analog]] of the [[electromagnetic field]]:

a field configuration over $\Sigma$ (a $\Sigma$-shaped _trajectory_) is a [[morphism]] $\phi : \Sigma \to \mathbf{B}G$, hence equivalently a $G$-[[principal bundle]] on $\Sigma$. The [[configuration space]] of fields over $\Sigma$ is the [[groupoid]] of $G$-[[principal bundle]]s over $\Sigma$.

The [[background gauge field]] is a [[morphism]] $\alpha : \mathbf{B}G \to \mathbf{B}^n U(1)$ -- hence a [[characteristic class]] for $G$: a [[cocycle]] of degree $n$ in the [[group cohomology]] of $G$.

The value of the [[Lagrangian]] $L(\phi)$ on a field configuration $\phi$ is the [[characteristic class]] of this bundle with respect to the universal characteristic class of the given [[principal infinity-bundle|circle n-bundle]]:

$$
  L : (\Sigma \stackrel{\phi}{\to} \mathbf{B}G)
  \mapsto
  (\alpha(\phi) : \Sigma \stackrel{\phi}{\to} \mathbf{B}G \stackrel{\alpha}{\to} \mathbf{B}^n U(1)).
$$

This is the [[classical field theory]] input of the model. The [[extended quantum field theory]] defined by this is supposed to be a rule that assigns space of [[state]]s to lower dimensional pieces of $\Sigma$ and to $n$-dimensional $\Sigma$s a propagator.

The space of [[state]]s assigned to a $\Sigma$ of [[dimension]] $n-k$ for $k \in \mathbb{N}$ is the [[k-groupoid]] of [[section]]s of the higher line bundle [[associated infinity-bundle|associated]] to the [[circle n-bundle|circle (n-k)-bundle]] $\tau_\Sigma \alpha$ obtained by [[transgression]] of $\alpha$ to the [[mapping space]] $\mathbf{H}(\Sigma, \mathbf{B}G)$.

The propagator on $\Sigma$ of dimension $n$ is given by the [[path integral]] computed with [[measure]] the [[groupoid cardinality]] of $\mathbf{B}G$ and  [[integral kernel]] given by the [[action functional]]

$$
  \exp(i S(-)) : G Bund(\Sigma) \to U(1)
$$

that sends a field $\phi$ to the evaluation of $\alpha(\phi)$ on the [[fundamental class]] of $\Sigma$

$$
  \exp(i S(\phi)) = \int_\Sigma \alpha(\phi)
  \,.
$$

### Gentle exposition

(...)

## Details of DW-theory as an extended TQFT

The Dijkgraaf-Witten model is an example of (fully) [[extended topological quantum field theory]]. Namely, the above data not only assign an element in $U(1)$ to any closed $n$-dimensional manifold, but also a vector space to any closed $(n-1)$-dimensional manifold, a [[2-vector space]] to any closed $(n-2)$ manifold, and so on, ending with an [[n-vector space]] assigned to the point. Also, manifolds with boundary corresponds to (higher) linear operators between these (higher) vector spaces. According to the [[cobordism hypothesis]], the whole structure of the Dijkgraaf-Witten model as an fully extended  TQFT is contained in the datum of the $n$-Vector space it assigns to the point. This is the space of [[section|sections]] of the flat $n$-vector bundle $\mathbf{B}G\to n Vect$ induced by the background field $\mathbf{B}G\to \mathbf{B}^n U(1)$.

### Finite Group Cohomology

Since the target space of Dijkgraaf-Witten theory is the [[delooping]] groupoid $\mathbf{B}G$ of a [[group]] $G$ ([[internalization|internal]] to [[Set]]), any background field given by a morphism $\alpha : \mathbf{B}G \to A$ in [[∞Grpd]] is a [[cocycle]] in the [[group cohomology]] of $G$, as described there.

Here we have a finite (or discrete) group $G$, and a discrete abelian group $A$, and we want to define $H^n(G;A)$. A way of doing this is to realize everything topologically: from $G$ we build the classifying space $\mathcal{B}G$, and from $A$ the [[Eilenberg-MacLane space]] $\mathcal{B}^n A=K(A,n)$. Then we consider the space of maps $hom(\mathcal{B}G,\mathcal{B}^n A)$ (these are our [[cocycle]]s) and take its $\pi_0$.

This way we have a familiar description, in a certain sense (topological spaces, continuous maps, homotopies,..), of the set $H^n(G;A)$. The drawback is that the topological spaces involved here are "gigantic" (infinite dimensional [[CW-complex]]es), where we had started with a very "little" datum: a finite group. So one can wonder if there is a finite model for the above construction, and the homotopy hypothesis serves it on a silver plate. Namely, since $G$ is discrete, $\mathcal{B}G$ is a [[homotopy n-type|1-type]], and nothing but the [[geometric realization]] of the [[delooping]] [[groupoid]] $\mathbf{B}G$ (boldface $B$ here); similarly $\mathcal{B}^n A$ is the topological geometric realization of the $n$-groupoid $\mathbf{B}^n A$, and the space of cocycles is $hom(B G,B^n A)$. since $G$ is a finite group, $B G$ is a finite groupoid, and so $hom(B G,B^n A)$ is a finite set. This set is the finite model for $hom(\mathcal{B}G,\mathcal{B}^n A)$ we were looking for.

To be continued...

### The $k$-vector spaces of states in codimension $k$
 {#StatesInCodimensionk}

The [[n-vector space|k-vector space]] associated with a closed oriented $(n-k)$-dimensional manifold $X_{n-k}$ admits a simple description in terms of [[section]]s:

The background field $\alpha : \mathbf{B}G \to A$ is [[transgression|transgressed]] to the mapping space $[\Pi(X_{n-k}), \mathbf{B}G]$ by forming the [[internal hom]]

$$
  [\Pi(X_{n-k}), \mathbf{B}G]
  \stackrel{[\Pi(X_{n-k}), \alpha]}{\to}
  [\Pi(X_{n-k}), A]
  \stackrel{\tau_{\leq k}}{\to}
  \tau_{\leq k} [\Pi(X_{n-k}), A]
  \,,
$$

where the last morphism is the projection on the [[truncated|k-truncation]]. This defines a [[cocycle]] on the _space of fields_ $[\Pi(X_{n-k}), \mathbf{B}G]$ over $X_{n-k}$, which classifies some [[principal ∞-bundle]] on this space. Given a canonical [[representation]] of the _spaces of phases_ $\tau_k [\Pi(X_{n-k}), A]$ on a [[n-vector space|k-vector space]] we obtain the corresponding [[associated bundle]] over the space of fields. The $(k-1)$-category assigned by the [[extended topological quantum field theory]] to the closed $X_{n-k}$ is the category of sections of this $k$-vector bundle.

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

We have

$$
  \tau_k [\Pi(X_{n-k}), \mathbf{B}^n U(1)]
  \simeq
  \mathbf{B}^k U(1)
$$

=--

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

By general abstract reasoning (recalled at [[cohomology]] and [[fiber sequence]], for instance) we have for the [[homotopy group]]s that

\[
  \pi_i[\Pi(X_{n-k}),\mathbf{B}^n U(1)]
  \simeq
  H^{n-i}(X_{n-k}, U(1))
  \,.
\]

Now use the [[universal coefficient theorem]], which asserts that we have an [[exact sequence]]

\[
  0
  \to
  Ext^1(H_{n-i-1}(X_{n-k},\mathbb{Z}),U(1))
  \to
  H^{n-i}(X_{n-k},U(1))
  \to
  Hom(H_{n-i}(X_{n-k},\mathbb{Z}),U(1))
  \to 0
  \,.
\]

Since $U(1)$ is an [[injective object|injective]] $\mathbb{Z}$-[[module]] we have

$$
  Ext^1(-,U(1))=0
  \,.
$$

Together this means that we have an [[isomorphism]]

\[
  H^{n-i}(X_{n-k},U(1))
  \simeq
  Hom_{Ab}(H_{n-i}(X_{n-k},\mathbb{Z}),U(1))
\]

that identifies the [[cohomology group]] in question with the [[internal hom]] in [[Ab]] from the integral [[homology]] group of $X_{n-k}$ to $U(1)$.

For $i\lt k$, the right hand side is zero, and so

$$
  \pi_i[\Pi(X_{n-k}),\mathbf{B}^n U(1)]=0 \;\;\;\;
  for i\lt k
  \,.
$$

For $i=k$, instead, $H_{n-i}(X_{n-k},\mathbb{Z})\simeq \mathbb{Z}$, since $X_{n-k}$ is a closed $(n-k)$-manifold and so

$$
  \pi_k[\Pi(X_{n-k}),\mathbf{B}^n U(1)]\simeq U(1)
  \,.
$$

=--

+-- {: .num_prop}
###### Remark
Another proof of the isomorphism $H^{n-k}(X_{n-k},U(1))\cong U(1)$ and of the identities $H^{n-i}(X_{n-k},U(1))=0$ for $i\lt k$ can be obtained as follows. Consider the short exact sequence of locally constant sheaves of abelian groups
$$
0\to \mathbb{Z}\to \mathbb{R}\to U(1)\to 1.
$$
This induces a long exact sequence in cohomology
$$
\cdots \to H^{n-i-1}(X_{n-k},U(1))\to H^{n-i}(X_{n-k},\mathbb{Z})\to H^{n-i}(X_{n-k},\mathbb{R}) \to H^{n-i}(X_{n-k},U(1))\to H^{n-i+1}(X_{n-k},\mathbb{Z})\to \cdots
$$
For $i\lt k$ we have $H^{n-i}(X_{n-k},U(1))=0$ by dimensional reasons, while for $i=k$ we find the exact sequence
$$
\cdots \to H^{n-k}(X_{n-k},\mathbb{Z})\to H^{n-k}(X_{n-k},\mathbb{R}) \to H^{n-k}(X_{n-k},U(1))\to 0.
$$
Since $X_{n-k}$ is a closed oriented manifold, we have $H^{n-k}(X_{n-k},\mathbb{Z})=\mathbb{Z}$, $H^{n-k}(X_{n-k},\mathbb{R})=\mathbb{R}$, and the map $H^{n-k}(X_{n-k},\mathbb{Z})\to H^{n-k}(X_{n-k},\mathbb{R})$ is the inclusion of $\mathbb{Z}$ into $\mathbb{R}$. Hence $H^{n-k}(X_{n-k},U(1))\cong \mathbb{R}/\mathbb{Z}\cong U(1)$.
=--

This means that the [[transgression]] of the Dijkgraaf-Witten background field

$$
  \alpha : \mathbf{B}G \to \mathbf{B}^n U(1)
$$

to the _space of field configurations_ $[\Pi(X_{n-k}), \mathbf{B}G]$ over $X_{n-k}$ is a [[cocycle]] of the form

$$
  [\Pi(X_{n-k}), \alpha] : [\Pi(X_{n-k}), \mathbf{B}G] \to \mathbf{B}^k U(1)
  \,.
$$

This classifies a $\mathbf{B}^{k-1} U(1)$-[[principal ∞-bundle]] $P$ over the space of field configurations, given by the [[pullback]]

$$
  \array{
    P &\to & \mathbf{E} \mathbf{B}^{k-1} U(1)
    \\
    \downarrow && \downarrow
     \\
     [\Pi(X_{n-k}), \mathbf{B}G] &\stackrel{[\Pi(X_{n-k}), \rho]}{\to}& \mathbf{B}^k U(1)
  }
  \,.
$$

(Here $\mathbf{E} \mathbf{B}^{k-1} U(1)$ is as described at [[universal principal ∞-bundle]].)

By the canonical $k$-[[representation]] $\rho : \mathbf{B}^k U(1) \to k Vect_{\mathbb{C}}$ of $\mathbf{B}^{k-1}U(1)$ on complex [[n-vector space|k-vector space]]s, we have [[associated bundle|associated]] to this canonically a $k$-vector bundle $E$, which may be realized as the pullback

$$
  \array{
    E &\to & k Vect_*
    \\
    \downarrow && \downarrow
     \\
     [\Pi(X_{n-k}), \mathbf{B}G] &\stackrel{\rho \circ [\Pi(X_{n-k}), \rho]}{\to}& k Vect
  }
  \,.
$$

Here $k Vect_*$ is the [[n-category|k-category]] of pointed $k$-vector bundles, see again [[generalized universal bundle]] for more.

If $X_{n-k}$ is closed then the $k$-vector spaces associated by the TFT to $X_{n-k}$ is the [[n-category|(k-1)-category]] of [[section]]s of this bundle $E$.

...

## Relation to Chern-Simons theory

Dijkgraaf-Witten theory is to be thought of as the finite group version of [[Chern-Simons theory]]. Chern-Simons theory looks formally just as the above, only that all finite $n$-groupoids appearing here are replaced by [[Lie ∞-groupoid]]s ([[∞-stack]]s on [[CartSp]]).

## Related concepts

* [[transgression in group cohomology]]

* [[schreiber:∞-Chern-Simons theory]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

    * [[1d Dijkgraaf-Witten theory]]

  * [[2d Chern-Simons theory]]

  * [[3d Chern-Simons theory]]

    * **Dijkgraaf-Witten theory**

    * [[2d Wess-Zumino-Witten theory]]

  * [[4d Chern-Simons theory]]

  * [[5d Chern-Simons theory]]

  * [[6d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

  * [[infinite-dimensional Chern-Simons theory]]

  * [[AKSZ sigma-model]]

## References

The idea originates, of course, in

* {#DijkgraafWitten90} [[Robbert Dijkgraaf]], [[Edward Witten]], _[[DW.pdf:file]]_, Commun. Math. Phys. __129__ (1990), 393 ([euclid:cmp/1104180750](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-129/issue-2/Topological-gauge-theories-and-group-cohomology/cmp/1104180750.full))

The discussion of the [[quasi-Hopf algebra]] associated with a [[group cohomology]] 3-[[cocycle]] $c \colon \mathbf{B}G \to \mathbf{B}^3 U(1)$ originates in

* [[Robbert Dijkgraaf]], V. Pasquier, P. Roche, _QuasiHopf algebras, group cohomology and orbifold models_, Nucl. Phys. B Proc. Suppl. __18B__ (1990), 60-72; _Quasi-quantum groups related to orbifold models_, Modern quantum field theory (Bombay, 1990), 375–383, World Sci. 1991

A review is in

*  A. Coste, J-M. Maillard, _Representation Theory of Twisted Group Double_, Annales Fond.Broglie 29 (2004) 681-694, ([arXiv:hep-th/0309257](http://arxiv.org/abs/hep-th/0309257))

and conceptual clarifications were established in

* [[Simon Willerton]], _The twisted Drinfeld double of a finite group via gerbes and finite groupoids_, [arXiv/math.QA/0503266](http://arxiv.org/abs/math.QA/0503266)

and, earlier, in an unpublished manuscript of [[Paul Bressler]] (2002-2004). See at _[[Drinfeld double]]_ for more on this.

A first comprehensive structural account of DW theory as a [[FQFT|functorial QFT]] was given in

* [[Daniel Freed]], [[Frank Quinn]], _Chern-Simons theory with finite gauge group_ Commun.Math.Phys. __156__:435-472, 1993, ([arXiv:hep-th/9111004](http://de.arxiv.org/abs/hep-th/9111004))
 {#FreedQuinn93}

A review is given on p. 68 of

* [[Bruce Bartlett]], _Categorical aspects of topological quantum field theory_, [arXiv/math.QA/0512103](http://de.arxiv.org/abs/math.QA/0512103)

First steps towards understand DW theory as an [[extended TQFT]] appear in

* [[Daniel Freed]], _Higher Algebraic Structures and Quantization_ Commun.Math.Phys. 159 (1994) 343-398 ([arXiv:arXiv:hep-th/9212115](http://arxiv.org/abs/arXiv:hep-th/9212115))
 {#Freed94}

Discussion aiming towards a refinement of DW theory to an [[extended TQFT]] is in

* Kevin Wray, _Extended topological gauge theories in codimension 0 and higher_ ([pdf](http://math.berkeley.edu/~kwray/papers/thesis.pdf))

Further conceptual refinement of this is indicated in section 3 and section 8 of

* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_ (2010)

This proposes a general abstract way to construct [[path integral]] quantizations for finite group theories such as DW, see also at _[[prequantum field theory]]_. More along these lines is in

* [[Jacob Lurie]], _Finiteness and ambidexterity in $K(n)$-local stable homotopy theory_, talk at _Notre Dame Graduate Summer School on Topology and Field Theories_ and _Harvard lecture_ 2012 ([video part 1](http://www.youtube.com/watch?v=eQayYLDw1VA), [part 2](http://www.youtube.com/watch?v=OEShrQyvmS4), [part 3](http://www.youtube.com/watch?v=nOIcdn1iUR4) [part 4](http://www.youtube.com/watch?v=ZwnClYedaYM), [pdf lecture notes](http://www.math.northwestern.edu/~celliott/notre_dame_notes/Lurie_notes.pdf) by Chris Elliott)

See also

* [[Thomas Nikolaus]], _Equivariant Dijkgraaf-Witten theory_, a talk at Muenster ([pdf](http://www.math.uni-hamburg.de/home/nikolaus/Muenster.pdf))

For more on this see the discussion on the [n-Forum](http://nforum.mathforge.org/discussion/1046/path-category-vs-cobordisms-for-bundles/?Focus=8337#Comment_8337).

## Dijkgraaf–Witten theory via Kan extensions

* [[Jeffrey C. Morton]], _Cohomological Twisting of 2-Linearization and Extended TQFT_, [arXiv:1003.5603](https://arxiv.org/abs/1003.5603).

* Section 2 and 3 of [[Gijs Heuts]] and [[Jacob Lurie]]'s Ambidexterity, in: Topology and Field Theories, [doi](https://doi.org/10.1090/conm/613).

* Section 3 of [[Daniel Freed]], [[Michael Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], Topological quantum field theories from compact Lie groups, [arXiv:0905.0731](https://arxiv.org/abs/0905.0731).

[[!redirects Dijkgraaf-Witten theories]]

[[!redirects Dijkgraaf--Witten theory]]
[[!redirects Dijkgraaf–Witten theory]]
[[!redirects Dijkgraaf-Witten model]]
[[!redirects Dijkgraaf-Witten models]]


dilatation operator

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[conformal field theory]]

* [[single trace operator]]/[[BMN operator]]

* [[spin chain]]

## References

### In $D=4$ $\mathcal{N} = 4$ super Yang-Mills theory

In [[D=4 N=4 super Yang-Mills theory]]:

* Abhishek Agarwal, Sarada Rajeev, _The Dilatation Operator of $\mathcal{N} =- 4$ SYM and Classical Limits of Spin Chains and Matrix Models_, Mod. Phys. Lett. A19 (2004) 2549 ([arXiv:v](https://arxiv.org/abs/hep-th/0405116))

* Asger C. Ipsen, Kasper E. Vardinghus, _The dilatation operator for defect conformal $\mathcal{N} = 4$ SYM_ ([arXiv:1909.12181](https://arxiv.org/abs/1909.12181))

[[!redirects dilatation operators]]


dilatino

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##Idea

By _dilatino_ one refers to  a [[superpartner]] of the [[dilaton]].

## Ambiguity of the nomenclature

In [[supergravity]] literature there is an ambiguity in the nomenclature relatively to the [[dilatino]] field.

* Part of the literature uses the name "dilatino" to mean the superpartner $\lambda_\alpha \coloneqq D_\alpha\phi$ of the scalar [[dilaton]] field $\phi$, which comes from the circular [[Kaluza-Klein compactification]] of the bosonic [[supervielbein]] given by $e^n = \phi(\mathrm{d}x^n+A)$ in a $n$-dimensional [[spacetime]].

* Part of the literature uses it to mean, instead, the spinor which comes from the circular [[Kaluza-Klein compactification]] of the fermionic [[supervielbein]] (i.e. of the [[gravitino]]) given by $\psi^\alpha_{n\text{d}} = \psi^\alpha_{(n-1)\text{d}} + \chi^\alpha e^n$ in a $n$-dimensional [[spacetime]]. See for instance ([DFGT08](#DFGT08)).

##References

* {#DFGT08} [[Riccardo D'Auria]], [[Pietro Fré]], [[Pietro Grassi]], Trigiante, _Pure Spinor Superstrings on Generic type IIA Supergravity Backgrounds_ [arXiv:0803.1703](https://arxiv.org/abs/0803.1703)

[[!redirects gravitello]]


dilaton

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## Idea

Generally in a context of [[Kaluza-Klein compactification]] a _dilaton_ is a [[field (physics)|field]]  on a lower-dimensional [[spacetime]] which is a component of the field of [[gravity]] on a higher dimensional spacetime, in that it is part of the [[pseudo-Riemannian metric|metric]] of the [[fiber]]-spaces on which the KK-compactification takes place. Specifically for KK-compactification on a [[circle]] fiber "the dilaton" (or "radion") is the lowest [[Fourier series|Fourier]] mode of the metric of the circle, hence is the [[length]] (circumference) (or [[radius]], up to a factor) of the circle fiber.

The subtlety in [[Kaluza-Klein theory]] is that the dilaton should have small but approximately constant value in order to yields [[effective field theory]] gravity coupled to [[gauge theory]] in lower dimensions from pure gravity in higher dimensions. This is the problem of _[[moduli stabilization]]_.

Specifically in [[string theory]], together with the field of [[gravity]] and the [[Kalb?Ramond field]], the _dilaton field_ is one of the three massless [[boson|bosonic fields]] that appears in [[effective QFT|effective background]] [[quantum field theory|quantum field theories]]. For [[type IIA string theory]] this may be interpreted as the Kaluza-Klein dilaton in the above sense, arising from [[11-dimensional supergravity]] ([[M-theory]]) compactified on a circle. Similarly for [[heterotic string theory]] and [[Horava-Witten theory]].

## Details

### Action functional of dilaton gravity

Let $X$ be a [[compact space|compact]] [[smooth manifold]]. Write $Conf$ for the [[configuration space]] of [[pseudo-Riemannian metric]]s $g$ (the [[graviton]]) and of [[smooth function]]s $f$ (the _dilaton_ ) on $X$.

The [[action functional]] for [[dilaton gravity]] is

$$
  S : Conf \to \mathbb{R}
$$

$$
  S : (g,f) \mapsto \int_X e^{-f}(R_g dvol_g+ d f \wedge \star_g d f)
  \,,
$$

where $R_g$ is the [[Riemann curvature]] scalar of $g$ and $\star_g$ the [[Hodge star operator]] and $dvol_g$ is the [[volume form]] of $g$.

For $f = 0$ this reduces to the [[Einstein-Hilbert action]]. For $f = const$ it is still a multiple of the Einstein-Hilbert action functional.

The [[gradient flow]] of this functional is [[Ricci flow]].

### Global cohomological description

The global nature of the gravitational field and the Kalb--Ramond field are well understood conceptually: the gravitational field is a [[pseudo-Riemannian metric]] and the Kalb--Ramond field is a cocycle in third integral [[differential cohomology]] (for instance realized by a cocycle in [[Deligne cohomology]] or by a [[bundle gerbe]] with connection).

In [[generalized complex geometry]], both these fields are shown to be unified as one single [[schreiber:∞-Lie algebroid valued differential forms|∞-Lie algebroid valued]] form field: a connection on a [[standard Courant algebroid]] (as described in more detail there).

While it was clear that the diaton field is locally just a real-valued function, is formal global identification has not been understood in an analogous manner for a long time.

But a proposal for a precise conceptual identification of the dilaton as a structure appearing in the context of [[generalized complex geometry]] is in

* Mariana Graña, Ruben Minasian, Michela Petrini, Daniel Waldram, _T-duality, generalized geometry and non-geometric backgrounds_ ([arXiv](http://arxiv.org/abs/0807.4527))

## Applications

The [[gradient flow]] of the [[action functional]] for [[dilaton gravity]] is essentially [[Ricci flow]].

## Related concepts

* [[Liouville theory]]

* [[JT-gravity]]

[[!include fields and quanta - table]]

## References

The derivation of dilaton gravity as part of the [[effective QFT]] of [[string theory]] is discussed for instance aroung page 911 of

* [[Eric D'Hoker]], _String theory_ in [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey,
[[David Kazhdan]], [[John Morgan]], D.R. Morrison and [[Edward Witten]], (eds.) _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

Relation of the [[string theory]]-dilaton to the [[double dimensional reduction]] of the quantum [[M2-brane]]:

* [[Krzysztof A. Meissner]], [[Hermann Nicolai]], *Fundamental Membranes and the String Dilaton* [[arXiv:2208.05822](https://arxiv.org/abs/2208.05822)]

[[!redirects dilaton field]]

[[!redirects dilaton gravity]]


dilepton

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## Idea

A _dilepton_ is a [[pair]] consisting of a [[lepton]] and its [[antiparticle]], for example an [[electron]]/[[positron]]-[[pair]], specifically as they arise jointly in a [[decay]]-process.

## Related concepts

* [[diquark]]

* [[Dalitz decay]], [[Dalitz pair]]

* [[vector meson dominance]]

[[!redirects dileptons]]


dilogarithm

Euler's __dilogarithm__ is a complex valued function $Li_2$ given by

$$
Li_2(x) = \sum_{n=1}^\infty \frac{x^n}{n^2}
$$

The dilogarithm is a special case of the [[polylogarithm]] $Li_n$. The __Bloch--Wigner dilogarithm__ is defined by

$$ D(z) := Im(Li_2(z)) + arg(1-z) log |z| $$

The dilogarithm has remarkable relations to many areas of mathematics and [[mathematical physics]] including [[scissors congruence]], [[Reidemeister torsion]], [[regulator]]s in higher [[algebraic K-theory]], the Bloch group, [[CFT]], [[Liouville gravity]], [[hyperbolic geometry]] and [[cluster transformation]]s.

See also the references at [mathworld](http://mathworld.wolfram.com/Dilogarithm.html)
and P.P. Cook's [blog](http://ppcook.blogspot.com/2005/12/dilogarithms.html) and the related entry [[quantum dilogarithm]].

* Don Zagier, _The dilogarithm function_, in Frontiers in Number Theory, Physics, and Geometry II, pp. 3--35 (2007) [MR2290758](http://www.ams.org/mathscinet-getitem?mr=2290758) [doi:10.1007/978-3-540-30308-4](http://dx.doi.org/10.1007/978-3-540-30308-4) [preprint pdf](http://maths.dur.ac.uk/~dma0hg/dilog.pdf); _Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields_, in Arithmetic Algebraic Geometry, Progr. Math. __89__, Birkhäuser, Boston, 1990, 391–430 [MR1085270](http://www.ams.org/mathscinet-getitem?mr=1085270)

* A. N. Kirillov, _Dilogarithm identities_, Progr. Theoret. Phys. Suppl. 118 (1995), 61–142 [hep-th/9408113](http://arxiv.org/abs/hep-th/9408113)
[MR1356515](http://www.ams.org/mathscinet-getitem?mr=1356515) [doi](http://dx.doi.org/10.1143/PTPS.118.61); _Identities for the Rogers dilogarithm function connected with simple Lie algebras_, J. Soviet Math. 47 (1989), 2450–2458.

* R.M. Kashaev, _Heisenberg double and the pentagon relation_, St. Petersburg Math. J. 8 (1997) 585– 592 [q-alg/9503005](http://arxiv.org/abs/q-alg/9503005)
* Rinat M. Kashaev, Tomoki Nakanishi, _Classical and quantum dilogarithm identities_, [arxiv/1104.4630](https://arxiv.org/abs/1104.4630) [doi](https://doi.org/10.3842/SIGMA.2011.102) (treatment in terms of [[cluster algebra]]s)
* Tomoki Nakanishi, _Dilogarithm identities for conformal field theories and cluster algebras: Simply laced case_, Nagoya Math. J. __202__ (2011), 23-43, [MR2804544](http://www.ams.org/mathscinet-getitem?mr=2804544) [doi](http://dx.doi.org/10.1215/00277630-1260432)

* B. Richmond, G. Szekeres, _Some formulas related to dilogarithm, the zeta function and the Andrews-Gordon identities_, J. Aust. Math. Soc. __31__ (1981), 362–373 [MR633444](http://www.ams.org/mathscinet-getitem?mr=633444) [doi](http://dx.doi.org/10.1017/S1446788700019492)

* S. Bloch, _Applications of the dilogarithm function in algebraic K-theory and algebraic geometry_, in: Proc. Int. Symp. on Alg. Geometry, Kinokuniya, Tokyo 1978.

* W. Nahm, _Conformal field theory and torsion elements of the Bloch group_, in Frontiers in Number Theory, Physics, and Geometry, II, Springer, Berlin, 2007, 67–132 [MR2290759](http://www.ams.org/mathscinet-getitem?mr=2290759) [doi](http://dx.doi.org/10.1007/978-3-540-30308-4_2)

* Marco Aldi, Reimundo Heluani, _Dilogarithms, OPE, and twisted T-duality_, International Mathematics Research Notices 2014:6, 1528–1575, [doi](https://doi.org/10.1093/imrn/rns258)

* Andreas Deser, _Lie algebroids, non-associative structures and non-geometric fluxes_, [arXiv]()

* Vladimir V. Fock, Alexander B. Goncharov, _Cluster ensembles, quantization and the dilogarithm_, Annales scientifiques de l'École Normale Supérieure, Serie 4, __42__:6 (2009) 865-930 [doi](https://doi.org/10.24033/asens.2112)

* E. Aldrovandi, _On hermitian-holomorphic classes related to uniformization, the dilogarithm, and the Liouville action_, Commun. Math. Phys. (2004) 251: 27. [doi](https://doi.org/10.1007/s00220-004-116

category: analysis, physics


Dima Arinkin

* [webpage](http://www.math.wisc.edu/~arinkin/)

## Related entries

* [[Cartier duality]]

category: people


dimension

* table of contents
{: toc}

## Idea

A notion of _dimension_ is a notion of "size" of [[objects]]. There are various variations of what exactly this means, applicable in various contexts, that tend to agree when they jointly apply.

### Of spaces

There are many notions of _dimension_ of [[space|spaces]].  What they all have in common, is that the [[cartesian space]] $\mathbb{R}^n$ has dimension $n$.

*  The __dimension__ of a [[vector space]] $V$ is the [[cardinality]] of any [[basis of a vector space|linear basis]], hence of any [[set]] $B$ such that $V$ is the $B$-indexed [[direct sum]] ([[coproduct]] in [[Vect]])  of the [[ground field]].  (The *[[basis theorem]]*, equivalent to the [[axiom of choice]], says that every vector space has a unique dimension.)  For [[modules]] over [[rings]] that are not [[fields]] (for which the theorem above does not hold, neither existence nor uniqueness) the term used is [[rank]].

   The [[trace]] of a the [[identity]] on a  vector space coincides with its categorical trace in the [[symmetric monoidal category]] [[Vect]] of [[vector spaces]].

* Generalizing from this example, the [[trace]] of an object in a [[spherical category|spherical]] [[fusion category]] is called its *[[quantum dimension]]*.

*  A [[manifold]] has [[dimension of a manifold]] equal to $n$ if it is locally isomorphic to the [[Cartesian space]] $\mathbb{R}^n$.  A [[complex manifold]] is of __complex dimension__ $n$ if it is locally isomorphic to $\mathbb{C}^n$, hence has (real) dimension $2 n$.

*  A [[topological space]] $X$ has (Lebesgue) __[[covering dimension]]__ less than $n$ if every [[open cover]] $U$ of $X$ has a [[refinement]] $V$ such that every element of $X$ belongs to fewer than $n + 1$ elements of $V$.  (Then $X$ has dimension $n$ if it has dimension less than $n + 1$ but not less than $n$.)  By [[negative thinking]], this makes sense for $n \geq -1$; precisely the [[empty space]] has dimension $-1$, and precisely the [[point]] (of course) has dimension $0$.

* A [[CW-complex]] has [[dimension of a CW-complex]], this being the largest $n$ for which there are nontrivial $n$-[[cells]].

*  A [[metric space]] has a __[[Hausdorff dimension]]__ which may be any non-negative real number.

* A space in [[noncommutative geometry]] ([[spectral geometry]] via [[spectral triples]]) may have a notion of [[KO-dimension]].

See also

  * [[homotopy dimension]]

  * [[cohomology dimension]]

  * [[Heyting dimension]]

### Of objects in an abelian category: Length

See at _[[length of an object]]_.

### Of objects in a symmetric monoidal category: Euler characteristic

The dimension of a (finite dimensional) [[vector space]] $V$ over a [[field]] $k$ is equivalently the [[trace]] of the [[identity]] [[morphism]] in the [[symmetric monoidal category]] [[Vect]] (which is a linear map $k \to k$, canonically identified with an element in $k$)

$$
  tr(V) := tr(Id_V) : k \to V \otimes V^* \stackrel{\simeq}{\to} V^* \otimes V \to k
  \,.
$$

Therefore it makes sense for any [[symmetric monoidal category]] $C$ and every [[dualizable object]] $V$ to call $tr(Id_V) : 1 \to 1$ the _categorical trace of $V$.

This definition subsumes standard notions of _[[Euler characteristic]]_ and hence may also be thought of as generalizing that notion.

### Of objects in an $(\infty,1)$-topos

The following notions of dimension capture aspects of the concept for objects in a [[topos]] or more generally an [[(∞,1)-topos]]:

* [[homotopy dimension]]

* [[cohomological dimension]]

* [[covering dimension]]

* [[Heyting dimension]]

### Frobenius-Perron dimension

* [[Frobenius-Perron dimension]]

## Properties

* [[global dimension theorem]]

* [[topological invariance of dimension]]

## Related concepts

[[!include notions of dimension -- contents]]

## References

For the dimension in symmetric monoidal categories see the references at [[Euler characteristic]].

A general abstract [[(∞,1)-topos theory|(∞,1)-topos theoretic]] discussion of notions of homotopy/cohomology/covering _dimension_ is in section 7.2 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

[[!redirects dimension]]
[[!redirects dimensions]]

[[!redirects dimensional]]

[[!redirects dimension of a vector space]]
[[!redirects dimensions of vector spaces]]


dimension of a cell complex

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## Idea

The _[[dimension]]_ of a [[cell complex]] $X$ is the largest [[natural number]] $n \in \mathbb{N}$, if such exists, for which there are non-trivial $n$-[[cells]] in $X$. If there is no such largest $n$ the dimension is also said to be _[[infinity|infinite]]_.

## Definition

Specifically, if $X$ is a [[CW-complex]]

$$
  X \;\simeq\; \underset{\longrightarrow_{\mathrlap{n}}}{\lim} X_n
$$

where each $X_n$ is a [[pushout]] of the form

$$
  \array{
    \underset{ i \in I_n }{\sqcup} S^{n-1}
    &\longrightarrow&
    X_{n-1}
    \\
    \big\downarrow &(po)& \big\downarrow
    \\
    \underset{ i \in I_n }{\sqcup} D^{n}
    &\longrightarrow&
    X_n
  }
$$

the dimension of $X$ is the largest $n$ for which the indexing sets $I_n$ are [[inhabited set|non-empty]].

## Related concepts

[[!include notions of dimension -- contents]]

[[!redirects dimension of cell complexes]]

[[!redirects dimension of a CW-complex]]
[[!redirects dimension of CW-complexes]]

[[!redirects dimension of a CW complex]]
[[!redirects dimension of CW complexes]]


dimension of a manifold

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## Definition

A [[manifold]] whose [[charts]] are [[open subsets]] of the [[Cartesian space]] $\mathbb{R}^n$ is said to have _[[dimension]]_ equal to $n \in \mathbb{N}$.

## Related concepts

[[!include notions of dimension -- contents]]

[[!redirects dimensions of manifolds]]


dimension of a separable metric space

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## Definition

The [[dimension]] of a [[separable metric space]] is taken to be its [[covering dimension]], which coincides with its (small and large) [[inductive dimension]] (e.g., [Engelking 78, Theorem 1.7.7](#Engelking78))

## Related concepts

[[!include notions of dimension -- contents]]

## References

* {#Engelking78} [[Ryszard Engelking]], _Dimension Theory_, Mathematical Library **19**, North-Holland Publishing/Polish Scientific Publishers 1978 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/engelking.pdf))

* {#Engelking95} [[Ryszard Engelking]], _Theory of Dimensions -- Finite and Infinite_, Sigma Series in Pure Mathematics **10**, Helderman 1995 ([pdf](http://www.gbv.de/dms/goettingen/187241074.pdf))

[[!redirects dimensions of separable metric spaces]]


dimensional regularization

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## Idea

In [[perturbative quantum field theory]], _dimensional regularization_ is a method for [[renormalization]]. Here [[Feynman amplitudes]] are expressed via [[Schwinger parameters]] in [[worldline formalism]] making the resulting expression a [[function]] of [[spacetime]] [[dimension]] $d$ which may formally be set to any [[real number]]. Then one regularizes via taking the limit $d \to 4$.

## References

Formulation in the rigorous context of [[causal perturbation theory]]/[[perturbative AQFT]] is due to

* {#DuetschFredenhagenKellerRejzner14} [[Michael Dütsch]], [[Klaus Fredenhagen]], [[Kai Keller]], [[Katarzyna Rejzner]], _Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization_, J. Math. Phy.
55(12), 122303 (2014) ([arXiv:1311.5424](https://arxiv.org/abs/1311.5424))

See also

* Wikipedia, _Dimemensional regularization_

[[!redirects dimensional regularizations]]


dimer

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* table of contents
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## Idea

In [[solid state physics]] and [[quantum many-body physics]], and specifically in the context of [[topological phases of matter]],
by a *dimer* one means an [[edge]] $a \to b$ in some [[graph]] that models a given [[physical system]].

Typically this goes along with some prescription for what is to be associated with/modeled by this dimer.

For example a [[Majorana dimer]] encodes an pair of [[entanglement|entangled]] [[qbits]] ("spins"). In [[Majorana dimer codes]] these dimers arrange as the [[chords]] of a [[chord diagram]], see at *[[holographic entanglement entropy]]* the section *[Chord diagram, representation](holographic+entanglement+entropy#ChordDiagramRepresentation)*.

## References

Survey:

* [[Richard Kenyon]], [[Andrei Okounkov]], *What is... a dimer?*, Notices of the AMS, Vol. 52 No 3 (2005) ([pdf](https://www.ams.org/notices/200503/what-is.pdf))

* [[Richard Kenyon]], *An introduction to the dimer model* ([arXiv:math/0310326](https://arxiv.org/abs/math/0310326))

* [[Subir Sachdev]], *[Quantum dimer model](http://qpt.physics.harvard.edu/qdimer.html)*

Original articles:

As a model for [[superconductivity]]:

* Daniel S. Rokhsar and Steven A. Kivelson, *Superconductivity and the Quantum Hard-Core Dimer Gas*, Phys. Rev. Lett. 61, 2376 – Published 14 November 1988 ([arXiv:10.1103/PhysRevLett.61.2376](https://doi.org/10.1103/PhysRevLett.61.2376))

[[!redirects dimers]]

[[!redirects dimer model]]
[[!redirects dimer models]]

[[!redirects quantum dimer model]]
[[!redirects quantum dimer models]]


Dimitri Ara

* [webpage](http://iml.univ-mrs.fr/~ara/)

## related entries

* [[Grothendieck-Maltsiniotis infinity-category]]

* [[n-quasicategory]], [[model structure on cellular sets]]

category: people


Dimitri Nanopoulos

* [webpage](http://people.physics.tamu.edu/dimitri/)

## Selected writings

On realistic [[Yukawa couplings]] and [[fermion]] [[masses]] in an [[MSSM]] [[Pati-Salam GUT model]] with 3 [[generations of fermions]] realized on [[intersecting D-brane model|intersecting]] [[D6-branes]] [[KK-compactification|KK-compactified]] on a [[toroidal orbifold]] $T^6\sslash (\mathbb{Z}_2 \times \mathbb{Z}_2)$ are claimed in

* {#ChenLiMayesNanopoulos07a} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _A Realistic World from Intersecting D6-Branes_, Phys.Lett.B665:267-270, 2008 ([arXiv:hep-th/0703280](https://arxiv.org/abs/hep-th/0703280), [doi:10.1016/j.physletb.2008.06.024](https://doi.org/10.1016/j.physletb.2008.06.024))

* {#ChenLiMayesNanopoulos07b} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _Realistic Yukawa Textures and SUSY Spectra from Intersecting Branes_, Phys.Rev.D77:125023, 2008 ([arXiv:0711.0396](https://arxiv.org/abs/0711.0396))

On the [[Hopf-Wess-Zumino term]] in the [[M5-brane]] [[Green-Schwarz sigma-model]]:

* Shan Hu, [[Dimitri Nanopoulos]], _Hopf-Wess-Zumino term in the effective action of the 6d, $(2, 0)$ field theory revisted_, JHEP 1110:054 (2011) [[arXiv:1110.0861](https://arxiv.org/abs/1110.0861), doi:10.1007/JHEP10(2011)054]

On the [[Starobinsky model of cosmic inflation]] and its embedding into [[supergravity]]:

* {#EllisCarciaNanopoulosOlive15} [[John Ellis]], Marcos A. G. Garcia, [[Dimitri Nanopoulos]], [[Keith Olive]], _Phenomenological Aspects of No-Scale Inflation Models_ ([arXiv:1503.08867](http://arxiv.org/abs/1503.08867))

* [[John Ellis]], [[Dimitri Nanopoulos]], [[Keith Olive]], _No-Scale Supergravity Realization of the Starobinsky Model of Inflation_ Phys. Rev. Lett. 111, 111301 (2013) ([arXiv:1305.1247](http://arxiv.org/abs/1305.1247))

* [[John Ellis]], [[Dimitri Nanopoulos]], [[Keith Olive]], _Starobinsky-like Inflationary Models as Avatars of No-Scale Supergravity_ JCAP 1310, 009 (2013) ([arXiv:1307.3537](http://arxiv.org/abs/1307.3537))

* {#ENOV19} [[John Ellis]], [[Dimitri Nanopoulos]], [[Keith Olive]], Sarunas Verner, _A Unified No-Scale Model of Modulus Fixing, Inflation, Supersymmetry Breaking and Dark Energy_, Phys. Rev. D 100, 025009 (2019) ([arXiv:1903.05267](https://arxiv.org/abs/1903.05267), [doi:10.1103/PhysRevD.100.025009](https://doi.org/10.1103/PhysRevD.100.025009))

## Related entries

* [[Starobinsky model of cosmic inflation]]

category: people


Dimitrios Toulikas

* [InSpire page](https://inspirehep.net/authors/2029224)

## Selected writings

On [[M2-M5-brane bound states]]:

* [[Iosif Bena]], [[Anthony Houppe]], [[Dimitrios Toulikas]], [[Nicholas P. Warner]], *Maze Topiary in Supergravity* [[arXiv:2312.02286](https://arxiv.org/abs/2312.02286)]

category: people


Dimitrios Tsimpis

* [webpage](http://www.ipnl.in2p3.fr/perso/tsimpis/cv.html)

## Selected writings

On [[massive type IIA supergravity]]:

* {#Tsimpis05} [[Dimitrios Tsimpis]], _Massive IIA supergravities_, JHEP 0510:057, 2005 ([arXiv:hep-th/0508214](https://arxiv.org/abs/hep-th/0508214))

On [[M2-M5 brane bound states]]:

* [[George Papadopoulos]], [[Dimitrios Tsimpis]], _The holonomy of the supercovariant connection and Killing spinors_, JHEP 0307:018, 2003 ([arXiv:hep-th/0306117](https://arxiv.org/abs/hep-th/0306117))

On [[higher curvature corrections]] in [[11-dimensional supergravity]]:

* [[Dimitrios Tsimpis]], _11D supergravity at $\mathcal{O}(\ell^3)$_, JHEP0410:046, 2004 ([arXiv:hep-th/0407271](https://arxiv.org/abs/hep-th/0407271))

* [[Martin Cederwall]], [[Ulf Gran]], [[Bengt Nilsson]], [[Dimitrios Tsimpis]], _Supersymmetric Corrections to Eleven-Dimensional Supergravity_, JHEP0505:052, 2005 ([arXiv:hep-th/0409107](https://arxiv.org/abs/hep-th/0409107))

and specificallyt the on the [[I8]]-term

* {#SoueresTsimpis17} [[Bertrand Souères]], [[Dimitrios Tsimpis]], _The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity_, Phys. Rev. D 95, 026013 (2017) ([arXiv:1612.02021](https://arxiv.org/abs/1612.02021))

On [[M-theory on 8-manifolds]]:

* [[Dimitrios Tsimpis]], _M-theory on eight-manifolds revisited: N=1 supersymmetry and generalized $Spin(7)$ structures_, JHEP 0604 (2006) 027 ([arXiv:hep-th/0511047](https://arxiv.org/abs/hep-th/0511047))

and specifically in terms of [[G-structures]]:

* [[Daniël Prins]], [[Dimitrios Tsimpis]], _IIA supergravity and M-theory on manifolds with SU(4) structure_, PhysRevD.89.064030 ([arXiv:1312.1692](https://arxiv.org/abs/1312.1692))

More on [[G-structures]] in [[supergravity]]:

* [[Robin Terrisse]], [[Dimitrios Tsimpis]], _$SU(3)$ structures on $S^2$ bundles over four-manifolds_, JHEP09 (2017) 133 ([arXiv:1707.04636](https://arxiv.org/abs/1707.04636))

On [[topological twists]]:

* [[Jeong-Hyuck Park]], [[Dimitrios Tsimpis]], _Topological twisting of conformal supercharges_, Nucl. Phys. B776:405-430, 2007 ([arXiv:hep-th/0610159](http://arxiv.org/abs/hep-th/0610159))

Discussion of *asymptotic* [[de Sitter spacetimes]] from time-dependent [[KK-compactification]] of [[type II supergravity]]:

* Paul Marconnet, [[Dimitrios Tsimpis]], *Universal accelerating cosmologies from 10d supergravity*,  J. High Energ. Phys. **2023** 33 (2023) [[arXiv:2210.10813](https://arxiv.org/abs/2210.10813), doi:10.1007/JHEP01(2023)033]

reviewed in:

* [[Dimitrios Tsimpis]], *Universal accelerating cosmologies from 10d supergravity*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (Jan 2023) [[web](/nlab/show/M-Theory+and+Mathematics#Tsimpis2023)]

and further discussed in:

* David Andriot, [[Dimitrios Tsimpis]], Timm Wrase, *Accelerated expansion of an open universe, and string theory realizations* [[arXiv:2309.03938](https://arxiv.org/abs/2309.03938)]

## Related entries

* [[supergravity torsion constraints]]

* [[higher curvature corrections]]

* [[M-theory on 8-manifolds]]

category: people


Dimitris Tsementzis

* [webpage](http://rci.rutgers.edu/~dt506/)

## Publications

On [[structuralism]] and the [[structure identity principle]] in [[univalent foundations of mathematics]]:

* [[Dimitris Tsementzis]], *Univalent foundations as structuralist foundations*, Synthese **194** 9 (2017) 3583–3617 [[jstor:26748765](https://www.jstor.org/stable/26748765), [doi:10.1007/s11229-016-1109-x](https://doi.org/10.1007/s11229-016-1109-x), [pdf](https://core.ac.uk/download/pdf/157866891.pdf)]

On the [[univalence axiom]]:

* [[Benedikt Ahrens]], [[Paige Randall North]], [[Michael Shulman]], [[Dimitris Tsementzis]], *The Univalence Principle* ([arXiv:2102.06275](https://arxiv.org/abs/2102.06275))

## related $n$Lab entries

* [[meaning explanation]]

category: people


dinatural transformation

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## Idea

**Dinatural transformations** are a generalization of ordinary [[natural transformations]] and also of [[extranatural transformations]].  The differences can be summarized thus:

* In an ordinary *natural transformation* $F\to G$, both $F$ and $G$ involved depend on some variable $x$ with the same variance (covariant or contravariant).
* In an *extranatural transformation* $F\to G$, either $F$ depends on $x$ both covariantly *and* contravariantly and $G$ does not depend on $x$ at all, or vice versa.
* In a **dinatural transformation** $F\to G$, both $F$ and $G$ can depend on $x$ both covariantly and contravariantly.

If the dependence of $F$ or $G$ on $x$ in a dinatural transformation is trivial, it reduces to an extranatural one.  Similarly, if the contravariant (or, dually, the covariant) dependence of $F$ and $G$ on $x$ are trivial, it reduces to an ordinary natural one.

Arguably, most dinatural transformations which arise in practice are ordinary natural, extranatural, or [[paranatural]].

## Definition

Let $F, G: C^{op} \times C \to D$ be functors. A __dinatural transformation__ from $F$ to $G$, sometimes written

$$\alpha: F \stackrel{\bullet}{\to} G,$$

consists of a collection of morphisms

$$\alpha_{c}: F(c, c) \to G(c, c)$$

such that for every morphism $f: c \to c'$ in $C$,

\[ \label{hexagon} G(c, f)\alpha_c F(f, c) = G(f, c')\alpha_{c'}F(c', f): F(c', c) \to G(c, c') \]

If drawn out as a commutative diagram, this becomes a "hexagon identity":

\[
\begin{array}{ccccccc}
 &  & F(c,c) & \overset{\alpha_{c}}{\to} & G(c,c)\\
 & \overset{\mathclap{F(f,c)}}{\nearrow} &  &  &  & \overset{\mathclap{G(c,f)}}{\searrow}\\
F(c',c) &  &  &  &  &  & G(c,c')\\
 & \underset{\mathclap{F(c',f)}}{\searrow} &  &  &  & \underset{\mathclap{G(f,c')}}{\nearrow}\\
 &  & F(c',c') & \underset{\alpha_{c'}}{\to} & G(c',c')
\end{array}\,.
\]

### Special cases

If $F$ and $G$ both factor through the projection $C^{op}\times C \to C$, then the notion reduces to an ordinary natural transformation, and similarly if they both factor through $C^{op}$.

If $F$ factors through $C$ while $G$ factors through $C^{op}$, then we obtain a notion of natural transformation from a covariant functor to a contravariant one, and dually.

If $G$ is constant, the hexagon identity reduces to the "domain" version of extranaturality, involving a commutative square of the form

\[ \label{domain} \alpha_c F(f, c) = \alpha_{c'} F(c', f): F(c', c) \to G \]

where $G$ is constant with respect to the argument $c$.  Similarly, if $F$ is constant, it yields the "codomain" version with a commutative square of the form

\[ \label{codomain} G(c, f) \alpha_c = G(f, c')\alpha_{c'}: F \to G(c, c') \]

when $F$ is constant with respect to the argument $c$.

## Examples

Of course, all ordinary natural transformations, and also all [[extranatural transformations]], are also dinatural ones.  Here we will confine ourselves to examples that do not reduce to either of these.

* Perhaps the most well known example is the [[Church numerals]]: for any category $C$, we have a class of dinatural transformations from the hom-functor $hom: C^{op}\times C\to Set$

  $$\hom(x, x) \stackrel{\alpha_n}{\to} \hom(x, x)$$

  defined by the rule $\alpha_n(f) = f^{(n)}$.

* Not every endo-dinatural-transformation of a hom-functor is of this form; several other examples are given in [Pare-Roman](#PareRoman).  For instance, if $C=FinSet$ then the operation sending each endomorphism to its [[eventual image]] is dinatural.

* The following example of a dinatural transformation from a covariant functor to a contravariant one is found in [Dubuc-Street](#DubucStreet) and attributed to [[Mac Lane]].  Let $C$ be the category of [[inner product spaces]] and isometries, and $D$ the category of [[vector spaces]] and linear transformations (all over a fixed [[field]]).  Let $F:C\to D$ be the forgetful functor and let $G:C^{op}\to D$ be the functor sending each inner product space to its dual.  Define $\alpha_V : F(V) \to G(V)$ to send $v\in V$ to the operation $\langle v,-\rangle$; then $\alpha$ is dinatural.

By a yoneda-like argument, dinatural transformations $\alpha : F \xrightarrow{\bullet} G$ are in bijection with natural transformations $\eta_{x,y} : \hom(x, y) \to \hom(F(y,x), G(x,y))$. The corresponding
transformations are related by

* $\eta_{x,y}(f) = G(x, f) \alpha_x F(f, x) = G(f, y) \alpha_y F(x, f)$

* $\alpha_x = \eta_{x,x}(id)$

## Dinaturality versus extranaturality

Many people who encounter the notion of dinaturality through the general definition (as in equation (eq:hexagon)) have subsequent difficulty grokking it.  It is the opinion of at least one author of this article ([[Todd Trimble]]), and it was certainly the opinion of Max Kelly, that this "efficient" definition is not the most useful or intuitive one.  Rather, one may be better off grokking the separate squares (eq:domain) and (eq:codomain) -- that is, the notion of [[extranaturality]] -- and how they arise in practice.

One could try to argue against that by pointing to dinatural transformations which do not reduce to extranatural ones.

A counterargument, however, is that *any* dinatural transformation between functors $F,G:C^{op}\times C\to D$ can be "bent" into domain extranaturality by defining

$$C^{op} \times C \stackrel{[F,G]}{\to} Set: (x, y) \mapsto D(F(y, x), G(x, y)).$$

Then a dinatural transformation $F\to G$ can be identified with an extranatural transformation from the constant $1$ (the terminal set) to $[F,G]$. Such tricks support the counterargument that the extra generality of the traditional definition is largely spurious, and not particularly helpful in terms of comprehension.

A further argument for the relative importance of extranatural transformations over dinatural ones is that extranatural transformations can be defined for any sort of [[enriched categories]], whereas dinatural ones (including the other special case of natural transformations from covariant functors to contravariant ones) only make sense when the enriching category is [[cartesian monoidal category|cartesian]].

## Composition

Dinatural transformations cannot, in general, be composed with each other, although there are certain circumstances when they can be.

* One such case is when certain squares are pushouts or pullbacks.

* Of course, ordinary natural transformations can be composed.

* In fact, there is a category whose objects consist of both covariant functors $C\to D$ and contravariant ones $C^{op}\to D$, and whose morphisms are dinatural transformations.

In general, what we can say is that for two fixed categories $C$ and $D$, the functors $C^{op}\times C \to D$ and the dinatural transformations between them form a [[paracategory]].

## Related concepts

* [[homotopy]]

* [[contravariant functor]], [[2-category with contravariance]]

* [[transfor]]

  * [[natural transformation]]

    * [[extranatural transformation]], **dinatural transformation**

  * [[pseudonatural transformation]]

  * [[lax natural transformation]]

  * [[2-dinatural transformation]]

## References

* [[Eduardo Dubuc]] and [[Ross Street]], *Dinatural transformations*, Reports of the Midwest Category Seminar IV, Lecture Notes in Math., vol. 137, Springer, Berlin (1970), pp. 126–137 (doi:[10.1007/BFb0060443](https://doi.org/10.1007/BFb0060443))
 {#DubucStreet}

* [[Robert Paré]] and Leopoldo Román, *Dinatural numbers*, [JPAA](http://www.sciencedirect.com/science/article/pii/S0022404997000364)
 {#PareRoman}

* Guy McCusker, Alessio Santamaria, _A Calculus of Substitution for Dinatural Transformations, I_, ([arXiv:2007.07576](https://arxiv.org/abs/2007.07576))

Here is a blog post inspired by the above discussion that discusses these concepts in the context of the programming language Haskell:

* Dan Piponi, [Dinatural transformations and coends](http://blog.sigfpe.com/2009/03/dinatural-transformations-and-coends.html)

[[!redirects dinaturality]]
[[!redirects dinatural transformations]]


Diogenes Laërtius

* [Wikipedia entry](https://en.wikipedia.org/wiki/Diogenes_Laertius)

## Selected writings

On ancient Greek philosophers:

* [[Diogenes Laërtius]], (translated by Robert Drew Hicks): *Lives of the Eminent Philosophers* (~200 AD) [[wikisource](https://en.m.wikisource.org/wiki/Lives_of_the_Eminent_Philosophers), [perseus.tufts](http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0258%3Abook%3D1%3Achapter%3Dprologue)]

## Related entries

* [[Heraclitus]]

[[!redirects Diogenes Laertius]]


Diophantine equation

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## Idea

A _Diophantine equation_ is a [[polynomial]] [[equation]] over the [[integers]], hence an equation of the form

$$
  P(x_1, x_2, \cdots , x_n) = 0
$$

where $P(x_1, x_2, \cdots , x_n) \in \mathbb{Z}[x_1, \cdots, x_m]$ is a [[polynomial]] in $n$ [[variables]] with [[integer]] [[coefficients]], and where solutions are considered for these variables taking values in the integers.

These are equations of central interest in [[number theory]].

Spaces of solutions of Diophantine equations may be thought of as [[varieties]] in [[arithmetic geometry]].

## References

* Wikipedia, _[Diophantine equation](http://en.wikipedia.org/wiki/Diophantine_equation)_

[[!redirects Diophantine equations]]

[[!redirects diophantine equation]]
[[!redirects diophantine equations]]


dipole moment

## Related entries

* [[electromagnetic field]], [[electric charge]]

* [[quadrupole moment]]

## References

* Wikipedia, _[Dipole moment](https://en.wikipedia.org/wiki/Dipole_moment)_

[[!redirects dipole moments]]


diquark

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## Idea

By a _diquark_ one refers to a [[bound state]] of two [[constituent quarks]] _inside_ a larger [[bound state]] of a total of three [[constituent quarks]] constituting a [[baryon]].

## Related concepts

* [[dilepton]]

* [[hadron supersymmetry]]

## References

* Mauro Anselmino, Enrico Predazzi, Svante Ekelin, Sverker Fredriksson, [[Don B. Lichtenberg]], *Diquarks*, Rev. Mod. Phys. **65** (1993) 1199 ([doi:10.1103/RevModPhys.65.1199](https://doi.org/10.1103/RevModPhys.65.1199))

* Martin Oettel, _Baryons as Relativistic Bound States of Quark and Diquark_ ([arXiv:nucl-th/0012067](https://arxiv.org/abs/nucl-th/0012067), [spire:538966](https://inspirehep.net/literature/538966))

See also

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

[[!redirects diquark]]
[[!redirects diquarks]]


Dirac charge quantization

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## Idea

### For monopole charges in electromagnetism

If the field of [[electromagnetism]] serves as a [[background gauge field]] for electrically charged [[quantum mechanics|quantum]] [[particles]] it is subject to a _quantization condition_: Outside the locus of any [[magnetic charge]] -- for instance a magnetic [[monopole]] [[topological defect]] -- the [[electromagnetic field]] must be a [[circle bundle with connection|connection on a principal U(1) bundle]] whose [[first Chern class]] is the discrete measure for the units of magnetic charge. Equivalently this means that the [[electromagnetic field]] is a [[cocycle]] in [[ordinary differential cohomology]] of degree 2.

In the underlying topological sector ("[[monopole]]"/"[[instantons]]"-sector) this is [[integral cohomology]] in degree-2, whose [[classifying space]] is equivalently the [[infinite complex projective space]] $B U(1) \simeq \mathbb{C}P^\infty$:

{#Illustration} $\,$

This goes back to an insight due to [Dirac 31](#Dirac31). See [Heras 18](#Heras18) for traditional elementary review, but see [Alvarez 85](#Alvarez85),  [Frankel](#Frankel) and [Mangiarotti-Sardanashvily 00](#MangiarottiSardanashvily00) for exposition of the modern picture in terms of [[fiber bundles in physics]]. See [Freed 00, Section 2](#Freed00) for review in terms of [[differential cohomology]] with outlook to generalization to [[higher gauge fields]] in [[string theory]] (more on which in the references [below](#ReferencesForBFieldAndRRFields)).

A closely related phenomenon is the magnetic flux quantization in [[type II superconductors]], see [there](superconductivity#MagneticFluxQuantizationInTypeIISuperconductors).

On the locus of the magnetic charge itself the situation is more complex. There the [[magnetic current]] is given by a cocycle in [[ordinary differential cohomology]] of degree 3 (with compact support) and now the electromagnetic field is a connection on a [[twisted bundle]] ([Freed 00, Section 2](#Freed00)).

\linebreak

### For monopole charges in non-abelian Yang-Mills theory

A similar charge quantization condition govers [[monopoles]] in [[SU(2)]]-[[Yang-Mills theory]], see at _[[moduli space of monopoles]]_. Here the [[Atiyah-Hitchin charge quantization]] ([Atiyah-Hitchin 88, Theorem 2.10](#AtiyahHitchin88)) says that the [[moduli space]] of [[monopoles]] is the [[holomorphic function|complex]]-[[rational function|rational]] 2-[[Cohomotopy]] of an asymptotic 2-sphere enclosing the monopoles:

{#IllustrationAtiyahHitchinQuantization} $\,$

### For RR-field D-brane charges in K-theory

See at _[[D-brane charge quantization in K-theory]]_

### For the C-field in J-twisted Cohomotopy

See at _[supergravity C-field -- shifted flux quantization condition](supergravity+C-field#ShiftedCFieldFluxQuantizationCondition)_

## Related entries

* [[flux]], [[flux tube]], [[flux compactification]], [[uncertainty of fluxes]]

* [[geometry of physics -- flux quantization]]

* [[fiber bundles in physics]]

* [[non-perturbative quantum field theory]]

* [[monopole]]

  * [[Dirac monopole]], [[Yang monopole]]

* [[moduli space of monopoles]]

\linebreak

## References

See also the references at *[[fiber bundles in physics]]*.

### For the electromagnetic field

The original argument for charge quantization of the [[electromagnetic field]]:

* {#Dirac31} [[P.A.M. Dirac]], _Quantized Singularities in the Electromagnetic Field_,  Proceedings of the Royal Society A **133** (1931) 60-72 [[doi:10.1098/rspa.1931.0130](http://rspa.royalsocietypublishing.org/content/133/821/60.short)]

Further early discussion

* [[Julian Schwinger]], *Magnetic Charge and Quantum Field Theory*, Phys. Rev. **144** (1966) 1087 [[doi:10.1103/PhysRev.144.1087](https://doi.org/10.1103/PhysRev.144.1087)]

* [[Daniel Zwanziger]], *Quantum Field Theory of Particles with Both Electric and Magnetic Charges*, Phys. Rev. **176** (1968) 1489 [[doi:10.1103/PhysRev.176.1489](https://doi.org/10.1103/PhysRev.176.1489)]

Understanding of charge quantization in terms of [[Čech cohomology]]:

* {#Alvarez85a} [[Orlando Alvarez]], *Cohomology and Field Theory*, talk at: *Symposium on Anomalies, Geometry, Topology*, Argonne IL (28-30 March 1985) [[inspire:965785](https://inspirehep.net/conferences/965785), [pdf](https://lib-extopc.kek.jp/preprints/PDF/1985/8507/8507262.pdf), [[Alvarez-CohomologyAndFieldTheory.pdf:file]]]

* {#Alvarez85} [[Orlando Alvarez]], _Topological quantization and cohomology_, Comm. Math. Phys. **100** 2 (1985) 279-309 [[euclid:cmp/1103943448](https://projecteuclid.org/euclid.cmp/1103943448)]

reviewed in:

* Guillermo A. González, *Cohomologa de Cĕch y Cuantización Topológica de Parámetros Físicos*, Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales **47** 182 (2023) 51-71 [[doi:10.18257/raccefyn.1782](https://doi.org/10.18257/raccefyn.1782), [[Gonzales-TopologicalQuantization.pdf:file]]]

General review and further discussion:

* {#Brylinski93} [[Jean-Luc Brylinski]], §7.1 in: *Loop spaces, characteristic classes and geometric quantization*, Birkhäuser (1993) [[doi:10.1007/978-0-8176-4731-5](https://link.springer.com/book/10.1007/978-0-8176-4731-5)]

* {#Frankel} [[Theodore Frankel]], section 16.4e of: _[[The Geometry of Physics - An Introduction]]_, Cambridge University Press (1997, 2004, 2012) [[doi:10.1017/CBO9781139061377](https://doi.org/10.1017/CBO9781139061377)]

* [[Luis Alvarez-Gaumé]], Frederic Zamora, Section II in:  *Duality in Quantum Field Theory (and String Theory)*, AIP Conference Proceedings **423** (1998) 46–83 [[arXiv:hep-th/9709180](https://arxiv.org/abs/hep-th/9709180), [doi:10.1063/1.55087](https://doi.org/10.1063/1.55087)]

* {#Freed00} [[Daniel Freed]], Section 2 of: *[[Dirac charge quantization and generalized differential cohomology]]*, Surveys in Differential Geometry **7**, Int. Press (2000) 129-194 [[arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220), [doi:10.4310/SDG.2002.v7.n1.a6](https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6), [spire:537392](https://inspirehep.net/literature/537392)]

* {#MangiarottiSardanashvily00} L. Mangiarotti, [[Gennadi Sardanashvily]], _Connections in Classical and Quantum Field Theory_, World Scientific, 2000 ([doi:10.1142/2524](https://doi.org/10.1142/2524))

* [[Yakov Shnir]], Section 2.1 of: *Magnetic Monopoles*, Springer 2005 ([ISBN:978-3-540-29082-7](https://www.springer.com/gp/book/9783540252771))

* {#FreedMooreSegal07} [Freed, Moore & Segal 2007, p. 7](#FreedMooreSegal07)

* {#Heras18} Ricardo Heras, _Dirac quantisation condition: a comprehensive review_, Contemp. Phys. 59, 331 (2018) ([arXiv:1810.13403](https://arxiv.org/abs/1810.13403))

See also

* Wikipedia, _[Dirac's quantization](https://en.wikipedia.org/wiki/Magnetic_monopole#Dirac%27s_quantization)_

Arguments for electric charge quantization from [[anomaly cancellation]] in the [[standard model of particle physics]]:

* K. S. Babu and Rabindra N. Mohapatra, *Quantization of electric charge from anomaly constraints and a Majorana neutrino*, Phys. Rev. D **41** (1990) 271 [[doi:10.1103/PhysRevD.41.271](https://doi.org/10.1103/PhysRevD.41.271)]

Consideration of combined magnetic *and* electric charge quantization in [[vacuum]] (cf. *[[duality-symmetric electromagnetism]]*):

* {#FreedMooreSegal07} [[Daniel S. Freed]], [[Gregory W. Moore]], [[Graeme Segal]], *The Uncertainty of Fluxes*, Commun. Math. Phys. **271** (2007) 247-274 [[arXiv:hep-th/0605198](https://arxiv.org/abs/hep-th/0605198), [doi:10.1007/s00220-006-0181-3](https://doi.org/10.1007/s00220-006-0181-3)]

* {#BBSS15} [[Christian Becker]], [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], Rem. 2.3 in: _Abelian duality on globally hyperbolic spacetimes_, Commun. Math. Phys. **349** (2017) 361-392 [[arXiv:1511.00316](https://arxiv.org/abs/1511.00316), [doi:10.1007/s00220-016-2669-9](https://doi.org/10.1007/s00220-016-2669-9)]

* {#LazaroiuShahbazi22} [[Calin Lazaroiu]], [[Carlos S. Shahbazi]], around Def. 1.16 in: *The duality covariant geometry and DSZ quantization of abelian gauge theory*, Advances in Theoretical and Mathematical Physics **26** (2022) 2213–2312 [[arXiv:2101.07236](https://arxiv.org/abs/2101.07236), [doi:10.4310/ATMP.2022.v26.n7.a5](https://dx.doi.org/10.4310/ATMP.2022.v26.n7.a5)]

* {#LazaroiuShahbazi23} [[Calin Lazaroiu]], [[Carlos S. Shahbazi]], around (3) in: *The geometry and DSZ quantization of four-dimensional supergravity*, Letters in Mathematical Physics  **113** 4 (2023) [[arXiv:2101.07778](https://arxiv.org/abs/2101.07778), [doi:10.1007/s11005-022-01626-y](https://doi.org/10.1007/s11005-022-01626-y)]

### For the weak nuclear force field

Discussion of the [[moduli space of monopoles]] for [[SU(2)]]-[[Yang-Mills theory]] ([[weak nuclear force]]):

* {#AtiyahHitchin88} [[Michael Atiyah]], [[Nigel Hitchin]], _The geometry and dynamics of magnetic monopoles_  M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988 ([jstor:j.ctt7zv206](https://www.jstor.org/stable/j.ctt7zv206))

* {#Segal79} [[Graeme Segal]], _The topology of spaces of rational functions_, Acta Math. Volume 143 (1979), 39-72 ([euclid:1485890033](https://projecteuclid.org/euclid.acta/1485890033))

[[!include D-brane charge quantization in topological K-theory -- references]]

### For the C-field in F-theory

Discussion of [[shifted C-field flux quantization]] of the [[C-field]]  in [[F-theory]]:

* [[Andres Collinucci]], [[Raffaele Savelli]], *On Flux Quantization in F-Theory*,  J. High Energ. Phys. **2012** 15 (2012)   [[arXiv:1011.6388](https://arxiv.org/abs/1011.6388), doi:10.1007/JHEP02(2012)015]

* [[Andres Collinucci]], [[Raffaele Savelli]], *On Flux Quantization in F-Theory II: Unitary and Symplectic Gauge Groups*,  J. High Energ. Phys. **2012** 94 (2012) [[arXiv:1203.4542](https://arxiv.org/abs/1203.4542), doi:10.1007/JHEP08(2012)094]

* [[Raffaele Savelli]], *Flux Quantization in F-theory and Freed-Witten anomaly*, talk at StringMath (2012) [[pdf](https://www.hcm.uni-bonn.de/fileadmin/stringmath2012/parallel_talks/savelli.pdf), [[Savelli-FTheoryFluxQuantization.pdf:file]]]

### For the C-field in M-theory

Discussion of [[shifted C-field flux quantization]] of the [[C-field]] in [[D=11 supergravity]]/[[M-theory]]:

* {#DFM} E. Diaconescu, [[Dan Freed]], [[Greg Moore]],  _The $M$-theory 3-form and $E_8$-gauge theory_, chapter in [[Haynes Miller]], [[Douglas Ravenel]] (eds.) _Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues_, Cambridge University Press 2007 ([arXiv:hep-th/0312069](http://arxiv.org/abs/hep-th/0312069), [doi:10.1017/CBO9780511721489](https:
//doi.org/10.1017/CBO9780511721489))

* {#FreedMoore04} [[Dan Freed]], [[Greg Moore]], _Setting the [[quantum integrand]] of M-theory_, Communications in Mathematical Physics, Volume 263, Number 1, 89-132, ([arXiv:hep-th/0409135](http://arxiv.org/abs/hep-th/0409135), [doi:10.1007/s00220-005-1482-7](https://doi.org/10.1007/s00220-005-1482-7))

* [[Greg Moore]], _Anomalies, Gauss laws, and Page charges in M-theory_ ([arXiv:hep-th/0409158](http://arxiv.org/abs/hep-th/0409158))

* {#FSS12} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]],  _[[schreiber:The moduli 3-stack of the C-field]]_, Communications in Mathematical Physics, Volume 333, Issue 1 (2015), Page 117-151, ([arXiv:1202.2455](http://arxiv.org/abs/1202.2455), [DOI 10.1007/s00220-014-2228-1](http://link.springer.com/article/10.1007%2Fs00220-014-2228-1))

* {#FiSaSc} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane|M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory]]_ Advances in Theoretical and Mathematical Physics, Volume 18, Number 2 (2014) p. 229?321   ([arXiv:1201.5277](http://arxiv.org/abs/1201.5277), [doi:10.4310/ATMP.2014.v18.n2.a1](https://dx.doi.org/10.4310/ATMP.2014.v18.n2.a1))

Discussion in [[twisted Cohomotopy]] ("[[Hypothesis H]]"):

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]_ ([arXiv:1904.10207](https://arxiv.org/abs/1904.10207))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]_ ([arXiv:1906.07417](https://arxiv.org/abs/1906.07417))

and in [[equivariant Cohomotopy]]:

* [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], _[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]_ ([arXiv:1909.12277](https://arxiv.org/abs/1909.12277))

[[!redirects Dirac's charge quantization]]

[[!redirects Dirac flux quantization]]

[[!redirects charge quantization]]
[[!redirects charge quantizations]]

[[!redirects flux quantization]]
[[!redirects flux quantizations]]

[[!redirects flux quantization law]]
[[!redirects flux quantization laws]]


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

This entry collects links related to the article:

* [[Daniel Freed]]:

  \linebreak

  **Dirac charge quantization and generalized differential cohomology_

  \linebreak

  Surveys in Differential Geometry **7** (2002) 129-194

  [arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220)

  [doi:10.4310/SDG.2002.v7.n1.a6](https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6)

  [spire:537392](https://inspirehep.net/literature/537392)

This discusses how [[differential cohomology]] -- specifically [[ordinary differential cohomology]] and [[differential K-theory]] -- is the correct formalism for describing [[gauge fields]] and [[higher gauge fields]] in [[physics]] and [[string theory]] (such as [[higher U(1)-gauge fields]] like the [[electromagnetic field]], the [[Kalb-Ramond field]] and the [[supergravity C-field]], and also the [[RR-field]]) and the [[quantum anomaly]]  of  their [[action functionals]].

## Related entries

* [[differential cohomology]]

  * [[ordinary differential cohomology]]

    * [[Deligne complex]], [[Cheeger-Simons differential character]]

    * [[electromagnetism]], [[B-field]], [[C-field]]

      * [[Dirac charge quantization]]

  * [[differential K-theory]], [[twisted K-theory]], [[twisted differential K-theory]]

    * [[RR-fields]]

* [[quantum anomaly]], [[Green-Schwarz mechanism]]

* [[electric-magnetic duality]], [[self-dual higher gauge theory]]

* _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_

* [[twisted smooth cohomology in string theory]]

  * [[Chern-Simons circle 3-bundle]], [[Chern-Simons circle 7-bundle]]

category: reference


Dirac cohomology

### Idea

Dirac cohomology appears in the effort to refine the nilpotent Lie algebra cohomology as it appears in the context of [[representation theory]] (in work of [[Bertram Kostant]] (Lie algebra cohomology and the generalized Borel-Weil theorem, Annals Math. 74 (1961) 329--387), related to [[Borel-Weil-Bott theorem]]).

### References

* Jing-Song Huang, [[Pavle Pandžić]], _Dirac cohomology, unitary representations and a proof of a conjecture of Vogan_, J. Amer. Math. Soc. __15__:1 (2002) 185--202

* [[Bertram Kostant]], _Dirac cohomology for the cubic Dirac operator_, in Studies in Memory of Issai Schur, Birkhauser, 2003; arXiv:[math.RT/0208048](https://arxiv.org/abs/math/0208048); reprinted in collected works ([pdf](https://link.springer.com/content/pdf/10.1007/978-0-387-09591-2_2.pdf))

* Pavle Pandžić, [[Petr Somberg]], _Higher Dirac cohomology of modules with generalized infinitesimal character_, [arxiv/1310.3570](http://arxiv.org/abs/1310.3570)

* Jing-Song Huang, Pavle Pandžić, _Dirac operators in representation theory_, Representations of real and $p$-adic groups, 163--219, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 2, Singapore Univ. Press, 2004.

* J_S. Huang, _Dirac cohomology, elliptic representations and endoscopy_, in: Nevins, M., Trapa, P. (eds) Representations of Reductive Groups. Progress in Mathematics, vol 312. Birkhäuser 2015 [doi](https://doi.org/10.1007/978-3-319-23443-4_9)

Woit wrote about relation to a case of Hilbert space from BRST cohomology:

* [[Peter Woit]], _Quantization and Dirac cohomology_, [pdf](http://www.math.columbia.edu/~woit/qmdirac.pdf);  _BRST and Dirac cohomology_, slides, [pdf](http://www.math.columbia.edu/~woit/brst-dartmouth.pdf), _BRST and Dirac cohomology_, preprint [pdf](http://www.math.columbia.edu/~woit/brstdirac.pdf), 2009

* Spyridon Afentoulidis-Almpanis, _Dirac cohomology for the BGG category $\mathcal{O}$_, [arXiv:2209.12566](https://arxiv.org/abs/2209.12566)

category: algebra


Dirac conjugate

## Idea

For the moment see at _[[geometry of physics -- supersymmetry]]_: [this def.](geometry+of+physics+--+supersymmetry#DiracConjugate)

## Related concepts

* [[Dirac spinor]], [[Majorana spinor]]

* [[Dirac field]]

[[!redirects Dirac conjugates]]

[[!redirects Dirac conjugation]]
[[!redirects Dirac conjugations]]


Dirac current

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the [[Lagrangian field theory]] of the [[Dirac field]], the _Dirac current_ is a [[conserved current]] whose interpretation is literally the current of the [[spinor]] [[particles]] that are the [[quanta]] of the Dirac field. Therefore the corresponding [[charge]] is _fermion number_. As such the Dirac current (or rather its chiral version) plays a key role for instance in [[baryogenesis]].

The Dirac current is the [[conserved current]] which is associated via [[Noether's theorem|Noether's theorem I]] to the [[infinitesimal symmetry of the Lagrangian]] given by multiplying the [[Dirac field]] by a complex phase.

## Definition

In the usual standard coordinates, the Dirac current is of the form

$$
  \overline{\psi}\gamma^\mu \psi \, \iota_{\partial_\mu} dvol_\Sigma
   \;\in\;
   \Omega^{p,0}
   \,.
$$

For details see at _[[geometry of physics -- A first idea of quantum field theory]]_ [this example](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#DiracCurrent).

## Related concepts

* [[baryon current]]

* [[chiral anomaly]]

* [[baryogenesis]]

* [[energy-momentum tensor]]

[[!redirects Dirac currents]]

[[!redirects fermion current]]
[[!redirects fermion currents]]


Dirac distribution

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **Dirac distribution** or **Dirac $\delta$-distribution** $\delta(p)$ is the [[distribution]] that is given by [[evaluation|evaluating]] a [[function]] at a point $p$.

It is closely related to [[Dirac measures]], in the language of [[measure theory]].

## Properties

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

The [[derivative of distributions|distributional derivative]] of the [[Heaviside distribution]] $\Theta \in \mathcal{D}'(\mathbb{R})$ is the [[delta distribution]] $\delta \in \mathcal{D}'(\mathbb{R})$:

$$
  \partial \Theta = \delta
  \,.
$$

=--

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

For $b \in C^\infty_c(\mathbb{R})$ any [[bump function]] we compute:

$$
  \begin{aligned}
    \int \partial\Theta(x) b(x) \, d x
    & =
    - \int \Theta(x) \partial b(x)\, dx
    \\
    & =
    - \int_0^\infty \partial b(x) d x
    \\
    & =
    - \left( b(x)\vert_{x \to \infty} - b(0) \right)
    \\
    & =
    b(0)
    \\
    &
    =
    \int \delta(x) b(x) \, dx
    \,.
  \end{aligned}
$$

=--

### Fourier transform
 {#FourierTransform}

+-- {: .num_defn #FourierTransformOfDeltaDistribution}
###### Example
**([[Fourier transform of distributions|Fourier transform]] of the [[delta-distribution]])**

The [[Fourier transform of distributions|Fourier transform]] ([this def.](Fourier+transform#FourierTransformOnTemperedDistributions)) of the [[delta distribution]], via [this example](Fourier+transform#CompactlySupportedDistributionFourierTransform), is the [[constant function]] on 1:

$$
  \begin{aligned}
    \widehat {\delta}(k)
    & =
    \underset{x \in \mathbb{R}^n}{\int} \delta(x) e^{- 2\pi i k x} \, d x
    \\
    & =
    1
  \end{aligned}
$$

This implies by the [[Fourier inversion theorem]] ([this prop.](Fourier+transform#FourierInversionTheoremForDistributions)) that the [[delta distribution]] itself has equivalently, in [[generalized function]]-notation, the expression

$$
  \label{FourierModeExpansionOfDeltaDistribution}
  \begin{aligned}
    \delta(x)
    & =
    \widehat{\widehat{\delta}}(-x)
    \\
    & =
    \int_{k \in \mathbb{R}^n} e^{2 \pi i k \cdot x} \, d k
  \end{aligned}
$$

in the sense that for every [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$ we have

$$
  \begin{aligned}
    f(x)
    & =
    \underset{y \in \mathbb{R}^n}{\int}
    f(y) \delta(y-x)  \, dvol(y)
    \\
    & =
    \underset{y \in \mathbb{R}^n}{\int}
    \underset{k \in \mathbb{R}^n}{\int}
    f(y) e^{2 \pi i k \cdot (y-x)}
    \, dvol(k)\, dvol(y)
    \\
    & =
    \underset{k \in \mathbb{R}^n}{\int}
    e^{- 2 \pi i k \cdot x}
    \underset{= (-1)^n\widehat{f}(-k)}{
    \underbrace{
      \underset{y \in \mathbb{R}^n}{\int}
      f(y) e^{2 \pi i k \cdot y}
      \, dvol(y)
    }
    }
    \, dvol(k)
    \\
    & =
    \widehat{\widehat{f}}(-x)
  \end{aligned}
$$

which is just the statement of the [[Fourier inversion theorem]] for smooth functions ([this prop.](Fourier+transform#FourierInversion)).

=--

### Relation to point-supported distributions

It is clear that:

The delta distribution is a [[compactly supported distribution]], and in fact a [[point-supported distribution]].

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

Every [[point-supported distribution]] $u$  with $supp(u) = \{p\}$ is a [[finite set|finite]] [[sum]] of multiplies of [[derivative of a distribution|derivatives]] of the delta distribution at that point:

$$
  u =
  \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq k } }{\sum}  c^\alpha \partial_\alpha \delta(p)
$$

where $\{c^\alpha \in \mathbb{R}\}_\alpha$,
and for $k \in \mathbb{N}$ the [[order of a distribution|order]] of $u$.

=--

(e.g. [Hörmander 90, theorem 2.3.4](#Hoermander90))

## Related entries

* [[Green's function]]

* [[Kronecker delta]]

* [[Dirac measure]]

## References

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

[[!redirects Dirac distribution]]
[[!redirects Dirac distributions]]
[[!redirects Dirac delta-distribution]]
[[!redirects Dirac delta-distributions]]
[[!redirects Dirac delta distribution]]
[[!redirects Dirac delta distributions]]
[[!redirects delta-distribution]]
[[!redirects delta-distributions]]
[[!redirects delta distribution]]
[[!redirects delta distributions]]

[[!redirects ∞-distribution]]
[[!redirects ∞-distributions]]
[[!redirects Dirac ∞-distribution]]
[[!redirects Dirac ∞-distributions]]

[[!redirects Dirac delta]]
[[!redirects Dirac ?]]

[[!redirects delta-function]]
[[!redirects delta-functions]]
[[!redirects delta function]]
[[!redirects delta functions]]
[[!redirects ∞-function]]
[[!redirects ∞-functions]]
[[!redirects Dirac delta-function]]
[[!redirects Dirac delta-functions]]
[[!redirects Dirac delta function]]
[[!redirects Dirac delta functions]]
[[!redirects Dirac ∞-function]]
[[!redirects Dirac ∞-functions]]

[[!redirects Dirac functional]]


Dirac equation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[differential equation]] encoded by a [[Dirac operator]].

The [[equations of motion]] of the [[Dirac field]].

## Related concepts

* [[Dirac spinor]], [[Dirac conjugate]]

* [[Dirac field]]

* [[heat equation]]

* [[wave equation]], [[Klein-Gordon equation]]

* [[Schrödinger equation]]

## References

Textbook account on the relativistic Dirac equation in 3+1d Minkowski spacetime:

* [[Bernd Thaller]], *The Dirac Equation*,  Texts and Monographs in Physics, Springer (1992) ([doi:10.1007/978-3-662-02753-0](https://link.springer.com/book/10.1007/978-3-662-02753-0))

See also:

* Wikipedia, _[Dirac equation](https://en.wikipedia.org/wiki/Dirac_equation)_

The [[path integral]] approach to the Dirac equation:

* Takashi Ichinose, Hiroshi Tamura, _Path Integral Approach to Relativistic Quantum Mechanics: Two-Dimensional Dirac Equation_, Progress of Theoretical Physics Supplement, Volume 92, April 1987, Pages 144–175, [doi](https://doi.org/10.1143/PTPS.92.144).

* Pierre Gosselin, Janos Polonyi, _Path Integral for Relativistic Equations of Motion_, [arXiv:hep-th/9708121](https://arxiv.org/abs/hep-th/9708121).

* Janos Polonyi, _Path Integral for the Dirac Equation_, [arXiv:hep-th/9809115](https://arxiv.org/abs/hep-th/9809115).

* Wataru Ichinose, _On the Feynman Path Integral for the Dirac Equation in the General Dimensional Spacetime_, Communications in Mathematical Physics 329, 83–508 (2014), [doi](https://doi.org/10.1007/s00220-014-1997-x).

* Wataru Ichinose, _Notes on the Feynman path integral for the Dirac equation_, Journal of Pseudo-Differential Operators and Applications 9, 789–809 (2018), [doi](https://doi.org/10.1007/s11868-017-0227-7).

The Dirac equation in a [[gravity|gravitational]] [[Schwarzschild spacetime]] [[background field|background]]:

* Paul M. Alsing, *Bound states of the Dirac equation in Schwarzschild spacetime: an exploration of intuition for the curious student* [[arXiv:2207.00905](https://arxiv.org/abs/2207.00905)]

[[!redirects Dirac equations]]