Chiara Nappi

* [webpage](https://phy.princeton.edu/people/chiara-nappi)

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

## Selected writings

On [[baryons]] as [[Skyrmions]]:

* {#AdkinsNappiWitten83} [[Gregory Adkins]], [[Chiara Nappi]], [[Edward Witten]], _Static Properties of Nucleons in the Skyrme Model_, Nucl. Phys. B228 (1983) 552 ([spire:190174](http://inspirehep.net/record/190174), doi:10.1016/0550-3213(83)90559-X)

Original proposal for inclusion of the [[ω-meson]] into the [[Skyrme model]]:

* {#AdkinsNappi84} [[Gregory Adkins]], [[Chiara Nappi]], _Stabilization of Chiral Solitons via Vector Mesons_, Phys. Lett. 137B (1984) 251-256 ([spire:194727](http://inspirehep.net/record/194727), doi:10.1016/0370-2693(84)90239-9)

The [[DBI action]] for single [[D-branes]]:

* A. Abouelsaood, [[Curtis Callan]], [[Chiara Nappi]], S. A. Yost, _Open strings in background gauge fields_, Nuclear Physics B Volume 280, 1987, Pages 599-624 (doi:10.1016/0550-3213(87)90164-7)

Proposal for a non-abelian [[DBI-action]] for [[intersecting brane|coincident]] [[D-branes]]:

* [[Philip Argyres]], [[Chiara Nappi]], _Spin-1 effective actions from open strings_, Nuclear Physics B Volume 330, Issue 1, 22 January 1990, Pages 151-173 Nuclear Physics B (doi:10.1016/0550-3213(90)90305-W)

category: people


child's drawing

# Children's drawings (Dessins d'enfants)
* automatic table of contents goes here
{: toc}

## Idea and prehistory

It is an old result that any (finite) [[graph]] can be embedded in an orientable [[surface]]. (The result can be found, for instance in [White 1984](#White1984), in which he give an algorithm, calling it the _Edmonds algorithm_ and giving a date of 1960, (see [Edmonds 1960](#Edmonds1960)).)

In 1879 [[Felix Klein]] (see [Klein's dessins d'enfants](#KleinsDessins)) in his paper "Über die Transformationen elfter Ordnung der elliptischen Funktionen" ("On the transformations of order eleven of elliptic functions" ) classified some [[Riemann surfaces]] of a certain [[ramification]] type by associating to [[covering space|covers]] of the surface with certain [[monodromy]] group their "dessins d'enfants" which he called "Linienzüge".

The modern perception of dessins d'enfants begins with [[Grothendieck]], who showed how to describe, [[category theory|category theoretically]], the geometric structure corresponding to such embedded graphs, namely by the [[action]] of a certain [[cartographic group]] on sets.  His motivation was the way in which this shed light on the structure of [[Riemann surfaces]], and through that to the [[absolute Galois group]] $G$ of the [[rational numbers]]. The technical vehicle explicating this relation is [[Belyi's theorem]] from which follows that $G$ has a faithful [[action]] on Grothendieck's dessins d'enfants.

> A _dessin d'enfant_ (French for a "child's drawing", plural "dessins d'enfants", ''children's drawings") is a type of [[graph]] drawing used to study [[Riemann surfaces]] and to provide combinatorial [[Invariant (mathematics)|invariants]] for the action of the [[absolute Galois group]] of the [[rational numbers]].

> "Intuitively, a dessin d'enfant is simply a [[undirected graph|graph]], with its [[vertex|vertices]] colored alternating black and white, [[graph embedding|embedded]] onto an [[Orientability|oriented surface]] that, in many cases, is simply a [[plane]]. For the coloring to exist, the graph must be [[bipartite graph|bipartite]]. The faces of the embedding must be topological disks. The surface and the embedding may be described combinatorially using a [[rotation system]], a [[cyclic order]] of the edges surrounding each vertex of the graph that describes the order in which the edges would be crossed by a path that travels clockwise on the surface in a small loop around the vertex. Any dessin can provide the surface it is embedded on with a structure as a Riemann surface. It natural to ask which Riemann surfaces arise in this way. The answer is provided by [[Belyi's theorem]], which states that these are precisely those that can be defined over the field of algebraic numbers (when considered as algebraic curves). The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins." (wikipedia)

The original development does not, however, explicitly use bipartite graphs, rather it used a form of 1-complex, embedded in a surface (see below). The use of bipartite graphs simplifies the exposition.

## Definition

A [[topological map]] is an embedding of a connected graph $G$ (loops and multiple edges allowed) into a compact oriented surface $X$, such that it fills the surface, i.e. $X \setminus G$ is a disjoint union of open cells (referred to as the _faces_ of the map). Two maps $(X_1 , G_1 )$ and $(X_2 , G_2 )$ are called _equivalent_ if there exists a homeomorphism $f : X_1 \to X_2$
such that $f (G_1 ) = G_2$ .

A **hypermap** is a map together with a coloring of the vertices in black and white such that any two adjacent vertices have different colors.  Observe that any map (with vertices all colored in black) gives rise to a hypermap by placing a white vertex in the middle of every edge—so, hypermaps can be seen as maps with extra information, but also conversely as a generalization of maps (see Walsh 1975, "Hypermaps Versus Bipartite Maps").

Finally, a **dessin d'enfant** is a hypermap seen as a representation of a _Belyi function_: a [[meromorphic function]] $f : X \to \bar{\mathbb{C} }$ whose only critical values are 0, 1, and $\infty$.  Such a function determines a hypermap on the Riemann surface $X$ by considering the inverse image of the unit interval $H = f^{-1}([0,1])$, and coloring the preimages of 0 and 1 as black and white vertices, respectively.  Moreover, each face of $H$ contains exactly one _pole_, that is, one preimage of $\infty$.

Note that it is also possible to associate a dessin more generally to any function $f : X \to \bar{\mathbb{C}}$ with at most three critical values, since any such function may be normalized to have critical values in $\{0,1,\infty\}$ by applying a [[linear fractional transformation]].  Likewise, the dessin can also equivalently be seen as a representation of a [[branched cover of the Riemann sphere]], ramified over at most three points.

## Examples

* Any polynomial $f : \bar{\mathbb{C}} \to \bar{\mathbb{C}}$ with no critical values other than 0 or 1 (and more generally, after normalization, any polynomial with at most two critical values) may be associated with a dessin d'enfant which is a one-face hypermap, i.e., a plane [[tree]].  (This is because any polynomial has a unique pole, at $\infty$.)  For example, the dessin d'enfant representing $f(x) = x^n$ is the "star-tree", containing one black vertex surrounded by $n$ white vertices.

* It is well known that the bipartite graph $K_{3,3}$ is non-planar, so does not embed in the 2-sphere, but will embed in the torus, in such a way that its complement is a disjoint union of discs.

##Remark

The original work  was stated in terms of objects called [[cellular maps]], and the category of these cellular maps and isotopy classes of morphisms between them was shown to correspond to a category of finite $\mathbb{G}$-sets where $\mathbb{G}$ is a group called the [[cartographic group]], which has a simple presentation in terms of operations of a geometric /  combinatorial nature.

## A translated extract

(from the [[Esquisse d'un programme]] describing the idea as seen by Grothendieck:)

"My interest in topological surfaces began to appear in 1974, when I proposed to Yves Ladegaillerie the theme of the isotopic study of embeddings
of a topological 1-complex into a compact surface. Over the two following
years, this study led him to a remarkable isotopy theorem, giving a complete algebraic description of the isotopy classes of embeddings of such
1-complexes, or compact surfaces with boundary, in a compact oriented
surface, in terms of certain very simple combinatorial invariants, and the
fundamental groups of the protagonists. This theorem, which should be
easily generalisable to embeddings of any compact space (triangulable to
simplify) in a compact oriented surface, gives as easy corollaries several
deep classical results in the theory of surfaces, and in particular Baer's isotopy theorem.

 In the work of Ladegaillerie there is also a purely algebraic description, in terms
of fundamental groups, of the "isotopic" category of compact surfaces X ,
equipped with a topological 1-complex K embedded in X . This description,
which had the misfortune to run counter to ''today's taste'' and because of
this appears to be unpublishable, nevertheless served (and still serves) as a
precious guide in my later reflections, particularly in the context of absolute  algebraic geometry in characteristic zero."

## Related concepts

* [[adinkra]]

## References

The Wikipedia page on this is:

* [wikipedia](http://en.wikipedia.org/wiki/Dessin_d'enfant)

Fairly elementary sources for background material include:

* {#White1984} Arthur T. White: 1984 _Graphs, groups and surfaces_

This gives the Edmonds algorithm for the embedding. the original source for that is:

* {#Edmonds1960} J. Edmonds (1960), A combinatorial representation for polyhedral surfaces, Notices Amer. Math. Soc. 7, 646.

References on the deeper theory as developed by Grothendieck and his students, and including summaries of subsequent work,  include:

*  [[L. Schneps]], ed., 1994, _The Grothendieck theory of dessins d'enfants_, volume 200 of London Mathematical Society Lecture Note Series , Cambridge University Press, Cambridge, papers from the Conference on Dessins d'Enfant held in Luminy, April 19–24, 1993

* Marco Robalo, _Galois Theory towards Dessins d'Enfants_ ([pdf](https://dspace.ist.utl.pt/bitstream/2295/575330/1/dissertacao.pdf))

*  F. Herrlich, G. Schmithüsen: [Dessins d'enfants and Origami curves](http://www.math.kit.edu/iag3/~schmithuesen/media/dessins.pdf)

The original work on the elementary theory was contained in

* [[Christine Voisin]], [[Jean Malgoire]], Cartes cellulaires, Cahiers Mathématiques, 12, Montpellier, 1977.

* [[Christine Voisin]], [[Jean Malgoire]], Factorisation de Stein topologique, découpe, Cahiers Mathématiques, 15, Montpellier, 1979.

* [[Christine Voisin]], [[Jean Malgoire]], Cartes topologiques infinies et revêtements ramifiés de la sphère, Cahiers Mathématiques, 19, Montpellier, 1980.

Other modern textbook accounts include:

* Sergei K. Lando, Alexander K. Zvonkin, _Graphs on Surfaces and Their Applications_, Springer, 2004.

* E. Girondo, G. González-Diez, _Introduction to compact Riemann surfaces and dessins d'enfants_,
London Mat. Soc. Student Texts __79__,
Cambridge Univ. Press 2012. xii+298 pp.

References on the history of dessins d'enfants

* Lieven Le Bruyn, Klein's dessins d'enfant and the buckyball, 2008. [web](http://www.neverendingbooks.org/index.php/kleins-dessins-denfant-and-the-buckyball.html){#KleinsDessins}

Connection to [[matrix model]]s and integrable hierarchies is discussed in

* Jan Ambjørn, Leonid Chekhov, _The matrix model for dessins d'enfants_, [arxiv/1404.4240](http://arxiv.org/abs/1404.4240)

The relation to [[super multiplet]] classification via [[adinkras]] is due to

* {#DoranIgaLandweberMendez-Diez13} [[Charles Doran]], Kevin Iga, [[Greg Landweber]], Stefan Mendez-Diez, _Geometrization of $N$-Extended 1-Dimensional Supersymmetry Algebras_ ([arXiv:1311.3736](https://arxiv.org/abs/1311.3736))

[[!redirects child's drawing]]
[[!redirects child's drawing]]
[[!redirects child's drawings]]
[[!redirects children's drawing]]
[[!redirects children's drawings]]
[[!redirects dessin d'enfant]]
[[!redirects dessins d'enfant]]
[[!redirects dessin d'enfants]]
[[!redirects dessins d'enfants]]
[[!redirects dessin d'enfant]]
[[!redirects dessins d'enfant]]
[[!redirects dessin d'enfants]]
[[!redirects dessins d'enfants]]


Ching-Kai Chiu

* [GoogleScholar entry](https://scholar.google.com/citations?user=I-1I1BsAAAAJ&hl=en)

## Selected writings

On [[topological semi-metals]]:

* {#ChiuSchnyder14} [[Ching-Kai Chiu]], [[Andreas P. Schnyder]], *Classification of reflection-symmetry-protected topological semimetals and nodal superconductors*, Phys. Rev. B **90** 205136 (2014) $[$[doi:10.1103/PhysRevB.90.205136](https://doi.org/10.1103/PhysRevB.90.205136)$]$

On ([[symmetry protected topological phase|symmetry protected]]) [[topological phases of matter]]:

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

category: people


chiral algebra

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#### Algebra
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#Contents#
* table of contents
{:toc}

## Idea

The syntagm *chiral algebra* has two meanings: one in physics and a related but more specific one after the work of Beilinson-Drinfeld. In QFT the chiral operator has two classes of eigenmodes left-handed
and right-handed; they give rise to left and right chiral fields. CFT in 2d, in particular has two chiral parts expressed via holomorphic and antiholomorphic functions in OPEs.

[[Beilinson]] and [[Drinfel'd]] were unhappy with non-illuminating definition of [[vertex operator algebra]] and invented a mathematical definition of version of chiral [[conformal field theory]] on an algebraic curve, under the name __chiral algebra__; their manuscript has being circulating from around 1995 as a long preprint and being more recently published by Amer. Math. Soc.

## Related concepts

* [[factorization algebra]]

* [[full field algebra]]

## References

The standard textbook is

* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _[[Chiral Algebras]]_, Colloqium Publications __51__,  Amer. Math. Soc. 2004, [gbooks](http://books.google.hr/books?id=yHZh3p-kFqQC&lpg=PP1&vq=%22Two-dimensional%20conformal%20geometry%20and%20vertex%20operator%20algebras%22&dq=isbn%3A0817638296&pg=PP1#v=onepage&q=%22Two-dimensional%20conformal%20geometry%20and%20vertex%20operator%20algebras%22&f=false)

An "example-driven digest" with applications in [[string theory]] is

* [[Dennis Gaitsgory]], _Notes on 2d conformal field theory and string theory_, from p. 1017 on in

  [[Pierre Deligne]], [[P. Etingof]], [[D. Freed]], L. Jeffrey, [[D. Kazhdan]], J. 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))

Generalization to [[higher algebra]] is discussed in

* [[Dennis Gaitsgory]], [[John Francis]] _Chiral Koszul duality_ ([arXiv:1103.5803](http://arxiv.org/abs/1103.5803))
 {#GaitsgoryFrancis}

Other references

* N. Rozenblyum, _Modules over a chiral algebra_, [arxiv/1010.1998](http://arxiv.org/abs/1010.1998)

[[!redirects chiral algebras]]


Chiral Algebras

This entry is about the book

* [[Alexander Beilinson]] and [[Vladimir Drinfeld]]

  _Chiral Algebras_, Amer. Math. Society 2004

  ([webpage](http://www.ams.org/publications/authors/books/postpub/coll-51))

  * [Contents](http://www.math.uchicago.edu/~mitya/langlands/chiral/cha_toc.pdf), [Introduction](http://www.math.uchicago.edu/~mitya/langlands/chiral/cha_intro.pdf), [Chapter 1](http://www.math.uchicago.edu/~mitya/langlands/chiral/cha_ch1.pdf), [Chapter 2](http://www.math.uchicago.edu/~mitya/langlands/chiral/cha_ch2.pdf), [Chapter 3](http://www.math.uchicago.edu/~mitya/langlands/chiral/cha_ch3.pdf), [Chapter 4](http://www.math.uchicago.edu/~mitya/langlands/chiral/cha_ch4.pdf), [References](http://www.math.uchicago.edu/~mitya/langlands/chiral/cha_ref_new.pdf), [Index](http://www.math.uchicago.edu/~mitya/langlands/chiral/cha_ind.pdf)

on [[chiral algebra]]s/[[factorization algebra]]s in the language of [[D-scheme]]s.

#Contents#
* table of contents
{:toc}

## Introduction

## 1. Axiomatic patterns

## 2. The Geometry of $\mathcal{D}$-Schemes

### 2.1 $\mathcal{D}$-modules: recollection and basics

* [[D-module]]

### 2.2 The compound tensor structure

### 2.3 $\mathcal{D}_X$-Schemes

* [[D-scheme]]

#### 2.3.1 $\mathcal{D}_X$-algebras

#### 2.3.2 Jet schemes

* [[jet bundle]]

#### 2.3.20 The calculus of variations

* [[variational calculus]]

  * [[Lagrangian]]

  * [[action functional]]

### 2.4 The space of horizontal sections

### 2.5 $Lie^\ast$-algebras and algebroids

### 2.6 Coisson algebras

* [[Coisson algebra]]s

### 2.7 The Tate extension

### 2.8 Tate structures and characteristic classes

#### 2.8.11

* [[vertical differential form]]

* [[variational bicomplex]]

### 2.9 The Harish-Chandra setting and the setting of $c$-stacks

## 3. Local theory: chiral basics

## 4. Global theory: chiral homology

## Related references

Much complementary content by the same authors is also in the developing book on the geometric Langlands program

* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _Quantization of the Hitchin system and Hecke eigensheaves_, draft, [pdf](http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf)

A goood accessible survey of or introduction into some of the developments in this book can be found around section 8.3

* [[Frédéric Paugam]], _Towards the mathematics of quantum field theory_ ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics-impa-v01-2011.pdf))

and section 3 of

* [[Frédéric Paugam]], _Histories and observables in covariant field theory_ Journal of Geometry and Physics (2010) ([arXiv](http://arxiv.org/abs/1010.3210))

with more details in

* [[Frédéric Paugam]], _Homotopical Poisson Reduction of gauge theories_ ([pdf](http://people.math.jussieu.fr/~fpaugam/documents/homotopical-poisson-reduction-of-gauge-theories.pdf))

In the context of [[factorization algebra of observables]]:

* [[Kevin Costello]], [[Owen Gwilliam]], _[[Factorization algebras in perturbative quantum field theory]]_

category: reference


chiral anomaly

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#### Physics
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* table of contents
{:toc}

## Idea

In [[quantum field theory]] with chiral [[fermions]] ([[spinor fields]]) $\psi$ with _chiral version of the [[Dirac current]]_ $J = i \bar \Psi \Gamma \Psi$, a _chiral anomaly_ is a non-[[conserved current|conservation]] of this current

$$
  div J = Anomaly
  \,.
$$

(See at _[[Ward identity]]_.)

In the [[standard model of particle physics]] this happens and plays a role for [[pion]] decay and for [[baryogenesis]]. Here the current is the [[baryon current]] and the anomaly term is the Pontryagin 4-form $Anomaly = \langle F_\nabla \wedge F_\nabla\rangle$ of the [[gauge field]] $\nabla$, hence the [[curvature]] 4-form of the corresponding [[Chern-Simons line 3-bundle]].

If there are [[instantons]], i.e. if the [[gauge field]] [[principal connection]] $\nabla$ has a nontrivial underlying [[principal bundle]], then also the [[Chern-Simons line 3-bundle]] is topologically nontrivial the anomaly term $\langle F_\nabla \wedge F_\nabla\rangle$ is a non-exact integral form, hence the above equation is to be read as the _local_ expression identifying $\star j$ with the local [[3-connection]] on the CS 3-bundle.

## Properties

### Relation between Chiral anomaly, Skyrme model, Baryon current and WZ-term

The "topological"-part $B_{top}$ of the [[baryon]] [[current]] (the piece that is not generally [[conserved current|conserved]], reflecting the [[chiral anomaly]]), is the [[Wess-Zumino-Witten term]] of the [[exponentiated pion field]]:

\[
  \label{ExponentiatedPionField}
  e^{i \vec \pi/f_\pi}
  \;\colon\;
  \mathbb{R}^{0,1} \times (\mathbb{R}^3)^{cpt}
  \;=\;
  \mathbb{R}^{0,1} \times S^3
  \longrightarrow
  S^3
  \simeq
  SU(2)
\]

$$
  \begin{aligned}
  B_{top}
  & \coloneqq \;
  Tr
  \big(
    ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} )
    \wedge
    ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} )
    \wedge
    ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} )
  \big)
  \\
  & =\;
  \big\langle
    (e^{i \vec \pi/f_\pi})^\ast(\theta)
    \wedge
    (e^{i \vec \pi/f_\pi})^\ast(\theta)
    \wedge
    (e^{i \vec \pi/f_\pi})^\ast(\theta)
  \big\rangle
  \;\;\in\;
  \Omega^3(\mathbb{R}^{3,1})
  \end{aligned}
$$

Here the expression in the first line uses the fact that [[SU(2)]] is a [[matrix group]], while the second line exporesses the same via [[pullback of differential forms|pullback]] of the [[Maurer-Cartan form]] $\theta$ from the group manifold.

The [[homotopy class]] of the [[exponentiated pion field]] (eq:ExponentiatedPionField), as a [[continuous function]], is an element of the ([[cohomotopy|co-]])[[homotopy group of spheres]] $\pi_3(S^3) \simeq \pi^3(S^3) \simeq \mathbb{Z}$, is the [[Skyrmion]]-number, or, in fact, the [[baryon]]-number, encoded in the [knotted stucture](On+Vortex+Atoms#RanadaTrueba01) of the pion field.

\linebreak

See also [physics.stackexchange.com/a/306242/5603](https://physics.stackexchange.com/a/306242/5603)

## Related concepts

* [[chiral perturbation theory]]

## References

### General

The orginal observation is due to

* [[Stephen Adler]]. _Axial-Vector Vertex in Spinor Electrodynamics_, Physical Review 177 (5): 2426. (1969)

* [[John Bell]], [[Roman Jackiw]], _A PCAC puzzle: $\pi^0 \to \gamma \gamma$ in the σ-model". Il Nuovo Cimento A 60: 47. (1969)

A detailed mathematical derivation is in

* [[Luis Alvarez-Gaumé]], [[Paul Ginsparg]], section 3 of: _The structure of gauge and gravitational anomalies_ , Ann. Phys. 161 (1985) 423. ([spire](http://inspirehep.net/record/202565/?ln=en))

See also:

* Valentin Benedetti, Horacio Casini, Javier M. Magan, *ABJ anomaly as a $U(1)$ symmetry and Noether's theorem* [[arXiv:2309.03264](https://arxiv.org/abs/2309.03264)]

Detailed argument for the [[theta vacuum]] ([[Yang-Mills instanton]] vacuum) from [[chiral symmetry breaking]] :

* [[Curtis Callan]], R.F. Dashen, [[David Gross]], _The Structure of the Gauge Theory Vacuum_, Phys.Lett. 63B (1976) 334-340 ([spire](http://inspirehep.net/record/3673?ln=en))

* G. Morchio, [[Franco Strocchi]], _Chiral symmetry breaking and theta vacuum structure in QCD_, Annals Phys. **324** (2009) 2236-2254 [[arXiv:0907.2522](https://arxiv.org/abs/0907.2522), [doi:10.1016/j.aop.2009.07.005](https://doi.org/10.1016/j.aop.2009.07.005)]

Textbook account:

* [[Robert E. Marshak]], Chapter 7 of: *Conceptual Foundations of Modern Particle Physics*, World Scientific 1993 ([doi:10.1142/1767](https://doi.org/10.1142/1767))

Further review:

* [[Raphael Flauger]], _Anomalies and the Atiyah-Singer Index Theorem_ ([pdf](http://www.ma.utexas.edu/~dafr/Index/Flauger.pdf))

* [[Jeffrey Harvey]], section 1 and 3.6 of _TASI 2003 Lectures on Anomalies_ ([arXiv:hep-th/0509097](http://arxiv.org/abs/hep-th/0509097))

* B.L.Ioffe, _Axial anomaly: the modern status_, Int. J. Mod. Phys. A21:6249-6266,2006 ([arXiv:hep-ph/0611026](http://arxiv.org/abs/hep-ph/0611026))

* {#Jackiw08} [[Roman Jackiw]], _Axial anomaly_, (2008), Scholarpedia, 3(10):7302. ([web](http://www.scholarpedia.org/article/Axial_anomaly))

* Wikipedia, _[Chiral anomaly](http://en.wikipedia.org/wiki/Chiral_anomaly)_

Discussion in the rigorous context of [[causal perturbation theory]]/[[perturbative AQFT]] is (for $m \gt 0$) in

* {#Scharf95} [[Günter Scharf]], section 5.3 of _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

and (for $m = 0$) in

* {#Duetsch92} [[Michael Dütsch]], F. Krahe, [[Günter Scharf]], _Axial Anomalies in Massless Finite QED_, N. Cimento A 105 (3) (1992), 399–422.

and reviewed in the context the [[master Ward identity]] in

* [[Michael Dütsch]], around p. 331 in _[[From classical field theory
to perturbative quantum field theory]]_, 2018

Application to [[baryogenesis]] is due to

* {#Hooft76} [[Gerard 't Hooft]], _Symmetry Breaking through Bell-Jackiw Anomalies_ Phys. Rev. Lett. 37 (1976) ([pdf](http://www.staff.science.uu.nl/~hooft101/gthpub/symm_br_bell_jackiw.pdf))

* [[Gerard 't Hooft]], _Computation of the quantum effects due to a four-dimensional pseudoparticle_, Phys. Rev. D14:3432-3450 (1976).

[[!include WZW term of QCD chiral perturbation theory -- references]]

[[!redirects chiral anomalies]]

[[!redirects axial anomaly]]
[[!redirects axial anomalies]]


chiral de Rham complex

#Contents#
* table of contents
{:toc}

## Idea

A [[sheaf of vertex operator algebras]] (in fact a [[vertex operator algebroid]]) which is naturally associated with a [[complex manifold]] whose first and second [[Chern class]] vanishes.

## Properties

### Fine resolution

The chiral de Rham complex [[sheaf of vertex operator algebras]] as a [[resolution]] to a [[fine sheaf]] by the _[[chiral Dolbeault complex]]_ ([Cheung 10](#Cheung10)).

### Relation to 2d (2,0)-superconformal QFT

The chiral de Rham complex of $X$ arises as the [[quantum observables]] of the [[topological twist|topologically twisted]] [[2d (2,0)-superconformal QFT]] [[sigma-model]] with [[target space]] $X$.

Under suitable geometric conditions (a version of [[string structure]]) the local chiral de Rham complexes glue together to a [[sheaf of vertex operator algebras]] and serves to compute the [[Witten genus]].

### Related concepts

*  [[chiral differential operator]], [[vertex operator algebra]], [[chiral algebra]]

* [[Berkovits superstring]]

* [[formal loop space]]

## References

* [[Vassily Gorbounov]], [[Fyodor Malikov]], [[Vadim Schechtman]], _Gerbes of chiral differential operators_  Math. Res. Lett. __7__ (2000),  no. 1, 55--66, [MR2002c:17040](http://www.ams.org/mathscinet-getitem?mr=2002c:17040), [math.AG/9906117](http://arxiv.org/abs/math/9906117); _Gerbes of chiral differential operators. II. Vertex algebroids_, Invent. Math. __155__ (2004), no. 3, 605--680, [MR2005e:17047](http://www.ams.org/mathscinet-getitem?mr=2005e:17047), [math.AG/0003170](http://arxiv.org/abs/math/0003170), [doi](http://dx.doi.org/10.1007/s00222-003-0333-4); _Gerbes of chiral differential operators. III_, in: The orbit method in geometry and physics (Marseille, 2000),  73--100, Progr. Math. __213__, Birkhäuser 2003, [MR2005a:17028](http://www.ams.org/mathscinet-getitem?mr=2005a:17028), [math.AG/0005201](http://arxiv.org/abs/math/0005201), _On chiral differential operators over homogeneous spaces_, Int. J. Math. Math. Sci. __26__ (2001), no. 2, 83--106, [MR2002g:14020](http://www.ams.org/mathscinet-getitem?mr=2002g:14020), [math.AG/0008154](http://arxiv.org/abs/math/0008154), [doi](http://dx.doi.org/10.1155/S0161171201020051)

The resolution by the [[chiral Dolbeault complex]] is due to

* {#Cheung10} Pokman Cheung, _The Witten genus and vertex algebras_ ([arXiv:0811.1418](http://arxiv.org/abs/0811.1418))

* [[Urs Schreiber]] at string cafe:

  * _[Nekrasov on pure spinor superstring](http://golem.ph.utexas.edu/string/archives/000661.html)_

  * _[Sheaves of CDOs](http://golem.ph.utexas.edu/string/archives/000668.html)_

* Yuly Billig, Vyacheslav Futorny, _Representations of Lie algebra of vector fields on a torus and chiral de Rham complex_, [arxiv/1108.6092](http://arxiv.org/abs/1108.6092)

* A. Linshaw, [[Varghese Mathai]], _Twisted chiral de Rham complex, generalized geometry, and T-duality_, Commun. Math. Phys. __339__ (2015) 663--697 [doi](https://doi.org/10.1007/s00220-015-2403-z)

* [[Mikhail Kapranov]], [[Eric Vasserot]], _Vertex algebras and formal loop space_, Publ. Math., Inst. Hautes Étud. Sci. 100 (2004) 209--269 (2004) [doi](https://doi.org/10.1007/s10240-004-0023-9) [arXiv:math/0107143](https://arxiv.org/pdf/math/0107143.pdf)

> We construct a certain algebro-geometric version $\mathcal{L}(X)$ of the free loop space for a complex algebraic variety $X$.
This is an ind-scheme containing the scheme $\mathcal{L}_0(X)$ of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on $\mathcal{L}(X)$ supported in $\mathcal{L}_0(X)$. We also show that $\mathcal{L}(X)$ possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic principle that “all” linear constructions applied to the free loop space produce vertex algebras.


chiral differential operator

#Contents#
* table of contents
{:toc}

## Idea

(...) see at _[[chiral de Rham complex]]_ (...)

## Properties

### Relation to the Witten genus
 {#RelationToTheWittenGenus}

For $U \subset \mathbb{C}$ an [[open subset]] of the [[complex plane]] then the space $\mathcal{D}^{ch}(U)$ of chiral differential operators on $U$ is naturally a [[super vertex operator algebra]]. For $X$ a [[complex manifold]] such that its [[first Chern class]] and [[second Chern class]] vanish over the [[rational numbers]], then this assignment gives a [[sheaf of vertex operator algebras]] $\mathcal{D}^{ch}_X(-)$ on $X$. Its [[cochain cohomology]] $H^\bullet(\mathcal{D}^{ch}_X)$ is itself a [[super vertex operator algebra]] and its super-[[Kac-Weyl character]] is proportional to the [[Witten genus]] $w(X)$ of $X$:

$$
  char H^\bullet(\mathcal{D}^{ch}_X)\propto w(X)
  \,.
$$

Physically this result is understood by observing that $\mathcal{D}^{ch}_X$is the sheaf of [[quantum observables]] of the [[topological twist|topologically twisted]] [[2d (2,0)-superconformal QFT]] (see there for more on this) of which the [[Witten genus]] is (the [[large volume limit]] of) the [[partition function]].

See ([Cheung 10](#Cheung10)) for a brief review (where the problem of generalizing of this construction to [[sheaves of vertex operator algebras]] over more general [[string structure]] manifolds is addressed.)

### Relation to formal loop space
 {#RelationToFormalLoopSpace}

With $X$ a suitable scheme, its [[formal loop space]] $L_inf X$ in the sense of ([Kapranov-Vasserot I](#KapranovVasserotI)) has a [[Tate structure]] and hence an associated [[determinantal gerbe]] $Det_{L_{inf} X}$ with [[band]] $\mathcal{O}^\ast_{L_{inf} X}$. According to
([Kapranov-Vasserot IV](#KapranovVasserotIV)) this [[gerbe]] is essentially identified with the [[gerbe]] $CDO_X$ of chiral differential operators on $X$.

## Related concepts

* [[chiral de Rham complex]], [[vertex operator algebra]], [[chiral algebra]]

* [[formal loop space]]

## References

The original articles are

* {#GMS} [[Vassily Gorbounov]], [[Fyodor Malikov]], [[Vadim Schechtman]], _Gerbes of chiral differential operators_  Math. Res. Lett. __7__ (2000),  no. 1, 55--66, [MR2002c:17040](http://www.ams.org/mathscinet-getitem?mr=2002c:17040), [math.AG/9906117](http://arxiv.org/abs/math/9906117); _Gerbes of chiral differential operators. II. Vertex algebroids_, Invent. Math. __155__ (2004), no. 3, 605--680, [MR2005e:17047](http://www.ams.org/mathscinet-getitem?mr=2005e:17047), [math.AG/0003170](http://arxiv.org/abs/math/0003170), [doi](http://dx.doi.org/10.1007/s00222-003-0333-4); _Gerbes of chiral differential operators. III_, in: The orbit method in geometry and physics (Marseille, 2000),  73--100, Progr. Math. __213__, Birkhäuser 2003, [MR2005a:17028](http://www.ams.org/mathscinet-getitem?mr=2005a:17028), [math.AG/0005201](http://arxiv.org/abs/math/0005201), _On chiral differential operators over homogeneous spaces_, Int. J. Math. Math. Sci. __26__ (2001), no. 2, 83--106, [MR2002g:14020](http://www.ams.org/mathscinet-getitem?mr=2002g:14020), [math.AG/0008154](http://arxiv.org/abs/math/0008154), [doi](http://dx.doi.org/10.1155/S0161171201020051)

Surveys and further developments include

* Andrew R. Linshaw, _Introduction to invariant chiral differential operators_, in: Vertex operator algebras and related areas,  157--168, Contemp. Math. __497__, Amer. Math. Soc. 2009; _Invariant chiral differential operators and the $W_3$ algebra _, J. Pure Appl. Algebra __213__ (2009), 632-648, [arxiv/0710.0194](http://arxiv.org/abs/0710.0194), [MR2010b:17035](http://www.ams.org/mathscinet-getitem?mr=2010b:17035), [doi](http://dx.doi.org/10.1016/j.jpaa.2008.08.006)

* {#Cheung10} [[Pokman Cheung]], _Chiral differential operators and topology_, [arxiv/1009.5479](http://arxiv.org/abs/1009.5479)

The relation to [[formal loop space]] geometry is discussed in

* {#KapranovVasserotI} [[Mikhail Kapranov]], E. Vasserot, _Vertex algebras and the formal loop space_ Publications Mathématiques de l'IHÉS,
100 (2004), 209–269. ([arXiv:math/0107143](http://arxiv.org/abs/math/0107143))

* [[Mikhail Kapranov]], E. Vasserot, _Formal Loops II : the local Riemann-Roch theorem for determinantal gerbes_, Ann. Sci. ENS, ([arXiv:math/0509646](http://arxiv.org/abs/math/0509646))

* [[Mikhail Kapranov]], E. Vasserot, _Formal loops III: Factorizing functions and the Radon transform_ ([arXiv:math/0510476](http://arxiv.org/abs/math/0510476))

* {#KapranovVasserotIV} [[Mikhail Kapranov]], E. Vasserot, _Formal loops IV: Chiral differential operators_ ([arXiv:math/0612371](http://arxiv.org/abs/math/0612371))

Tentative aspects of a generalization to [[differential geometry]] are discussed in

* [[Pokman Cheung]], _Chiral differential operators: formal loop group actions and associated modules_ ([arXiv:1205.7089](http://arxiv.org/abs/1205.7089))

[[!redirects chiral differential operators]]


chiral Dolbeault complex

#Contents#
* table of contents
{:toc}

## Idea

The [[chiral de Rham complex]] [[sheaf of vertex operator algebras]] of a [[complex manifold]] admits a [[resolution]] by a [[fine sheaf]] which in degree-0 is the Dolbeault-resolution of the [[structure sheaf]]. This is the  _chiral Dolbeault complex_ ([Cheung 10](#Cheung10)).

Where the construction of the [[chiral de Rham complex]] as a [[sheaf of vertex operator algebras]] requires the complex analog of a [[string structure]] in that the [[second Chern class]] vanishes, the construction of the chiral Dolbeault complex requires an analog of a [[differential string structure]] where a differential 3-form is chosen such that $\mathbf{d}H \propto \langle R \wedge R\rangle$.

## Related concepts

* [[Witten genus]]

## References

* {#Cheung10} Pokman Cheung, _The Witten genus and vertex algebras_ ([arXiv:0811.1418](http://arxiv.org/abs/0811.1418))


chiral fermion

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#### Physics
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[[!include physicscontents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

A [[fermion]] in a chiral [[spin representation]].

A key feature of the [[standard model of particle physics]] is that its fermions are chiral, in that the left and right [[Weyl spinor]]-components of the would-be [[Dirac spinor]]-representation of the [[quarks]], [[electrons]] and [[neutrinos]] couple _differently_ to the [[gauge fields]] (for review see e.g. [Ibanez-Uranga 12, section 1.1](#IbanezUranga12)).

Possible realization of chiral fermions in [[string phenomenology]] is given by [[intersecting D-brane models]] ([here](intersecting+D-brane+model#ChiralFermions)) lifting to [[M-theory on G2-manifolds]] ([here](M-theory+on+G2-manifolds#EnhancedGaugeGroups)) at [[ADE-singularities]]

## Related entries

* [[standard model of particle physics]]

* [[quantum anomaly]]

## References

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge 2012

On [[chiral fermions]] in [[intersecting D-brane models]]:

* {#BerkoozDouglasLeigh96} [[Micha Berkooz]], [[Michael Douglas]], [[Robert Leigh]], _Branes Intersecting at Angles_, Nucl.Phys.B480:265-278, 1996 ([arXiv:hep-th/9606139](https://arxiv.org/abs/hep-th/9606139))

[[!redirects chiral fermions]]


Chiral Nuclear Dynamics

This page collects material related to the book

* [[Maciej Nowak]], [[Mannque Rho]], [[Ismail Zahed]],

  _Chiral Nuclear Dynamics_,

  World Scientific 1996

  [doi:10.1142/1681](https://doi.org/10.1142/1681)

on [[nuclear physics]] via [[chiral perturbation theory]] ([[quantum hadrodynamics]]).

#Contents#
* table of contents
{:toc}

## ...

## Chapter 7 -- Chiral solitons

* [[Skyrmion]]

* [[spin-isospin transmutation]]

## ...

## Related entries

* [[Mannque Rho]], [[Ismail Zahed]] (eds.)

  _The Multifaceted Skyrmion_,

  World Scientific

  Second edition, 2016

  [doi:10.1142/9710](https://doi.org/10.1142/9710)

on [[Skyrmions]], particularly as a model for [[confinement]] in [[QCD]] of [[quarks]] within [[hadrons]] (but also in [[solid state physics]]) and their origin as [[instantons]] of a [[AdS/CFT|holographic]] 5d [[Yang-Mills theory]] ([[holographic QCD]], [[AdS/QCD correspondence]]).

category: reference


chiral partner

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#Contents#
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{:toc}

## Idea

In [[chiral perturbation theory]]/[[quantum hadrodynamics]] the _chiral partner_ of a [[meson]] is that with the same [[Wigner classification]] except for parity: The chiral partner of a [[Lorentz group|Lorentz]] [[scalar]] is a [[pseudoscalar]] and that of a [[vector representation]] a [[pseudovector representation]].

## References

Survey for [[rho-meson]]/[[a1-meson]] and [[omega-meson]]/[[f1-meson]]:

* Su Houng Lee, _Observing chiral partners in nuclear  medium_ ([[LeeChiralPartners.pdf:file]])

Specifically on the [[rho-meson]]/[[a1-meson]] [[chiral partner]]-ship:

* [[Robert Pisarski]], _Where does the Rho Go? Chirally Symmetric Vector Mesons in the Quark-Gluon Plasma_, Phys. Rev. D52 (1995) 3773-3776 ([arXiv:hep-ph/9503328](https://arxiv.org/abs/hep-ph/9503328))

* [[Stefan Leupold]], Markus Wagner, _Chiral Partners in a Chirally Broken World_, International Journal of Modern Physics AVol. 24, No. 02n03, pp. 229-236 (2009) ([arXiv:0807.2389](https://arxiv.org/abs/0807.2389))


chiral perturbation theory

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[[!include AQFT and operator algebra contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _[[chiral perturbation theory]]_ in [[quantum field theory]] of [[nuclear physics]] is the [[effective field theory]] of [[quantum chromodynamics]] in the [[confinement|confined]] sector, where the [[effective field theory|effective]] [[field (physics)|fields]] are [[hadrons]]:

* [[light meson|light]]$\;$[[scalar mesons]] appear as the [[Goldstone bosons]] of the [[spontaneous symmetry breaking|spontaneously broken]] [[chiral symmetry]] (see below),

* [[vector mesons]] appear as [[gauge fields]] of a [[hidden local symmetry]],

* [[baryons]] (specifically [[nuclei]]) appear as [[solitons]] in the meson-fields (called [[Skyrmions]]),

* [[heavy mesons]] appear as further [[field (physics)|fields]].

Hence some authors also speak of _[[quantum hadrodynamics]]_.

More concretely, chiral perturbation theory is [[perturbation theory]] not in the [[coupling constant]] of [[QCD]], but in the [[masses]] of the [[light quarks]], which in practice means: of the [[up quark]] and the [[down quark]] and possibly also of the [[strange quark]], with masses $m_u$, $m_d$ and $m_s$, respectively.

As such, the point about which [[perturbation theory|perturbations]] are considered in chiral perturbation theory of QCD is that of vanishing [[quark]] masses, hence the [[limit of a sequence|limit]] $ m_u, m_s, m_d \rightarrow 0 $.

{#ProtonMass} Notice that the [[proton]] [[mass]] (the characteristic scale of [[QCD]] witnessing its [[confinement]]/[[mass gap]]-property, see e.g. [Roberts 21](proton#Roberts21)) is much larger than the [[sum]] of its [[constituent quarks]]

$$
  m_p \gg 2 m_u + m_d
  \,,
$$

suggesting that $m_u, m_d \to 0$ is a good limit around which to inspect the [[hadron|hadronic]] [[bound states]] of [[QCD]].

{#LagrangianInCPTLimit} But since only the quark mass term in the [[Lagrangian density]] of [[QCD]] mixes the two [[Weyl spinor|chiralities]] of quarks, its [[Lagrangian density]] in this massless limit is the sum of a term that only contains the left-chiral quark spinors, and an analogous summand that only contains the right-chiral quark spinors (e.g. [Ecker 95, (2.1)](#Ecker95)):

$$
  \mathcal{L}^0_QCD
  \;=\;
  \sum_{l=u,d,s}
  \big(
    \bar{q}^l_L i (\slash{D} + \slash{A}) q^l_L
    +
    \bar{q}^l_R i (\slash{D} + \slash{A}) q^l_R
  \big)
  -
  \frac{1}{4}G_{a \mu v}G^{\mu v}_a
$$

(here $q$ denotes the [[quark]] [[Dirac field]] with left and right [[chiral fermion|chiral]] [[Weyl spinor]] components $q_L$ and $q_R$, $\slash{D}$ denotes the [[Dirac operator]], $A$ the [[gluon]] [[field (physics)|field]] (in [[Feynman slash notation]]), and $G$ the [[field strength]] of the [[gluon]] [[field]] $A$ -- see at *[[Yang-Mills theory]]* for more).

Moreover, all two ([[up quark|up]] and [[down quark|down]]) or even all three (if including [[strange quark|strange]]) [[flavour (particle physics)|flavours]] of quarks enter each of these two spinor-chiral summands symmetrically, such that each of them has [[SU(2)]] or respectively [[SU(3)]] [[global symmetry]], [[action|acting]] by mixing the quark flavours. This yields a total [[direct product group|direct product]] [[symmetry group]]

$$
  G = SU(N_f)_L \times SU(N_f)_R
  \,,
$$

where $N_f \in \{ 2,3 \}$ is the [[number]] of [[flavour (particle physics)|flavours]] of [[light quarks]] that are being considered, and where the subscripts indicate the left/right-handed [[chiral spinor]]-components of quarks whose [[flavour (particle physics)|flavours]] are being mixed by these [[group]] [[actions]]. This [[symmetry group]] is hence also called _chiral symmetry_. Accordingly, its [[symmetry breaking]] to the [[diagonal morphism|diagonal]] [[subgroup]]

$$
  SU(N_f)_V
  \;\subset\;
  SU(N_f)_L \times SU(N_f)_R
$$

(the subscript "$V$" is for "vector") either explicitly by [[positive number|positive]] [[quark]] [[masses]] or [[spontaneous symmetry breaking|spontaneously]] by a [[vacuum expectation value]] $\langle \bar q q\rangle = \langle \bar q_L q_R\rangle + \langle \bar q_R q_L\rangle$, is called _[[chiral symmetry breaking]]. This is the origin of the term _chiral perturbation theory_.

Additionally, the application of [[noether's theorem]] to the classical global symmetry of the lagrangian density in the chiral limit yields $ 2 \times (8+1) = 18 $ conserved currents, via the method of [[Gell-Mann]] and [[Levy]]. Namely, the promotion of the [[global symmetry]] inherent in the lagrangian density to a [[local symmetry]] allows us to identify these conserved [[noether currents]]

$$
  J^\mu_a = \frac{\partial \delta \mathcal{L}}{\partial \partial_\mu \epsilon_a}
  \partial_{\mu} J^{\mu}_a = \frac{\partial \delta \mathcal{L}}{\partial \epsilon_a}
$$

(...)

## Properties

### Relation to the Skyrme model

* [[Dmitri Diakonov]], [[Victor Petrov]], _Exotic baryon resonances in the Skyrme model_ ([arXiv:0812.1212](https://arxiv.org/abs/0812.1212), [doi:10.1142/9789814704410_0004](https://doi.org/10.1142/9789814704410_0004)), Chapter 3 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

> It is astounding that [[Tony Skyrme|Skyrme]] had suggested [[Skyrmion|his model]] as early as in 1961 before it has been generally accepted that [[pions]] are (pseudo) [[Goldstone bosons]] associated with the [[spontaneous symmetry breaking|spontaneous breaking]] of [[chiral symmetry]], and of course long before [[QCD|Quantum Chromodynamics (QCD)]] has been put forward as the microscopic theory of [[strong nuclear force|strong interactions]].

> The revival of the Skyrme idea in 1983 is [due to Witten](Skyrmion#Witten83) who explained the _raison d'ˆetre_ of the [[Skyrmion|Skyrme model]] from the viewpoint of [[QCD]]. In the [[chiral perturbation theory|chiral limit]] when the [[light quark]] masses $m_u$, $m_d$, $m_s$ tend to zero, such that the octet of the pseudoscalar mesons [[pion|π]], [[kaon|K]] , η become nearly massless (pseudo) [[Goldstone bosons]], they are the lightest degrees of freedom of [[QCD]]. The [[effective field theory|effective]] chiral Lagrangian (EχL) for pseudoscalar mesons, understood as an infinite expansion in the derivatives of the pseudoscalar (or chiral) fields, encodes, in principle, full information about QCD. The famous two-term Skyrme Lagrangian can be understood as a low-energy truncation of this infinite series. Witten has added an important four-derivative [[WZW-term|Wess–Zumino term]] to the original Skyrme Lagrangian and pointed out that the overall coefficient in front of the EχL is proportional to the number of quark [[color charge|colours]] $N_c$.

> $[...]$

> Soon after Witten's work it has been realized that it is possible to bring the Skyrme model and the Skyrmion even closer to QCD and to the more customary language of constituent quarks. It has been first noticed $[$[6](#DiakonovEides83), [7a](#DharWadia84), [7b](#DarShankarWadia85), [8](#DiakonovPetrov86)$]$ that a simple chiral invariant Lagrangian for massive (constituent) quarks $Q$ interacting with the octet chiral field $\pi_A$ $(A = 1, ..., 8)$,

> $$\mathcal{L} = \overline{Q} \left( \partial\!\!\!\!/ - M e^{ \tfrac{i \pi^A \lambda^A \gamma_5}{F_\pi}  }  \right) Q$$

> induces, via a quark loop in the external pseudoscalar fields (see Fig. 3.1), the EχL whose lowest-derivative terms coincide with the Skyrme Lagrangian, including
automatically the Wess–Zumino term, with the correct coefficient!

> $[...]$

> The condition that the winding number of the trial field is unity needs to be imposed to get a deeply bound state, that is to guarantee that the baryon number is unity. $[$[9](#DiakonovPetrovPobylitsa88)$]$ The Skyrmion is, thus, nothing but the **mean chiral field binding quarks in a baryon**.

\linebreak

* {#DiakonovEides83} [[Dmitri Diakonov]], Michael I. Eides, _Chiral Lagrangian from a functional integral over quarks_, JETP Letters 38.7 (1983): 433-436 ([pdf](http://jetpletters.ac.ru/ps/1483/article_22635.pdf), [[DiakonovEides83.pdf:file]])

* {#DharWadia84} A. Dhar, Spenta R. Wadia, _Nambu—Jona-Lasinio Model: An Effective Lagrangian for Quantum Chromodynamics at Intermediate Length Scales_, Phys. Rev. Lett. 52, 959 (1984) ([doi:10.1103/PhysRevLett.52.959](https://doi.org/10.1103/PhysRevLett.52.959))

* {#DarShankarWadia85} Avinash Dhar, R. Shankar, Spenta R. Wadia, _Nambu–Jona-Lasinio–type effective Lagrangian: Anomalies and nonlinear Lagrangian of low-energy, large-N QCD_, Phys. Rev. D 31, 3256 (1985) ([doi:10.1103/PhysRevD.31.3256](https://doi.org/10.1103/PhysRevD.31.3256))

* {#DiakonovPetrov86} [[Dmitri Diakonov]], [[Victor Petrov]], _A theory of light quarks in the instanton vacuum_, Nuclear Physics B Volume 272, Issue 2, 21 July 1986, Pages 457-489 (doi:10.1016/0550-3213(86)90011-8)

* {#DiakonovPetrovPobylitsa88} [[Dmitri Diakonov]], [[Victor Petrov]], P.V. Pobylitsa, _A Chiral Theory of Nucleons_, Nucl. Phys. B306 (1988) 809 ([spire:247700](http://inspirehep.net/record/247700), doi:10.1016/0550-3213(88)90443-9)

### Baryon Chiral Perturbation Theory

The QCD vacuum breaks chiral symmetry down to the diagonal subgroup of SU(3), $U(3)_v$, whereby 8 massless [[Goldstone boson | goldstone bosons]] appear, each of which are coupled via $F_0$ to the conserved axial-vector current. The physics of these [[Goldstone boson | goldstone bosons]], which are themselves the pion fields, describe the low energry [[effective field theory]] known as [[chiral perturbation theory]]. In this low energy structure of QCD, heavy quarks do not play a role since their degrees of freedom are frozen at low energies. However, when the baryon fields are treated as heavy static [[fermions]] in the expansion of the physical vacuum state of QCD, one can write the effective theory in terms of baryon fields with a definite velocity, $B_v$. In the heavy baryon limit, the most general Lagrangian at leading-order is
$$
  L_v^0 = iTr\overline{B}_v(v*\mathscr{D})B_v + 2DTr\overline{B}_v S_v^\mu \{A_\mu,B_v\} + 2FTr\overline{B}_v S_v^\mu \{A_\mu,B_v\} + \frac{1}{4}f^2Tr\partial_{\mu}\Sigma\partial^\mu \Sigma^\dagger + aTrM(\Sigma + \Sigma^\dagger)
$$

### Partially Quenched Chiral Perturbation Theory

In [[Lattice QCD]], it is notoriously difficult to include the loops of [[light quarks]]. One remedy to this problem is to approximate the fermion determinant of a Dirac operator $D$, which is proportional to the summation of external sources interacting with one internal [[fermion loop]]. The partially quenched prescription includes the determinant but with [[sea quark]] masses $\ggg$ those of the valence quarks. The simulation of partially quenched QCD (PQQCD) on the lattice can yield information about QCD itself by utilizing [[chiral perturbation theory]]. The formulation of XPT in PQQCD is difficult compared to the unquenched case (standard formulation) due to the lack of a physical [[hilbert space]].

* S.R.Sharpe and N.Shoresh, _Partially quenched chiral perturbation theory without Phi0_, Phys. Rev. D 64, 114510 (2001), doi:10.1103/PhysRevD.64.114510, ([arXiv:hep-lat/0108003](https://arxiv.org/abs/hep-lat/0108003v1))

* Claude W. Bernard and Maarten F. L. Golterman, _Partially quenched gauge theories and an application to staggered fermions_, Phys. Rev. D 49, 486 – Published 1 January 1994, ([doi:10.1103/486] (https://doi.org/10.1103/PhysRevD.49.486))

### Random Matrix Theory and Chiral Symmetry

It has been posited that the statistics of the low-lying eigenvalues of the QCD [[Dirac operator]] can be described by a [[random matrix theory]] with the global symmetries of the QCD [[partition function]].

* J.J.M. Verbaarschot and T.Wettig, _Random matrix theory and chiral symmetry in QCD_, Ann. Rev. Nucl. Part. Sci. 50, 343-410 (2000), doi:10.1146/annurev.nucl.50.1.343, ([arXiv:hep-ph/0003017 [hep-ph]](https://arxiv.org/abs/hep-ph/0003017v2)).

* J.C. Osborn, D.Toublan and J.J.M.Verbaarschot, _From chiral random matrix theory to chiral perturbation theory_, Nucl. Phys. B 540, 317-344 (1999), doi:10.1016/S0550-3213(98)00716-0 ([arXiv:hep-th/9806110](https://arxiv.org/abs/hep-th/9806110v2)).

## Related concepts

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

* [[confinement]]

* [[chiral symmetry breaking]]

* [[hadron]]

  * [[meson]]

  * [[baryon]]

* [[chiral partner]]

* [[quark bag model]], [[Cheshire cat principle]]

* [[hadron supersymmetry]], [[hadron Kaluza-Klein theory]]

  * [[flavor brane]]

* [[leptonic decay]]

* [[holographic QCD]]

## References

### Introduction and review

* [[Jürg Gasser]], _The QCD Vacuum and Chiral Symmetry_, In: Vautherin D., Lenz F., Negele J.W. (eds.) _Hadrons and Hadronic Matter_, NATO ASI Series (Series B: Physics), vol 228. Springer 1990 ([doi:10.1007/978-1-4684-1336-6_4](https://doi.org/10.1007/978-1-4684-1336-6_4))

* [[Gerhard Ecker]], _The standard model at low energies_, Czech. J. Phys. 44:405-430, 1995 ([arXiv:hep-ph/9309268](https://arxiv.org/abs/hep-ph/9309268))

* {#Ecker95} [[Gerhard Ecker]], *Chiral Perturbation Theory*, Prog. Part. Nucl. Phys. 35:1-80, 1995 ([arXiv:hep-ph/9501357](https://arxiv.org/abs/hep-ph/9501357))

* [[Antonio Pich]], _Chiral Perturbation Theory_ ([arXiv:hep-ph/9502366](https://arxiv.org/abs/hep-ph/9502366))

* [[Maciej Nowak]], [[Mannque Rho]], [[Ismail Zahed]], _[[Chiral Nuclear Dynamics]]_,  World Scientific 1996 ([doi:10.1142/1681](https://doi.org/10.1142/1681))

* Volker Koch, _Aspects of Chiral Symmetry_, Int. J. Mod. Phys. E6 (1997) 203-250 ([arXiv:nucl-th/9706075](https://arxiv.org/abs/nucl-th/9706075))

* Barry R. Holstein, _A Brief Introduction to Chiral Perturbation Theory_, Czech. J. Phys. 50S4:9-23, 2000 ([arXiv:hep-ph/9911449](https://arxiv.org/abs/hep-ph/9911449))

* [[Stefan Scherer]], _Introduction to Chiral Perturbation Theory_, Adv. Nucl. Phys. 27 (2003) 277 ([arXiv:hep-ph/0210398](https://arxiv.org/abs/hep-ph/0210398))

* [[Stefan Scherer]], Matthias R. Schindler, _A Chiral Perturbation Theory Primer_ ([arXiv:hep-ph/0505265](https://arxiv.org/abs/hep-ph/0505265))

* [[Véronique Bernard]], [[Ulf-G. Meissner]], _Chiral perturbation theory_, Ann. Rev. Nucl. Part. Sci.57:33-60, 2007 ([arXiv:hep-ph/0611231](https://arxiv.org/abs/hep-ph/0611231))

* [[Gerhard Ecker]], _Status of chiral perturbation theory_, PoS Confinement 8:025, 2008 ([arXiv:0812.4196](https://arxiv.org/abs/0812.4196))

* Johan Bijnens, _Chiral perturbation theory in the meson sector_, PoS CD09:031, 2009 ([arXiv:0909.4635](https://arxiv.org/abs/0909.4635))

* Johan Bijnens, _Status of Strong ChPT_, PoS EFT09 (2009) 022 ([spire:818610](http://inspirehep.net/record/818610/))

* {#MachleidtEntem11} [[Rupert Machleidt]], [[David Rodríguez Entem]], _Chiral effective field theory and nuclear forces_, Phys. Rept. 503:1-75, 2011 ([arXiv:1105.2919](https://arxiv.org/abs/1105.2919), [doi:10.1016/j.physrep.2011.02.001](https://doi.org/10.1016/j.physrep.2011.02.001))

* [[Franco Strocchi]], §8 of: _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

  (in a context of [[non-perturbative quantum field theory]])

* {#Brambilla14} Brambilla et al., Sections 3-4 in: _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981  ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723), [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

* Yong-Liang Ma, Masayasu Harada, Section II.B of: _Lecture notes on the Skyrme model_ ([arXiv:1604.04850](https://arxiv.org/abs/1604.04850), [spire:1448311](https://inspirehep.net/literature/1448311))

* [[Franz Gross]], [[Eberhard Klempt]] et al., Section 6.2 of: *50 Years of Quantum Chromodynamics*, EJPC [[arXiv:2212.11107](https://arxiv.org/abs/2212.11107)]

See also:

* Wikipedia, _[Chiral perturbation theory](https://en.wikipedia.org/wiki/Chiral_perturbation_theory)_

As [[quantum hadrodynamics]]:

* Brian D. Serot, _Quantum hadrodynamics_, Rept.Prog.Phys. 55 (1992) 1855-1946, ([spire:333292](http://inspirehep.net/record/333292), [doi:10.1088/0034-4885/55/11/001](https://doi.org/10.1088/0034-4885/55/11/001))

* Brian D. Serot, John Dirk Walecka, _Recent Progress in Quantum Hadrodynamics_, Int. J. Mod. Phys. E6:515-631, 1997 ([arXiv:nucl-th/9701058](https://arxiv.org/abs/nucl-th/9701058))

Via the [[S-matrix]]/[[scattering amplitudes]] of [[mesons]]:

* Andrea Guerrieri, Joao Penedones, Pedro Vieira, _S-matrix Bootstrap for Effective Field Theories: Massless Pions_ ([arXiv:2011.02802](https://arxiv.org/abs/2011.02802))

* Eef van Beveren, George Rupp, _Modern meson spectroscopy: the fundamental role of unitarity_ ([arXiv:2012.03693](https://arxiv.org/abs/2012.03693))

### Original articles

* [[Sidney Coleman]], [[Julius Wess]], [[Bruno Zumino]], _Structure of Phenomenological Lagrangians. I_, Phys. Rev. 177, 2239 (1969) ([doi:10.1103/PhysRev.177.2239](https://doi.org/10.1103/PhysRev.177.2239))

* [[Curtis Callan]], Jr., [[Sidney Coleman]], [[Julius Wess]], [[Bruno Zumino]], _Structure of Phenomenological Lagrangians. II_, Phys. Rev. 177, 2247 (1969) ([doi:10.1103/PhysRev.177.2247](https://doi.org/10.1103/PhysRev.177.2247))

* {#Weinberg79} [[Steven Weinberg]], _Phenomenological Lagrangians_, Physica A: Statistical Mechanics and its Applications Volume 96, Issues 1–2, April 1979, Pages 327-340 (doi:10.1016/0378-4371(79)90223-1)

* {#tHooft1980} [[Gerard 't Hooft]], *Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking* In: G. 'Hooft et al., *Recent Developments in Gauge Theories*, NATO Advanced Study Institutes Series **59**, Springer (1980) [[doi:10.1007/978-1-4684-7571-5_9](https://doi.org/10.1007/978-1-4684-7571-5_9)]

* [[Jürg Gasser]], [[Heinrich Leutwyler]], _Quark masses_, Physics Reports Volume 87, Issue 3, July 1982, Pages 77-169 (doi:10.1016/0370-1573(82)90035-7)

* [[Jürg Gasser]], [[Heinrich Leutwyler]], _Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark_, Nucl. Phys. B 250 (1985) 465-516, ([spire:200027](https://inspirehep.net/literature/200027), doi:10.1016/0550-3213(85)90492-4))

* [[Jürg Gasser]], [[Heinrich Leutwyler]], _Chiral perturbation theory to one loop_, Annals of Physics Volume 158, Issue 1, November 1984, Pages 142-210 (doi:10.1016/0003-4916(84)90242-2)

* [[Gerard 't Hooft]], _How instantons solve the $U(1)$ problem_, Physics Reports Volume 142, Issue 6, September 1986, Pages 357-387 (doi:10.1016/0370-1573(86)90117-1)

* [[Heinrich Leutwyler]], _On the Foundations of Chiral Perturbation Theory_, Annals Phys. 235 (1994) 165-203 ([arXiv:hep-ph/9311274](https://arxiv.org/abs/hep-ph/9311274))

At higher order:

* Nils Hermansson-Truedsson, _Chiral Perturbation Theory at NNNLO_ ([arXiv:2006.01430](https://arxiv.org/abs/2006.01430))

In relation to the [[theta vacuum]] ([[Yang-Mills instanton]] vacuum):

* [[Giovanni Morchio]], [[Franco Strocchi]], _Chiral symmetry breaking and theta vacuum structure in QCD_, Annals Phys. **324** (2009) 2236-2254 [[arXiv:0907.2522](https://arxiv.org/abs/0907.2522), [doi:10.1016/j.aop.2009.07.005](https://doi.org/10.1016/j.aop.2009.07.005)]

Specifically for [[kaon]] decay:

* V. Cirigliano, [[Gerhard Ecker]], [[Helmut Neufeld]], [[Antonio Pich]], J. Portoles, _Kaon Decays in the Standard Model_, Rev. Mod. Phys. 84 (2012) 399 ([arXiv:1107.6001](https://arxiv.org/abs/1107.6001))

Further discussion of [[phenomenology]]:

* Prabal Adhikari, Jens O. Andersen, _Quark and pion condensates at finite isospin density in chiral perturbation theory_ ([arXiv:2003.12567](https://arxiv.org/abs/2003.12567))

* Bryan W. Lynn, Brian J. Coffey, Kellen E. McGee, Glenn D. Starkman, _Nuclear matter as a liquid phase of spontaneously broken semi-classical $SU(2)_L \times SU(2)_R$ chiral perturbation theory: Static chiral nucleon liquids_ ([arXiv:2004.01706](https://arxiv.org/abs/2004.01706))

See also

* Yu-Jia Wang, Feng-Kun Guo, Cen Zhang, Shuang-Yong Zhou, _Generalized positivity bounds on chiral perturbation theory_ ([arXiv:2004.03992](https://arxiv.org/abs/2004.03992))

* Lukas Graf, Brian Henning, Xiaochuan Lu, Tom Melia, Hitoshi Murayama, _2, 12, 117, 1959, 45171, 1170086, ...: A Hilbert series for the QCD chiral Lagrangian_ ([arXiv:2009.01239](https://arxiv.org/abs/2009.01239))

[[!include Skyrme hadrodynamics with vector mesons -- references]]

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

[[!include WZW term of QCD chiral perturbation theory -- references]]

[[!include baryon chiral perturbation theory -- references]]

### Applications to lattice QCD

Application to [[lattice QCD]]:

* {#Sharpe:2006pu} S.R.Sharpe, Applications of Chiral Perturbation theory to lattice QCD, ILFTN Workshop on "Perspectives in Lattice QCD", Nara, Japan, Oct 31-Nov 11 2005, 2006 ([arXiv:hep-lat/0607016](https://arxiv.org/abs/hep-lat/0607016v2))

### Inclusion of leptons

On [[chiral perturbation theory]] including [[leptons]]:

* M. Knecht, [[Helmut Neufeld]], H. Rupertsberger, P. Talavera, _Chiral Perturbation Theory with Virtual Photons and Leptons_, Eur. Phys. J. C12:469-478, 2000 ([arXiv:hep-ph/9909284](https://arxiv.org/abs/hep-ph/9909284))

* [[Helmut Neufeld]], _Chiral Perturbation Theory with Photons and Leptons_, PiN Newslett. 15:189-192, 1999 ([arXiv:hep-ph/9912462](https://arxiv.org/abs/hep-ph/9912462))

Specifically in relation to the [[Skyrme model]]:

* [[Eric D'Hoker]], Edward Farhi, _The decay of the skyrmion_, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 86-90 (doi:10.1016/0370-2693(84)90991-2)

### Antisymmetric tensor representation for vector mesons
 {#ReferencesAntisymmetricTensorRepOfVectorMesons}

The formulation of vector mesons in "antisymmetric tensor representation":

Original articles:

* [[Jürg Gasser]], [[Heinrich Leutwyler]], Appendix C of: _Chiral perturbation theory to one loop_, Annals of Physics Volume 158, Issue 1, November 1984, Pages 142-210 (doi:10.1016/0003-4916(84)90242-2)

* [[Gerhard Ecker]], [[Jürg Gasser]], [[Antonio Pich]], E. DeRafael, Sec 3 and Appendix A in: _The role of resonances in chiral perturbation theory_, Nuclear Physics B
Volume 321, Issue 2, 24 July 1989, Pages 311-342 (doi:10.1016/0550-3213(89)90346-5)

Relation between the vector- and the antisymmetric tensor-representation of vector mesons:

* Karol Kampf, Jiri Novotny, Jaroslav Trnka, _On different lagrangian formalisms for vector resonances within chiral perturbation theory_, Eur. Phys. J. C50:385-403, 2007 ([arXiv:hep-ph/0608051](https://arxiv.org/abs/hep-ph/0608051))

See also

* V. Dmitrašinović, around (11) in: _Axial baryon number nonconserving antisymmetric tensor four-quark effective interaction_, Phys. Rev. D 62, 096010 (2000) ([doi:10.1103/PhysRevD.62.096010](https://doi.org/10.1103/PhysRevD.62.096010))

Further application of the antisymmetric tensor representation of vector mesons to [[quantum hadrodynamics]]:

* [[Stefan Leupold]], Carla Terschlusen, _Towards an effective field theory for vector mesons_, Talk presented at the 50th International Winter Meeting on Nuclear Physics, 23-27 January 2012, Bormio ([arXiv:1206.2253](https://arxiv.org/abs/1206.2253))

* Carla Terschlusen, [[Stefan Leupold]], M. F. M. Lutz, _Electromagnetic transitions in an effective chiral Lagrangian with the eta-prime and light vector mesons_, Eur. Phys. J. A 48, 190 (2012) ([arXiv:1204.4125](https://arxiv.org/abs/1204.4125))

[[!redirects chiral symmetry breaking]]

[[!redirects chiral symmetry]]
[[!redirects chiral symmetries]]


chiral ring

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#### Physics
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#Contents#
* table of contents
{:toc}

## Idea

The [[cohomology groups]] of the twisting operator in [[topologically twisted D=4 super Yang-Mills theory|topologically twisted]] [[N=2 D=4 super Yang-Mills theory]] form a [[ring]]. This is called the _chiral ring_ of the [[theory (physics)|theory]].

For one of the possible twists this is the [[ordinary cohomology]] of a corresponding [[Kähler manifold]]. For the other it is a deformation thereof by [[instanton]]-correction, also called a _[[quantum cohomology]]_.

## References

Lecture notes are in

* [[Edward Witten]], _Dynamical aspects of QFT_, Lecture 14: _$N=2$ SUSY Theories in Dimension Two, Part II: Chiral Rings and Twisted Theories_,  in Part IV of _[[Quantum Fields and Strings]]_.

Original articles include

* Freddy Cachazo, [[Michael Douglas]], [[Nathan Seiberg]], [[Edward Witten]]. _Chiral Rings and Anomalies in Supersymmetric Gauge Theory_, JHEP 0212:071,2002 ([arXiv:hep-th/0211170](http://arxiv.org/abs/hep-th/0211170))

[[!redirects chiral rings]]


choice function

# Choice functions
* table of contents
{: toc}

## Idea

Given a set $A$, a choice function for $A$ chooses an element of every subset of $A$, except for the [[empty subset]], of course.

## Definition

Let $A$ be a [[set]].  Let $\mathcal{P}^+A$ be the collection of [[inhabited subsets]] of $A$.  (So if $\mathcal{P}A$ is the [[power set]] of $A$, then $\mathcal{P}^+A$ is contained in $\mathcal{P}A$, and $\mathcal{P}A \setminus \mathcal{P}^+A = \{\empty\}$.)

A __choice function__ for $A$ is a [[function]] $c$ from $\mathcal{P}^+A$ to $A$ such that $c(x) \in x$ (for every inhabited subset $x$).

## Properties

$A$ is a __[[choice object|choice set]]__ if it has a choice function.  (The choice function need not be unique, and rarely is.)

The __[[axiom of choice]]__ states that every set has a choice function.

## Internalisation

In any [[topos]], the definition above can be [[internalisation|internalised]]; we get the notion of a __choice morphism__.

[[!redirects choice function]]
[[!redirects choice functions]]
[[!redirects choice morphism]]
[[!redirects choice morphisms]]


choice object

# Choice objects
* table of contents
{: toc}

## Definitions and terminology

As it is used here, and by (at least some) [[topos]] theorists, a **choice object** is an object $B$ such that the [[axiom of choice]] holds when making choices _from_ $B$.

By contrast, a [[projective object]] is an object $A$ such that AC holds when making choices _indexed by_ $A$.

Confusingly, some constructivists use "choice set" to mean what we call a "[[projective set]]".

## Properties

One way to state the axiom of choice is that every [[entire relation]] from $A$ to $B$ (so that every element of $A$ is related to some element of $B$) contains (the graph of) a [[function]] $A\to B$.  With this terminology, we may say that

* $A$ is _projective_ iff every entire relation from $A$ to $B$, for any $B$, contains a function $A\to B$, while
* $B$ is _choice_ iff every entire relation from $A$ to $B$, for any $A$, contains a function $A\to B$.

Equivalently (at least, in a topos) $B$ is choice iff it has a _[[choice function]]_: a function $c\colon P^+B \to B$ such that $c(x)\in x$ for all $x\in P^+B$.  Here $P^+B$ is the object of all [[inhabited set|inhabited]] subsets of $B$.  We can also say that an object is choice if and only if it is [[well-order|classically well-orderable]] (that is, it admits a [[total order]] where every inhabited subset has a least element); see D4.5.13 in the [[Elephant]].

In [[constructive mathematics]], the principle of [[excluded middle]] is equivalent to the statement that the set $2 = \{a \ne b\}$ is a choice set.

## Related entries

* [[projective object]]

* [[axiom of choice]]

* [[well-ordering theorem]]

## References

* [[Peter Freyd]], _Choice and Well-ordering_ , APAL **35** (1987) pp.149-166.

* M.-M. Mawanda, _Well-ordering and Choice in Toposes_ , JPAA **50** (1988) pp.171-184.

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. II_, Oxford UP 2002. (D4.5, pp.987-998)

[[!redirects choice object]]
[[!redirects choice objects]]
[[!redirects choice set]]
[[!redirects choice sets]]
[[!redirects well-orderable set]]
[[!redirects well-orderable sets]]


choice operator

# Choice operators
* table of contents
{: toc}

## Definitions

A **(global) choice operator** is structure included in some [[foundations]] of [[mathematics]] which provides a uniform way to choose an element of every [[inhabited set|inhabited]] [[set]] or [[class]].

In [[David Hilbert|Hilbert]]'s original formulation, for any property $P(x)$ we can form a term $\varepsilon x.P(x)$ such that
$$\exists x.P(x) \;\Rightarrow\; P\big(\varepsilon x.P(x)\big).$$
In other words, if $P$ holds of anything at all, then it holds of the particular term $\varepsilon x.P(x)$.  A similar operator was used by [[Bourbaki]] and appears in [[FMathL]].

### In dependent type theory

In dependent type theory with [[propositional truncations]], the naive translation of the global choice operator results in the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:[A] \vdash \varepsilon_A(x):A}$$

The above rule could be simplified to the following rule using [[function types]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \varepsilon_A:[A] \to A}$$

This rule is an [[axiom of set truncation]], because given type $A$ and elements $x:A$ and $y:A$, one has a function $\varepsilon_{x =_A y}:[x =_A y] \to x =_A y$. Since $[x =_A y]$ is an [[equivalence relation]], the fact that there is a function from $[x =_A y]$ to $x =_A y$ implies that $A$ is a [[set]].

One could try to set-truncate the target, or restrict the global choice operator to inhabited [[h-sets]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:[A] \vdash \varepsilon_A(x):[A]_0} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, p:\mathrm{isSet}(A), x:[A] \vdash \varepsilon_A(p, x):A}$$

However, this inference rule implies the untruncated [[axiom of choice]]:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A, y:B(x) \vdash P(x, y) \; \mathrm{type}}{\Gamma \vdash \mathrm{ac}_{A, B, P}:\Pi x:A.\exists y:B(x).[P(x, y)] \to \exists g:(\Pi x:A.B(x)).\Pi x:A.[P(x, g(x))]}$$

which is inconsistent with a univalent Tarski universe with [[h-proposition|non-propositional]] [[h-sets]]. (See the [[HoTT book]], section 3.8).

There is also a version of the type-theoretic choice operator local to a [[Tarski universe]] $(U, T)$, given by:

$$\frac{\Gamma \vdash U \; \mathrm{type} \quad \Gamma, X:U \vdash T \; \mathrm{type}}{\Gamma, A:U, x:[T(A)] \vdash \varepsilon_U(A, x):T(A)}$$

$U$ having a choice operator implies that $U$ satisfies [[axiom K]] or [[UIP]]. If $U$ is also univalent, then it is an [[h-groupoid]].

### Relation to double negation

The existence of a choice operator implies [[excluded middle]] and the [[double negation law]], and in particular, means that in [[dependent type theory]] the [[propositional truncation]] of a type is given by the [[double negation]] [[modality]]:

$$[A] \coloneqq \neg \neg A$$

This means that the rules for choice operators can be rewritten using double negation rather than propositional truncations:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:\neg \neg A \vdash \varepsilon_A(x):A}$$

equivalently, that

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \varepsilon_A:\neg \neg A \to A}$$

It then becomes clear that the [[law of double negation]] is simply the choice operator only for [[h-propositions]]. Thus, choice operators could be called the **type-theoretic law of double negation** under the [[propositions as types]] interpretation of [[dependent type theory]].

One could equivalently define the set-theoretic choice operator using double negations, by one of these inference rules:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \varepsilon_A:\neg \neg A \to [A]_0} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \varepsilon_A:\mathrm{isSet}(A) \times \neg \neg A \to A}$$

By [[currying]] the conclusion of the second rule, one gets

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \varepsilon_A:\mathrm{isSet}(A) \to (\neg \neg A \to A)}$$

which is a version of the [[law of double negation]] for [[h-sets]].

### Relation to the axiom of choice

The usual [[axiom of choice]] is the choice operator rule limited to [[fiber types]] of [[surjections]] between [[sets]] at elements of the [[codomain]].

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isSet}(A) \quad \Gamma \vdash q:\mathrm{isSet}(B) \quad \Gamma \vdash f:A \to B \quad \Gamma \vdash r:\mathrm{isSurjection}(f) \quad \Gamma \vdash b:B}{\Gamma \vdash \varepsilon_{\mathrm{fiber}_{A, B}(f, b)}:|\mathrm{fiber}_{A, B}(f, b)| \to \mathrm{fiber}_{A, B}(f, b)}$$

where

$$\mathrm{fiber}_{A, B}(f, b) \equiv \sum_{x:A} f(x) =_B b$$

If we remove the requirement that the domain is a set, we get the [[axiom of infinity-choice]]:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash q:\mathrm{isSet}(B) \quad \Gamma \vdash f:A \to B \quad \Gamma \vdash r:\mathrm{isSurjection}(f) \quad \Gamma \vdash b:B}{\Gamma \vdash \varepsilon_{\mathrm{fiber}_{A, B}(f, b)}:|\mathrm{fiber}_{A, B}(f, b)| \to \mathrm{fiber}_{A, B}(f, b)}$$

Similarly, the [[axiom of countable choice]] is the the choice operator rule limited to [[fiber types]] of [[surjections]] from [[sets]] to the [[natural numbers type]] at natural numbers.

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isSet}(A) \quad \Gamma \vdash f:A \to \mathbb{N} \quad \Gamma \vdash r:\mathrm{isSurjection}(f) \quad \Gamma \vdash n:\mathbb{N}}{\Gamma \vdash \varepsilon_{\mathrm{fiber}_{A, \mathbb{N}}(f, n)}:|\mathrm{fiber}_{A, \mathbb{N}}(f, n)| \to \mathrm{fiber}_{A, \mathbb{N}}(f, n)}$$

If we remove the requirement that the domain is a set, we get the [[axiom of countable infinity-choice]]:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash f:A \to \mathbb{N} \quad \Gamma \vdash r:\mathrm{isSurjection}(f) \quad \Gamma \vdash n:\mathbb{N}}{\Gamma \vdash \varepsilon_{\mathrm{fiber}_{A, \mathbb{N}}(f, n)}:|\mathrm{fiber}_{A, \mathbb{N}}(f, n)| \to \mathrm{fiber}_{A, \mathbb{N}}(f, n)}$$

## Foundational status

It should be noted that Hilbert and Bourbaki take the $\varepsilon$-operator, called $\tau$ in "Theory of Sets" to be a primitive symbol in a mathematical theory.  This has the advantage of also allowing the existential and universal [[quantifiers]] to be constructed explicitly, since $\exists x$ translates to $P(\varepsilon_x(P(x)))$.  The intuitive meaning behind the operator is that it returns a distinguished object for which the proposition is true, or if no such object exists, it returns any object for which it is not true.  Of course, the intuitive meaning can be misleading since the properties of the epsilon operator are governed by the axiom schema of existence and $\varepsilon$-extensionality, without which, the symbol has no meaning.  The __axiom scheme of existence__ is a statement of Hilbert's axiom that avoids mention of the existential quantifier.

A wonderful usage of the global choice operator in Bourbaki is that we can easily show that paradoxical sets  do not exist, since the definition of a set is given in terms of whether or not the relation defining the set is collectivizing.  The example of [[Russell's paradox]] is computed in (Theory of Sets, Chapter II, §1, no.4).  A consequence of this formulation is that the epsilon calculus is quantifier-free, since $\varepsilon$ subsumes their roles from the predicate calculus.  If we adjoin $\varepsilon$ to intuitionistic logic with the axiom scheme of existence, we obtain a non-conservative extension of intuitionistic logic in which the law of excluded middle still does not hold.  However, the assumption of the axiom scheme of $\varepsilon$-extensionality implies the axiom of global choice and therefore the law of excluded middle.

One use of the choice operator is to eliminate undesirable details of implementation.  For example, if $P(x)$ is "$x$ is a Dedekind-complete totally ordered field," then we can define "the real numbers" to be $\varepsilon x.P(x)$.  Any of the numerous constructions of the [[real numbers]] can then be used to show that there exists an $x$ such that $P(x)$, after which point we can discard whichever explicit construction and work only with $\mathbb{R}=\varepsilon x.P(x)$.  This way we can no longer assert that real numbers (elements of $\mathbb{R}$) *are* Dedekind cuts, or equivalence classes of Cauchy sequences, or anything in particular, since the axioms provide no rules about how $\varepsilon x.P(x)$ must be chosen other than that it must satisfy $P$.  So $\mathbb{R}$ *might* consist of Cauchy sequences, or Dedekind cuts, or any other way to construct the reals, but we have no way of knowing, and thus we cannot make use of any such definition in a proof about $\mathbb{R}$; we are forced to use only its formal properties. (For a type-theoretic treatment of this situation, see [generalized the](generalized+the#formalization)).

In many cases, the assumption of a global choice operator implies the [[axiom of choice]], since for any family $(A_u)$ of inhabited sets, a choice function is given by $u\mapsto (\varepsilon x.x\in A_u)$.  In fact, in the form given above it often implies the stronger *global* axiom of choice, which applies to proper classes as well as sets.

> "Here, moreover, we come upon a very remarkable circumstance, namely, that all of these transfinite axioms are derivable from a single axiom, one that also contains the core of one of the most attacked axioms in the literature of mathematics, namely, the axiom of choice: $A(a) \rightarrow A(\varepsilon(A))$, where $\varepsilon$ is the transfinite logical choice function."

> ---Hilbert (1925), "On the Infinite", excerpted in Jean van Heijenoort, _From Frege to Gödel_, p. 382.

However, in some [[foundations]] it seems to be possible to avoid this conclusion (if it is unwanted) by not requiring the choice operator to be extensional, i.e. it may not be the case that $A = B$ implies $\varepsilon x.A = \varepsilon x.B$.

Like the [[axiom of choice]], the existence of a global choice operator is consistent with the other axioms of most foundations.  For example, in ZF, the [[constructible universe]] (which models $ZF + (V=L)$, the [[axiom of constructibility]]) admits a natural classical [[well-ordering]] of the entire universe, giving rise to a naturally defined global choice operator (namely, $\varepsilon x.P$ = the smallest $x$ such that $P$ in the global well-ordering).

On the other hand, a type-theoretic global choice operator is inconsistent with the [[univalence axiom]] of [[homotopy type theory]]. For univalence requires that all operations on the [[type of types]] must be natural with respect to equivalences, which no global choice operator can be.

## Related concepts

* [[split support]]
* [[law of double negation]]
* [[propositions as types]]
* [[axiom of choice]]

## Readings

* [Hilbert's $\varepsilon$-Operator](https://planetmath.org/hilbertsvarepsilonoperator), _PlanetMath_.

* Stanford Encyclopedia of Philosophy article on [The Epsilon Calculus](http://plato.stanford.edu/entries/epsilon-calculus/).

* See also Hilbert (1927), "The Foundations of Mathematics", pp. 464–479 in JvH.

category: foundational axiom

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choice sequence

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## Idea

In [[intuitionistic mathematics]], the concept of _choice sequence_ introduced by [[L.E.J. Brouwer]] is a [[constructive mathematics|constructive]] notion of _[[sequence]]_ that admits "free choices" (i.e. not necessarily "determined by a law").

This has been formalized in various ways, sometimes by writing down axioms that follow from Brouwer's philosophical description, other times by constructing models such as sheaf [[toposes]].

## References

* Wikipedia, _[Choice sequence](http://en.wikipedia.org/wiki/Choice_sequence)_

Discussion of models for choice sequences in various sheaf toposes (usually involving [[Baire space]] or [[Cantor space]]) include:

* [[Gerrit Van Der Hoeven]] and [[Ieke Moerdijk]], *Sheaf models for choice sequences*, Annals of Pure and Applied Logic, Volume 27, Issue 1, August 1984, Pages 63-107, [DOI](https://doi.org/10.1016/0168-0072(84%2990035-6)

* [[Michael Fourman]], *Notions of Choice Sequence*, Studies in Logic and the Foundations of Mathematics, Volume 110, 1982, Pages 91-105, [DOI](https://doi.org/10.1016/S0049-237X(09%2970125-9)

* [[Thierry Coquand]], *On a model of choice sequences*, [pdf](http://www.cse.chalmers.se/~coquand/choice.pdf)

[[!redirects choice sequences]]


Chomsky hierarchy

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# Contents
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[[formal grammar|Formal grammars]] come in a hierarchy of expressive power, first introduced by [[Noam Chomsky]].

* **Type-0 grammars**, also known as [[formal grammar|unrestricted grammars]] generate the [[decidability|recursively enumerable]] languages, they correspond to [[Turing machine|Turing machines]].
* **Type-1 grammars** are also called [[context sensitive grammar|context sensitive grammars]], they correspond to linear-bounded [[nondeterministic automaton |nondeterministic]] Turing machines.
* **Type-2 grammars** are also called [[context free grammar|context free grammars]], they correspond to [[nondeterministic automaton |nondeterministic]] pushdown [[automaton|automata]].
* **Type-3 grammars** generate regular languages, they are equivalent to [[regular language|regular languages]] and finite-state [[automaton|automata]].


Chong-Sun Chu

* [webpage](http://www.phys.nthu.edu.tw/~cschu/)

## Selected writings

On [[M-waves]] in [[intersecting brane|intersecting]] [[M5-brane]] systems:

* {#ChuIsono13} [[Chong-Sun Chu]], Hiroshi Isono, _Instanton String and M-Wave in Multiple M5-Branes System_ ([arXiv:1305.6808](http://arxiv.org/abs/1305.6808), slides: [[IsonoSlides.pdf:file]])

On [[self-dual strings]] on [[M5-branes]] in the [[ABJM model]]:

* {#ChuSmith09} [[Chong-Sun Chu]], Douglas J. Smith, _Multiple Self-Dual Strings on M5-Branes_, JHEP 1001:001, 2010 ([arXiv:0909.2333](https://arxiv.org/abs/0909.2333))

On the non-abelian [[D=6 N=(2,0) SCFT]] on [[M5-branes]] and [[double dimensional reduction]] to [[D4-branes]]:

* [[Chong-Sun Chu]], _A Theory of Non-Abelian Tensor Gauge Field with Non-Abelian Gauge Symmetry $G \times G$_ ([arXiv:1108.5131](http://arxiv.org/abs/1108.5131))

* [[Chong-Sun Chu]], Sheng-Lan Ko, _Non-abelian Action for Multiple Five-Branes with Self-Dual Tensors_, ([arXiv:1203.4224](http://arxiv.org/abs/1203.4224)) JHEP05(2012)028

* [[Chong-Sun Chu]], _A proposal for the worldvolume action of multiple M5-branes_, 2013 ([pdf](http://hep.phy.ntnu.edu.tw/old_version/talks/2013/2013-05-09Chong-SunChu.pdf))

category: people


Chongying Dong

* [institute page](https://www.math.ucsc.edu/people/faculty.php?uid=dong)

## Selected writings

On classification of strongly rational holomorphic [[vertex operator algebras]] of [[central charge]] $\leq 24$ (of relevance in [[heterotic string theory]] and [[monstrous moonshine]]):

* [[Chongying Dong]], [[Geoffrey Mason]], *Holomorphic Vertex Operator Algebras of Small Central Charges*, Pacific Journal of Mathematics **213** 2 (2004) 253–266 [[arXiv:math/0203005](https://arxiv.org/abs/math/0203005), [doi:10.2140/pjm.2004.213.253](http://dx.doi.org/10.2140/pjm.2004.213.253)]

category: people