characteristic class

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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# Contents
* table of contents
{:toc}

## Idea

In the general context of [[cohomology]], as described there, a [[cocycle]] representing a cohomology class on an object $X$ with coefficients in an object $A$ is a [[morphism]] $c : X \to A$ in a given ambient [[(∞,1)-topos]] $\mathbf{H}$.

The same applies with the object $A$ taken as the domain object: for $B$ yet another object, the $B$-valued cohomology of $A$ is similarly $H(A,B) = \pi_0 \mathbf{H}(A,B)$. For $[k] \in H(A,B)$ any cohomology class in there, we obtain an [[∞-functor]]

$$
  [k(-)] : \mathbf{H}(X,A) \to \mathbf{H}(X,B)
$$

from the $A$-valued cohomology of $X$ to its $B$-valued cohomology, simply from the composition operation

$$
  \mathbf{H}(X,A) \times \mathbf{H}(A,B) \to \mathbf{H}(X,B)
  \,.
$$

Quite generally, for $[c] \in H(X,A)$ an $A$-cohomology class, its image $[k(c)] \in H(X,B)$ is the corresponding **characteristic class**.

Notice that if $A = \mathbf{B}G$ is [[connected]], an $A$-cocycle on $X$ is a $G$-[[principal ∞-bundle]]. Hence characteristic classes are equivalently characteristic classes of principal $\infty$-bundles.

+-- {: .standout}

From the [[nPOV]], where [[cocycle]]s are elements in an [[derived hom space|(∞,1)-categorical hom-space]], forming **characteristic classes** is nothing but the _composition_ of cocycles.

=--

In practice one is interested in this notion for particularly simple objects $B$, notably for $B$ an [[Eilenberg-MacLane object]] $\mathbf{B}^n K$ for some component $K$ of a [[spectrum object]]. This serves to **characterize** cohomology with coefficients in a complicated object $A$ by a collection of cohomology classes with simpler coefficients. Historically the name _characteristic class_ came a little different way about, however (see also [[historical note on characteristic classes]]).

In that case, with the usual notation $H^n(X,K) := H(X, \mathbf{B}^n K)$, a given characteristic class in degree $n$ assigns

$$
  [k(-)] : \mathbf{H}(X,A) \to H^n(X,K)
  \,.
$$

Moreover, recall from the discussion at [[cohomology]] that to every [[cocycle]] $c : X \to A$ is associated the object $P \to X$ that it classifies -- its [[homotopy fiber]] -- which may be thought of as an $A$-[[principal ∞-bundle]] over $X$ with classifying map $X \to A$. One typically thinks of the characteristic class $[k(c)]$ as characterizing this [[principal ∞-bundle]] $P$.

## Examples

### Characteristic classes of principal bundles

This is the archetypical example: let $\mathbf{H}  =$ [[Top]] $\simeq$  [[∞Grpd]], the canonical [[(∞,1)-topos]] of [[discrete ∞-groupoid]]s, or more generally let $\mathbf{H} = $ [[ETop∞Grpd]], the [[cohesive (∞,1)-topos]] of [[Euclidean-topological ∞-groupoid]]s.

For $G$ [[topological group]] write  $B G$ for its [[classifying space]]: the ([[geometric realization]] of its) [[delooping]].

For $A$ any other [[abelian group|abelian]] [[topological group]], similarly write $B^n A$ for its $n$-fold [[delooping]]. If $A$ is a [[discrete group]] then this is the [[Eilenberg-MacLane space]] $K(A,n)$.

Generally,

$$
  H^n(B G, A) = \pi_0 \mathbf{H}(B G, B^n A)
$$

is the [[cohomology]] of $B G$ with coefficients in $A$. Every [[cocycle]] $k : B G \to B^n A$ represents a characteristic class $[k]$ on $B G$ with coefficients in $A$.

A $G$-[[principal bundle]] $P \to X$ is classified by some map $c: X \to B G$. For any $k \in H^n(B G, A)$ a degree $n$ cohomology class of the classifying space, the corresponding composite map
$X \stackrel{c}{\to} B G \stackrel{k}{\to} B^n A$ represents a class $[k(c)] \in H^n(X,A)$. This is the corresponding characteristic class of the bundle.

Notable families of examples include:

* for $G = O$ the [[orthogonal group]]:

  [[Pontryagin classes]], [[Stiefel-Whitney classes]], [[Wu classes]];

  [[one-loop anomaly polynomial I8]]

* for $G = SO$ the [[special orthogonal group]]:

  [[Euler classes]];

* for $G = U$ the [[unitary group]]:

  [[Chern classes]], [[Conner-Floyd Chern classes]];

  [[Todd class]];

* for $G = \mathbf{B}U(1)$ the [[circle n-group|circle 2-group]]:

  the [[Dixmier-Douady class]].

### Of line bundles

* [[canonical characteristic class]], [[Theta characteristic]]

### Of linear representations

* [[characteristic class of a linear representation]]

### Chern character

The [[Chern character]] is a natural characteristic class with values in real cohomology. See there for more details.

### $k$-Invariants

* [[k-invariants]]

### Level in $\infty$-Chern-Simons theory

* [[level (Chern-Simons theory)]]

### Of Lagrangian submanifolds

A characteristic class of [[Lagrangian submanifolds]] is the _[[Maslov index]]_.

### Classes in the sense of Fuks

In ([Fuks (1987), section 7](#Fuks)) an axiomatization of characteristic classes is proposed. We review the definition and discuss how it is a special case of the one given above.

#### Fuks's definition

Fuks considers a base category $\mathcal{T}$ of "spaces" and a category $\mathcal{S}$ of spaces with a structure (for example, space together with a [[vector bundle]] on it), this category should be a category over $\mathcal{T}$, i.e. at least equipped with a [[functor]] $U : \mathcal{S}\to\mathcal{T}$.

A morphism of categories with structures is a morphism in the [[overcategory]] [[Cat]]$/\mathcal{T}$, i.e. a morphism $U\to U'$ is a functor $F: dom(U)\to dom(U')$ such that $U' F = U$.

Suppose now the category $\mathcal{T}$ is equipped with a [[cohomology theory]] which is, for purposes of this definition, a functor of the form $H : \mathcal{T}^{op} \to A$ where $A$ is some [[concrete category]], typically category of [[algebra over an algebraic theory|T-algebras]] for some [[algebraic theory]] in [[Set]], e.g. the category of [[abelian group]]s. Define
$\mathcal{H} = \mathcal{H}_H$ as a category whose objects are pairs $(X,a)$ where $X$ is a space (= object in $\mathcal{T}$) and $a\in H(X)$. This makes sense as $A$ is a [[concrete category]]. A morphism $(X,a)\to (Y,b)$ is a morphism $f: X\to Y$ such that $H(f)(b) = a$. We also denote $f^* = H(f)$, hence $f^*(b) = a$.

A __characteristic class of structures of type $\mathcal{S}$ with values in__ $H$ in the sense of ([Fuks](#Fuks)) is a morphism of structures $h: \mathcal{S}\to\mathcal{H}_H$ over $\mathcal{T}$. In other words, to each structure $S$ of the type $\mathcal{S}$ over a space $X$ in $\mathcal{T}$ it assigns an element $h(S)$ in $H(X)$ such that for a morphism $t: S\to T$ in $\mathcal{S}$ the homomorphism $(U(t))^* : H(Y)\to H(X)$, where $Y = U(T)$, sends $h(S)$ to $h(T)$.

#### Discussion

Notice that $\mathcal{H}_H \to \mathcal{T}$ in the above is nothing but the [[fibered category]] that under the [[Grothendieck construction]] is an equivalent incarnation of the [[presheaf]] $H$. In fact, since $A$ in the above is assume to be just a 1-category of sets with structure, $\mathcal{H}_H$ is just its [[category of elements]] of $H$.

Similarly in all applications that arise in practice (for instance for the structure of [[vector bundle]]s) that was mentioned, the functor $\mathcal{S} \to \mathcal{T}$ is a [[fibered category]], too, corresponding under the inverse of the [[Grothendieck construction]] to a [[prestack]] $F_{\mathcal{S}}$.

Therefore morphisms of fibered categories over $\mathcal{T}$

$$
  c : \mathcal{S} \to \mathcal{H}_H
$$

are equivalently morphisms of [[prestack|(pre)]][[stack]]s

$$
  c : F_{\mathcal{S}} \to H
  \,.
$$

In either picture, these are morphism in a [[2-topos]] over the [[site]] $\mathcal{T}$.

So, as before, for $X \in \mathcal{T}$ some [[space]], a $\mathcal{S}$-structure on $X$ (for instance a [[vector bundle]]) is a moprhism in the topos

$$
  g : X \to F_{\mathcal{S}}
$$

(in this setup simply by the [[2-Yoneda lemma]]) and the characteristic class $[c(g)]$ of that bundle is the bullback of that [[universal characteristic class|universal class]] $c$, hence the class represented by the composite

$$
  c(g) : X \stackrel{g}{\to} F_{\mathcal{S}} \stackrel{c}{\to} H
  \,.
$$

## Related concepts

* [[characteristic differential form]]

* [[universal characteristic class]], [[universal characteristic map]];

* [[differential characteristic class]]

* [[characteristic class of a linear representation]]

## References

Textbook accounts include

* {#MilnorStasheff74} [[John Milnor]], [[Jim Stasheff]], _Characteristic classes_, Princeton Univ. Press (1974) [[ISBN:9780691081229](https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76), [doi:10.1515/9781400881826](https://doi.org/10.1515/9781400881826), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf), [pdf](https://aareyanmanzoor.github.io/assets/books/characteristic-classes.pdf)]

* {#Kochmann96} [[Stanley Kochmann]], section 2.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], _[[Basic Bundle Theory and K-Cohomology Invariants]]_,
Lecture Notes in Physics, Springer 2008 ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf))

* [[Peter May]], chapter 23 of _A concise course in algebraic topology_ ([pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

With an eye towards application in [[mathematical physics]]:

* {#Zhang11} [[Yang Zhang]], _A brief introduction to characteristic classes from the differentiable viewpoint_, 2011 ([pdf](http://staff.ustc.edu.cn/~yzhphy/notes/A%20brief%20introduction%20to%20characteristic%20classes%20from%20the%20differentiable%20viewpoint.pdf), [[Zhang_CharacteristicClasses.pdf:file]])

* [[Mikio Nakahara]], Chapter 11 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

Further texts include

* Jean-Pierre Schneiders, _Introduction to characteristic classes and  index theory_ (book), Lisboa (Lisbon) 2000

* [[Johan Dupont]], _Fibre bundles and Chern-Weil theory_, Lecture Notes Series __69__, Dept. of Math., University of Aarhus, 2003, 115 pp. [pdf](http://data.imf.au.dk/publications/ln/2003/imf-ln-2003-69.pdf)

* Shigeyuki Morita, _Geometry of characteristic classes_, Transl. Math. Mon. __199__, AMS 2001

* [[Raoul Bott]], L. W. Tu, _Differential forms in algebraic topology_, GTM __82__, Springer 1982.

* D. B. Fuks, _Непреривные когомологии топологических групп и характеристические классы_ , appendix to the Russian translation of K. S. Brown, _Cohomology of groups_, Moskva, Mir 1987.
 {#Fuks}

[[!redirects characteristic class]]
[[!redirects characteristic classes]]


characteristic class of a linear representation

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### Context
#### Representation theory
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[[!include representation theory - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
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## Idea

Given a [[linear representation]] of a [[finite group]] $G$ on a [[finite-dimensional vector space]] $V$, this induces an [[associated bundle|associated]] [[vector bundle]] over the [[classifying space]] $B G$. The [[characteristic classes]] of this vector bundle, notably it [[Chern classes]] or [[Pontryagin]], are hence entirely determined by the linear representation, and may be associated with it.

## Definition

Under the identification of the [[representation ring]] with the [[equivariant K-theory]] of the [[point]] (see [there](equivariant+K-theory#eq:RepresentationRingAsEquivariantKTheoryOfThePoint)) and the [[Atiyah-Segal completion]] map

$$
  R_{\mathbb{C}}(G)
  \simeq
  KU_G^0(\ast)
    \overset{ \widehat{(-)} }{\longrightarrow}
  KU(BG)

$$

one may ask for Chern classes of the K-theory class $\widehat{V} \in KU(B G)$ expressed in terms of the actual [[character]] of the [[representation]] $V$.

(...)

## Examples

### First Chern class

There is a closed formula at least for the [[first Chern class]] ([Atiyah 61, appendix](#Atiyah61)):

For 1-dimensional representations $V$ their [[first Chern class]]  $c_1(\widehat{V}) \in H^2(B G, \mathbb{Z})$ is their image under the canonical [[isomorphism]] from 1-[[dimension|dimensional]] characters in $Hom_{Grp}(G,U(1))$ to the [[group cohomology]] $H^2_{grp}(G, \mathbb{Z})$ and further to the [[ordinary cohomology]] $H^2(B G, \mathbb{Z})$ of the [[classifying space]] $B G$:

$$
  c_1\left(\widehat{(-)}\right)
  \;\colon\;
  Hom_{Grp}(G, U(1))
  \overset{\simeq}{\longrightarrow}
  H^2_{grp}(G,\mathbb{Z})
  \overset{\simeq}{\longrightarrow}
  H^2(B G, \mathbb{Z})
  \,.
$$

More generally, for $n$-[[dimension|dimensional]] [[linear representations]] $V$ their [[first Chern class]] $c_1(\widehat V)$ is the previously defined first Chern-class of the [[line bundle]] $\widehat{\wedge^n V}$ corresponding to the  $n$-th [[exterior power]] $\wedge^n V$ of $V$. The latter is a 1-dimensional representation, corresponding to the [[determinant line bundle]] $det(\widehat{V}) = \widehat{\wedge^n V}$:

$$
  c_1(\widehat{V})
    \;=\;
  c_1(det(\widehat{V}))
    \;=\;
  c_1( \widehat{\wedge^n V} )
  \,.
$$

([Atiyah 61, appendix, item (7)](#Atiyah61))

More explicitly, via the formula for the [[determinant]] as a [[polynomial]] in [[traces]] of powers (see [there](determinant#eq:DeterminantAsPolynomialInTracesOfPowers)) this means that the first Chern class of the $n$-dimensional representation $V$ is expressed in terms of its [[character]] $\chi_V$ as

\[
  \label{FirstChernClassOfRepresentationInTermsOfTheCharacter}
  c_1(V)
  =
  \chi_{\left(\wedge^n V\right)}
  \;\colon\;
  g
  \;\mapsto\;
  \underset{
     { k_1,\cdots, k_n \in \mathbb{N} }
     \atop
     { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n }
  }{\sum}
  \underoverset{ l = 1 }{ n }{\prod}
  \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l !  }
  \left(\chi_V(g^l)\right)^{k_l}
\]

For example, for a representation of dimension $n = 2$ this reduces to

$$
  c_1(V)
  =
  \chi_{V \wedge V}
  \;\colon\;
  g
  \;\mapsto\;
  \frac{1}{2}
  \left(
    \left( \chi_V(g)\right)^2
    -
    \chi_V(g^2)
  \right)
$$

(see also e.g. [tom Dieck 09, p. 45](representation+theory#tomDieck09))

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

Let $G =\mathbb{Z}_{2n+1}$ be a [[finite group|finite]] [[cyclic group]] of [[odd number|odd]] [[order of a group|order]] and let $k[\mathbb{Z}_{2n+1}]$ be its [[regular representation]]. Then the first Chern class vanishes:

$$
  c_1\big( k[\mathbb{Z}_{2n+1}]\big)
  \;=\;
  0
$$

=--

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

The underlying [[set]] of $\mathbb{Z}_{2n+1}$ constitutes the canonical [[linear basis]] of $k[\mathbb{Z}_{2n+1}]$. Moreover, this carries a canonical [[linear order]] $(e, g_1, g_2, \cdots, g_{2n+1})$. With respect to this ordering, the [[action]] of each group element $g \in \mathbb{Z}_n$ is by a [[cyclic permutation]]. Since for [[odd number]] of elements the [[signature of a permutation|signature]] of a [[cyclic permutation]] is $+1$, it follows that for every group element

$$
  g(e \wedge g_1 \wedge \cdots \wedge g_{2n+1})
  =
  + e \wedge g_1 \wedge \cdots \wedge g_{2n+1}
  \,.
$$

This shows that the [[character of a linear representation|character]] of $\wedge^{2n+1}k[\mathbb{Z}_{2n+1}]$ equals that of the [[trivial representation]] $\mathbf{1}$

=--

## Properties

### Splitting principle
 {#SplittingPrinciple}

> see at [Symonds' explicit Brauer induction](Brauer+induction+theorem#SymondExplicitBrauerInduction)

Let $G$ be a [[finite group]], let the [[ground field]] to be the [[complex numbers]].

Then by the [[Brauer induction theorem]] every [[virtual representation]]

$$
  [V] \in R_\mathbb{C}(G)
$$

has a presentation as a virtual combination of [[induced representations]] of 1-[[dimension|dimensional]] representations:

$$
 [V]
 \;=\;
 \underset{
     \mathclap{
         {H_i \hookrightarrow G}
           \atop
         {
           { W_i \in Rep(H_i) \,, }
           \atop
           { dim(W_i) =1 }
         }
     }
   }{\sum}
   n_i
  \left[
     ind_{H_i}^G W_i
  \right]
$$

Of course this expansion is not unique.

According to [Symonds 91, p. 4 & Prop. 2.4](#Symonds91), there is a natural choice for this expansion, and for this there holds a [[splitting principle]] for the corresponding [[Chern classes]] summarized in the [[total Chern class]] (formal sum of all [[Chern classes]])

$$
  c(V)
  \;\coloneqq\;
  1 + c_2(V) + c_2(V) + \cdots
  \;\in\;
  \underset{k}{\prod} H^{2 k}\big( B G , \mathbb{Z}\big)
$$

as follows:

\[
  \label{SplittingFormulaForChernCharacterOfLinearRepresentation}
  ch
  \left(
     V
  \right)
  \;=\;
  \underset{
     \mathclap{
         {H_i \hookrightarrow G}
           \atop
         {
           { W_i \in Rep(H_i)\,, }
           \atop
           { dim(W_i) = 1 }
         }
     }
   }{\prod}
   \mathcal{N}_{H_i}^G
   \Big(
     \overset{
        \mathclap{
          = (1 + c_1(W_i))
        }
      }{
      \overbrace{
        ch\left(W_i\right)
      }
     }
     {}^{\alpha(W_i)}
   \Big)
   \;\;
   \in
   \underset{ k \in \mathbb{N} }{\prod}
   H^{2k}\big(B G, \mathbb{Z} \big)
\]

where

1. the transfer maps

   \[
     \label{MultiplicativeTransfer}
     \mathcal{N}_H^G
     \;\colon\;
     H^\bullet(B H, \mathbb{Z})
       \longrightarrow
     H^\bullet( B G, \mathbb{Z} )
   \]

   are from [Evens 63, bottom of p. 7](#Evens63),

1. the $\alpha(W_i)$-s are the [[Euler characteristics]] of certain [[CW-complexes]], described in [Symonds 91, p. 3](#Symonds91).

Here over the brace we used that the $W_i$ are 1-dimensional, so that at most their [[first Chern class]] may be non-vanishing.

Notice that the transfer maps (eq:MultiplicativeTransfer) are multiplicative under [[cup product]] ([Evens 63, prop. 4](#Evens63)), whence [Symonds 91](#Symonds91) refers to them as the "mutliplicative transfer".

## References

* {#Atiyah61} [[Michael Atiyah]], Appendix of _Characters and cohomology of finite groups_, Publications Mathématiques de l'IHÉS, Volume 9 (1961) , p. 23-64 ([numdam]( http://www.numdam.org/item?id=PMIHES_1961__9__23_0))

* {#Evens63} [[Leonard Evens]], _A Generalization of the Transfer Map in the Cohomology of Groups_, Transactions of the American Mathematical Society Vol. 108, No. 1 (Jul., 1963), pp. 54-65 ([doi:10.1090/S0002-9947-1963-0153725-1](https://doi.org/10.1090/S0002-9947-1963-0153725-1), [jstor:1993825](https://www.jstor.org/stable/1993825))

* {#Evens65} [[Leonard Evens]], _On the Chern Classes of Representations of Finite Groups_, Transactions of the American Mathematical Society, Vol. 115 (Mar., 1965), pp. 180-193 ([doi:10.2307/1994264](https://www.jstor.org/stable/1994264))

* {#LamberTondeur67} F. Kamber, [[Philippe Tondeur]], _Flat Bundles and Characteristic Classes of Group-Representations_, American Journal of Mathematics, Vol. 89, No. 4 (Oct., 1967), pp. 857-886 ([doi:10.2307/2373408](https://www.jstor.org/stable/2373408))

* {#Kroll87} Ove Kroll, _An Algebraic Characterisation of Chern Classes of Finite Group Representations_, Bulletin of the LMS, Volume19, Issue3 May 1987 Pages 245-248 ([doi:10.1112/blms/19.3.245](https://doi.org/10.1112/blms/19.3.245))

* J. Gunarwardena, B. Kahn, C. Thomas, _Stiefel-Whitney classes of real representations of finite groups_, Journal of Algebra Volume 126, Issue 2, 1 November 1989, Pages 327-347 (doi:10.1016/0021-8693(89)90309-8)

* {#Beauville01} Arnaud Beauville, _Chern classes for representations of reductive groups_ ([arXiv:math/0104031](https://arxiv.org/abs/math/0104031))

[[splitting principle]]:

* {#Symonds91} [[Peter Symonds]], _A splitting principle for group representations_, Comment. Math. Helv. (1991) 66: 169 ([dml:140229](https://eudml.org/doc/140229), [doi:10.1007/BF02566643](https://doi.org/10.1007/BF02566643))

[[!redirects characteristic classes of a linear representation]]
[[!redirects characteristic classes of linear representations]]

[[!redirects Chern class of a linear representation]]
[[!redirects Chern classes of a linear representation]]
[[!redirects Chern classes of linear representations]]

[[!redirects total Chern class of a linear representation]]
[[!redirects total Chern classes of a linear representation]]
[[!redirects total Chern classes of linear representations]]


characteristic class of a structure

Entry [[characteristic class]] defines characteristic classes from [[nPOV]], within $(\infty,1)$-topos and relates the characteristic classes to $(\infty,1)$-principal bundles. Here we present an older axiomatics, where characteristic classes *characterize* any structure on spaces (example: characteristic classes of foliations, bundles, bundles with structure etc).

Thus one considers a base category $\mathcal{T}$ of "spaces" and a category $\mathcal{S}$ of spaces with a structure (for example, space together with a vector bundle on it), this category should be a category over $\mathcal{T}$, i.e. equipped with a functor $U : \mathcal{S}\to\mathcal{T}$. A morphism of categories with structures is a morphism in the overcategory $Cat/\mathcal{T}$, i.e. a morphism $U\to U'$ is a functor $F: dom(U)\to dom(U')$ such that $U' F = U$.

Suppose now the category $\mathcal{T}$ is equipped with a cohomology theory which is, for purposes of this definition, a functor of the form $H : \mathcal{T}\to A$ where $A$ is some concrete category, typically category of $T$-algebras for some algebraic theory in $Set$, e.g. the category of abelian groups. Define
$\mathcal{H} = \mathcal{H}_H$ as a category whose objects are pairs $(X,a)$ where $X$ is a space (= object in $\mathcal{T}$) and $a\in H(X)$. This makes sense as $A$ is a concrete category. A morphism $(X,a)\to (Y,b)$ is a morphism $f: X\to Y$ such that $H(f)(b) = a$. We also denote $f^* = H(f)$, hence $f^*(b) = a$.

A __characteristic class of structures of type $\mathcal{S}$ with values in__ $H$ is a morphism of structures $h: \mathcal{S}\to\mathcal{H}_H$ over $\mathcal{T}$. In other words, to each structure $S$ of the type $\mathcal{S}$ over a space $X$ in $\mathcal{T}$ it assigns an element $h(S)$ in $H(X)$ such that for a morphism $t: S\to T$ in $\mathcal{S}$ the homomorphism $(U(t))^* : H(Y)\to H(X)$, where $Y = U(T)$, sends $h(S)$ to $h(T)$.

This axiomatics can be found e.g. in the section 7 of

* D. B. Fuks, Непреривные когомологии топологических групп и характеристические классы, appendix to the Russian translation of K. S. Brown, _Cohomology of groups_, Moskva, Mir 1987.


characteristic element

* [[characteristic element of a bilinear form]]

[[!redirects characteristic elements]]


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### Context
#### Linear algebra
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[[!include homotopy - contents]]
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# Contents
* table of contents
{: toc}

## Definition

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

For $A$ an [[abelian group]] and given a symmetric [[bilinear form]]

$$
  \langle-,-\rangle \;\colon\; A\times A \longrightarrow \mathbb{Z}
$$

then a _characteristic element_ of the bilinear form is an element $\lambda\in A$ such that for all $x\in A$ the [[equation]]

$$
  \langle x,x\rangle = \langle x,\lambda \rangle\; mod\;2
$$

holds, or in other words such that

$$
  \langle x,x\rangle \pm \langle x,\lambda \rangle \in \mathbb{Z}
$$

is divisible by $2\in \mathbb{Z}$.

=--

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

Often this is considered for the bilinear form being an [[intersection product]] in which case $A$ is a [[cohomology group]] and so in this case one also often speaks of _characteristic cohomology classes_.

=--

## Properties

### Relation to quadratic refinements
 {#RelationToQuadraticRefinements}

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

A characteristic element determines a [[quadratic refinement]] $q_\lambda$ of $\langle-,-\rangle$ by

$$
  q_\lambda
    \colon
  x \mapsto
  \tfrac{1}{2}\left(
     \langle x,x\rangle
     +
     \langle x,\lambda\rangle
  \right)
  \,.
$$

=--

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

This follows by trivial direct computation:

$$
  \begin{aligned}
    q_\lambda(x  + y) - q_\lambda(x)- q_\lambda(y)
    & =
    \tfrac{1}{2}
    \left(
      \left(x^2 + y^2 + 2 x y + \lambda x + \lambda y\right)
      -
      \left(x^2 + \lambda x\right)
      -
      \left(y^2 + \lambda y\right)
    \right)
    \\
    & =
    x y
  \end{aligned}
$$

the only point being that the prefactor $\tfrac{1}{2}$ indeed makes sense, by def. \ref{CharElement}.

=--

## Examples

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

Characteristic elements for the [[intersection pairing]] on integral [[ordinary cohomology]] are integral lifts of the [[Wu classes]], hence [[integral Wu structures]].

=--

## References

Discussion of the refinement of [[integral Wu structures]] as the characteristic elements of the [[intersection product]] from [[ordinary cohomology]] to [[ordinary differential cohomology]] is in

* {#HopkinsSinger02} [[Michael Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_, ([arXiv:math.AT/0211216](http://arxiv.org/abs/math.AT/0211216))

Further discussion of the use of characteristic elements in the construction of [[Theta characteristics]] is in

* {#PoonenRains11} Bjorn Poonen, Eric Rains, _Self cup products and the theta characteristic torsor_ ([arXiv:1104.2105](http://arxiv.org/abs/1104.2105))

[[!redirects characteristic element of a bilinear form]]
[[!redirects characteristic elements of a bilinear form]]
[[!redirects characteristic elements of bilinear forms]]


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

This entry is to record the reference

* [[Shiing-shen Chern]], [[James Simons]], _Characteristic forms and geometric invariants_     The Annals of Mathematics, Second Series, Vol. 99, No. 1 (1974) ([jstor](http://www.jstor.org/stable/1971013)).

This is the origin of the notion of [[Chern-Simons form]]s and their conception as [[secondary characteristic class]]es, which led to the refinement of the [[Chern-Weil homomorphism]] by [[Cheeger-Simons differential character]]s.

This is also where _[[Chern-Simons theory]]_ derives its name from.

category: reference


characteristic function

#Contents#
* table of contents
{:toc}

## Definition

### Of a subset

The __characteristic function__ of a [[subset]] $U$ of some [[set]] $X$ is a [[function]] from $X$ to the set $TV$ of [[truth values]] (which classically is $TV = \{\bot,\top\}$) that takes $a$ in $X$ to the truth value of the statement that $a \in U$.  That is,
$$ \chi_U(a) \;\Leftrightarrow\; a \in U ,$$
where $\chi_U$ (also often $1_U$) is the characteristic function of $U$.

### Of a subobject

More generally, the __characteristic morphism__ of a [[subobject]] $U$ of some objects $X$ in a category with a [[subobject classifier]] $\Omega$ is the morphism from $X$ to $\Omega$ that classifies $U$; we have that
$$ \array {
  U          & \hookrightarrow    & X \\
  \downarrow &                    & \downarrow & \chi_U \\
  1          & \underset{\top}\to & \Omega
} $$
is a [[pullback]] square.

### See also
* [[classifying morphism]]
* [[subobject classifier]]

[[!redirects characteristic functions]]
[[!redirects characteristic morphism]]
[[!redirects indicator functions]]
[[!redirects indicator function]]

[[!redirects characteristic map]]
[[!redirects characteristic function]]

[[!redirects characteristic maps]]
[[!redirects characteristic morphisms]]


characteristic ideal

Related $n$Lab entries: [[characteristic variety]], [[D-module]], [[Iwasawa theory]], [[characteristic polynomial]], [[Alexander polynomial]] (support of a module,
characteristics of a PDE, annihilators, exterior differential systems...)

* Andrea Bandini, Francesc Bars, Ignazio Longhi, _Characteristic ideals and Iwasawa theory_, [arxiv/1310.0680](http://arxiv.org/abs/1310.0680); _Characteristic ideals and Selmer groups_, [arxiv/1404.2788](http://arxiv.org/abs/1404.2788)


characteristic polynomial

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

#Contents#
* table of contents
{:toc}

## Characteristic polynomial and Cayley-Hamilton theorem.

See also _[[determinant]]_.

+-- {: .num_lemma}
###### Lemma

Let $R$ be a [[commutative ring]], and let $A$ be an $n \times n$ [[matrix]] with entries in $R$. Then there exists an $n \times n$ matrix $\tilde{A}$ with entries in $R$, called the **adjugate** of $A$, such that $A \tilde{A} = \tilde{A} A = \det(A) \cdot I_n$.

=--

+-- {: .proof}
###### Proof
We may as well take $R$ to be the [[polynomial ring]] $\mathbb{Z}[a_{i j}]_{1 \leq i, j \leq n}$, since we are then free to interpret the indeterminates $a_{i j}$ however we like along a ring map $\mathbb{Z}[a_{i j}] \to R$. Let $A = (a_{i j})$ denote the corresponding generic matrix.

Guided by Cramer's rule (see [[determinant]]), put

$$\tilde{A}_{j i} = \det(a_1, \ldots, e_i, \ldots a_n),$$

the $a_i$ being columns of $A$ and $e_i$, the column vector with $1$ in the $i^{th}$ row and $0$'s elsewhere, appearing as the $j^{th}$ column. If we pretend $A$ is invertible, then we know $A \tilde{A} = \det(A) \cdot I_n = \tilde{A} A$ by [[Cramer's rule]]. We claim this holds for general $A$.

Indeed, we can interpret this as a [[polynomial]] equation in $\mathbb{C}[a_{i j}]$ and check it there. As an equation between polynomial functions on the space of matrices $A \in Mat_n(\mathbb{C}) = Spec(\mathbb{C}[a_{i j}])$, it holds on the dense subset $GL_n(\mathbb{C}) \hookrightarrow Mat_n(\mathbb{C})$. Therefore, by continuity, it holds on all of $Mat_n(\mathbb{C})$. But a polynomial function equation with coefficients in $\mathbb{C}$ implies the corresponding polynomial identity, and the proof is complete.
=--

An alternative proof that avoids appeal to a continuity
argument may be given in terms of [[exterior algebra]]. Start by observing that the equation $\widetilde{A} A = \det(A) \cdot 1_n$ holds over $R = \mathbb{Z}[a_{i j}]$ if it holds over a field $k$ in which $R$ embeds. If $V$ is an $n$-dimensional vector space over $k$ and $A\colon V \to V$ is $k$-linear, then the natural transformation

$$\wedge \colon V \otimes_k \Lambda^{n-1} V \to \Lambda^n V$$

gives an exact pairing $V \otimes_k \Lambda^{n-1} V \to k$ which induces an isomorphism $V \cong (\Lambda^{n-1} V)^\ast$, and the adjugate $\widetilde{A}: V \to V$ is defined to be the evident composite

$$V \cong (\Lambda^{n-1} V)^\ast \stackrel{(\Lambda^{n-1} A)^\ast}{\to} (\Lambda^{n-1} V)^\ast \cong V.$$

The equation $\widetilde{A} A = \det A \cdot 1_V$ holds iff the perimeter of the diagram

\begin{tikzcd}
V \ar[r, "A"] \ar[dr, swap, "\det A \cdot 1_V"] & V \ar[r, "\sim"] \ar[d, "\widetilde{A}"] & (\Lambda^{n-1} V)^\ast \ar[d, "(\Lambda^{n-1} A)^\ast"] \\
& V \ar[r, "\sim"] & (\Lambda^{n-1} V)^\ast
\end{tikzcd}

commutes, but this is equivalent to commutativity of the naturality square

\begin{tikzcd}
V \otimes_k \Lambda^{n-1} V \ar[r, "\wedge"] \ar[d, swap, "A \otimes_k \Lambda^{n-1} A"] & \Lambda^n V \ar[d, "\det A \cdot Id"] \\ V \otimes_k \Lambda^{n-1} V \ar[r, "\wedge"] & \Lambda^n V
\end{tikzcd}

and this completes the proof.

+-- {: .num_theorem}
###### Theorem
(**Cayley-Hamilton**)
Let $V$ be a finitely generated free module over a commutative ring $R$, and let $f \colon V \to V$ be an $R$-module map. Let $p(t) \in R[t]$ be the **characteristic polynomial** $\det(t \cdot 1_V - f)$ of $f$, and let $\phi_f \colon R[t] \to Mod_R(V, V)$ be the unique $R$-algebra map sending $t$ to $f$. Then $p(f) \coloneqq \phi_f(p)$ is the zero map $0 \colon V \to V$.

=--

+-- {: .proof}
###### Proof
Via $\phi_f$, regard $V$ as an $R[t]$-module, and with regard to some $R$-basis $\{v_i\}_{1 \leq i \leq n}$ of $V$, represent $f$ by a matrix $A$. Now consider $t \cdot I_n - A$ as an $n \times n$ matrix $B(t)$ with entries in $R[t]$. By definition of the module structure, this matrix $B(t)$, seen as acting on $V^n$, annihilates the length $n$ column vector $c$ whose $i^{th}$ row entry is $v_i$.

By the previous lemma, there is $\tilde{B}(t)$ such that $\tilde{B}(t) B(t)$ is $\det(t \cdot I_n - A)$ times the identity matrix. It follows that

$$\det(t \cdot I_n - A) c = \tilde{B}(t) B(t) c = \tilde{B}(t) 0 = 0$$

i.e., $\det(t \cdot I_n - A) \cdot v_i = 0$ for each $i$. Since the $v_i$ form an $R$-basis, the $R[t]$-scalar $\det(t \cdot I_n - A)$ annihilates the $R[t]$-module $V$, as was to be shown.
=--

The Cayley-Hamilton theorem easily generalizes to finitely generated $R$-modules (not necessarily free) as follows. Let $f \colon V \to V$ be a module endomorphism, and suppose $\pi \colon R^n \to V$ is an epimorphism. Since $R^n$ is projective, the map $f \circ \pi$ can be lifted through $\pi$ to a map $A \colon R^n \to R^n$. Let $P(t)$ be the characteristic polynomial of $A$.

+-- {: .num_prop #lem}
###### Proposition
$P(f) = 0$.
=--

+-- {: .proof}
###### Proof
Write $P(t) = \sum_i a_i t^i$. We already know $P(A) = 0$. From $f \circ \pi = \pi \circ A$, it follows that $f^i \circ \pi = \pi \circ A^i$ for any $i \geq 0$. Hence $P(f) \circ \pi = \pi \circ P(A) = 0$. Since $\pi$ is epic, $P(f) = 0$ follows.
=--

We give an interesting and perhaps surprising consequence of the Cayley-Hamilton theorem below, after establishing a lemma close in spirit to [[Nakayama's lemma]].

+-- {: .num_lemma}
###### Lemma
Suppose $V$ is a finitely generated $R$-module, and $g \colon V \to V$ is a module map such that $g(V) \subseteq I V$ for some ideal $I$ of $R$. Then there is a polynomial $p(t) = t^n + a_1 t^{n-1} + \ldots + a_n$, with all $a_i \in I$, such that $p(g) = 0$.
=--

+-- {: .proof}
###### Proof
For some finite $n \geq 0$, we have a surjective map $p: R^n \to V$. Tensoring $p$ with $I$, we obtain a surjective map $I^n \cong I \otimes_R R^n \stackrel{I \otimes_R p}{\twoheadrightarrow} I \otimes_R V \stackrel{mult}{\twoheadrightarrow} I V$, fitting in a commutative diagram

$$\array{
 & & & & & & I^n & \hookrightarrow & R^n \\
 & & & & & & \downarrow & & \downarrow _\mathrlap{p} \\
R^n & \stackrel{p}{\to} & V & \stackrel{g}{\to} & im(g) & \stackrel{i}{\hookrightarrow} & I V & \hookrightarrow & V
}$$

By projectivity of $R^n$, we can lift $i g p: R^n \to I V$ to a map $h: R^n \to I^n$ making the diagram commute. Let $A$ be the $R$-module map $R^n \stackrel{h}{\to} I^n \hookrightarrow R^n$, regarded as a matrix. Then the characteristic polynomial of $A$ satisfies the conclusion, by the Cayley-Hamilton theorem and Proposition \ref{lem}.
=--

+-- {: .num_prop #surj}
###### Proposition
Let $V$ be a finitely generated module over a commutative ring $R$, and let $f \colon V \to V$ be a surjective module map. Then $f$ is an isomorphism.
=--

+-- {: .proof}
###### Proof
Regard $V$ as a finitely generated $R[t]$-module via $\phi_f \colon R[t] \to Mod_R(V, V)$. Since $f$ is assumed surjective, we have $I V = V$ for the ideal $I = (t)$ of $R[t]$. Now take $g = 1_V$ as in the preceding lemma, a module map over the ring $R' = R[t]$. By the lemma, we see that
$g^n + a_1 g^{n-1} + \ldots + a_n = 0$ where $a_i \in (t)$, in other words the $R[t]$-scalar

$$(1 + a_1 + \ldots + a_n)1_V = 0$$

as an operator on $V$. Write $a_i = b_i(t) t$ for polynomials $b_i(t) \in R[t]$. Now we may rewrite the previous displayed equation as

$$1_V(v) = -(\sum_i b_i(t)) t \cdot v$$

for all $v \in V$, which translates into saying that $1_V = -\sum_i b_i(f) f$, i.e., that $-\sum_i b_i(f)$ is a retraction of $f$. Since $f$ is epic, we now see $f$ is an isomorphism.
=--

## Examples

* Characteristic polynomials of [[Frobenius homomorphisms]] acting via [[Galois representations]] constitute [[Artin L-functions]].

## References

The proof of the Cayley-Hamilton theorem follows the treatment in

* Serge Lang, _Algebra_ ($3^{rd}$ edition), Addison-Wesley, 1993.

The proof of Proposition \ref{surj} on surjective endomorphisms of finitely generated modules was extracted from

* Stacks Project, Commutative Algebra, section 15 ([pdf](http://math.columbia.edu/algebraic_geometry/stacks-git/algebra.pdf))

[[!redirects Cayley-Hamilton theorem]]

[[!redirects characteristic polynomials]]


characteristic series

Different things are called _characteristic series_ in mathematics. For instance

* the [[formal power series]] characterizing a [[genus]] (see there) of oriented or stable almost complex manifolds

* in group theory, a [[subgroup series]] of a group $G$ is termed a _characteristic series_ if all members of the series are [[characteristic subgroup]]s of $G$. For instance, a [[central series]] is a characteristic series.


characteristic variety

* McKay, _Notes on characteristic variety_, [pdf](http://www.math.utah.edu/~mckay/charvrty.pdf)

* [[J. Bernstein]], [[Valery Lunts|V. Lunts]], _On nonholonomic irreducible D-modules_, Invent. Math. __94__, (1988), no. 2, 223–243.

* P. Maisonobe, C. Sabbah, _D-modules cohérents et holonomes_, Hermann, Paris 1993.

* [[Ofer Gabber]], _The integrability of characteristic variety_

* J. V. Pereira, _The characteristic variety of a generic foliation_, [arxiv/0912.1779](http://arxiv.org/abs/0912.1779)

* [[Victor Ginzburg|V. Ginsburg]], _Characteristic varieties and vanishing cycles_, Invent. Math. __84__:327–402, 1986, [MR87j:32030](http://www.ams.org/mathscinet-getitem?mr=833194), [doi](http://dx.doi.org/10.1007/BF01388811)


characteristic zero

#Contents#
* table of contents
{:toc}

## Definition

Every [[commutative ring]] $R$ has a [[characteristic]]: it is zero if equivalently

* its underlying $\mathbb{Z}$-[[module]] is a [[flat module]],

* its underlying [[abelian group]] is a [[torsion-free group]].

* The unique [[ring homomorphism]] from $\mathbb{Z}$ to $R$ is an [[injection]].

If a mathematical construct involves a "base ring", e.g. an algebraic variety, then we say that it is in characteristic zero, if its base ring is.

## Examples

The basic example of a ring of characteristic zero is the field $\mathbb{Q}$ of [[rational numbers]].  Therefore one often says that any [[ring]] (or even [[super ring]]) $R\supset \mathbb{Q}$ containing the rationals is also of characteristic zero.

The basic example of an [[algebraically closed field]] of characteristic zero is the field $\mathbb{C}$ of [[complex numbers]].

In [[model theory]], there is a first-order theory of fields: every (commutative) field is a model. There is a transfer principle called __Lefschetz principle__ which says: every sentence expressed in the first order theory of fields which is true for complex numbers is true for every algebraically closed field of characteristic zero. It is named after Solomon Lefschetz who used it in algebraic geometry, reasoning topologically for other algebarically closed fields of characteristic zero. The formalization and its proof are due [[Alfred Tarski]].

## Related concepts

* [[positive characteristic]]

[[!redirects characteristic 0]]


characters are cyclotomic integers

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

#Contents#
* table of contents
{:toc}

## Statement

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

Let $G$ be a [[finite group]], $k$ a [[field]], and $V \in G Rep_k$ a [[finite-dimensional vector space|finite-dimensional]] $k$-[[linear representation]] of $G$.

Then the [[character of a linear representation|character]] $\chi_{V} \colon G \to k$ of $V$ takes values inside the [[cyclotomic integers]] for some [[root of unity]].

(e.g. [Naik](#Naik))

It follows that

1. characters take values in [[algebraic integers]];

1. if the [[ground field]] $k$ has [[characteristic zero]], then a [[character]] with values in the [[rational numbers]] in fact already takes values in the [[integers]];

1. in particular if the [[ground field]] $k =\mathbb{Q}$ is the [[rational numbers]], then all characters take values in the actual [[integers]].

   (e.g. [Yang, lemma 2](#Yang))

=--

## Examples
 {#Examples}

### For cyclic groups

The following example ovbserves that for [[cyclic groups]] the general fact that characters are cyclotomic integers reduces to the [[trigonometry|trigonometric]] statement that if the [[cosine]] of a rational [[angle]] is itself rational, then it is in fact integer:

+-- {: .num_example #RationalCharactersForCyclicGroups}
###### Example
**([[rational number|rational]] [[characters]] for [[cyclic groups]])**

For $n \in \mathbb{N}$, $n \geq 2$, consider the [[cyclic group]] $G = \mathbb{Z}/n$. Its [[irreducible representation|irreducible]] [[linear representations]] over the [[real numbers]] are, up to [[isomorphism]],

1. the 1-[[dimensional]] [[trivial representation]] $\mathbf{1}$;

1. the 1-[[dimensional]] [[sign representation]] $\mathbf{1}_{sgn}$;

1. the 2-[[dimensional]] [[rotation]]-representation $\mathbf{2}_k$ for $0 \lt k \lt n/2$, generated by the [[matrix]]

   $$
    \rho_{\mathbf{2}_k}(1)
    \;=\;
     \left(
       \array{
         cos(\theta) & -sin(\theta)
         \\
         sin(\theta) & \phantom{-}cos(\theta)
       }
     \right)
     \phantom{AA}
     \text{with}
     \;
     \theta \coloneqq 2 \pi \tfrac{k}{n}
     \,,
   $$

(by [this Example](cyclic+group#IrreducibleRealRepresentationsOfCyclicGroups)).

Now the characters $\chi_{\mathbf{1}}$ and $\chi_{\mathbf{1}_{sgn}}$ clearly take values in $\{\pm 1\} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$.

But the characters of the 2-dimensional irreps $\mathbf{2}_{k}$ are given by [[traces]] of $2 \times 2$ [[rotation]] [[matrices]] by [[rational number|rational]] [[angles]]:

$$
  \chi_{\mathbf{2}_k}
  \;\colon\;
  \underset{
    \in \mathbb{Z}/n
  }{
    \underbrace{
      [q]
    }
  }
  \;\mapsto\;
  \mathrm{tr}
   \left(
     \left(
       \array{
         cos(\theta) & -sin(\theta)
         \\
         sin(\theta) & \phantom{-}cos(\theta)
       }
     \right)^q
  \right)
   \;=\;
  2
  \,
  \cos
  \left(
    2 \pi
    \tfrac{q k}{n}
  \right)
$$

hence are two times the [[cosine]] of a [[rational number|rational]] [[angle]].

This means that for [[cyclic groups]] the statement of Prop. \ref{CharactersAreCyclotomicIntegers} says equivalently that if the [[cosine]] of a [[rational number|rational]] [[angle]] is a [[rational number]], then it is in fact a half-integer

\[
  \label{RationalCosineValuesOfRationalAngles}
  cos
  \big(
    2 \pi \tfrac{k}{n}
  \big)
  \;\in\;
  \mathbb{Q}
  \phantom{AA}
  \Leftrightarrow
  \phantom{AA}
  cos
  \big(
    2 \pi \tfrac{k}{n}
  \big)
  \;\in\;
  \tfrac{1}{2}\mathbb{Z}
  \phantom{AA}
  \Leftrightarrow
  \phantom{AA}
  cos
  \big(
    2 \pi \tfrac{k}{n}
  \big)
  \;\in\;
  \big\{
     -1, -1/2, 0, +1/2, +1
  \big\}
\]

(remember that $k,n \in \mathbb{Z}$)

A direct [[proof]] of this fact, using identities of [[trigonometric functions]], is given in [Jahnel, Sec. 3](#Jahnel).

=--

## References

* {#Naik} V. Naik, _[Characters are cyclotomic integers](https://groupprops.subwiki.org/wiki/Characters_are_cyclotomic_integers)_

* {#Yang} Kay Yang, _Rational valued characters_, ([pdf](http://www.math.toronto.edu/murnaghan/courses/mat445/kayyang.pdf), [[RationalValuedCharacters.pdf:file]])

* {#Jahnel} Jörg Jahnel, _When is the (co)sine of a rational angle equal to a rational number?_ ([pdf](https://www.uni-math.gwdg.de/jahnel/Preprints/cos.pdf), [[JahnelCosineRational.pdf:file]])

[[!redirects characters in characteristic zero are integers]]


charge

* [[charge of a conserved current]]

* [[electric charge]], [[magnetic charge]]

* [[D-brane charge]], [[M5-brane charge]]

* [[central charge]]

* [[Cohomotopy charge]]

[[!redirects charges]]

[[!redirects charged field]]

[[!redirects charged particle]]
[[!redirects charged particles]]

[[!redirects charge (physics)]]


charge conjugation matrix

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

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]], the [[matrix]] representing a [[bilinear form]] on a real [[spin representation]], i.e. on [[Majorana spinors]] is also called the _charge conjugation matrix_ on spinors.

For details see at _[Majorana spinor -- Charge conjugation matrix](Majorana+spinor#ChargeConjugationMatrix)_.

## References

* {#FigueroaOFarrill} [[José Figueroa-O'Farrill]], p. 12 of _Majorana spinors_ ([pdf](http://www.maths.ed.ac.uk/~jmf/Teaching/Lectures/Majorana.pdf))

The abstract relation to bilinear forms on spinors is commented on for instance around (3.4) of

* {#Freed99} [[Daniel Freed]], _Lecture 3 of [[Five lectures on supersymmetry]]_ 1999

and in appendix B.1 of

* {#ACDP} [[Dmitry Vladimirovich Alekseevsky]], [[Vicente Cortez]], C. Devchand, [[Antoine Van Proeyen]], _Polyvector Super-Poincaré Algebras_ Commun.Math.Phys. 253 (2004) 385-422 ([arXiv:hep-th/0311107](http://arxiv.org/abs/hep-th/0311107))

The matrix component yoga used in physics is summarized for instance in

* [[Antoine Van Proeyen]], section 3.1 of _Tools for supersymmetry_ ([arXiv:hep-th/9910030](http://arxiv.org/abs/hep-th/9910030))

* [[Joseph Polchinski]], volume II, appendix B of  _[[String theory]]_

* {#Brandt12-13} [[Friedemann Brandt]], section 2.1, 2.3 of _Supersymmetry algebra cohomology_  _I: Definition and general structure_  J. Math. Phys.51:122302, 2010, ([arXiv:0911.2118](http://arxiv.org/abs/0911.2118))

[[!redirects charge conjugation matrices]]


charge of a conserved current

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Variational calculus
+--{: .hide}
[[!include variational calculus - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $X$ be a ([[spacetime]]) [[smooth manifold]] of [[dimension]] $n$ , $E \to X$ a field bundle and $L \in \Omega^{n,0}(j_\infty E)$ a [[Lagrangian]].

For $j \in \Omega^{n-1,p}$ a [[conserved current]] and $\Sigma \subset X$ a submanifold of dimension $n-1$, the **charge** of $j$ relative to $\Sigma$ is the [[integral]]

$$
  Q_\Sigma = \int_\Sigma j
  \,.
$$

## Examples

* [[electric charge]], [[magnetic charge]]

* [[mass]], [[spin]]

* [[D-term]]

* [[supercharge]]

## Related concepts

* [[Noether's theorem]]

* [[central charge]]

[[!include gauge field - table]]

[[!redirects conserved charge]]
[[!redirects conserved charges]]


charge sector

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An irreducible representation in the _[[DHR category]]_.

## References

The formalization in the context of [[AQFT]] is for instance in section 1.4 of

* [[Hans-Jürgen Borchers]], [[Jakob Yngvason]], *On the PCT--Theorem in the Theory of Local Observables*, in: *Mathematical physics in mathematics and physics: Quantum and operator algebraic aspects*, Fields Inst. Commun. **30** (2001) 39-64  [[arXiv:math-ph/0012020](http://arxiv.org/abs/math-ph/0012020), [spire:538524](https://inspirehep.net/literature/538524)]

(which is otherwise about the [[PCT theorem]]).

[[!redirects charge sectors]]


charged and rotating black holes -- table

| **[[black hole]] [[spacetimes]]** | vanishing [[angular momentum]] | [[positive number|positive]] [[angular momentum]] |
|-|--------------------------------|-------------------------------|
| **vanishing [[charge]]** | [[Schwarzschild spacetime]] | [[Kerr spacetime]] |
| **[[positive number|positive]] [[charge]]** | [[Reissner-Nordstrom spacetime]] | [[Kerr-Newman spacetime]] |


charged current

[[!redirects charged currents]]

#Contents#
* table of contents
{:toc}

## Idea

In the [[standard model of particle physics]]: an [[interaction]] via the [[weak nuclear force]] mediated by [[W-bosons]].

## References

* Wikipedia, _[Charged current](http://en.wikipedia.org/wiki/Charged_current)_


Charles A. S. Young

* [institute page](https://researchprofiles.herts.ac.uk/en/persons/charles-young)

* [InSpire page](https://inspirehep.net/authors/1045680)

## Selected writings

On [[higher Kac-Moody algebras]]:

* [[Luigi Alfonsi]], [[Charles A. S. Young]], *Higher current algebras, homotopy Manin triples, and a rectilinear adelic complex*, Journal of Geometry and Physics **191** (2023) 104903 [[doi:10.1016/j.geomphys.2023.104903](https://doi.org/10.1016/j.geomphys.2023.104903), [math.QA/2208.06009](https://arxiv.org/abs/2208.06009)]

On [[derived differential geometry]] and [[higher Lie theory]] cast in ([[differential cohesion|differential]]) [[cohesive (infinity,1)-topos|cohesive $(\infty,1)$-topos]] theory:

* [[Luigi Alfonsi]], [[Charles A. S. Young]], *Towards non-perturbative BV-theory via derived differential cohesive geometry* [[arXiv:2307.15106](https://arxiv.org/abs/2307.15106)]

category: people

[[!redirects Charles Young]]


Charles Angas Hurst

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

## Selected writings

Observation that [[vacua]]/[[ground states]] of the [[free field|free]] [[Dirac field]] in a [[electromagnetic field|Coulomb]] [[background field]] are characterized by [[Fredholm operators]] whose [[index]] is the total charge in the vacuum:

* {#CareyHurstOBrien82} [[Alan Carey]], [[Charles Angas Hurst]], [[Denis O'Brien]], *Automorphisms of the canonical anticommutation relations and index theory*, Journal of Functional Analysis **48** 3 (1982) 360-393 $[$doi:10.1016/0022-1236(82)90092-1$]$

[[!redirects Charles Hurst]]
[[!redirects Charles A. Hurst]]
[[!redirects C. A. Hurst]]


Charles Boyer

* [webpage](https://en.wikipedia.org/wiki/Charles_P._Boyer)

## Selected writings

On the [[moduli space of Yang-Mills monopoles]] and [[rational maps]] to [[complex projective spaces]]:

* [[Charles P. Boyer]], [[B. M. Mann]], *Monopoles, non-linear $\sigma$-models, and two-fold loop spaces*,  Commun. Math. Phys. 115, 571–594 (1988) ([arXiv:10.1007/BF01224128](https://doi.org/10.1007/BF01224128))

On [[homotopy type of spaces of rational maps]]:

* [[Charles P. Boyer]], [[B. M. Mann]], [[J. C. Hurtubise]], [[R. J. Milgram]], *The topology of the space of rational maps into generalized flag manifolds*, Acta Mathematica. 1994 173(1):61-101 ([doi:10.1007/BF02392569](https://projecteuclid.org/journals/acta-mathematica/volume-173/issue-1/The-topology-of-the-space-of-rational-maps-into-generalized/10.1007/BF02392569.full))

On [[Sasakian geometry]]

* {#BoyerGalicki07} [[Charles Boyer]], [[Krzysztof Galicki]], _Sasakian Geometry_, Oxford Mathematical Monographs, Oxford University Press, 2007 ([doi:10.1093/acprof:oso/9780198564959.001.0001](https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198564959.001.0001/acprof-9780198564959))

* [[Charles Boyer]], _Sasakian geometry: The recent work of Krzysztof Galicki_ ([arXiv:0806.0373](https://arxiv.org/abs/0806.0373))

category: people

[[!redirects Charles P. Boyer]]
[[!redirects C. P. Boyer]]


Charles Doran

Charles Doran is a professor of mathematics at the University of Alberta.

* [webpage](https://www.charlesdoran.net)

## Selected writings

On [[orientifold]] [[Gepner models]]:

* Brandon Bates, [[Charles Doran]], Koenraad Schalm, _Crosscaps in Gepner Models and the Moduli space of $T^2$ Orientifolds_, Advances in Theoretical and Mathematical Physics, Volume 11, Number 5, 839-912, 2007 ([arXiv:hep-th/0612228](https://arxiv.org/abs/hep-th/0612228))

On [[D-brane charge]] in [[orientifolds]] formulated in [[KR-theory]]:

* {#DMR13} [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], _T-duality For orientifolds and twisted KR-theory_ ([arXiv:1306.1779](https://arxiv.org/abs/1306.1779))

* {#DMR14} [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], _String theory on elliptic curve orientifolds and KR-theory_ ([arXiv:1402.4885](https://arxiv.org/abs/1402.4885))

On the [[mirror map]] and related [[Picard-Fuchs equation]]s

* Charles F. Doran, _Picard-Fuchs uniformization: modularity of the mirror map and mirror-moonshine_, arxiv:[math.AG/9812162](https://arxiv.org/pdf/math/9812162)

## Related $n$Lab entries

* [[adinkra]]
* [[mirror symmetry]]

category: people


Charles Ehresmann

Charles Ehresmann was a pioneer in investigating [[groupoids]] and then [[category|categories]] for their applications in [[geometry|geometric]] problems. It is notable that he was a student of [[Elie Cartan]], famous for his work in Analysis. As a consequence, Ehresmann was fascinated by local-to-global problems, which are among the key problems in mathematics and science. This is one reason for his approach to [[category theory]] being different from that in the USA, where category theory was founded.

He had a succession of influential students, and among the concepts which he initiated are:  [[fibre bundles]], [[foliations]], [[germs]], [[gerbes]], [[double categories]], [[sketches]], [[topological groupoids]], [[Lie groupoids]], [[holonomy]], [[structured categories]].

The journal he founded and edited, [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]], has been continued by his widow, [[Andree Ehresmann]].

* [English Wikipedia page](http://en.wikipedia.org/wiki/Charles_Ehresmann)

* [Wikipédia français](http://fr.wikipedia.org/wiki/Charles_Ehresmann)

* [MacTutor biography](http://www.gap-system.org/~history/Biographies/Ehresmann.html)

* "The mathematical legacy of Charles Ehresmann", conference "Geometry and Topology of Manifolds" Bedlewo 2005, Editors: Jan Kubarski, Jean Pradines, Tomasz Rybicki and Robert Wolak, Institute of mathematics, Polish academy of sciences, Banach center publications 76 (2007).

## Selected writings

Collected works:

* *[Œuvres complètes et commentées](https://mes-ehres.fr/ChEh/oeuvres.htm)*

  I-1,2. Topologie  algébrique et géométrie différentielle, With commentary by [[Willem van Est]], [[Michel Zisman]], Georges Reeb, Paulette Libermann, [[René Thom]], [[Jean Pradines]], Robert Hermann, [[Anders Kock]], [[André Haefliger]], [[Jean Bénabou]], René Guitart, and [[Andrée Charles Ehresmann]], Edited by Andrée Charles Ehresmann,  Cahiers Topologie Géométrie Différentielle, 24, (1983) suppll. 1.

Introducing the notion of [[Ehresmann connections]] and proving [[Ehresmann's theorem]]:

* [[Charles Ehresmann]], *Les connexions infinitésimales dans un espace fibré différentiable*,

   Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris (1951) 29–55

  [[Séminaire Bourbaki]], **1** 24 (1952) 153–168 [[numdam:SB_1948-1951__1__153_0](http://www.numdam.org/item/?id=SB_1948-1951__1__153_0), [pdf](http://www.numdam.org/article/SB_1948-1951__1__153_0.pdf)]

Early proposal to grasp the notion of *[[mathematical structure]]* via [[category theory]], specifically in terms of [[forgetful functors]] between the [[groupoids]] which they form (cf. *[[stuff, structure, property]]*):

* {#Ehresmann57} [[Charles Ehresmann]], *Gattungen in Lokalen Strukturen*, Jahresbericht der Deutschen Mathematiker-Vereinigung **60** (1958) 49-77 [[dml:146434](https://eudml.org/doc/146434)]

* {#Ehresmann65} [[Charles Ehresmann]], _Catégories et Structures_, Séminaire Ehresmann. Topologie et géométrie différentielle **6** (1964) 1-31 [[numdam:SE_1964__6__A8_0](http://www.numdam.org/item?id=SE_1964__6__A8_0), [dml:112200](https://eudml.org/doc/112200)]

Introduction of the notion of *[[double category]]*:

* [[Charles Ehresmann]], *Catégories double et catégories structurées*, C.R. Acad. Paris 256 (1963) 1198-1201 [[[Ehresmann-CategoriesDoubles.pdf:file]], [gallica](https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1246)]

Introducing the notion of [[topological groupoids]] and [[Lie groupoids]]:

* [[Charles Ehresmann]], _Catégories topologiques et categories différentiables_, Colloque de Géométrie différentielle globale, Bruxelles, C.B.R.M., (1959) pp. 137-150 ([[EhresmannCategoriesTopologiques.pdf:file]], [zbMath:0205.28202](https://zbmath.org/?q=an:0205.28202))

Introducing the notion of [[internal categories]] (or at least something in this direction):

* [[Charles Ehresmann]], _Catégories structurées_, Annales scientifiques de l'École Normale Supérieure, Série 3, Tome 80 (1963) no. 4, pp. 349-426 ([numdam:ASENS_1963_3_80_4_349_0](http://www.numdam.org/item/ASENS_1963_3_80_4_349_0))

Introducing [[sketches]]:

* {#Ehresmann68} [[Charles Ehresmann]], _Esquisses et types de structures algébriques_, Bul. Inst. Polit. Iasi **14** 1–2 (1968) 1-14 [[[Ehresmann-EsquisseEtTypes.pdf:file]]]

On [[internalization]] of [[mathematical structures]] via [[sketches]]:

* [[Andrée Bastiani]], [[Charles Ehresmann]], _Categories of sketched structures_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 13 (1972) no. 2, pp. 104-214 ([numdam:CTGDC_1972__13_2_104_0](http://www.numdam.org/item/?id=CTGDC_1972__13_2_104_0))

category: people

[[!redirects Ehresmann]]
[[!redirects C. Ehresmann]]


Charles Fefferman

* [webpage](http://en.wikipedia.org/wiki/Charles_Fefferman)

## related $n$Lab entries:

* [[Fefferman-Graham ambient construction]]

category: people


Charles H. Bennett

* Wikipedia entry

## Selected writings

On [[reversible computing]]:

* [[Charles H. Bennett]], *Logical Reversibility of Computation*, IBM Journal of Research and Development  **17** 6  (1973) [[doi:10.1147/rd.176.0525](https://doi.org/10.1147/rd.176.0525)]

* [[Charles H. Bennett]], *The thermodynamics of computation -- a review*, International Journal of Theoretical Physics **21** (1982) 905–940 [[doi:10.1007/BF02084158](https://doi.org/10.1007/BF02084158)]

* [[Rolf Landauer]], [[Charles H. Bennett]]: *[The Fundamental Physical Limits of Computation](https://www.scientificamerican.com/article/the-fundamental-physical-limits-of-computation/)*, Scientific American **253** 1 (1985)

* [[Charles H. Bennett]], *Notes on the history of reversible computation*,  IBM Journal of Research and Development *32* 1 (1988) [[doi:10.1147/rd.321.0016](https://doi.org/10.1147/rd.321.0016)]

* [[Charles H. Bennett]], *Notes on Landauer's principle, reversible computation, and Maxwell's Demon*, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, **34** 3 (2003) 501-510 (doi:10.1016/S1355-2198(03)00039-X)

Introducing the notion of [[quantum key distribution]]:

* [[Charles H. Bennett]], [[Gilles Brassard]], *Quantum cryptography: Public key distribution and coin tossing*, International Conference on Computers, Systems & Signal Processing, Bangalore, India (1984) 175-179 [scan: [arXiv:2003.06557](https://arxiv.org/abs/2003.06557)]

  reprinted in: 	Theoretical Computer Science **560** 1 (2014) 7-11 [[doi:10.1016/j.tcs.2014.05.025](https://doi.org/10.1016/j.tcs.2014.05.025)]

Introducing the notion of [[quantum teleportation]]:

* {#BBCJPW93} [[Charles H. Bennett]], [[Gilles Brassard]], [[Claude Crépeau]], [[Richard Jozsa]], [[Asher Peres]], [[William K. Wootters]]: *Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels*, Phys. Rev. Lett. **70** 1895 (1993) [[doi:10.1103/PhysRevLett.70.1895](https://doi.org/10.1103/PhysRevLett.70.1895)]

Influential early discussion of ([[controlled quantum gate|controlled]]) [[quantum logic gates]]:

* {#BBCDMSSSW95} [[Adriano Barenco]], [[Charles H. Bennett]], [[Richard Cleve]], [[David P. DiVincenzo]], [[Norman Margolus]], [[Peter Shor]], [[Tycho Sleator]], [[John A. Smolin]], [[Harald Weinfurter]], *Elementary gates for quantum computation*, Phys. Rev. A**52** (1995) 3457 [[arXiv:quant-ph/9503016](https://arxiv.org/abs/quant-ph/9503016), [doi:10.1103/PhysRevA.52.3457](https://doi.org/10.1103/PhysRevA.52.3457)]

On ([[quantum entanglement|entangled]]) [[quantum states]] [as resources in](quantum+information#ReferencesQuantumResources) [[quantum information theory]]/[[quantum computation]], not unlike the idea of resources in [[linear logic]]:

* [[Charles H. Bennett]], *A resource-based view of quantum information*, Quantum Information & Computation **4** 6 (2004) 460–466 [[doi:10.5555/2011593.2011598](https://dl.acm.org/doi/abs/10.5555/2011593.2011598)]

category: people

[[!redirects Charles Bennet]]


Charles Hermite

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

## Selected writings

On [[quadratic forms]] and apparently introducing what is now known as *[[Hermitian forms]]*:

* [[Charles Hermite]], *Sur la théorie des formes quadratiques. Premier mémoire*,  Journal für die reine und angewandte Mathematik **47** (1854) 313-342 [[doi:10.1515/crll.1854.47.343](https://doi.org/10.1515/crll.1854.47.343)]

## Related entries

* [[Hermite polyomial]]

* [[Euler number]]

* [[countable set]]

category: people


Charles Kane

* [webpage](https://live-sas-physics.pantheon.sas.upenn.edu/people/standing-faculty/charles-kane)

## Selected writings

Observation that [[graphene]] appears as a [[topological insulator]] when its [[spin-orbit coupling]] is resolved, due to a [[quantum spin Hall effect]]:

* {#KaneMele05a} [[Charles Kane]], [[Eugene Mele]], _Quantum Spin Hall Effect in Graphene_, Phys. Rev. Lett. 95, 226801 (2005) ([arXiv:cond-mat/0411737](https://arxiv.org/abs/cond-mat/0411737), [doi:10.1103/PhysRevLett.95.226801](https://doi.org/10.1103/PhysRevLett.95.226801))

Characterization of the resulting [[topological phase of matter]] via introduction of the [[Kane-Mele invariant]]:

* {#KaneMele05b} [[Charles Kane]], [[Eugene Mele]],  *$Z_2$ Topological Order and the Quantum Spin Hall Effect*, Phys. Rev. Lett. **95** (2005) 146802 ([doi:10.1103/PhysRevLett.95.146802](https://doi.org/10.1103/PhysRevLett.95.146802))

On [[topological insulators]]:

* [[Liang Fu]], [[Charles Kane]], [[Eugene Mele]], *Topological Insulators in Three Dimensions*, Phys. Rev. Lett. **98** (2007) 106803 $[$[doi:10.1103/PhysRevLett.98.106803](https://doi.org/10.1103/PhysRevLett.98.106803)$]$

* [[M. Zahid Hasan]], [[Charles Kane]], *Colloquium: Topological insulators*, Rev. Mod. Phys. **82** (2010) 3045 $[$[doi:10.1103/RevModPhys.82.3045](https://doi.org/10.1103/RevModPhys.82.3045)$]$

On realizing [[Majorana zero modes]] via [[topological insulators]]:

* {#FuKane08} [[Liang Fu]], [[Charles Kane]], *Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator*, Phys. Rev. Lett. **100** (2008) 096407 [[doi:10.1103/PhysRevLett.100.096407](https://doi.org/10.1103/PhysRevLett.100.096407)]

On [[topological crystalline insulators]] and their classification by [[equivariant K-theory]]:

* Jorrit Kruthoff, [[Jan de Boer]], Jasper van Wezel, [[Charles L. Kane]], [[Robert-Jan Slager]], *Topological Classification of Crystalline Insulators through Band Structure Combinatorics*, Phys. Rev. X **7** (2017) 041069 $[$[doi:10.1103/PhysRevX.7.041069](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.7.041069)$]$

category: people

[[!redirects Charles L. Kane]]


Charles Kittel

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

## Selected writings

Classical textbook on [[solid state physics]]:

* [[Charles Kittel]], *Introduction to Solid State Physics*, Wiley (1953-) $[$[ISBN:978-0-471-41526-8](https://www.wiley.com/en-us/Introduction+to+Solid+State+Physics%2C+8th+Edition-p-9780471415268), [pdf](http://metal.elte.hu/~groma/Anyagtudomany/kittel.pdf),  [Wikipedia entry](https://en.wikipedia.org/wiki/Introduction_to_Solid_State_Physics)$]$

category: people


Charles Misner

* [Wikipedia entry](http://en.wikipedia.org/wiki/Charles_W._Misner)

## Selected writings

On [[geometrodynamics]]:

* [[Charles W. Misner]], [[John A. Wheeler]], *Classical physics as geometry*, Annals of Physics **2** 6 (1957) 525-603 \[doi:10.1016/0003-4916(57)90049-0\]

On the [[classical field theory]] of [[Einstein gravity]] ("[[general relativity]]"):

* [[Charles Misner]], [[Kip Thorne]], [[John Wheeler]]: *[[Gravitation]]* W. H. Freeman, San Francisco (1973) [ISBN:9780716703440, 9780691177793, [inspire:95654](https://inspirehep.net/literature/95654), Wikipedia entry]

## Related entries

* _[[Gravitation]]_

[[!redirects Charles W. Misner]]

category: people


Charles Rezk

Charles Waldo Rezk is a mathematician at the University of Illinois Urbana–Champaign.

He got his PhD degree in 1996 at MIT, advised by [[Michael J. Hopkins]].

His PhD students include [[Nathaniel Stapleton]] and [[Nima Rasekh]].

* [website](https://rezk.web.illinois.edu/)

## Selected writings

Proof of the [[canonical model structure]] on [[Cat]]:

* [[Charles Rezk]], *A model category for categories*, 1996 ([dvi](http://www.math.uiuc.edu/~rezk/cat-ho.dvi), [[Rezk_ModelCategoryForCategories.pdf:file]])

On the [[homotopy theory of homotopy theories]] (the [[(infinity,1)-category of (infinity,1)-categories]]) in terms of [[complete Segal spaces]]:

* {#Rezk98} [[Charles Rezk]], _A model for the homotopy theory of homotopy theory_, Trans. Amer. Math. Soc. 353 (2001), 973-1007 ([arXiv:math/9811037](https://arxiv.org/abs/math/9811037), [doi:10.1090/S0002-9947-00-02653-2](https://doi.org/10.1090/S0002-9947-00-02653-2))

Introducing [[model toposes]]/[[(∞,1)-toposes]]:

* {#Rezk} [[Charles Rezk]], _Toposes and homotopy toposes_, 2010 ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf), [[Rezk_HomotopyToposes.pdf:file]])

Introducing [[quasi-elliptic cohomology]]:

* {#Rezk14} [[Charles Rezk]], *Quasi-Elliptic Cohomology* (2014) [[[Rezk-QuasiEllipticCohomology.pdf:file]]]

  > (not publicly available until upload here, in 2023)

Review of [[higher topos theory]]:

* [[Charles Rezk]], _Lectures on Higher Topos Theory_, Leeds 2019 ([pdf](https://rezk.web.illinois.edu/leeds-lectures-2019.pdf), [[RezkHigherToposTheory2019.pdf:file]])

On [[complete Segal spaces]]:

* {#Rezk} [[Charles Rezk]], _A model for the homotopy theory of homotopy theory_ , Trans. Amer. Math. Soc., 353(3), 973-1007 ([pdf](http://www.math.uiuc.edu/~rezk/rezk-ho-models-final-changes.pdf))

On [[power operations]]:

* [[Charles Rezk]], _Lectures on power operations_ ([dvi](http://www.math.uiuc.edu/~rezk/power-operation-lectures.dvi)) (2006)

* [[Charles Rezk]], _Power operations in Morava E-theory --  a survey_ (2009) ([pdf](http://www.math.uiuc.edu/~rezk/midwest-2009-power-ops.pdf))

* {#Rezk14} [[Charles Rezk]], _Isogenies, power operations, and homotopy theory_, article ([pdf](http://www.math.uiuc.edu/~rezk/rezk-icm-talk-posted.pdf)) and talk at ICM 2014 ([pdf](http://www.math.uiuc.edu/~rezk/rezk-icm-2014-slides.pdf))

Expressing [[logarithmic cohomology operations]] in terms of the [[Bousfield-Kuhn functor]] and  [[power operations]]:

* {#Rezk06} [[Charles Rezk]], _The units of a ring spectrum and a logarithmic cohomology operation_, J. Amer. Math. Soc. 19 (2006), 969-1014 ([arXiv:math/0407022](http://arxiv.org/abs/math/0407022))

On [[∞-groups of units]], [[Thom spectra]] and [[twisted generalized cohomology]]:

* {#ABGHR08} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra and Thom spectra_ ([arXiv:0810.4535](http://arxiv.org/abs/0810.4535))

* [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory_, Journal of Topology, Volume7, Issue 4, December 2014 ([arXiv:1403.4320](https://arxiv.org/abs/1403.4320), [arXiv:10.1112/jtopol/jtu009](https://doi.org/10.1112/jtopol/jtu009))

* {#ABGHR14} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology_, Journal of Topology Volume 7, Issue 3 2014 Pages 869–893  ([arXiv:1403.4325](http://arxiv.org/abs/1403.4325), [doi:10.1112/jtopol/jtt035](https://doi.org/10.1112/jtopol/jtt035))

On [[Theta-spaces]]:

* [[Charles Rezk]], _A cartesian presentation of weak $n$-categories_ Geom. Topol. 14 (2010), no. 1, 521–571 ([arXiv:0901.3602](http://arxiv.org/abs/0901.3602))

  _Correction to "A cartesian presentation of weak $n$-categories"_ Geom. Topol. 14 (2010), no. 4, 2301–2304. MR 2740648 ([pdf](http://www.math.uiuc.edu/~rezk/cs-objects-correction.pdf))

* [[Charles Rezk]], _Cartesian presentations of weak $n$-categories -- An introduction to $\Theta_n$-spaces_ (2009) ([pdf](http://www.math.uiuc.edu/~rezk/northwestern-2009-n-cat-handout.pdf))

On the [[string orientation of tmf]]:

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Michael Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))

On [[slice (infinity,1)-topos|slices]] of [[global equivariant homotopy theory]] as a [[cohesive (∞,1)-topoi]] over $G$-[[equivariant homotopy theory]] ([[cohesion of global- over G-equivariant homotopy theory]]):

* [[Charles Rezk]], _[[Global Homotopy Theory and Cohesion]]_, 2014 ([pdf](http://www.math.uiuc.edu/~rezk/global-cohesion.pdf), [[Rezk14.pdf:file]])

On sufficient conditions for [[geometric realization of simplicial topological spaces]] to [[commutativity of limits and colimits|commute]] with [[homotopy pullback]]

* {#Rezk14} [[Charles Rezk]], _When are homotopy colimits compatible with homotopy base change?_, 2014 ([pdf](https://faculty.math.illinois.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf), [[RezkHomotopyColimitsBaseChange.pdf:file]])

On [[elliptic cohomology]]:

* [[Charles Rezk]], _Elliptic cohomology and elliptic curves_, Felix Klein Lectures, Bonn  (2015) $[$[web](http://www.hcm.uni-bonn.de/fkl-rezk), [pdf](https://faculty.math.illinois.edu/~rezk/felix-klein-lectures-notes.pdf), [[RezkElliptic2015.pdf:file]], video rec: [part 1](https://youtu.be/eFzB_2fkOxY), [2](https://youtu.be/4e9e3AtK7yg), [3](https://youtu.be/ueL5dMH9JoI), [4](https://youtu.be/r_7SsIoU9No)$]$

On [[classifying spaces]] for [[equivariant principal bundles]] with [[1-truncated]] [[compact Lie group|compact Lie]] [[structure group]]:

* [[Charles Rezk]], *Classifying spaces for 1-truncated compact Lie groups*, Algebr. Geom. Topol. **18** (2018) 525-546 ([arXiv:1608.02999](https://arxiv.org/abs/1608.02999))

On [[quasi-categories]]:

* {#Rezk16} [[Charles Rezk]], _Stuff about quasicategories_, Lecture Notes for course at University of Illinois at Urbana-Champaign, 2016, version May 2017 ([pdf](http://math.uiuc.edu/~rezk/595-fal16/quasicats.pdf), [[RezkQuasicategories.pdf:file]])

* [[Charles Rezk]], *Introduction to quasicategories* (2022) [[pdf](https://faculty.math.illinois.edu/~rezk/quasicats.pdf), [[Rezk-IntroToQuasicategories.pdf:file]]]

On [[compactly generated topological spaces]]:

* [[Charles Rezk]], *Compactly Generated Spaces*, 2018 ([pdf](https://faculty.math.illinois.edu/~rezk/cg-spaces-better.pdf), [[Rezk_CompactlyGeneratedSpaces.pdf:file]])

On [[accessible (infinity,1)-categories|accessible $(\infty,1)$-categories]]:

* [[Charles Rezk]], *Generalizing accessible ∞-categories*, 2021 ([pdf](https://faculty.math.illinois.edu/~rezk/accessible-cat-thoughts.pdf)).

On [[spectral algebraic geometry]]:

* [[Charles Rezk]], *Spectral algebraic geometry* ([pdf](https://faculty.math.illinois.edu/~rezk/sag-chapter-web.pdf)) in:  [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds.), *[[Stable categories and structured ring spectra]]*, MSRI Book Series, Cambridge University Press

## Related $n$Lab entries

in [[higher algebra]]/[[stable homotopy theory]]

* [[power operation]]

* [[logarithmic cohomology operation]]

* [[string orientation of tmf]]

* [[twisted cohomology]], [[twisted Umkehr map]]

in [[higher category theory]]:

* [[(∞,n)-category]]

* [[Theta space]]/[[(n,r)-category]]

in [[higher topos theory]]:

* [[(∞,1)-topos]]

* [[higher topos theory]]

* [[global equivariant homotopy theory]]

category: people


Charles Sanders Peirce

#Contents#
* table of contents
{:toc}

Charles Sanders Peirce (1839-1914), a philosopher, logician and scientist, was one of the founders of modern symbolic [[logic]]. In particular, he developed a form of [[predicate logic]]. Peirce is seen as the father of pragmatism, although he later distanced himself from other American philosophers who identified them as pragmatists, by adopting the term _pragmaticism_.

Peirce was a prolific thinker who left behind an enormous corpus of work, much unpublished. Where his _Collected Papers_ run to 8 volumes, an ongoing edition aims to publish 30 volumes.

Peirce's philosophy may be seen as Schellingism transformed in light of (in Peirce's time) modern physics, as Peirce himself notes in an 1894 letter to William James:

> My views were probably influenced by [[Schelling]] - by all stages of Schelling, but especially the _Philosophie der Natur_. I consider Schelling as enormous, and one thing I admire about him is his freedom from the trammels of system, and his holding himself uncommitted to any previous utterance. in that, he is like a scientific man. If you were to call my philosophy Schellingism transformed in the light of modern physics, I should not take it hard.

## Existential graphs

Peirce devised a graphical notation, known as **existential graphs**, to represent logical calculi. There were three systems of such graphs: the system _alpha_, to represent [[propositional logic]], the system _beta_, to represent [[predicate logic]], and the system _gamma_, to represent [[modal logic]] ([MaPiet 18](#MaPiet18)).

[[Geraldine Brady]] and [[Todd Trimble]] have given a category theoretic interpretation of the alpha and beta systems. The latter, a form of [[string diagram|string diagrammatic]] notation, was developed ([PontoShul](#PontoShul)) into a [[string diagram]] notation for [[indexed monoidal categories]]. A development also appears in [MellZeil](#MellZeil), see also [BSS18](#BSS18).

##References

Stanford Encyclopedia of Philosophy entries:

* [Charles Sanders Peirce](https://plato.stanford.edu/entries/peirce/)
* [Peirce's Deductive Logic](https://plato.stanford.edu/entries/peirce-logic/)
* [Peirce's Theory of Signs](https://plato.stanford.edu/entries/peirce-semiotics/)
* [Peirce's View of the Relationship Between His Own Work and German Idealism](https://plato.stanford.edu/entries/peirce/self-contextualization.html)
* [Peirce on Abduction](https://plato.stanford.edu/entries/abduction/peirce.html)

Charles S. Peirce studies

* [website](http://www.peirce.org/)

Other references

* [[Geraldine Brady]] and [[Todd Trimble]] (2000a). A Categorical Interpretation of C. S. Peirce's Propositional Logic Alpha. Journal of Pure and Applied Algebra 149 (2000): 213-239, ([pdf](https://core.ac.uk/download/pdf/82545173.pdf))

* {#BradyTrimble} [[Geraldine Brady]] and [[Todd Trimble]] (2000b). _[[A string diagram calculus for predicate logic|A String Diagram Calculus for Predicate Logic and C. S. Peirce's System Beta]]_, preprint

* [[Kate Ponto]] and [[Mike Shulman]], Duality and traces in indexed monoidal categories, ([web](http://www.sandiego.edu/~shulman/papers/indexed-color.pdf))
{#PontoShul}

* [[Fernando Zalamea]] (2012), Peirce's Logic of Continuity: A Conceptual and Mathematical Approach

* Frederik Stjernfelt (2014), Natural Propositions: The Actuality of Peirce's Doctrine of Dicisigns

* Rosa Maria Perez-Teran Mayorga (2008), From Realism to 'Realicism': The Metaphysics of Charles Sanders Peirce

* Andrew Reynolds (2002), Peirce's Scientific Metaphysics: The Philosophy of Chance, Law, and Evolution

* Matthew Moore (ed) (2010), New Essays on Peirce's Mathematical Philosophy

* [[Louis Kauffman]] (2001), [The Mathematics of Charles Sanders Peirce](http://homepages.math.uic.edu/~kauffman/CHK.pdf)

* C.S. Peirce & Matthew Moore (ed) (2010), Philosophy of Mathematics: Selected Writings

* {#MellZeil} [[Paul-André Melliès]] and [[Noam Zeilberger]] (2016) _A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine_, ([	arXiv:1601.06098, cs.LO](http://arxiv.org/abs/1601.06098))

* Fernando Zalamea, Peirce's logic of continuity, a conceptual and mathematical approach, [link](https://www.academia.edu/31948393/ZalameaPeirceCont.pdf)

* {#MaPiet18} Minghui Ma, Ahti-Veikko Pietarinen (2018), _Gamma graph calculi for modal logics_, Synthese 195: 3621, ([doi](https://doi.org/10.1007/s11229-017-1390-3))

* {#BSS18} Filippo Bonchi, Jens Seeber, Pawel Sobocinski, _Graphical Conjunctive Queries_, ([arXiv:1804.07626](https://arxiv.org/abs/1804.07626))

[[!redirects C.S. Peirce]]
[[!redirects Charles Peirce]]
[[!redirects Charles S. Peirce]]
[[!redirects Peirce]]

category: people, philosophy


Charles Sommerfield

* [webpage](http://physics.yale.edu/people/charles-sommerfield)

## related entries

* [[BPS state]]

category: people


Charles Thomas

* [Mathematics genealogy page](http://www.genealogy.ams.org/id.php?id=65752)

## Selected writings

On [[free group actions on n-spheres]]:

* {#MadsenThomasWall76} [[Ib Madsen]], [[Charles B. Thomas]], [[C. T. C. Wall]], *The topological spherical space form problem II: existence of free actions*, Topology Volume 15, Issue 4, 1976, Pages 375-382 (doi:10.1016/0040-9383(76)90031-8)

* {#MadsenThomasWall83} [[Ib Madsen]], [[Charles B. Thomas]], [[C. T. C. Wall]], *Topological spherical space form problem III: Dimensional bounds and smoothing*,  Pacific J. Math. 106(1): 135-143 (1983) ([pjm:1102721110](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-106/issue-1/Topological-spherical-space-form-problem-III-Dimensional-bounds-and-smoothing/pjm/1102721110.full))

On [[elliptic cohomology]] (and [[K3-cohomology]]):

* {#Thomas02} [[Charles Thomas]], _Elliptic cohomology_, Kluwer Academic, 2002 ([doi:10.1007/b115001](https://link.springer.com/book/10.1007/b115001), [pdf](https://link.springer.com/content/pdf/10.1007%2Fb115001.pdf))

## Related $n$Lab entries

* [[elliptic cohomology]]

category: people

[[!redirects Charles B. Thomas]]


Charles Thorn

[[!redirects Charles B. Thorn]]

* [webpage](http://www.phys.ufl.edu/~thorn/)

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

## Selected writings

Proving the [[Goddard-Thorn no-ghost theorem]]:

* {#GoddardThorn72} [[Peter Goddard]], [[Charles Thorn]], _Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model_, Phys. Lett., B 40, No. 2 (1972), 235-238 ([spire:74899](http://inspirehep.net/record/74899/))

Early discussion of [[quantization]] of the [[string]] [[sigma-model]] in [[light-cone gauge]]:

* [[Peter Goddard]], [[Jeffrey Goldstone]], [[Claudio Rebbi]], [[Charles B. Thorn]], *Quantum dynamics of a massless relativistic string*, Nuclear Physics B **56** 1 (1973) 109-135 [doi:10.1016/0550-3213(73)90223-X]

Early discussion of [[discretized light-cone quantization]]:

* {#Thorn77} [[Charles B. Thorn]], *On the derivation of dual models from field theory*, Physics Letters B **70** 1  (1977) 85-87 [doi:10.1016/0370-2693(77)90351-3]

* {#Thorn78} [[Charles B. Thorn]], *Derivation of dual models from field theory. II*, Phys. Rev. D **17** (1978) 1073 [[doi:10.1103/PhysRevD.17.1073](https://doi.org/10.1103/PhysRevD.17.1073)]

and on the issue of excluding the $P^+ = 0$-mode:

* [[Joel S. Rozowsky]], [[Charles B. Thorn]], *Spontaneous Symmetry Breaking at Infinite Momentum without $P^+$ Zero-Modes*, Phys. Rev. Lett. **85** (2000) 1614-1617 [[arXiv:hep-th/0003301](https://arxiv.org/abs/hep-th/0003301), [doi:10.1103/PhysRevLett.85.1614](https://doi.org/10.1103/PhysRevLett.85.1614)]

On [[conformal field theory|conformal]] [[quantization]] of [[Liouville theory]]:

* [[Thomas L. Curtright]], [[Charles B. Thorn]], *Conformally Invariant Quantization of the Liouville Theory*, Phys. Rev. Lett. **48** (1982) 1309; Erratum Phys. Rev. Lett. **48**,  (1982) 1768 [[doi:10.1103/PhysRevLett.48.1309](https://doi.org/10.1103/PhysRevLett.48.1309)]

## Related entries

* [[Goddard-Thorn no-ghost theorem]]

* [[string bit model]]

category: people


Charles Torre

* [webpage](http://www.physics.usu.edu/torre/cgt_home/cgt_home/C._G._Torre_%E2%80%A2_Physics_%E2%80%A2_Utah_State_University.html)

## related $n$Lab entries

* [[picture of mechanics]]

category: people


Charles Walter

* [webpage](https://math.unice.fr/~walter/)

## Selected writings

On [[MSp]] and [[quaternionic oriented cohomology theory]]:

* [[Ivan Panin]], [[Charles Walter]], _Quaternionic Grassmannians and Pontryagin classes in algebraic geometry_ ([hal:00531725](https://hal.archives-ouvertes.fr/hal-00531725), [pdf](https://www.math.uni-bielefeld.de/LAG/man/405.pdf))

* [[Ivan Panin]], [[Charles Walter]], _Quaternionic Grassmannians and Borel classes in algebraic geometry_ ([arXiv:1011.0649](https://arxiv.org/abs/1011.0649))

* [[Ivan Panin]], [[Charles Walter]], _On the algebraic cobordism spectra $MSL$ and $MSp$_ ([arXiv:1011.0651](https://arxiv.org/abs/1011.0651))

## Related $n$Lab entries

* [[algebraic cobordism]]

category: people


Charles Weibel

* [website](http://www.math.rutgers.edu/~weibel/)

## Selected writings:

On [[homological algebra]]

* Charles Weibel, _[[An introduction to homological algebra]]_,  Cambridge Studies in Adv. Math. 38, CUP 1994

On [[algebraic K-theory]]:

* Charles Weibel,  [The development of algebraic K-theory before 1980](http://www.math.rutgers.edu/~weibel/papers-dir/khistory.pdf), published in _Algebra, K-theory, groups, and education: on the occasion of Hyman Bass's 65th birthday_, a volume of _Contemporary Mathematics_, American Mathematical Soc, 1999.

On [[KR-theory]] of [[algebraic varieties]]:

* [[Max Karoubi]], [[Charles Weibel]], *Algebraic and Real K-theory of Algebraic varieties*, Topology **42** (2003) 715-742 [[arXiv:math/0509412](https://arxiv.org/abs/math/0509412), doi:10.1016/S0040-9383(02)00069-1]

On [[twisted K-theory]], [[Hermitian K-theory]] and [[KR-theory]]:

* [[Max Karoubi]], [[Marco Schlichting]], [[Charles Weibel]], *The Witt group of real algebraic varieties*, Journal of Topology **9** 4 (2016) 1257-1302 [[arXiv:1506.03862](https://arxiv.org/abs/1506.03862), [doi:10.1112/jtopol/jtw024]( https://doi.org/10.1112/jtopol/jtw024)]

* [[Max Karoubi]], [[Charles Weibel]], *Twisted $K$-theory, Real $A$-bundles and Grothendieck–Witt groups*, Journal of Pure and Applied Algebra **221** 7 (2017) 1629-1640 [[doi:10.1016/j.jpaa.2016.12.020](https://doi.org/10.1016/j.jpaa.2016.12.020)]

## Related entries

* [[homological algebra]]

* [[algebraic K-theory]]

* [[motivic cohomology]]

category: people

[[!redirects C. Weibel]]
[[!redirects Charles A Weibel]]
[[!redirects Charles A. Weibel]]
[[!redirects Chuck Weibel]]


Charles Wells

Charles Wells is Professor Emeritus of Mathematics at
[Case Western Reserve University](http://en.wikipedia.org/wiki/Case_Western_Reserve_University); his research interest has been in [[group theory]], [[category theory]] and its connections with theoretical computer science, in mathematical discourse, and in the problems of teaching and understanding mathematics.

* [website](http://www.cwru.edu/artsci/math/wells/home.html)

* [publications](http://www.cwru.edu/artsci/math/wells/pub/papers.html)

## Selected writings:

Exposition of [[sheaf toposes]]:

* [[Michael Barr]], [[Colin McLarty]], [[Charles Wells]],  *Variable set theory*, prepared for *Scientific American* but unpublished (~1985) [[pdf](https://www.math.mcgill.ca/barr/papers/vst.pdf), [[BarrMcLartyWells-VariableSetTheory.pdf:file]]]

On [[topos theory]], [[monads]] ("triples") and [[algebraic theories]] via [[sketches]]:

* [[Michael Barr]], and  [[Charles Wells]], *Toposes, Triples, and Theories*, Grundlehren der math. Wissenschaften 278 Springer 1983, republished in: Reprints in Theory and Applications of Categories, No. 12 (2005) pp. 1-287 ([tac:tr12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html) in [[TAC reprints series]])

On [[category theory]]:

* _Category theory for computing science_ with [[Michael Barr]] ([web](http://www.cwru.edu/artsci/math/wells/pub/ctcs.html))

On [[sketches]]:

* [[Charles Wells]], *Sketches: Outline with references*, 1993 ([citeseer:10.1.1.217.3531](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.217.3531), [[Wells_Sketches.pdf:file]])

[[!redirects C. Wells]]

category: people


Charles Zhaoxi Xiong

* [personal page](https://charleszxiong.com/)

## Selected writings

On [[symmetry protected topological phases]]:

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

On [[topological phases of matter]]:

* [[Charles Zhaoxi Xiong]], p. 13 of: *Classification and Construction of Topological Phases of Quantum Matter* $[$[arXiv:1906.02892](https://arxiv.org/abs/1906.02892)$]$

category: people


Charles-Michel Marle

* [webpage](http://marle.perso.math.cnrs.fr)

category: people


charm quark

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## Idea

In the [[standard model of particle physics]] the _charm quark_ is the up-type [[quark]] in the second [[generation of fermions]], a [[heavy quark]].

## Related concepts

* [[charmonium]]

[[!include flavors of fundamental particles -- table]]

## References

### General

* Marco Gersabeck, _Introduction to Charm Physics_, Proceedings of Science, Volume 220 - _Flavorful Ways to New Physics (FWNP)_ - Keynote Talks  ([arXiv:1503.00032](https://arxiv.org/abs/1503.00032), [spire:1346952](https://inspirehep.net/literature/1346952), [doi:10.22323/1.220.0001](https://doi.org/10.22323/1.220.0001))

See also

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

On indirect searches for New Physics (i.e. [[phenomenology]] beyond the [[standard model of particle physics]] via [[charm quark]] physics:

* [[Alexey Petrov]], _Searching for New Physics with Charm_ ([arXiv:1003.0906](https://arxiv.org/abs/1003.0906))

and via [[flavour anomalies]]:

* [[Alexey Petrov]], _Theory of rare charm decays into leptons_ ([arXiv:1704.03862](https://arxiv.org/abs/1704.03862))

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

[[!redirects charm quarks]]


Charmaine Sia

* [webpage](http://www.math.harvard.edu/~sia/)

## related $n$Lab entries

* [[Milnor's theorem on MU]]

category: people


charmonium

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## Idea

In [[quantum hadrodynamics]], _charmonium_ $J/\psi$ is the [[charm quark|charm]]-form of _[[quarkonium]]_: a [[meson]] which is a [[bound state]] of a [[charm quark]] and its [[antiparticle]].

## Related concepts

* [[XYZ particle]]

## References

* An-Ping Chen, Yan-Qing Ma, Hong Zhang, *A short theoretical review of charmonium production* ([arXiv:2109.04028](https://arxiv.org/abs/2109.04028))

See also

* Wikipedia, _[J/psi meson](https://en.m.wikipedia.org/wiki/J/psi_meson)_

On the [[Dalitz decay]] of charmonium:

* _Study of the Dalitz decay $J/\psi \to e^+e^- \eta$_, Phys. Rev. D 99, 012006 (2019) ([arXiv:1810.03091](https://arxiv.org/abs/1810.03091))

Discussion via [[AdS/QCD]]:

* Miguel Angel Martin Contreras, Saulo Diles, Alfredo Vega, _Heavy quarkonia spectroscopy at zero and finite temperature in bottom-up AdS/QCD_ ([arXiv:2101.06212](https://arxiv.org/abs/2101.06212))

Observation of a charmonium decay products interpreted as originating from a fully [[charm quark|charmed]] [[tetraquark]]:

* [[LHCb collaboration]], *Observation of structure in the $J/\psi$-pair mass spectrum*, Science Bulletin 65 (2020) 1983 ([arXiv:2006.16957](https://arxiv.org/abs/2006.16957), [doi:10.1016/j.scib.2020.08.032](https://doi.org/10.1016/j.scib.2020.08.032), [CERN news](https://phys.org/news/2020-07-cern-physicists-discovery-unique-particle.html))

reviewed in

* [[Jean-Marc Richard]], *About the $J/\psi J/\psi$ peak of LHCb: fully-charmed tetraquark?*, Science Bulletin Volume 65, Issue 23, 15 December 2020, Pages 1954-1955 ([arXiv:2008.01962](https://arxiv.org/abs/2008.01962), [doi:10.1016/j.scib.2020.08.020](https://doi.org/10.1016/j.scib.2020.08.020))

[[!redirects charmonia]]

[[!redirects J/ψ-meson]]
[[!redirects J/ψ-mesons]]

[[!redirects J/ψ meson]]
[[!redirects J/ψ mesons]]

[[!redirects psi-meson]]
[[!redirects psi-mesons]]
[[!redirects psi meson]]
[[!redirects psi mesons]]

[[!redirects psion]]
[[!redirects psions]]


charracteristic


Chee-Keng Yap

* [personal page](https://cs.nyu.edu/yap/)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=51193)

## Selected writings

On [[exact real computer arithmetic]]:

* [[Chee-Keng Yap]], [[Thomas Dubé]], *The exact computation paradigm*,  in: *Computing in Euclidean Geometry*, Lecture Notes Series on Computing, World Scientific (1995) 452-492  [[doi:10.1142/9789812831699_0011](https://doi.org/10.1142/9789812831699_0011), [[YapDube-ExactComputationParadigm.pdf:file]]]

category: people

[[!redirects Chee Yap]]


Cheeger-Gromoll splitting theorem

> This page is about the statement in Riemannian geometry. For the [[splitting principle]] (in algebraic topology) see there.

***

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{:toc}

## Idea

On the classification of  [[Riemannian manifolds]] with non-[[negative number|negative]] [[Ricci curvature]]

## References

* {#CheegerGromoll} [[Jeff Cheeger]], [[Detlef Gromoll]], _The splitting theorem for manifolds of nonnegative Ricci curvature_, J. Differential Geom. Volume 6, Number 1 (1971), 119-128 ([euclid:1214430220](https://projecteuclid.org/euclid.jdg/1214430220))

* Arthur E. Fischer, Joseph A. Wolf,  _The structure of compact Ricci-flat Riemannian manifolds_ J. Differential Geom. 10 (1975), no. 2, 277–288 ([euclid:jdg/1214432794](https://projecteuclid.org/euclid.jdg/1214432794))

See also

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

* [[Bobby Acharya]], Section 1.2 of _Supersymmetry, Ricci Flat Manifolds and the String Landscape_ ([arXiv:1906.06886](https://arxiv.org/abs/1906.06886))

[[!redirects Cheeger-Gromoll splitting theorems]]

[[!redirects Cheeger-Gromoll theorem]]
[[!redirects Cheeger-Gromoll theorems]]

[[!redirects splitting theorem]]
[[!redirects splitting theorems]]


Cheeger-Simons differential character

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## Idea

The notion of _differential character_ as introduced by [CheegerSimons](#CheegerSimons) is one geometric model for the [[differential cohomology]]-refinement of ordinary integral cohomology -- i.e. of the [[cohomology theory]] represented by the [[Eilenberg-MacLane spectrum]] $K(-,\mathbb{Z})$.

Accordingly, Cheeger-Simons differential characters model [[connection on a bundle|connection]]s on circle n-group-[[principal ∞-bundle]]s ( $U(1)$-$(n-1)$-[[gerbe]]s)  and as such are equivalent to other models for these structures, such as [[Deligne cohomology]]. For $n=1$ these are ordinary [[connection on a bundle|connections]] on ordinary [[circle group]]-[[principal bundle]]s.

The definition of CS-differential characters encodes rather directly the higher dimensional notion of [[parallel transport]] of such higher connections: a CS-character is a rule that assigns values in the [[circle group]] $U(1)$ (whence "[[character]]") to $n$-dimensional  surfaces $\Sigma_n \to X$ in a manifold $X$, such that whenever $\Sigma_n = \partial \Sigma_{n+1}$ is the boundary of a $\phi : \Sigma_{n+1} \to X$, this assignment coincides with the integral $\int_{\Sigma_{n+1}} \phi^* F$ of a smooth [[curvature]] $(n+1)$-form $F \in \Omega^{n+1}_{cl}(X)$.

## As secondary characteristic classes

Since Cheeger-Simons characters enocde information beyond the [[curvature characteristic form]] which represents the underlying [[characteristic class]] in de Rham cohomology, they are frequently called [[secondary characteristic class]]es, in particular if the curvature characteristic form vanishes so that the corresponding [[Chern-Simons form]] becomes exact.

## Related concepts

* [[ordinary differential cohomology]]

* [[Chern-Simons form]]

* [[circle n-bundle with connection]]

* [[Chern-Weil homomorphism]]

* [[secondary characteristic class]]

* [[Cheeger-Simons homomorphism]]

## References

The original article is

* {#CheegerSimons} [[Jeff Cheeger]],  [[James Simons]], _Differential characters and geometric invariants_, pages 50–80 in: _Geometry and Topology_, Lecture Notes in Mathematics, vol 1167. Springer (1985)  ([pdf](http://numr.wdfiles.com/local--files/differential-cohomology/cheeger-simons.pdf), [doi:10.1007/BFb0075212](https://doi.org/10.1007/BFb0075212))

building on

* [[Shiing-shen Chern]], [[James Simons]], _[[Characteristic forms and geometric invariants]]_, The Annals of Mathematics, Second Series, Vol. 99, No. 1 (1974) ([jstor](http://www.jstor.org/stable/1971013)).

Further developments are in

* [[James Simons]], [[Dennis Sullivan]], _Axiomatic characterization of ordinary differential cohomology_,  J Topology (2008) 1 (1): 45-56 ([arXiv:math/0701077](https://arxiv.org/abs/math/0701077), [doi:10.1112/jtopol/jtm006]( https://doi.org/10.1112/jtopol/jtm006) [web pdf](http://jtopol.oxfordjournals.org/content/1/1/45.full.pdf+html))

* Mark Brightwell, Paul Turner, _Relative differential characters_ ([arXiv:math.AT/0408333](http://arxiv.org/abs/math.AT/0408333))

See also:

* [[Christian Bär]], [[Christian Becker]], *Differential Characters and Geometric Chains*, in: *Differential Characters*, Lecture Notes in Mathematics **2112**, Springer (2014)  [[arXiv:1303.6457](https://arxiv.org/abs/1303.6457), [doi:10.1007/978-3-319-07034-6_1](https://doi.org/10.1007/978-3-319-07034-6_1)]

For a review in the broader context of [[differential cohomology]] see also

* [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_ (2005)

[[!redirects differential character]]
[[!redirects differential characters]]
[[!redirects Cheeger-Simons differential characters]]


Cheeger-Simons homomorphism

+-- {: .rightHandSide}
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#Contents#
* table of contents
{:toc}

## Idea

Where the [[Chern-Weil homomorphism]] assigns [[characteristic forms]] in [[de Rham cohomology]] to [[principal bundles]], the _Cheeger-Simons homomorphism_ refines this to an assignment of [[secondary characteristic classes]] in [[ordinary differential cohomology]] to [[principal connections]]:

$$
  \array{
    G Connections(X)_{/\sim}
    &\overset{
      \overset{
        \color{blue}
        \text{Cheeger-Simons homomorphism}
      }{
        cs_G
      }
    }{\longrightarrow}&
    Hom
    \Big(
      H^\bullet(B G;\, \mathbb{Z})
      \,,\,
      \widehat H^\bullet(X)
    \Big)
    \\
    \big\downarrow
    &&
    \big\downarrow
    \\
    G Bundles(X)_{/\sim}
    &\underset{
      \underset{
        \color{blue}
        \text{Chern-Weil homomorphism}
      }{cw_G}
    }{\longrightarrow}&
    Hom
    \Big(
      imv^\bullet(\mathfrak{g})
      \,,\,
      H_{dR}^\bullet(X)
    \Big)
  }
$$

## References

* {#CheegerSimons85} [[Jeff Cheeger]],  [[James Simons]], Section 2 of: _Differential characters and geometric invariants_, pages 50–80 in: _Geometry and Topology_, Lecture Notes in Mathematics, vol 1167. Springer (1985)  ([pdf](http://numr.wdfiles.com/local--files/differential-cohomology/cheeger-simons.pdf), [doi:10.1007/BFb0075212](https://doi.org/10.1007/BFb0075212))

* [[Michael Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_, J. Differential Geom. Volume 70, Number 3 (2005), 329-452 ([arXiv:math.AT/0211216](http://arxiv.org/abs/math.AT/0211216), [euclid:1143642908](https://projecteuclid.org/euclid.jdg/1143642908))


chemical element

[[!redirects chemical elements]]

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=--
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* table of contents
{:toc}

## Idea

In [[chemistry]] a type of [[atom]] is called a _chemical element_. The [[bound states]] of these are the [[molecules]].

The _[[atomic number]]_ of a [[chemical element]] is the [[natural number|number]] of [[protons]] in any [[atomic nuclei]] of this element.

The corresponding [[natural number|number]] of [[neutrons]] on the nucleus tends to be close to the number of protons, hence to the atomic number of the element, but need not equal it and my differ considerably for heavy elements. These variants of neutron number for give atomic number specify the _[[isotopes]]_ of the chemical element.

The [[periodic table of the elements]] organizes the elements according to their [[atomic number]]. This behaviour is explained by the [[quantum physics]] of [[atoms]]. This transition area between [[quantum physics]] and [[chemistry]] is called [[quantum chemistry]].

## Examples

* [[carbon]]

* [[nitrogen]]

## References

See also

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

Speculative analogy to classification of [[complex surfaces]] (different from but inspired by [[Skyrmion]] [[model (in theoretical physics)|models]] for [[atomic nuclei]]):

* {#AtiyahManton16} [[Michael Atiyah]], [[Nicholas Manton]], _Complex Geometry of Nuclei and Atoms_ ([arXiv:1609.02816](https://arxiv.org/abs/1609.02816))


chemical potential

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### Context
#### Physics
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## Idea

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## References

See also

* Wikipedia, *[Chemical potential](https://en.wikipedia.org/wiki/Chemical_potential)*

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