cup product

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## Idea

Recall from the discussion at [[cohomology]] that every notion of cohomology (e.g. [[group cohomology]], [[abelian sheaf cohomology]], etc) is given by Hom-spaces in an [[(∞,1)-topos]] $\mathbf{H}$. Cohomology on an object $X \in \mathbf{H}$ with coefficients in an object $A \in \mathbf{H}$ is

$$
  H(X,A) := \pi_0 \mathbf{H}(X,A)
  \,.
$$

The _cup product_ is an operation on cocycles with coefficients $A_1$ and $A_2$ that is induced from a pairing of coefficients given by some morphism

$$
  A_1 \times A_2 \longrightarrow A_3
$$

in $\mathbf{H}$. In applications this is often a pairing operation with $A_1 = A_2$, i.e. $A \times A \to A'$, and typically it is the product morphism $A \times A \to A$ for a [[ring object]] structure on the [[coefficients]] $A$. (See at _[[multiplicative cohomology theory]]_).

If $g_1 : X \to A_1$ and $g_2 : X \to A_2$ are two cocycles in $\mathbf{H}(X,A_1)$ and $\mathbf{H}(X,A_2)$, respectively, then their cup product with respect to this pairing is the cocycle

$$
  g_1 \cdot g_2
  \;\coloneq\;
  X \stackrel{(id,id)}{\longrightarrow}
  X \times X
  \stackrel{g_1 \times g_2}{\to}
  A_1 \times A_2
  \to
  A_3
$$

in $\mathbf{H}(X,A_3)$ obtained by combining the pairing with precomposition by the [[diagonal]] map $\Delta_X = (id_X, id_X)$.

## Examples

### Via the Dold-Kan correspondence
 {#ViaTheDoldKanCorrespondence}

When the coefficient object $A \in $[[∞Grpd]] is "sufficiently abelian" in that under the [[Dold-Kan correspondence]] it is represented by a [[chain complex]] then using the [[lax monoidal functor|lax monoidalness]] of the Dold-Kan correspondence (see at _[[monoidal Dold-Kan correspondence]]_) one obtains a chain complex model for the cup product which makes the origin of the typical grading shift manifest.

Write

* $(Ch_{\bullet \geq 0}, \otimes)$ for the [[category of chain complexes]] of [[abelian groups]], in non-negative degrees;, regarded as a [[symmetric monoidal category]] with the standard [[tensor product of chain complexes]] $\otimes$;

* $(sAb, \otimes)$ for the [[category]] of [[simplicial abelian groups]], regarded as a [[symmetric monoidal category]] with the degreewise [[tensor product of abelian groups]];

* $U \;\colon\; sAb \longrightarrow KanCplx \to sSet$ for the [[forgetful functor]] to the underlying [[simplicial sets]] (which happens to land in [[Kan complexes]]);

* $\Gamma \;\colon \; Ch_{\bullet \geq 0} \stackrel{\simeq}{\longrightarrow} sAb$ for the [[equivalence of categories]] given by the [[Dold-Kan correspondence]];

* $DK \;\colon\; Ch_{\bullet \geq 0} \underoverset{\simeq}{\Gamma}{\longrightarrow} sAb \stackrel{F}{\longrightarrow} KanCplx \hookrightarrow sSet$ for the composite.

Now:

* $\Gamma$ is a [[lax monoidal functor]], the lax monoidal structure $\gamma_{A,B} \;\colon\; \Gamma(A) \otimes \Gamma(B) \to \Gamma(A \otimes B)$ being induced [dually](oplax+monoidal+functor#OplaxAdjointToLax) by the [[Alexander-Whitney map]];

* $U$ is a [[strong monoidal functor]];

* for the [[tensor product of abelian groups]] $A \otimes B$ there are canonical natural [[bilinear maps]] of underlying sets

  $$
     p_{A,B}
      \;\colon\;
     U(A)\times U(B)
     \longrightarrow
     U(A \otimes B)
  $$

Using all this, then for

$$
  f \;\colon\; V_\bullet \otimes W_\bullet \longrightarrow Z_\bullet
$$

a given [[chain map]], this induces a map of the corresponding Kan complexes

$$
  \cup_{DK}
  \;\colon\;
  DK(V_\bullet) \times DK(W_\bullet)
  \longrightarrow
  DK(Z_\bullet)
$$

as the following [[composition|composite]]

$$
  \cup_{DK}
  \;\colon\;
  DK(V_\bullet)\times DK(W_\bullet)
  =
  U(\Gamma(V_\bullet)) \times U(\Gamma(W_\bullet))
  \stackrel{p_{\Gamma(V_\bullet), \Gamma(W_\bullet)} }{\to}
  U(\Gamma(V_\bullet)\otimes \Gamma(W_\bullet))
  \stackrel{U(\gamma)}{\longrightarrow}
  U(\Gamma(V_\bullet \otimes W_\bullet))
  \stackrel{U(\Gamma(f))}{\to}
  U(\Gamma(Z_\bullet))
  =
  DK(Z_\bullet)
  \,.
$$

With this in hand then for $X$ any [[homotopy type]], the cup product on its [[cohomology]] with [[coefficients]] in $DK(V_\bullet)$ and $DK(W_\bullet)$ is induced by just homming $X$ into this morphism:

$$
  \mathbf{H}(X, DK(V_\bullet)) \times
  \mathbf{H}(X, DK(W_\bullet))
  \simeq
  \mathbf{H}(X, DK(V_\bullet) \times DK(W_\bullet))
  \stackrel{\mathbf{H}(X,\cup_{DK})}{\longrightarrow}
  \mathbf{H}(X, DK(Z_\bullet))
  \,.
$$

For example if

$$
  V_\bullet = \mathbb{Z}[n_1]
  \,,
  \;\;\;
  W_\bullet = \mathbb{Z}[n_2]
$$

is the [[chain complex]] concentrated in degree $n_1$ and $n_2$, respectively, on the group of [[integers]], then

$$
  DK(V_\bullet) \simeq B^{n_1} \mathbb{Z} \simeq K(\mathbb{Z},n_1)
$$

is the corresponding [[Eilenberg-MacLane space]] which classifies [[ordinary cohomology]] ([[singular cohomology]]) with integral coefficients in the given degree. By the nature of the [[tensor product of chain complexes]] one has

$$
  V_\bullet \otimes W_\bullet \simeq \mathbb{Z}[n_1 + n_2]
  \,.
$$

Hence we may take $Z_\bullet \coloneqq \mathbb{Z}[n_1 + n_2]$ and $f = id$ and we get a cup product

$$
  \cup \;\colon\; H^{n_1}(X, \mathbb{Z}) \times H^{n_2}(X, \mathbb{Z})
  \to
  H^{n_1 + n_2}(X, \mathbb{Z})
  \,.
$$

### On Moore complexes of cosimplicial algebras

For $A = (A^\bullet)$ any [[cosimplicial algebra]], its dual [[Moore complex|Moore cochain complex]] $N^\bullet(A)$ naturally inherits the structure of a [[dg-algebra]] under the cup product.

> The general formula is literally the same as that for the case where $A^\bullet$ is functions on the singular complex of a space, which is discussed below. For the moment, see below.

This cup product operation on $N^\bullet(A)$ is not in general commutative. However, it is a standard fact that it becomes commutative after passing to [[chain homology and cohomology|cochain cohomology]].

This suggests that the cup product should be, while not commutative, homotopy commutative in that it makes $N^\bullet(A)$ a [[commutative algebra in an (∞,1)-category|homotopy commutative monoid object]].

This in turn should mean that $N^\bullet(A)$ is an [[algebra over an operad]] for the [[E-k-operad|E-∞ operad]].

That this is indeed the case is the main statement in ([Berger-Fresse 01](#BergerFresse01))

### In singular cohomology ###
 {#InSingularCohomology}

A special case of the cup product on Moore complexes is the complex of [[singular cohomology]], which is the Moore complex of the cosimplicial algebra of functions on the singular simplicial set of a topological space.

Often in the literature by _cup product_ is meant specifically the realization of the cup product on [[singular cohomology]].

For $X$ a [[topological space]], let $\Pi(X)_\bullet := X^{\Delta_{Top}^\bullet}$ be the [[simplicial set]] of $n$-[[simplex|simplices]] in $X$ -- the [[fundamental ∞-groupoid]] of $X$.

For $R$ some [[ring]], let $Maps(\Pi(X),R)^{\bullet}$ be the [[cosimplicial algebra|cosimplicial ring]] of $R$-valued functions on the spaces of $n$-simplices. The corresponding [[Moore complex|Moore cochain complex]] $C^\bullet(X)$ is the cochain complex whose [[chain homology and cohomology|cochain cohomology]] is the [[singular cohomology]] of the space $X$: a homogeneous element $\omega_p \in C^p(X)$ is a function on $p$-simplices in $X$.

Write, as usual, for $p \in \mathbb{N}$, $[p] = \{0 \lt 1 \lt \cdots \lt p\}$ for the [[poset|totally ordered set]] with $p+1$ elements. For $\mu : [p] \to [p+q]$ an injective order preserving map and $K$ some [[simplicial object|cosimplicial object]], write $d_\mu^* K : K^p \to K^{p+q}$ for the image of this map under $K$.

Specifically, for $p,q \in \mathbb{N}$ let $L : [p] \to [p+q]$ be the map that sends $i \in [p]$ to $i \in [p+q]$ and let $R : [q] \to [p+q]$ be the map that sends $i \in [q]$ to $i+q \in [p+q]$.

Then the **cup product**

$$
  \smile : C^\bullet(X) \otimes C^\bullet(X)
   \to C^\bullet(X)
$$

is the cochain map that on homogeneous elements $a \otimes b \in C^p(X) \otimes C^q(X) \subset C^\bullet(X) \otimes C^\bullet(X)$ is defined by the formula

$$
  a \smile b = (d_L^* a) \cdot (d_R^* b)
  \,.
$$

> There is some glue missing here to connect this back to the above general definition, something involving the [[Eilenberg-Zilber map]].

This means that $(a \smile b)_{i_0, \cdots, i_{p+q}} = a_{i_0, \cdots, i_p} \cdot b_{i_p, \cdots, i_{p+q}}$.

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###### Proposition

This cup product enjoys the following properties:

* it is indeed a cochain complex morphism as claimed, in that it respects the differential: for any homogeneous $a\otimes b \in C^p(X) \otimes C^q(X)$ as above we have

  $$
    d(a \smile b) = (d a) \smile b + (-1)^p a \smile (d b)
    \,.
  $$

* the image of the cup product on [[chain homology and cohomology|cochain cohomology]]

  $$
     \smile :  H^\bullet(C^\bullet(X))\otimes H^\bullet(C^\bullet(X)) \to H^\bullet(C^\bullet(X))
  $$

  is associative and distributes over the addition in $H^\bullet(C^\bullet(X))$.

=--

Accordingly, the cup product makes $H^\bullet(C^\bullet(X)) = H^\bullet(X,R)$ into a [[ring]]: the **cohomology ring** on the [[Eilenberg-MacLane spectrum|ordinary cohomology]] of $X$.

See for instance section 3.2 of

* Hatcher, _Algebraic Topology_ ([web](http://www.math.cornell.edu/~hatcher/AT/ATpage.html) [pdf](http://www.math.cornell.edu/~hatcher/AT/AT.pdf))

### In Whitehead-generalized cohomology
  {#CupProductInWhiteheadGeneralizedCohomologyTheory}

For the cup product in [[multiplicative cohomology theory|multiplicative]] [[Whitehead-generalized cohomology theories]] is defined by the same recipe of "cocycles pre-composed by diagonal and post-composed by product operation", one just has to observe that [[symmetric monoidal smash product|symmetric smash-]][[monoidal category with diagonals|monoidal diagonals]] on [[suspension spectra]]

\[
  \label{SmashMonoidalDiagonalOnSuspensionSpectra}
  \Sigma^\infty X
  \overset{ \;\; D_X \;\; }{\longrightarrow}
  \big(
    \Sigma^\infty X
  \big)
  \wedge
  \big(
    \Sigma^\infty X
  \big)
\]

do exists. This is discussed at _[suspension spectrum -- Smash-monoidal diagonals](suspension spectrum#SmashMonoidalDiagonals)_.

With this, given a [[Whitehead-generalized cohomology theory]] $\widetilde E$ [[Brown representability theorem|represented]] by a  [[ring spectrum]]

$$
  \big(E, 1^E, m^E \big)
  \;\;
  \in
  \;
  SymmetricMonoids
  \big(
    Ho(Spectra), \mathbb{S}, \wedge
  \big)
$$

the smash-monoidal diagonal structure (eq:SmashMonoidalDiagonalOnSuspensionSpectra) on suspension spectra serves to define the cup product $(-)\cup (-)$ in the corresponding [[multiplicative cohomology theory|multiplicative cohomology theory structure]] by:

$$
  \begin{aligned}
  &
  \big[
    \Sigma^\infty X
    \overset{c_i}{\longrightarrow}
    \Sigma^{n_i} E
  \big]
  \,\in\, {\widetilde E}{}^{n_i}(X)
  \\
  &
  \Rightarrow
  \;\;
  [c_1] \cup [c_2]
  \,
    \coloneqq
  \,
  \Big[
    \Sigma^\infty X
    \overset{
      \Sigma^\infty(D_X)
    }{\longrightarrow}
    \big(
      \Sigma^\infty X
    \big)
      \wedge
    \big(
      \Sigma^\infty X
    \big)
    \overset{
      ( c_1 \wedge c_2 )
    }{\longrightarrow}
    \big(
      \Sigma^{n_1} E
    \big)
      \wedge
    \big(
      \Sigma^{n_2} E
    \big)
    \overset{
      m^E
    }{\longrightarrow}
    \Sigma^{n_1 + n_2}E
  \Big]
  \;\;
  \in
  \,
  {\widetilde E}{}^{n_1+n_2}(X)
  \,.
  \end{aligned}
$$

## In abelian sheaf cohomology ##

Traditionally the cup product is considered for
abelian cohomology, such as [[generalized (Eilenberg-Steenrod) cohomology]] and more generally [[abelian sheaf cohomology]].

In that case all coefficient objects $A_i$ are complexes $(A_i)_\bullet$ of sheaves and the pairing that one usually considers is the [[tensor product]] of [[chain complex]]es

$$
  (A_1)_\bullet \times (A_2)_\bullet \to (A_1 \otimes A_2)_\bullet
$$

where

$$
  (A_1 \otimes A_2)_n = \oplus_p (A_1)_p \otimes (A_2)_{n-p}
  \,.
$$

with differential

$$
  d (a_1 \otimes a_2) = (d a_1) \otimes a_2 + (-1)^{|a_1|} a_1 \otimes d a_2
  \,.
$$

### In abelian Čech cohomology ###

The cup product has a simple expression in abelian [[?ech cohomology]].

For $A_1$ and $A_2$ two [[chain complex]]es (of [[sheaves]] of [[abelian group]]s) construct a morphism of Čech complexes

$$
  \phi : C^\bullet(\{U_i\}, A_1) \otimes
         C^\bullet(\{U_i\}, A_2)
         \to
         C^\bullet(\{U_i\}, A_1 \otimes A_2)
$$

by sending $\alpha \in C^p(U,A_1)_\bullet$ and $\beta \in C^q(U,A_2)_\bullet$ to

$$
  \phi(\alpha \otimes \beta)_{i_0, \cdots , i_{p + q}}
  \;:=\;
  \alpha_{i_0, \cdots, i_p} \otimes \beta_{i_p, \cdots i_{p+q}}
  \,.
$$

For instance ([Brylinski, section (1.3)](#Brylinski)) spring

### In Čech-Deligne cohomology (ordinary differential cohomology)

For the case that of [[Čech cohomology|Čech]] [[hypercohomology]] with coefficients in [[Deligne complex]]es the above yields the _[[Beilinson-Deligne cup-product]]_ for [[ordinary differential cohomology]].

## Related concepts

* [[cap product]]

* [[cup-i product]]

* [[intersection pairing]]

* [[functional cup product]]

* [[Baer sum]]

* [[cohomology operation]]

* [[power operation]]

* [[Massey product]]

* [[cup product in differential cohomology]]

  * [[cup product in ordinary differential cohomology]]

* [[Euler class]] takes [[Whitney sum]] to [[cup product]], see [this Prop.](Euler+class#EulerClassOfWhitneySumIsCupProductOfEulerClasses)

## References

* {#Adams74} [[Frank Adams]], part III, sections 2 and 3 of _[[Stable homotopy and generalised homology]]_, 1974

* {#May} [[Peter May]], 18.3 and 22.3 of _A concise course in algebraic topology_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/maybook.pdf))

The cup product in Čech cohomology is discussed for instance in section 1.3 of

* [[Jean-Luc Brylinski]], _Loop spaces and characteristic classes_

Recall from the discussion at [[models for ∞-stack (∞,1)-toposes]] that all [[hypercompletion|hypercomplete]] [[∞-stack]] [[(∞,1)-topos]]es are modeled by the [[model structure on simplicial presheaves]]. Accordingly understanding the cup product on simplicial presheaves goes a long way towards the most general description. For a bit of discussion of this see around page 19 of

* [[John Frederick Jardine]], _Lectures on simplicial presheaves_ ([pdf](http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf))

An early treatment of cup product can be found in this classic

* [[Hassler Whitney]], _On Products in a Complex_ ([JSTOR](http://www.jstor.org/pss/1968795))

See also

* {#BergerFresse01} [[Clemens Berger]], [[Benoit Fresse]] _Combinatorial operad actions on cochains_, Math. Proc. Cambridge Philos. Soc. 137 (2004), 135-174. ([arXiv:math/0109158](http://arxiv.org/abs/math/0109158))

[[!redirects cup products]]

[[!redirects cup product in abelian Cech cohomology]]
[[!redirects cup product in Cech cohomology]]


cup product in differential cohomology

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## Idea

[[cup product]] in [[differential cohomology]]

## Examples

* [[cup product in ordinary differential cohomology]]

[[!redirects cup product on differential cohomology]]

[[!redirects cup product in generalized differential cohomology]]
[[!redirects cup product on generalized differential cohomology]]

[[!redirects cup product in differential generalized cohomology]]
[[!redirects cup product on differential generalized cohomology]]


cup-i product

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## Idea

Cup-$i$ products extend [[cup products]].

They can be used to define [[Steenrod squares]]
in the same manner as ordinary [[cup products]] can be used to define the square of a cohomology class.

## Definition

Given a [[simplicial set]] $X$ and $i\ge0$, we define
the cup-$i$ product as the map on simplicial cochains on $X$
with coefficients in $\mathbf{Z}/2$ induced by the map on simplicial chains
$$\Delta_i: C(X,\mathbf{Z}/2) \to C(X,\mathbf{Z}/2)\otimes C(X,\mathbf{Z}/2),$$
where $\Delta_i$ evaluate on an $n$-simplex $x\in X_n$
is 0 if $i\gt n$ and
$$\sum_U d_{U^0}(x)\otimes d_{U^1}(x),$$
where $U\subset \{0,\ldots,n\}$ has cardinality $n-i$
and
$$U^k=\{u\in U\mid r(u)+k=u \pmod2\},$$
where $r(u)=\#\{v\in U\mid v\le u\}$.

Thus, we have
$$(x \cup_i y)(c) = (x\otimes y)(\Delta_i(c)).$$

## Properties

We have
$$Sq^i(x) = x \cup_i x,$$
where $Sq^i$ denotes the $i$th [[Steenrod square]].

## Related concepts

* [[cup product]]

* [[Steenrod square]]

## References

* [[Anibal M. Medina-Mardones]], _New formulas for cup-$i$ products and fast computation of Steenrod squares_, [arXiv](arXiv:2105.08025).

* [[Ralph M. Kaufmann]], [[Anibal M. Medina-Mardones]], _Cochain level May-Steenrod operations_, [arXiv](https://arxiv.org/abs/2010.02571).

* [[Anibal M. Medina-Mardones]], _An axiomatic characterization of Steenrod's cup-i products_, [arXiv](https://arxiv.org/abs/1810.06505).

[[!redirects cup-i products]]


Curci-Ferrari model

* G. Curci, R. Ferrari, _On a class of Lagrangian models for massive and massless Yang-Mills fields_, Nuovo Cim. A32 (1976) 151.
* G. Curci and R. Ferrari, _The unitarity problem and the zero-mass limit for a model of
massive Yang-Mills theory_, Nuovo Cim. A35 (1976) 1.
* J.de Boer, K. Skenderis, P. van Nieuwenhuzen, A. Waldron, _On the renormalizability
and unitarity of the Curci-Ferrari model for massive vector bosons_, Phys. Lett. B367 (1996) 175.
* Peter M. Lavrov, _Remarks on the Curci-Ferrari model_, [arxiv/1205.0620](http://arxiv.org/abs/1205.0620)


curl

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# Contents
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## Definitions
 {#Definition}

### In terms of differential forms
 {#InTermsOfDifferentialForms}

In [[Riemannian geometry]], the __curl__ or __rotation__ of a [[vector field]] $v$ on an [[orientation|oriented]] $3$-[[dimension|dimensional]] [[Riemannian manifold]] $(X,g)$ is the vector field denoted $curl(v)$ (or $rot(v)$ or $\Del \times v$) defined by

$$
  curl(v) \;\coloneqq\; g^{-1} \left(\star_g d_{dR}\, g(v) \right)
  \,.
$$

where

1. $\Gamma(T X) \underoverset{\underset{g}{\longrightarrow}}{\overset{g^{-1}}{\longleftarrow}}{\phantom{AA}\simeq\phantom{AA}} \Omega^1(X)$ is the [[linear isomorphism]] between [[vector fields]] and [[differential 1-forms]] given by the [[metric tensor]] $g$;

2. $d_{dR} \;\colon\; \Omega^n(X) \longrightarrow \Omega^{n+1}(X)$ is the [[de Rham differential]]

3. $\star_g \;\colon\; \Omega^n(X) \to \Omega^{dim(X)-n}(X)$ is the [[Hodge star operator]] (which uses the orientation of $X$).

Notice that for this to make sense it is crucial that the [[dimension]] of $X$ is $3$, for only then is the Hodge dual of the de Rham differential of a $1$-form again a $1$-form; that is, $n = 3$ is the unique solution of $n - (1 + 1) = 1$.

### Via integration {#ViaIntegration}

Alternatively, the curl/rotation of a vector field $\vec v$ at some point $x\in X$ may be defined as the [[limit of a sequence|limit]] [[integral]] formula

$$
  \vec{n}\cdot rot \vec v
  =
  \lim_{area S\to 0}
  \frac{1}{area S}
  \oint_{\partial S} \vec{t}\cdot \vec v d r
$$

where $D$ runs over the smooth oriented [[surfaces]] ([[submanifolds]] of [[dimension]] $2$) containing the point $x$ and with smooth [[boundary]] $\partial D$, $\vec{n}$ is the unit vector through the surface $S$, and $\vec{t}$ is the unit vector tangent to the [[curve]] $\partial S$. (We use the orientation of $X$ to determine the [[direction of a vector|direction]] of $\vec{n}$ from the [[orientation]] of $S$.)

This formula does not depend on the shape of boundaries taken in limiting process, so one can typically take a [[coordinate chart]] and [[discs]] with decreasing radius in this particular coordinate chart.  One can even use $\pi r^2$ in place of the actual area of the disc around $x$ with coordinate-radius $r$, to save on calculating this area, as long as the coordinate chart assigns the standard coordinates to the metric at $x$.

The proof that this definition is coherent and agrees with the previous one is essentially the [[Kelvin–Stokes Theorem]]; see [below](#Stokes) for discussion.

### Via cross products

More generally, if $(X,g)$ is a [[Riemannian manifold]] whose [[cotangent spaces]] (equivalently, [[tangent spaces]]) are smoothly equipped with a binary [[cross product]] $⨉\colon \Omega^2(X;\mathbb{R}) \to \Omega^1(X;\mathbb{R})$, then the __curl__ of any vector field $v$ is

$$
  curl(v)
  \;=\; g^{-1} ⨉ d_{dR} g(v)
$$

However, this is not as general as it may appear:

* in $0$ or $1$ [[dimension]], the cross product, hence the curl, must always be $0$;
* in $3$ dimensions, a smooth choice of cross product is equivalent to a smooth choice of [[orientation]], and we recover the previous formula;
* in $7$ dimensions, if a smooth choice of cross product is possible (as on the [[7-sphere]]), then uncountably many are possible, giving as many different notions of curl;
* in any other number of dimensions, no binary cross product exists at all, hence no curl.

That said, there are also cross products of other [[arities|arity]] in other dimensions; using essentially the same formula, we can take the curl of a $k$-[[multivector field]] if we have a smooth $(k+1)$-ary cross product.  Or if the cross product is other than vector-valued, then we can obtain a curl that is other than a vector field.

In particular, in $2$ dimensions, we have the __scalar curl__

$$
  curl(v)
  \;=\; ⨉ d_{dR} g(v)
$$

where $⨉\colon \Omega^2(X;\mathbb{R}) \to \Omega^0(X;\mathbb{R})$ is the [[volume form]] (or area form) on the $2$-dimensional Riemannian manifold $X$.

## Examples

If $(X,g)$ is $\mathbb{R}^3$ endowed with the canonical Euclidean metric, then the curl of a vector field $(v^1,v^2,v^3) = v^1\partial_1 + v^2\partial_2 + v^3\partial_3$ is

$$
curl(v)^1 = \frac{\partial v^3}{\partial x^2}-\frac{\partial v^2}{\partial x^3}
;\qquad
curl(v)^2 = \frac{\partial v^1}{\partial x^3}-\frac{\partial v^3}{\partial x^1}
;\qquad
curl(v)^3 = \frac{\partial v^2}{\partial x^1}-\frac{\partial v^1}{\partial x^2}
.$$

This is the classical curl from [[vector analysis]].

If $(X,g)$ is $\mathbb{R}^2$ endowed with the canonical Euclidean metric, then the curl of a vector field $(v^1,v^2) = v^1\partial_1 + v^2\partial_2$ is

$$
curl(v) = \frac{\partial v^2}{\partial x^1}-\frac{\partial v^1}{\partial x^2}
.$$

## Relation to the Stokes theorems {#Stokes}

Recall that if $X$ is an $n$-dimensional [[differentiable manifold]], $D$ is a $p$-dimensional [[submanifold]] [[manifold with boundary|with boundary]], and $\alpha$ is a [[differentiable function|differentiable]] $(p-1)$-rank [[exterior differential form]] on a [[neighbourhood]] of $D$ in $X$, then the generalized [[Stokes Theorem]] says that the [[integration of differential forms|integral]] of $\alpha$ on the boundary $\partial{D}$ equals the integral on $S$ of the [[de Rham differential]] $\mathrm{d}_{DR}\alpha$.

When $n=3$, $p=2$, and $X$ is equipped with an orientation and a metric, then this is equivalent to saying that the integral of a vector field $v$ along the boundary of an oriented surface $D$ in $X$ is equal to the integral of the vector field\'s curl across the surface:

\[ \int_{\partial{D}} v \cdot \mathrm{d}\mathbf{r} = \int_D curl v \cdot \mathrm{d}\mathbf{S} \label{KS}.\]

(In particular, when $X$ is $\mathbb{R}^3$ with its standard orientation and metric, then this is equivalent to the classical Kelvin--Stokes Theorem.)  At least, this is what it says if the curl is defined [in terms of differential forms](#InTermsOfDifferentialForms); if the curl is defined [via integration](#ViaIntegration) instead, then \eqref{KS} is immediate, and the Kelvin--Stokes Theorem says that this definition matches the other one.

When $n=2$, $p=2$, and $X$ is equipped with an orientation and a metric, then the Stokes Theorem is equivalent to saying that the integral of a vector field $v$ along a [[simple closed curve]] in $X$ is equal to the integral of the vector field\'s scalar curl on the region $D$ enclosed by the curve:

\[ \int_{\partial{D}} v \cdot \mathrm{d}\mathbf{r} = \int_D curl v \mathrm{d}A \label{Green}.\]

(In particular, when $X$ is $\mathbb{R}^2$ with its standard orientation and metric, then this is equivalent to the curl-circulation form of [[Green's Theorem]].)

## Remark

In many classical applications of the curl in [[vector analysis]], the Riemannian structure is actually irrelevant, and the gradient can be replaced with the [[exterior differential|deRham differential]] $d_{dR}$.  That is, $X$ is treated as the $1$-form $g(X)$, its curl is treated as the $2$-form $d_{dR} g(X)$, and once these identifications are made there is no need to involve $g$ or $X$ directly at all.

## Related concepts

* [[nabla]]

* [[gradient]]

  * [[gradient flow]]

  * [[symplectic gradient]]

* [[Hamiltonian flow]]

* **curl**

* [[divergence]]

## References

See also

* Wikipedia, _[Curl](https://en.wikipedia.org/wiki/Curl_%28mathematics%29)_

[[!redirects curl]]
[[!redirects curls]]
[[!redirects curl of a vector field]]
[[!redirects curls of vector fields]]
[[!redirects rotation of a vector field]]
[[!redirects rotations of vector fields]]


current

The term **current** in physics denotes either the gradient of the density of a field or of matter or the gradient of a wave function or of an order parameter. Most currents (though not all) in that sense satisfy some conservation laws, e.g. the current of electric charge, and so are known as [[conserved currents]]. This leads to the notion of [[conserved charge]], see there.

In analysis, and specially in geometric measure theory, there is a different notion also called 'current'. It is a [[generalized function]] (distribution in the sense of Schwartz and others) with values in [[differential form]]s.
See at [[current (distribution theory)]] also [[integral current]].

Regarding the material connotations it is not very surprising that the physics notion of current is related to distribution of stuff, and that analysis notion to distribution in Schwartz sense. Thus the two notions of distribution (and the third, distribution of hyperplanes or subspaces in tangent bundle on a manifold) and the two notions of current are related.

[[!redirects currents]]


current (distribution theory)

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## Idea

In [[differential geometry]], by a *current* on a [[smooth manifold]] one means a [[generalized function]]/[[distribution]] with values in [[differential forms]].

See *[[current]]* and *[[conserved current]]* for other notions of "currents".

## Related concepts

* [[geometric measure theory]]

## References

### General

The original article:

* [[Georges de Rham]], *Variétés différentiables, formes, courants, formes harmoniques*, Act. Sci. et Ind. **1222**, Hermann, Paris (1955) [[GBooks](https://books.google.de/books/about/Vari%C3%A9t%C3%A9s_diff%C3%A9rentiables.html?id=2lLvAAAAMAAJ&redir_esc=y)]

Textbook accounts:

* [[Herbert Federer]], Section 4.1 in: *Geometric measure theory*, Grundlehren **153**, Springer (1969, 1996) [[doi:10.1007/978-3-642-62010-2](https://doi.org/10.1007/978-3-642-62010-2)]

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

* [[Georges de Rham]], Chapter III of: *Differentiable Manifolds -- Forms, Currents, Harmonic Forms*, Grundlehren **266**, Springer (1984) [[doi:10.1007/978-3-642-61752-2](https://doi.org/10.1007/978-3-642-61752-2)]

* [[Steven G. Krantz]], [[Harold Parks]], Section 7 of: _Geometric integration theory_, Springer (2008) [[pdf](http://www.math.wustl.edu/~sk/books/root.pdf)]

* [[Frank Morgan]], *Geometric measure theory: a beginner's guide*, Academic Press (2016) [[doi:10.1016/C2015-0-01918-9](https://doi.org/10.1016/C2015-0-01918-9)]

See also:

* Wikipedia, *Current (mathematics)*

### Applications

Currents appear as part of "geometric cycles" for [[differential cobordism cohomology]] by actual [[cobordism]]-classes equipped with [[differential geometry|differential geometric]] data:

* [[Ulrich Bunke]], [[Thomas Schick]], Ingo Schroeder, Moritz Wiethaup, §4.1.4 in: *Landweber exact formal group laws and smooth cohomology theories*, Algebr. Geom. Topol. **9** (2009) 1751-1790 [[arXiv:0711.1134](https://arxiv.org/abs/0711.1134), [doi:10.2140/agt.2009.9.1751](https://doi.org/10.2140/agt.2009.9.1751)]

and in the [[Hodge-filtered differential cohomology|Hodge-filtered]] version:

* {#Haus22} [[Knut Bjarte Haus]], §2.3, §4.12 in: *Geometric Hodge filtered complex cobordism*, PhD thesis (2022) [[ntnuopen:3017489](https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/3017489)]

* {#HausQuick22} [[Knut Bjarte Haus]], [[Gereon Quick]], §2 of: *Geometric Hodge filtered complex cobordism* [[arXiv:2210.13259](https://arxiv.org/abs/2210.13259)]

[[!redirects currents (distribution theory)]]


current algebra

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## Idea

In the broad meaning a _current algebra_ is the [[Poisson bracket]] algebra of [[conserved currents]] of a [[prequantum field theory]], or else its quantized version of the corresponding [[quantum field theory]].

If the [[symmetry]] corresponding to the [[conserved currents]] via [[Noether's theorem]] preserves the given [[Lagrangian]] only up to a [[divergence]] term, then the current algebra is a [[central extension]] of the Lie algebra of the underlying symmetries. This effect makes current algebras tend to be subtle and of particular mathematical interest.

Particularly famous is the case of the [[WZW model|WZW sigma model]] field theory for a [[string]] propagating on a [[Lie group]] $G$. In this case one chiral half of the algebra of currents is the corresponding [[affine Lie algebra]]. In parts of the (mathematical) literature it is this special case which meant by default by the term _current algebra_.

In fact, taking into account that this [[divergence]] term itself in general has [[higher gauge symmetries]], current algebras are secretly [[strong homotopy Lie algebras]]/[[Lie n-algebras]]. This we discuss [below](#AsAHomotopyLieAlgebra).

## As a homotopy Lie algebra
 {#AsAHomotopyLieAlgebra}

With [[local prequantum field theory]] in [[dimension]] $d$ formulated as [[de Donder-Weyl field theory]], every [[local Lagrangian]] is incarnated as a  [[circle n-bundle with connection|principal d-form connection]] $\mathbf{L}$ on some [[space]] $X$, such that its [[curvature]] form $\omega$ determines by its [[kernel]] the tangents to the solutions of the [[equations of motion]]. (see [cftcht, sections 3.1, 3.2,  3.4](#cftcht)).

What is traditionally considered in the literature is the special case of this where $X$ is the (dualized, first) [[jet bundle]] of a [[field bundle]] over a [[smooth manifold]] (see at _[[multisymplectic geometry]]_) and the underlying [[principal infinity-bundle]] of $\mathbf{L}$ is trivial, so that $\mathbf{L}$ is incarnated as a globally defined [[differential n-form|differential d-form]]. Moreover, this is traditionally expressed as the product $L \, vol$ of a [[smooth function]] times a prescribed [[volume form]]. This function $L$ then is what in much of the traditional literure is referred to as the Lagrangian of the [[theory (physics)|theory]].

A [[symmetry]] of the theory which preserves the Lagrangian up to a [[divergence]] is then precisely a [[quantomorphism n-group|quantomorphism]] of $\mathbf{L}$, namely a pair consisting of the symmetry

$$
  \phi \colon X \stackrel{\simeq}{\longrightarrow} X
$$

together with an [[equivalence]]

$$
  \eta \colon \phi^\ast \mathbf{L} \stackrel{\simeq}{\longrightarrow} \mathbf{L}
  \,.
$$

When expressing $\mathbf{L}$ as a [[modulating morphism]] $\mathbf{L} \colon X \longrightarrow \mathbf{B}^d U(1)_{conn}$ into the [[smooth infinity-stack|smooth]] [[moduli infinity-stack]] of [[circle n-bundle with connection|circle d-bundles with connection]] then this is an equivalence in the [[slice (infinity,1)-topos]] over $\mathbf{B}^d U(1)_{conn}$:

$$
  \mathbf{QuantMorph}(\mathbf{L})
  =
  \left\{
  \array{
    X && \stackrel{\phi}{\longrightarrow} && X
    \\
    & \searrow &\swArrow_{\eta}& \swarrow
    \\
    && \mathbf{B}^d U(1)_{conn}
  }
  \right\}
  \,.
$$

(The detailed definition of this [[quantomorphism n-group|quantomorphism d-group]] involved [[differential concretification]] of the naive [[automorphism infinity-group]] in the slice.)

Under [[Lie differentiation]] this gives [^Lie] the [[Poisson bracket Lie n-algebra]] $\mathfrak{Pois}(\mathbf{L})$. In its [[dg-Lie algebra]] version spelled out [here](Poisson+bracket+Lie+n-algebra#PoissondgAlgebra) and restricted to the special case of globally defined Lagrangian form,  this has, in degree 0,  precisely the Lie bracket of conserved currents as known from traditional literature, for instance ([AGIT89, equations (13), (14)](#AGIT89)). Details are in [cftcht, 3.3](#cftcht).

But the $\mathfrak{Pois}(\mathbf{L})$ contains in addition the equivalence between two central terms, coming from [[higher gauge transformations]]. Taking this into account, the [extension theorem](Poisson+bracket+Lie+n-algebra#ExtensionTheorem) for $\mathfrak{Pois}(X,\omega)$ says that its truncation to a Lie algebra is an extension by $H^{d-1}_{dR}(X)$.  This has been informally argued for instance in [AGIT 89, p. 8](#AGIT89).

[^Lie This has strictly been proven only for low $n$, but is sort of obvious generally ]

## References

* Sam Treiman,  [[Roman Jackiw]], [[David Gross]],  _Lectures on current algebra and its applications_ , Princeton University Press. (1972)  ([publisher](http://press.princeton.edu/titles/2071.html))

* [[Steven Weinberg]], _Current Algebra and Gauge Theories. I_ Phys. Rev. D 8, 605–625 (1973)

* Herbert Pietschmann, _On the Early History of Current Algebra_ ([arXiv:1101.2748](http://arxiv.org/abs/1101.2748))

General discussion and application to the [[Green-Schwarz super p-brane sigma models]] is in

* {#AGIT89} [[José de Azcárraga]], [[Jerome Gauntlett]], J.M. Izquierdo, [[Paul Townsend]], _Topological Extensions of the Supersymmetry Algebra for Extended Objects_, Phys. Rev. Lett. 63 (1989) 2443 ([spire](http://inspirehep.net/record/26393?ln=en))

Discussion from an [[nPOV]] is in

* {#cftcht} _[[schreiber:Classical field theory via Cohesive homotopy types]]_

[[!redirects current algebras]]


Curry's paradox

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## Idea

In [[mathematical logic]] and [[mathematical foundations]] *Curry's paradox* is a [[paradox]] which is a version of [[Russell's paradox]] that does not involve the use of [[negation]].

Russell's paradox is a proof of [[falsehood]], rendering a certain class of formal systems (those including some kind of unlimited [[axiom of comprehension]]) "inconsistent" in the sense that they prove false, which by the law of [[ex falso quodlibet]] entails that they prove anything at all.  Curry's paradox is a *direct* proof of "anything at all" without the detour through falsehood.  Thus, it applies more generally than Russell's paradox, e.g. to theories lacking a notion of "false", or to [[paraconsistent logic|paraconsistent]] theories whose notion of "false" does not satisfy ex falso quodlibet.

Curry's paradox does, however, still depend on the structural rule of [[contraction rule|contraction]].  Thus, it can be avoided by systems based on [[linear logic]].

## Argument

Let $P$ be any statement at all, and consider the set

$$ C = \{ x \mid (x\in x) \Rightarrow P \}. $$

Then if $C\in C$, by definition we have $(C\in C) \Rightarrow P$, and hence by [[modus ponens]] we have $P$.  Therefore, $(C\in C) \Rightarrow P$.  But by definition this means that $C\in C$, and therefore (as we just proved) $P$.

Note that, if [[negation]] is defined as $\neg P = (P \Rightarrow \bot)$ for some notion of [[falsehood]] $\bot$, then Russell's paradox is the special case of Curry's paradox for $P=\bot$.

## Related concepts

* [[axiom of full comprehension]]

## References

In relation to the [[axiom of full comprehension]]:

* {#Fitch69} [[Frederic Fitch]], *A method for avoiding the Curry paradox*, in *Essays in Honor of Carl G. Hempel*, Reidel, Dordrecht, Holland 1969, pp. 255--265 ([doi:10.1007/978-94-017-1466-2](https://link.springer.com/book/10.1007/978-94-017-1466-2))


currying

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## Idea

Currying is a process of transforming an operation on two variables into an operation on one variable that returns a function taking the second variable as an argument.  The term is used in [[computer science]] and the [[lambda calculus]], where it is often technically important to have operations that act on only one variable at a time.  But [[category theory]] also recognises it as a [[natural isomorphism]] in a [[closed monoidal category]], namely the [[universal property]] of an [[exponential object]]; thus it is also known as taking the __exponential transpose__.

The [[inverse]] operation is, in the straightforward style of [[computer science]], called __uncurrying__.

Currying is named after [[Haskell Curry]], in accordance with [[Stigler's law of eponymy]], since it was invented by [[Moses Schonfinkel|Moses Schönfinkel]]. (Perhaps Curry helped popularize the application to lambda calculus?)

## Definitions

### In set theory

In [[set theory]], currying transforms a [[function]] defined on a [[product set]] to a function that takes values in a [[function set]]. That is, starting with
$$ f\colon X \times Y \to Z ,$$
we __curry__ $f$ to produce
$$ \hat{f}\colon X \to Z ^ Y $$
according to the formula
$$\hat{f}(x) = (y \mapsto f(x, y)).$$
This is simply applying the universal property of the [[function set]] $Z^Y$.

### In dependent type theory

In [[dependent type theory]], currying transforms a family of elements defined on a [[dependent pair type]] to a family of elements that takes values in a [[dependent function type]]. That is, starting with
$$z:\sum_{x:X} Y(x) \vdash f(z):Z(z)$$
we __curry__ $f$ to produce
$$x:X \vdash \hat{f}(x):\prod_{y:Y(x)} Z(x, y)$$
according to the formula
$$\hat{f}(x) \coloneqq (\lambda y.f(x, y)).$$
This is simply applying the universal property of the [[dependent function type]] $\prod_{y:Y(x)} Z(x, y)$.

### In lambda calculus and CCCs

Similarly, in [[lambda calculus]], currying is a device which reduces the study of functions in several arguments to functions in one argument. For example, if $\phi$ is a lambda term in which variables $x$ and $y$ occur freely (so that $\phi$ is effectively a "function" of $x$ and $y$), the lambda calculus syntax favors the currified expression $\lambda x. \lambda y. \phi$, which denotes the intuitive expression $x \mapsto (y \mapsto \phi)$, where one abstracts variables one at a time.

Later it was observed (by Lawvere) that this is just a special case of a more general "curryfication" for [[cartesian closed categories]] (such as [[Cat]] or a [[nice category of spaces]]), where one has a natural bijection of morphisms:

$$\frac{X \times Y \to Z}{X \to Z^Y}$$

Indeed, cartesian closed categories are models for lambda calculus.

### In general

In fact, currying can be done in any (right) [[closed monoidal category]].  In that case, currying transforms a [[morphism]] whose [[source]] is a [[tensor product]] to a morphism whose [[target]] is an [[internal hom]].  That is, starting with
$$ f\colon X \otimes Y \to Z ,$$
we __curry__ $f$ to produce
$$ \hat{f}\colon X \to [Y, Z] .$$

Currying is invertible and natural in $X, Y, Z$; that is, $f \mapsto \hat{f}$ is a [[natural isomorphism]] (in any closed monoidal category).

## Currying in which variable(s)?

By convention, currying is always done on the *last* variable.  This fits in very nicely with a convention that products associate on the left while internal homs associate on the right.  More explicitly, if we use (as is common in computer science) '$\times$' for the product (even if it is not cartesian) and '$to$' for the internal hom (so that we must use another symbol, say '$\vdash$', for the external hom), then we need no parentheses to generalise currying
$$ f\colon X \times Y \vdash Z $$
to produce
$$ \hat{f}\colon X \vdash Y \to Z ;$$
by currying several $n-1$ times in succession, we turn
$$ f\colon X_1 \times \cdots \times X_n \vdash Z $$
into
$$ \hat{\overset\vdots{\hat{f}}}\colon X_1 \vdash X_2 \to \cdots \to X_n \to Z .$$
This does not generalise to $n = 0$, which is one way to see that even the untyped lambda calculus actually has two objects, one of which is a [[terminal object]] $1$ and one of which can play every other role.

In a closed [[braided monoidal category]] (such as a closed [[symmetric monoidal category]], and including the cartesian closed categories such as $Set$ and the models of lambda calculus), we can also 'curry through the first variable' or 'co-curry' to produce a map
$$ \check{f}\colon Y \vdash X \to Z ,$$
but computer scientists (or even mathematicians who are being very careful) will see this as a composite operation whose first step is [[composition]] with the [[braiding]] $Y \times X \vdash X \times Y$.

In a left closed monoidal category, currying does not exist, but cocurrying does.  Perhaps the product should associate on the right in a left closed monoidal category?

## See also

* [[Cartesian product]], [[function set]]
* indexed [[disjoint union]], indexed [[Cartesian product]]
* [[pair type]], [[function type]]
* [[dependent pair type]], [[dependent function type]]
* [[product]], [[exponential object]]
* [[dependent sum]], [[dependent product]]

## References

For currying in [[dependent type theory]], see

* Section 1.2 and 1.4 in *Homotopy Type Theory: Univalent Foundations of Mathematics*, The [[Univalent Foundations Project]], Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

* [[Egbert Rijke]], Section 4.6 in *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

[[!redirects uncurrying]]
[[!redirects curried]]
[[!redirects uncurried]]
[[!redirects exponential transpose]]


Curtis algorithm

#Contents#
* table of contents
{:toc}

## Idea

The _Curtis algorithm_ is an [[algorithm]] for computing the second page of the [[classical Adams spectral sequence]] using an algebraic version of the [[EHP spectral sequence]] for the [[Lambda-algebra]].

## Related concepts

* [[May spectral sequence]]

## References

Named after [[Edward Curtis]].

See

* [[Doug Ravenel]], section 3.3 of _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986

* {#Kochmann96} [[Stanley Kochman]], around prop. 5.2.6  of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996


Curtis Callan

* [webpage](https://phy.princeton.edu/people/curtis-callan)

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

## Selected writings

On [[chiral perturbation theory]]:

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

Inclusion of heavy [[flavor (particle physics)|flavors]] ([[strange quarks]]/[[kaons]], etc.) into the [[Skyrme model]]:

* [[Curtis Callan]], [[Igor Klebanov]], _Bound-state approach to strangeness in the Skyrme model_, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (doi10.1016/0550-3213(85)90292-5)

* [[Curtis Callan]], K. Hornbostel, [[Igor Klebanov]], _Baryon masses in the bound state approach to strangeness in the skyrme model_, Physics Letters B Volume 202, Issue 2, 3 March 1988, Pages 269-275 (doi10.1016/0370-2693(88)90022-6)

On [[anomaly inflow]]:

* [[Curtis Callan]], [[Jeffrey Harvey]], _Anomalies and Fermion Zero Modes on Strings and Domain Walls_, Nucl. Phys. B250 (1985) 427-436 (doi:10.1016/0550-3213(85)90489-4, [spire:15691](http://inspirehep.net/record/15691))

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

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

On microscopic explanation of [[Bekenstein-Hawking entropy]] via [[geometric engineering of QFT|geometric engineering]] of [[black holes in string theory]] as [[bound states of D-branes]]:

* [[Curtis Callan]], [[Juan Maldacena]], _D-brane Approach to Black Hole Quantum Mechanics_, Nucl. Phys. B472:591-610, 1996 ([arXiv:hep-th/9602043](http://arxiv.org/abs/hep-th/9602043))

On [[D1-D3 brane intersections]] as spikes/[[BIons]] in the [[D3-brane]] [[DBI action|DBI-theory]]:

* [[Curtis Callan]], [[Juan Maldacena]], _Brane Dynamics From the Born-Infeld Action_, Nucl. Phys. B513 (1998) 198-212 ([arXiv:hep-th/9708147](https://arxiv.org/abs/hep-th/9708147))

On [[baryons]] in [[holographic QCD]]:

* {#CGS98} [[Curtis Callan]], Alberto Guijosa, Konstantin G. Savvidy, _Baryons and String Creation from the Fivebrane Worldvolume Action_ ([arxiv:hep-th/9810092](https://arxiv.org/abs/hep-th/9810092))

* [[Curtis Callan]], Alberto Guijosa, Konstantin G. Savvidy, Oyvind Tafjord, _Baryons and Flux Tubes in Confining Gauge Theories from Brane Actions_, Nucl. Phys. B555 (1999) 183-200 ([arxiv:hep-th/9902197](https://arxiv.org/abs/hep-th/9902197))

## Related $n$Lab entries

* [[instanton in QCD]]

* [[renormalization group flow]], [[beta function]]

category: people


curvature

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## Idea

The formal notion of _curvature_ is a formalization and generalization of the intuitive notion of the ("[extrinsic](#Extrinsic)") curvature of a [[surface]] embedded in a [[Cartesian space]] $\mathbb{R}^n$.

This extrinsic curvature of a surface is called [[Gaussian curvature]]. It may also be understood intrinsically as a property of just the surface without reference to the ambient [[Cartesian space]] that it is embedded in: the canonical [[metric]] on $\mathbb{R}^n$ induces a [[Riemannian metric]] on the surface and the surface's curvature is encoded in the [[Levi-Civita connection]] on the [[tangent bundle]] of the surface.

This notion of curvature of a [[Levi-Civita connection]] in turn generalizes straightforwardly to a notion of curvature of any [[connection on a bundle]] and thus gives the name to the general concept.

For instance in the [[first-order formulation of gravity]] the curvature of [[spacetime]] is literally the curvature of the [[Levi-Civita connection]] on spacetime in this sense. But also the [[Yang-Mills field]] is a connection on a [[principal bundle]] and its curvature encodes the [[field strength]] of the [[Yang-Mills field]], which is a concept rather remote from the intuition of a curved surface (thogh not unrelated).

Even more generally, the notion of a [[connection on a bundle]] and of a [[Lie algebra]]-valued 1-form generalizes to connections on [[principal 2-bundle]]s and [[principal ∞-bundle]]s and [curvature of ∞-Lie algebroid valued differential forms](#OfInfLieAlgebroidValuedForms).

In [[generalized (Eilenberg-Steenrod) cohomology|Eilenberg-Steenrod]]-type [[differential cohomology]] describing _abelian_ such higher connections these curvatures appear in the form of generalized [[Chern character]] [[curvature characteristic form]]s.

## Extrinsic curvature of embedded manifolds {#Extrinsic}

Curvature $\kappa(\gamma)$ of a smooth curve $\gamma$ at a point $p$ of a smooth curve is the (signed) inverse radius of the (oriented) circle having tangency of order 1 with the curve at the point $p$. Every smooth curve in a 3-dimensional space is determined up to isometry by its (parametrised) curvature and torsion. Intuitively, the curvature measures how much curves are bent, when measured in some metrics. [[Frenet-Serret formulas]] express the derivative of Frenet moving frame with respect to the parameter of a  [[naturally parametrized curve]] in $n$-dimensional Euclidean space as an antisymmetric matrix times the Frenet moving frame. The nonzero coefficients of the matrix are, up to the sign, the curvature of the curve, torsion and higher analogues.
For higher dimensional surfaces one can look at normal curvature which is the curvature of the curve which is intersection of a plane determined by the normal vector to the surface and a given tangent vector. For any 2-dimensional tangent plane, the normal curvature has two extreme values. Their product is called the **Gaussian curvature**.

**Sectional curvature** of a higher dimensional smooth surface at its point $p$ in an Euclidean space along a tangent 2-dimensional plane is the Gaussian curvature of the curve which is the intersection of the surface with the plane. As a plane is determined by two vectors, the sectional curvature is determined by a surface and a pair of vectors, and all possible sectional curvatures can be read form that 2-form; therefore we talk about the operator of the curvature.

Curvature can be described also intrinsically, without recourse to the ambient space and its metric. Therefore it makes sense in Riemannian geometry based on the [[metric]] tensor just on a manifold. In 1917, Herman Weyl postulated a more fundamental quantity than a [[Riemannian metric]], the [[connection on a bundle|connection]] on a fibre bundle, giving hence rise to a modern, generalized idea of the curvature. While Riemannian metric gives rise to a [[Levi-Civita connection]] on the [[tangent bundle]] of the Riemannian manifold, not every connection on a vector or principal bundles is induced by metrics. In that sense the connection is a more basic notion in geometry.

## Curvature of a connection

The _curvature_ of a [[connection on a bundle]] measures how the connection is _locally_ non-trivial.

In as far as the notion of connection on a bundle is generalized by the notion of a cocycle in [[differential cohomology]], curvature is essentially the [[Chern character]].

In as far as cocycles in differential cohomology represent [[gauge field]]s in [[physics]], the curvature is the [[field strength]] of these gauge fields.

After the conception of [[gauge theory]], the term _curvature_ was firmly established in its generalization from this special case to the case of connections on all kinds of [[bundle]]s and [[principal infinity-bundle|higher bundles]].

### Curvature of a Lie-algebra valued form {#OfLieAlgebraValuedForm}

* A connection on a trivial [[line bundle]] on a [[space]] $X$ is just a [[differential forms|1-form]]

  $$
    A \in \Omega^1(X)
    \,.
  $$

  The curvature in this case is the 2-form $F = d A$.

*  A [[connection on a bundle|connection]] on a trivial $G$-[[principal bundle]] for $G$ a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$ is a $\mathfrak{g}$-valued 1-forfm (see [[groupoid of Lie-algebra valued forms]])

   $$
     A \in \Omega^1(X) \otimes \mathfrak{g}
     \,.
   $$

   Its curvature is the Lie-algebra valued 2-form

   $$
     F_A = \mathbf{d} A + [A \wedge A]
     \,,
   $$

   where $[-,-]$ is the Lie bracket in $\mathfrak{g}$.

* According to the discussion at [[∞-Chern-Weil theory]], a connection on a trivial [[principal ∞-bundle]] is given by a collection of [[∞-Lie algebroid valued differential forms]]. The notion of curvature in this general context is discussed at [curvature of ∞-Lie algebroid valued differential forms](#OfInfLieAlgebroidValuedForms).

### Curvature characteristic forms

* [[curvature characteristic form]]

### Geometric interpretation of curvature 2-forms

For the geometric interpretation of the curvature 2-form of a $\mathfrak{g}$-valued 1-form

$$
  \Omega^\bullet(X)
  \leftarrow
  W(\mathfrak{g})
  :
  A
$$

it is useful to recall that both the [[deRham complex]] $\Omega^\bullet(X)$ as well as the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ are naturally interpreted as function algebras on [[infinitesimal object]]s, as discussed at [[∞-Lie algebroid]].

The [[deRham complex]] may be thought of as the algebra of functions on the infinitesimal path ∞-groupoid $\Pi^{inf}(X)$. This has as [[object]]s the points of $X$, as [[morphism]]s [[infinitesimal object|infinitesimal]] paths

$$
  x \to y
$$

in $X$

as [[2-morphism]]s infinitesimal little surfaces

$$
  \array{
     x &\to& y_1
     \\
     \downarrow & \searrow & \downarrow
     \\
     y_2 &\to& z
  }
$$

in $X$, and so on.

On the other hand, $CE(\mathfrak{g})$ is the algebra of functions on the infinitesimal version $\mathbf{B}G_{(1)}$ of what is called the [[delooping]] [[groupoid]] $\mathbf{B}G$ of the [[Lie group]] of which $\mathfrak{g}$ is a [[Lie algebra]]. This has a single [[object]] ${*}$, and a morphism is an infinitesimal group element

$$
  {*} \stackrel{e + \lambda^a t_a}{\to} {*}
$$

for $e$ the neutral element of the group (the identity), for $t_a$ an element of the [[nLab:Lie algebra|Lie algebra]] as before and for $\lambda^a$ some coefficient.

A 2-morphism is an infinitesimal surface bounded by such infinitesimal 1-morphisms such that going either way around the surface

$$
  \array{
     {*} &\stackrel{e + \lambda^a_1 t_a}{\to}& {*}
     \\
     \downarrow^{\rlap{\lambda_3^a t_a}}
     & \searrow & \downarrow^{\rlap{\lambda_2^a t_a}}
     \\
     {*} &\stackrel{\lambda_4^a t_a}{\to}& {*}
  }
$$

produces the same result when then morphisms are composed using the product in the Lie group: the top right way around the square here yields

$$
  (e + \lambda_1^a t_a) )(e + \lambda_2^b t_b) )
  =
  e + \lambda_1^a t_a + \lambda_2^b t_b
  +
  \lambda_1^a \lambda_2^b t_a t_b
$$

and the other way round yields

$$
  (e + \lambda_3^a t_a) )(e + \lambda_4^b t_b) )
  =
  e + \lambda_3^a t_a + \lambda_4^b t_b
  +
  \lambda_3^a \lambda_4^b t_a t_b
  \,.
$$

A morphism of [[nLab:dg-algebra|dg-algebra]]s of the form we have been considering

$$
  \Omega^\bullet(X) \leftarrow CE(\mathfrak{g})
  :
  A
$$

is now evidently equivalenty a morphism

$$
  A : \Pi^{inf}(X) \to \mathbf{B}G_{(1)}
$$

that sends infinitesimal paths in $X$ to infinitesimal group elements of the form $e + \lambda^a t_a$:

$$
 A : (x \to y) \;\;\mapsto\;\;
     ({*} \stackrel{e + A^a(x,y) t_a}{\to} {*})
  \,.
$$

If we denote by

$$
  v = y - x
$$

the [[nLab:tangent vector|tangent vector]] that connects the [[nLab:infinitesimal object|infinitesimally]] close points $x$ and $y$ and write $A(x,y) = A_x(v)$ as a function of the first point and the vector pointing away from it, then this reads

$$
 A : (x \to y) \;\;\mapsto\;\;
     ({*} \stackrel{e + A^a_x(v) t_a}{\to} {*})
  \,.
$$

We can now look at what this assignment $A$ of infinitesimal group elements to infinitesimal paths does to a little square in $X$ as above, with sides spanned by [[nLab:tangent bundle|tangent vector]]s $v_1$ and $v_2$. We find

$$
  A
  \;\;
  :
  \;\;
  \array{
     x &\stackrel{v_1}{\to}& y_1
     \\
     \downarrow^{v_2}
     & \searrow & \downarrow
     \\
     y_2 &\to & z
  }
  \;\;\mapsto\;\;
  \array{
     {*} &\stackrel{e + A_x(v_1) t_a}{\to}& {*}
     \\
     \downarrow^{\rlap{A_x(v_2)^a t_a}}
     & \searrow &
    \downarrow^{\rlap{A_{y_1}(v_2)^a t_a}}
     \\
     {*} &\stackrel{A_{y_2}(v_1)^a t_a}{\to}& {*}
  }
  \,.
$$

For the result on the right to qualify as a [[nLab:k-morphism|2-morphism]] in $\mathbf{B}G_{(1)}$ we need that

going around the top right edges, which yields

$$
  e + A_x^a(v_1) t_a + A^b_{y_1}(v_2) t_b
  +
  A_x^a(v_1) A^b_{y_1}(v_2) t_a t_b
$$

is the same as

$$
  e + A_x^a(v_2) t_a + A^b_{y_2}(v_1) t_b
  +
  A_{x}^a(v_2) A^b_{y_2}(v_1) t_a t_b
  \,.
$$

To express what this means as a condition at the point $x$, we may Taylor expand to first order

$$
  A_{y_1} = A_x + \partial_{v_1} A_x
$$

and

$$
  A_{y_2} = A_x + \partial_{v_2} A_x
  \,.
$$

Then some terms cancel and the above condition becomes, to second order

$$
  \partial_{v_1} A^a_x(v_2) t_a
  +
  A_x^a(v_1) A^b_{x}(v_2) t_a t_b
  =
  \partial_{v_2} A^a_x(v_1) t_a
  +
  A_x^a(v_2) A^b_{x}(v_1) t_a t_b
  \,.
$$

In other words, the expression

$$
  F_A(v_1,v_2)
  :=
  \partial_{v_1} A^a_x(v_2) t_a
  -
  \partial_{v_2} A^a_x(v_1) t_a
  +
  \frac{1}{2} A_x^a(v_1) A^b_{x}(v_2) [t_a, t_b]
$$

has to vanish. This is the curvature form that we already found above by more algebraic means.

If this does not vanish, then we don't really have a morphism $A : \Pi^{inf}(X) \to \mathbf{B}G_{(1)}$. But then we instead have some morphism that uses the 1-forms $A^a$ to assigns data to little edges, _and_ that uses the 2-forms $F_A^a$ to assign data to little surfaces. That morphism then will respect a condition as above, but now on little cubes. That condition is the **[[Bianchi identity]]**

$$
  d F_A + [A \wedge F_A] = 0
$$

on the curvature 2-form.

## Curvature of $\infty$-Lie algebroid-valued forms {#OfInfLieAlgebroidValuedForms}

The notion of curvature of a [[Lie-algebra valued 1-form]] discussed [above](#OfLieAlgebraValuedForm) generalizes to that of [[∞-Lie algebroid valued differential forms]].

### Definition

Let $\mathfrak{g}$ be an [[∞-Lie algebra]]. A $\mathfrak{g}$-valued differential form on a [[smooth manifold]] $X$ is a morphism

$$
  \Omega^\bullet(X) \leftarrow
  W(\mathfrak{g})
  : A
$$

of [[dg-algebra]]s, where $\Omega^\bullet(X)$ is the [[de Rham complex]] and $W(\mathfrak{g})$ is the [[Weil algebra]].

There is a canonical inclusion of [[graded vector space]]s

$$
  W(\mathfrak{g}) \leftarrow \mathfrak{g}^*[1] : F_{(-)}
  \,.
$$

The **curvature of the $\infty$-Lie algebroid valued form $A$** is the composite

$$
  \Omega^\bullet(X) \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{F_{(-)}}{\leftarrow}
  \mathfrak{g}^*[1]
  :
  F_A
  \,.
$$

This consists, in general, of a tower of components: write $\mathfrak{g}_n$ for the degree $n$-part of the $\infty$-Lie algebra, then we have the further restrictions

$$
  \Omega^\bullet(X) \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{F_{(-)}}{\leftarrow}
  \mathfrak{g}_{k}^*[1]
  :
  (F_A)_{k+1}
  \,.
$$

So

$$
  (F_A)_n
  \in
  \Omega^n(X, \mathfrak{g}_{n-1})
$$

is a $\mathfrak{g}_{n-1}$-valued $n$-form on $X$.

One may speak of the **2-form curvature**, the **3-form curvature**, the **4-form curvature** and so on.

If instead of just an $\infty$-Lie algebra $\mathfrak{g}$ we take more generally an [[∞-Lie algebroid]], then there is also a **1-form curvature component**.

**Remark** In some places in the literature, the lower curvature form components have been called _fake curvature_ ([BreenMessing](http://arxiv.org/abs/math/0106083)).

### Properties

#### Obstruction to flatness

Precisely if the curvatures $F_A$ vanish does the morphism $A : W(\mathfrak{g}) \to \Omega^\bullet(X)$ factor through the [[Chevalley-Eilenberg algebra]] $W(\mathfrak{g}) \to CE(\mathfrak{g})$.

$$
  (F_A = 0)
  \;\;\Leftrightarrow
  \;\;
  \left(
  \array{
     && CE(\mathfrak{g})
     \\
     & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow
     \\
     \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g})
  }
  \right)
$$

in which case we call $A$ **flat**.

#### Bianchi identity

By the fact that $A$ is a [[dg-algebra]] [[homomorphism]], its curvature forms satisfy

$$
  d_{dR} F_A + A(d_{W(\mathfrak{g})}(-)) = 0
  \,.
$$

This is the [[Bianchi identity]].

### Curvature characteristic forms

The algebra $inv(\mathfrak{g})$ of [[invariant polynomial]]s embeds into the Weil algebra

$$
  W(\mathfrak{g}) \leftarrow inv(\mathfrak{g})
  \,.
$$

For $A$ a $\mathfrak{g}$-valued form, and $\langle - \rangle \in inv(\mathfrak{g})$, the ordinary closed $n$-form

$$
  \Omega^\bullet(X)
  \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{}{\leftarrow}
  inv(\mathfrak{g})
  \stackrel{\langle - \rangle}{\leftarrow}
  CE(b^{n-1} \mathbb{R})
  :
  \langle F_A \rangle
$$

is the corresponding [[curvature characteristic form]].

## Curved dg-algebras

(TO ADD: Something about curved $A_\infty$ algebras and curved dg algebras.)

## Related concepts

* [[∞-Lie algebroid valued differential forms]]

* [[connection on an ∞-bundle]]

* [[Bianchi identity]]

* [[curvature characteristic form]]

* [[Chern-Simons form]]

[[!include curvature in Riemannian geometry -- table]]

$\,$

[[!include gauge field - table]]

## References

* [[Mikio Nakahara]], Section 10.3 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

[[!redirects curvatures]]

[[!redirects curvature form]]
[[!redirects curvature forms]]

[[!redirects curvature 2-form]]
[[!redirects curvature 2-forms]]

[[!redirects curvature differential form]]
[[!redirects curvature differential forms]]


curvature characteristic form

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## Idea

A _curvature characteristic form_ is a [[differential form]] naturally associated to a [[Lie algebra]]-valued 1-form that is a measure for the non-triviality of the [[curvature]] of the 1-form.

More generally, there is a notion of curvature characteristic forms of [[L-∞-algebra]]-valued differential forms and [[schreiber:∞-Lie algebroid valued differential forms]].

## Definition

### Of connection 1-forms

For $\mathfrak{g}$ a [[Lie algebra]], $\langle -,-, \cdots, -\rangle$ an [[invariant polynomial]] of $n$ arguments on the Lie algebra and $A \in \Omega^1(P,\mathfrak{g})$ a Lie-algebra-valued 1-form with [[curvature]] 2-form $F_A = d_{dR} A + [A \wedge A]$, the **curvature characteristic form** of $A$ with respect to $\langle \cdots \rangle$ is the [[differential form]]

$$
  \langle F_A \wedge \cdots \wedge F_A \rangle \in \Omega^{2 n}(P)
  \,.
$$

This form is always an [[exact form]].  The $(2 n -1)$-form trivializing it is called a [[Chern-Simons form]].

Notably if $G$ is a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$, $P$ is the total space of a $G$-[[principal bundle]] $\pi : P \to X$, and $A \in \Omega^1(P,\mathfrak{g})$ is an [[Ehresmann connection]] 1-form on $P$ then by the very definition of the $G$-equivariance of $A$ and the _invariance_ of $\langle \cdots \rangle$ it follows that the curvature form is invariant under the $G$-action on $P$ and is therefore the pullback along $\pi$ of a $2 n$-form $P_n \in \Omega^{2 n}(X)$ down on $X$. This form is in general no longer exaxt, but is always a [[closed form]] and hence represent a class in the [[de Rham cohomology]] of $X$. This establishes the [[Weil homomorphism]] from invariant polynomials to de Rham cohomology

### In terms of $\infty$-Lie algebroids

The above description of curvature characteristic forms may be formulated in terms of [[∞-Lie theory]] as follows.

For $P \to X$ a $G$-[[principal bundle]] write $T X$, $T P$ and $T_{vert} P$ for the [[tangent Lie algebroid]] of $X$, of $P$ and the [[vertical tangent Lie algebroid]] of $P$, respectively. Write $inn(\mathfrak{g})$ for the [[Lie 2-algebra]] given by the [[differential crossed module]] $\mathfrak{g}\stackrel{Id}{\to} \mathfrak{g}$ and finally $\prod_i b^{n_i} \mathbb{R}$ for the [[L-∞-algebra]] with one abelian generator for each generating [[invariant polynomial]] of $\mathfrak{g}$

From the discussion at [[invariant polynomial]] we have a canonical morphism $inn(\mathfrak{g}) \to \prod_i b^{n_i}\mathbb{R}$ that represents the generating invariant polynomials.

Recall that a morphism of [[∞-Lie algebroid]]s

$$
  T X \to b^n \mathbb{R}
$$

is equivalently a closed $n$-form on $X$. The data of an [[Ehresmann connection]] on $P$ then induces the following diagram of [[∞-Lie algebroid]]s

$$
  \array{
    T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g}
    &&&
    flat vertical form
    \\
    \downarrow && \downarrow
    &&&
    first Ehresmann condition
    \\
    T P &\stackrel{A}{\to}& inn(\mathfrak{g})
    &&&
    form on total space
    \\
    \downarrow && \downarrow
    &&&
    second Ehresmann condition
    \\
    T X &\stackrel{(P_n)}{\to}& \prod_i b^{n_i} \mathbb{R}
    &&&
    curvature characteristic forms
  }
  \,.
$$

## Examples

* The single curvature characteristic form of a complex [[line bundle]]/$U(1)$-[[principal bundle]] is the curvature 2-form itself.

[[!include Chern- and Pontrjagin forms -- section]]

## Related concepts

* [[characteristic class]]

* [[Chern-Weil homomorphism]]

* [[∞-Lie algebroid valued differential forms]]

* [[adjusted Weil algebra]]

* [[connection on an ∞-bundle]]

* [[curvature]]

* [[Bianchi identity]]

* [[Chern-Simons form]]

## References

[[!include Chern-Weil homomorphism -- references]]

[[!redirects curvature characteristic forms]]

[[!redirects characteristic form]]
[[!redirects characteristic forms]]

[[!redirects characteristic differential form]]
[[!redirects characteristic differential forms]]

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

[[!redirects total Chern form]]
[[!redirects total Chern forms]]

[[!redirects Chern form]]
[[!redirects Chern forms]]

[[!redirects Pontrjagin form]]
[[!redirects Pontrjagin forms]]

[[!redirects Pontryagin form]]
[[!redirects Pontryagin forms]]

[[!redirects total Pontrjagin form]]
[[!redirects total Pontrjagin forms]]

[[!redirects total Pontryagin form]]
[[!redirects total Pontryagin forms]]


curvature in Riemannian geometry -- table

| [[curvature]] in [[Riemannian geometry]] |
|------------------------------------------|
| [[Riemann curvature]]    |
| [[Ricci curvature]]  |
| [[scalar curvature]] |
| [[sectional curvature]] |
| [[p-curvature]] |


curve

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# Curves
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{: toc}

## In differential geometry

For $X$ a [[smooth manifold]], a (parametrized oriented) __smooth curve__ in $X$ is a [[smooth function]] $\gamma\colon \mathbb{R} \to X$ from the [[real line]] (or an [[interval]] therein) to $X$.  (Compare [[path]].)

For most purposes in [[differential geometry]] one needs to work with a __regular curve__, which is a parametrized smooth curve whose [[velocity]], i.e. the [[derivative]] with respect to the parameter, is never zero. For example, this is important if one wants to split curve into segments which have no self-intersections, which is important.

In the foundations of [[differential topology]], it is possible to define a [[tangent vector]] as an equivalence class of smooth curves at a given point in the image of the curve, effectively identifying a curve with its derivative at (say) $0$.

See also the [[fundamental theorem of differential geometry of curves]].

### Open and closed curves

In a [[Cartesian space]] $\mathbb{R}^n$, an __open smooth curve__ is a smooth curve $\gamma:\mathbb{R} \to \mathbb{R}^n$ which does not intersect itself: For every real number $a \in \mathbb{R}$ and $b \in \mathbb{R}$, the [[distance]] between the two points on the curve parameterized by $a$ and $b$ is greater than zero: $\rho(\gamma(a),\gamma(b)) \gt 0$. Equivalently, an open smooth curve is a smooth curve such that the [[shape]] of the [[image]] of $\gamma$ is [[contractible]]: $\esh \mathrm{im}(\gamma) \simeq *$.

A __closed smooth curve__ is a smooth curve $\gamma:\mathbb{R} \to \mathbb{R}^n$ which does intersect itself: for every real number $a \in \mathbb{R}$, there is a real number $b \in \mathbb{R}$, such that $\gamma(a) = \gamma(a + b)$. Equivalently, a closed smooth curve is a smooth curve such that the [[shape]] of the [[image]] of $\gamma$ is [[equivalent]] to the [[circle type]]: $\esh \mathrm{im}(\gamma) \simeq S^1$.

In classical mathematics, every smooth curve $\gamma:\mathbb{R} \to \mathbb{R}^n$ is either open or closed. In [[constructive mathematics]], there are smooth curves where it cannot be proved to be either open or closed, due to the failure of [[trichotomy]].

## In algebraic geometry

In [[algebraic geometry]], an [[algebraic curve]] is a $1$-dimensional [[algebraic variety]] over a field.

An example: [[elliptic curve]].

## Related concepts

* [[surface]]

* [[differential geometry of curves and surfaces]]

* [[singular point of a curve]]

* [[torsion of a curve]]

* [[surface]], [[hypersurface]]

* [[velocity]], [[acceleration]], [[jerk]]

## References

See also

+ Wikipedia, _[Curve](https://en.wikipedia.org/wiki/Curve)_

[[!include infinitesimal and local - table]]

[[!redirects curve]]
[[!redirects curves]]

[[!redirects smooth curve]]
[[!redirects smooth curves]]

[[!redirects open curve]]
[[!redirects open curves]]

[[!redirects closed curve]]
[[!redirects closed curves]]


curved A-infinity algebra

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## Idea

A _curved $A_\infty$_ is defined just like an [[A-infinity algebra]], but allowing in the definition also a 0-ary operation. $m_0$. Including this means thte unary operation no longer squares to 0 but to be proportinal to the binary product with this constant

$$
  m_1 \circ m_1 \propto m_2(m_0,-)
  \,.
$$

This $m_0$ is called the _curvature_ of the curved $A_\infty$-algebra.

## Examples

* The curved [[BGG sequences]] under cup product yield curved $A_\infty$-algebras ([Calderbank-Diemer 00, section 6](#CalderbankDiemer00)).

## Related concepts

* [[curved dg-algebra]]

* [[curved L-infinity algebra]]

## References

* {#CalderbankDiemer00} [[David Calderbank]], Tammo Diemer, _Differential invariants and curved Bernstein-Gelfand-Gelfand sequences_, J.Reine Angew.Math. 537 (2001) 67-103 ([arXiv:math/0001158](http://arxiv.org/abs/math/0001158))

[[!redirects curved A-infinity algebras]]

[[!redirects curved A-∞ algebra]]
[[!redirects curved A-∞ algebras]]


curved dg-algebra

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## Idea

A _curved dg-algebra_ is like a [[dg-algebra]], but instead of the [[differential]] squaring to 0, it squares to the graded commutator with a fixed element of the algebra: its "[[curvature]]".

This is like the [[covariant derivative]] on the [[section]]s of a [[vector bundle]] with [[connection on a bundle|connection]] satisfying $\nabla \circ \nabla = F_\nabla$, where $F_\nabla$ is the [[curvature]] 2-form of the connection (valued, here, in fiber [[endomorphism]])s.

Curved dg-algebras appear in the description of various [[TQFT]]s.

(...)

## Related concepts

* [[curved L-infinity algebra]]

* [[curved A-infinity algebra]]

## References

A basic exposition of the definition is in

* A. Polishchuk, _Introduction to curved dg-algebra_ , notes taken in a talk ([pdf](http://www.math.polytechnique.fr/SEDIGA/documents/polishchuk_ens100324.pdf))

For applications in [[derived categories]] of [[D-brane]]s in [[Landau-Ginzburg model]]s see

* [[Dmitri Orlov]], _Derived Categories of Singularities and D-branes in Landau-Ginzburg models_ , ([arXiv:math.ag/0503632](http://arxiv.org/abs/math.ag/0503632))

* Anatoly Preygel, _Thom-Sebastiani and duality for matrix factorizations_ [arXiv:1101.5834](http://arxiv.org/abs/1101.5834)

An natural construction of curved dg-algebras as de Rham / Dolbeault complexes on a [[circle n-bundle with connection|circle 2-bundle with connection]] is in

* [[Jonathan Block]], _Duality and equivalence of module categories in noncommutative geometry_, [pdf](http://www.math.upenn.edu/~blockj/papers/BottVolume.pdf), in R. Bott Memorial Volume

and with more details in section 2 of

* [[Jonathan Block]], Calder Daenzer, _Mukai duality for gerbes with connection_ ([arXiv:0803.1529v2](http://arxiv.org/abs/0803.1529v2))

[[!redirects curved dg-algebras]]


curved L-infinity algebra

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{:toc}

## Idea

A _curved $L_\infty$-algebra_ (e.g. [Markl 11, p. 100](#Markl11)) is just like an ordinary _[[L-∞ algebra]]_, but possibly including also a 0-ary bracket, i.e. a constant. Conversely an ordinary [[L-∞ algebra]] is a curved $L_\infty$-algebra for which the 0-ary operation happens to be zero.

Accordingly, a "strong homotopy [[homomorphism]]" of curved $L_\infty$-algebras is defined just as for ordinary $L_\infty$-algebras, but allowing also for a 0-ary component. Notice that such "curved sh-maps" may be non-trivial even between ordinary $L_\infty$-algebras (amplified e.g. in [Mehta-Zambon 12, below (2)](#MehtaZambon12)).

The dual [[Chevalley-Eilenberg algebras]] automatically capture curved $L_\infty$-algebras unless one imposes a constraint: the non-curved $L_\infty$-algebras correspond to the [[augmented algebra|augmented]] CE-algebras. Similarly in the [[dg-coalgebra]] description the restriction to non-curved $L_\infty$-algebras requires co-augmentation or else (this is what is commonly used) non-unital dg-coalgebras.

## Definition

Where an ordinary _[[L-infinity algebra]]_ is a $\mathbb{Z}$-[[graded vector space]] $\mathfrak{g}$ equipped for all $n \in \mathbb{N}$, $n \geq 1$ with $n$-ary brackets:

$$
  l_n \;\colon\; \mathfrak{g}^{\otimes^n} \longrightarrow \mathfrak{g}
$$

out of the [[tensor product]] of $n$-copies of $\mathfrak{g}$,
subject to some conditions,  for a curved $L_\infty$-algebra also a component

$$
  l_0 \;\colon\; \mathfrak{g}^{\otimes^0} \simeq \mathbb{R} \to \mathfrak{g}
$$

is allowed allowed. Since an $\mathbb{R}$-linear map out of $\mathbb{R}$ is uniquely fixed by a single element (the image of $1 \in \mathbb{R}$), this is "a constant", called the _curvature_ of the curved $L_\infty$-algebra.

Now the strong homotopy Jacobi identity

\[
  \label{LInfinityJacobiIdentity}
  \sum_{{i,j \in \mathbb{N}} \atop {i+j = n+1}}
  \sum_{\sigma \in UnShuff(i,j)}
  \chi(\sigma,v_1, \cdots, v_{n})
  (-1)^{i(j-1)}
   l_{j} \left(
     l_i \left( v_{\sigma(1)}, \cdots, v_{\sigma(i)} \right),
     v_{\sigma(i+1)} , \cdots , v_{\sigma(n)}
   \right)
  = 0
  \,,
\]

implies in particular that

$$
  l_1 \circ l_1 = \pm l_2(l_0, -)
$$

hence that the unary operation $l_1$ no longer necessarily squares to zero (no longer defines a [[chain complex]] $(\mathfrak{g}, l_1)$) but to the binary bracket with the curvature.

## Related concepts

* [[curved A-infinity algebra]]

* [[curved dg-algebra]]

## References

* {#Markl11} [[Martin Markl]], _Deformation Theory of Algebras and Their Diagrams_, Regional Conference Series in Mathematics Number 116, American Mathematical Society (2011)

* [[Andrey Lazarev]], [[Travis Schedler]], _Curved infinity-algebras and their characteristic classes_, J Topology (2012) 5 (3): 503-528 ([arXiv:1009.6203](https://arxiv.org/abs/1009.6203))

* {#MehtaZambon12} [[Rajan Mehta]], [[Marco Zambon]], _$L_\infty$-Actions_, Differential Geometry and its Applications 30 (2012), 576-587 ([arXiv:1202.2607](https://arxiv.org/abs/1202.2607))

[[!redirects curved L-infinity algebras]]

[[!redirects curved L-∞ algebra]]
[[!redirects curved L-∞ algebras]]

[[!redirects curved sh-map]]
[[!redirects curved sh-maps]]

[[!redirects curved L-infinity homomorphism]]
[[!redirects curved L-infinity homomorphisms]]

[[!redirects curved L-∞ homomorphism]]
[[!redirects curved L-∞ homomorphisms]]


curved spacetime

## Idea

A [[spacetime]] which as a [[pseudo-Riemannian manifold]] has non-vanishing [[Riemann curvature]] is called _curved_. This corresponds to a non-trivial [[field (physics)|field]] of [[gravity]].

## Related concepts

* [[AQFT on curved spacetimes]]

[[!redirects curved spacetimes]]


curvelet

* Emmanuel Candès, _What is a curvelet_, Notices Amer. Math. Soc. 2003, [pdf](http://www.ams.org/notices/200311/what-is.pdf)
* E. J. Candès, L. Demanet, _Curvelets and Fourier integral operators_, C. R. Math. Acad. Sci. Paris 336 (2003), 395–398 [MR1979352](http://www.ams.org/mathscinet-getitem?mr=1979352) doi


curving

#Contents#
* table of contents
{:toc}

## Idea

The [[differential 2-form]]-data in a [[circle 2-bundle with connection]] is sometimes called the "curving" of the connection data.

This terminology results from thinking of the [[circle 2-bundle]] equivalently as a [[bundle gerbe]] given by a transition-[[circle bundle]] on a suitable [[Cech groupoid]]. From this perspective the [[differential 1-form]] is an ordinary [[connection on a bundle]] for the transition bundles, and the 2-form data looks a bit like a variant of the _[[curvature]]_ of these bundles, which not exactly being this curvature. Hence "curving".

But of course the [[circle 2-bundle with connection]] has its own [[curvature]] [[differential 3-form]]. At least as soon as one passes to [[circle n-bundles with connection]] for arbitrary $n \in \mathbb{N}$, the term "curving" becomes inappropriate.

## Properties

### $(\mathbf{B}U(1))$-principal 2-connections without curving

Circle 2-bundles "with connection but without curving" have as [[moduli infinity-stack|moduli 2-stack]] the [[delooping]]

$\mathbf{B}(\mathbf{B}U(1)_{conn}) \in$ [[Smooth∞Grpd]]

of the [[moduli stack]] $\mathbf{B}U(1)_{conn}$ or [[circle bundles with connection]].

The [[higher Atiyah groupoid]] of such a "pseudo-2-connection" $\nabla_1 \colon X \to \mathbf{B}(\mathbf{B}U(1)_{conn})$ is the corresponding [[higher Courant groupoid]] which [[Lie integration|Lie integrates]] the corresponding [[standard Courant Lie 2-algebroid]]. (See at _[[higher Atiyah groupoid]]_ for details.)

## Related concepts

* [[curvature]]

* [[differential cohomology]]

[[!redirects curvings]]


cusp

#Contents#
* table of contents
{:toc}

## Idea

A kind of [[singular point of an algebraic variety|singularity]].

For [[elliptic curves]] see at _[elliptic curve -- Nodal curves and cuspidal curves](elliptic+curve#EllipticCurvesNodalCurvesCuspidalCurves)_

## Related concepts

* [[nodal curve]]

[[!redirects cusps]]

[[!redirects cuspidal curve]]
[[!redirects cuspidal curves]]

[[!redirects cuspidal cubic curve]]
[[!redirects cuspidal cubic curves]]

[[!redirects cuspidal cubic]]
[[!redirects cuspidal cubics]]


cusp form

## Definition

A _cusp form_ is a [[modular form]] for which the zeroth Fourier coefficient is zero.

## References

See also:

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

[[!redirects cusp forms]]


cut rule

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _cut rule_ in [[sequent calculus]] ([[formal logic]]) is the [[structural rule|structural]] [[inference rule]] that from [[sequents]] of the form

$$
  \Gamma \vdash A , \Delta
$$

and

$$
  \Pi, A \vdash \Lambda
$$

the new sequent

$$
  \Pi, \Gamma \vdash \Lambda, \Delta
$$

may be [[deduction|deduced]]. This is often written in the form

$$\frac{\Gamma \vdash A, \Delta \quad \Pi, A \vdash \Lambda}{\Pi, \Gamma \vdash \Lambda, \Delta} \; cut.$$

In the [[categorical semantics]] where each [[sequent]] here is interpreted as a [[morphism]] in a [[category]], the cut rule asserts the existence of [[composition]] of morphisms.

## Cut elimination

> "A sequent calculus without cut-elimination is like a car without an engine" -- Jean-Yves Girard (in [Linear logic](http://iml.univ-mrs.fr/~girard/Synsem.pdf.gz))

The _cut-elimination theorem_ ("[[Gerhard Gentzen]]'s [[Gentzen's Hauptsatz|Hauptsatz]]") asserts that every [[judgement]] which has a [[proof]] using the cut-rule also has a proof not using it (a "cut-free proof").  While Gentzen\'s original theorem was for a few particular sequent calculi that he was considering, it is true of many other sequent calculi and is generally seen as desirable.  (That said, there are some useful sequent calculi in which it fails.)

Intuitively, the problem in deciding whether a formula $B$ follows from a formula $A$, i.e., deriving $A \vdash B$, is that there could be very complicated steps in the middle, i.e., in typical mathematical arguments one puts together steps $A \vdash C$ and $C \vdash B$ where $C$ is potentially a complicated or large formula. For an automated theorem prover, the search space for such $C$ is potentially infinite. By establishing a cut-elimination theorem for formal systems, one circumvents this problem, and it is quite typical that cut-free proofs build up complex sequents from less complex sequents (cf. [[subformula property]]), so that one can decide whether a sequent is provable or derivable by following an inductive procedure.

Cut-elimination is also a key step in deciding whether two proofs of a sequent are the "same" in some suitable sense. In [[type theory]], for instance, the issue is not merely whether $A \vdash B$ is provable or whether the function type $A \multimap B$ is inhabited (has a proof or a term witnessing that fact), but also the nature of the space of such proofs. Since any proof has a trivial cut-free formulation in a system where all provable sequents in the original system are simply postulated as axioms, a cut-elimination result worthy of the name will not merely replace a proof with one which is cut-free, but with a cut-free proof which is _equivalent_ to the original. This idea is used for instance in proving [[coherence theorems]].

Cut-elimination may also be used to give independent proof-theoretic motivation of the definition of a category, and other basic category theoretic notions, eg. [[adjunction]] (see [Došen 99](#Dos99)).

## Connection to identities

In the analogy between the composition and the cut rule, the analogue of [[identity morphisms]] (or nullary compositions) is the identity rule

$$
  A \vdash A
$$

Typically, a cut-elimination algorithm goes hand-in-hand with an algorithm which eliminates the identity rule, or rather which pushes back identities as far as possible, down to identities for basic propositional variables (so for example, $p \wedge q \vdash p \wedge q$ may be proved using $p \vdash p$ and $q \vdash q$, in addition to the rules for $\wedge$, but $p \vdash p$ itself must be adopted as an axiom).

In fact, there is a sense in which elimination of cuts is seen as _dual_ to elimination of identities, analogous to the sense in which [[beta reduction]] is seen as dual to [[eta expansion]]. Very typically, a normalization scheme on terms first applies eta expansions are far as they will go, and then applies beta reductions as far as they will go, so as to at last reach a normal form. The same goes for rewrite systems on sequent deductions, which first eliminate identities, then eliminate cuts.

## Examples of elimination steps

The conversion

$$
\frac{\displaystyle \frac{\Gamma, A \vdash B, \Delta}{\Gamma \vdash A \multimap B,\Delta} \;\;\; \frac{\Pi_1 \vdash A,\Lambda_1 \;\;\; \Pi_2,B \vdash \Lambda_2}{\Pi_2,\Pi_1, A\multimap B \vdash \Lambda_2,\Lambda_1}}{\Pi_2,\Pi_1, \Gamma \vdash \Lambda_2,\Lambda_1, \Delta}
  \quad\to\quad
\frac{\displaystyle \frac{\Gamma, A \vdash B, \Delta \;\;\; \Pi_1 \vdash A,\Lambda_1}{\Pi_1,\Gamma \vdash \Lambda_1,B,\Delta} \;\;\; \Pi_2,B \vdash \Lambda_2}{\Pi_2,\Pi_1, \Gamma \vdash \Lambda_2, \Lambda_1,\Delta}
$$

replaces a single cut on the formula $A \multimap B$ with a pair of cuts on the formulas $A$ and $B$, in the process eliminating the use of the logical rules ${\multimap}R$ and ${\multimap}L$.

Although this step replaces one cut by two, the cuts have been in effect pushed up the proof tree, to formulas of lower complexity. Cuts are finally eliminated when they have been pushed all the way up to identity axioms on propositional variables, by applying conversions of type

$$\frac{\displaystyle \frac{}{x \vdash x}\; axiom \;\;\;\; \frac{}{x \vdash x}\; axiom}{x \vdash x} \; cut \quad\to\quad \frac{}{x \vdash x}\; axiom.$$

Likewise, the conversion

$$A \wedge B \vdash A \wedge B
\quad\to\quad
\frac { \displaystyle \frac { A \vdash A } { A \wedge B \vdash A } \;\;\; \frac { B \vdash B } { A \wedge B \vdash B} } { A \wedge B \vdash A \wedge B } $$

reconstructs the identity on $A \wedge B$ from identities on $A$ and on $B$, by first applying the ${\wedge}R$ rule followed by the two ${\wedge}L$ rules (reading the derivation on the right bottom-up).

(Compare these two conversions arising from cut- and identity-elimination to the lambda calculus conversions $(\lambda x.t_1) t_2 \to t_1[t_2/x]$ and $t \to \langle\pi_1t,\pi_2 t\rangle$, i.e., a $\beta$ reduction and an $\eta$ expansion respectively.)

## Alternative forms of the cut rule

In [[linear logic]] (for instance), one sometimes sees sequents written in one-sided form:

$$\; \vdash \Gamma.$$

Here the [[negation]] operator is used to mediate between classical two-sided sequents and one-sided sequents, according to a scheme where a sequent $\Gamma, A \vdash \Delta$ is associated with a sequent $\Gamma \vdash \Delta, \neg A$ (each being derivable from the other). Thus one can contemplate sequents where all formulae have been pushed to the right of the entailment symbol $\vdash$.

For such one-sided sequents, say in multiplicative linear logic, the cut rule may be expressed in the form

$$\frac{\vdash \Gamma, \neg A \;\;\; \vdash \Delta, A}{\vdash \Gamma, \Delta} \; cut$$

and this rule is 'dual' to one which introduces an identity:

$$\frac{}{\vdash \neg A, A} \; identity.$$

Categorically, the cut rule in this form corresponds to the arrow $\neg A \otimes A \to \bot$ that implements an evaluation, and the identity rule corresponds to an arrow $\top \to \neg A \wp A = A \multimap A$ that names an identity morphism. These two arrows are de Morgan dual to one another.

## Cut elimination as a computation

[[Gerhard Gentzen]]'s [[Gentzen's Hauptsatz|Hauptsatz]] is often formulated under the form: "The cut rule is an [[admissible rule]]"- ie. the logical system obtained by removing the cut rule proves exactly the same sequents that the original system with the cut rule, it is important that very often, a stronger statement is true.

The theorem of Gentzen was formulated for [[classical logic]] and is also true in the other logics with a cut rule such as [[intuitionistic logic]] or [[linear logic]]. In these two latter systems, one has more, the cut elimination can be formulated as "There exists a cut elimination algorithm specified like this:

* Entry: proof $\pi_{1}$ of a sequent $\Gamma \vdash \Delta$
* Output: proof $\pi_{2}$ of the sequent $\Gamma \vdash \Delta$ which doesn't use the cut rule"

A better way to present things would be: "Here is an algorithm which takes in entry a proof $\pi_{1}$ of $\Gamma \vdash \Delta$ and returns a proof $\pi_{2}$ of $\Gamma \vdash \Delta$ which doesn't use the cut rule", because an actual algorithm is actually given and it is important to know what is this algorithm, we're not interested only by its existence.

The best way to tell it would be in fact as:

"Here is a [[rewriting system]] $\rightarrow$ on the set of proofs of this logic such that if $\pi_{1} \rightarrow \pi_{2}$ then $\pi_{1}, \pi_{2}$ are proofs of the same sequent, and for every proof $\pi$ of $\Gamma \vdash \Delta$, there exists a finite sequence $\pi_{0}, \pi_{1}, \ldots, \pi_{n}$ of proofs of $\Gamma \vdash \Delta$ such that $\pi = \pi_{0} \rightarrow \pi_{1} \rightarrow ... \rightarrow \pi_{n}$
and $\pi_{n}$ doesn't use the cut rule."

One can after this study the properties of this particular rewriting system: is it [[confluent]]? Is it [[normalizing]]? [[strongly normalizing]]? Does it provide a [[deterministic algorithm]] or a [[nondeterministic algorithm]]? What is the [[computational complexity]] of the algorithms it provides to eliminate cuts?

Unfortunately, the algorithmic aspect of cut elimination is rarely presented as explicitly, and most of the time the cut elimination is presented as the admissibility of the cut rule and the concrete algorithm to eliminate the instance of cuts in a proof $\pi$ must be extracted by the reader from this proof (of the cut elimination) which nevertheless use it in order to do the proof (of the cut elimination).

We're not sure whether there isn't such an algorithm of cut elimination for [[classical logic]] and this an ongoing subject of research. We can think that the usual sequent calculus of classical logic must be modified by using ideas such as [[polarity in type theory|polarization]] and [[focusing]] in order to understand the computational content of the cut elimination of classical logic.

## References

The notion of the cut rule as a [[structural inference rule]] originates (under the German name *Schnitt*) with:

* {#Gentzen35} [[Gerhard Gentzen]], §1.2.1 in: _Untersuchungen über das logische Schließen I_ _Mathematische Zeitschrift_ **39** 1 (1935) [[doi:10.1007/BF01201353](http://dx.doi.org/10.1007/BF01201353)]

* {#Gentzen69} [[Gerhard Gentzen]], §1.21 in: *Investigations into Logical Deduction*, in M. E. Szabo (ed.), *The Collected Papers of Gerhard Gentzen*, Studies in Logic and the Foundations of Mathematics **55**, Springer (1969) 68-131 [[ISBN:978-0-444-53419-4](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/55), [pdf](https://logic-teaching.github.io/prop/texts/Gentzen%201969%20-%20Investigations%20into%20Logical%20Deduction.pdf)]

* Wikipedia, _[Cut-elimination theorem](http://en.wikipedia.org/wiki/Cut-elimination_theorem)_

* {#Dos99} Kosta Došen, _Cut Elimination in Categories_ Dordrecht: Springer Netherlands, 1999.

[[!redirects cut-elimination]]
[[!redirects cut elimination]]

[[!redirects cut-elimination theorem]]
[[!redirects cut elimination theorem]]

[[!redirects cut-admissibility]]
[[!redirects cut admissibility]]

[[!redirects cut]]
[[!redirects cuts]]

[[!redirects Gentzen's Hauptsatz]]
[[!redirects Gentzen Hauptsatz]]