kernel

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#### Category theory
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[[!include category theory - contents]]
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#### Homological algebra
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[[!include homological algebra - contents]]
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#Contents#
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## Idea

The _kernel_ of a [[morphism]] is that part of its [[domain]] which is sent to [[zero]].

## Definition

There are various definitions of the notion of kernel, depending on the properties and structures available in the ambient category. We list a few definitions and discuss (in parts) when they are equivalent.

### As a pullback

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

In a [[category]] with an [[initial object]] $0$ and [[pullbacks]], the __kernel__ $ker(f)$ of a [[morphism]] $f: A \to B$ is the [[pullback]] $ker(f) \to A$ along $f$ of the unique morphism $0 \to B$

$$
  \array{
    ker(f)
    &\to&
    0
    \\
    {}^{\mathllap{p}}\downarrow && \downarrow
    \\
    A &\stackrel{f}{\to}& B
  }
  \,.
$$

=--

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

More explicitly, this characterizes the object $ker(f)$ as [[generalized the|the]] object (unique up to unique [[isomorphism]]) that satisfies the following [[universal property]]:

for every object $C$ and every morphism $h : C \to A$ such that $f\circ h = 0$ is the [[zero morphism]], there is a unique morphism $\phi : C \to ker(f)$ such that $h = p\circ \phi$.

=--

### As an equalizer

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

In a [[category]] with [[zero morphism|zero morphisms]] (meaning: [[enriched category|enriched]] over the [[category of pointed sets]]), [[generalized the|the]] **kernel** $ker(f)$ of a [[morphism]] $f : c \to d$ is, if it exists, the [[equalizer]] of $f$ and the zero morphism $0_{c,d}$.

=--

### As a weighted limit

In any category enriched over pointed sets, the kernel of a morphism $f:c\to d$ is the universal morphism $k:a\to c$ such that $f \circ k$ is the basepoint.  It is a [[weighted limit]] in the sense of [[enriched category theory]].  This applies in particular in any (pre)-[[additive category]].

This is a special case of the construction of [[generalized kernels]] in enriched categories.

### As a representing object
 {#AsARepresentingObject}

Let [[Ab]] be the [[category]] of [[abelian groups]]. This has all kernels, in particular (it is the archetypical [[abelian category]]).

In  every [[Ab]]-[[enriched category]] $A$, for every [[morphism]] $f \colon X\to Y$ in $A$ there is a [[subfunctor]]

$$ker f \colon A^{op}\to Ab$$

of the [[representable functor]] $hom(-,X)$, defined on [[objects]] by

$$ (ker f)(Z) = ker\big(hom(Z,X)\to hom(Z,Y)\big), $$

where $ker$ on the right-hand side is the kernel in [[Ab]].

If the category is in fact [[preabelian category|preabelian]], $ker f$ is also representable with [[representing object]] $Ker f$. One has to be careful with the [[cokernel]] $Coker f$, which does not represent the functor naive  $coker f$ defined as $(coker f)(Z) = coker\big(hom(Z,X)\to hom(Z,Y)\big)$ in [[Ab]],
which is often not representable at all, even in the simple example of the category of [[abelian groups]].  Instead, as a [[colimit]] construction, one should [[corepresentable functor|co-represent]] another functor, namely, the [[covariant functor]] $Z\mapsto ker\big(hom(Y,Z) \to hom(X,Z)\big)$ (which is a [[quotient object|quotent]] of the corepresentable functor $hom(X,-)$). In short, $Coker f$ is defined by the "double dualization" (passing to [[opposite categories]]) of the kernel in [[Ab]]: $Coker f = (Ker f^{op})^{op}$.  This is a particular case of the dualization involved in defining any [[colimit]] from its corresponding [[limit]].

### In an $(\infty,1)$-category

The kernel of a morphism in an [[(∞,1)-category]] with $\infty$-categorical [[zero object]] is the [[homotopy pullback]] as in the pullback definition above: the [[homotopy fiber]].

See also [[stable (∞,1)-category]].

### Other meanings

In some fields, the term 'kernel' refers to an [[equivalence relation]] that category theorists would see as a [[kernel pair]].  This is especially important in fields such as [[monoid]] theory where both notions exist but are not equivalent (while in [[group]] theory they are equivalent).

In [[ring]] theory, even when one assumes that rings have units preserved by ring homomorphisms, the traditional notion of kernel (an [[ideal]]) exists in the category of non-unital rings (and is not itself a unital ring in general).  A purely category-theoretic theory of unital rings can be recovered either by using the kernel pair instead or (to fit better the usual language) moving to a category of [[modules]].

In [[universal algebra]], this may be handled in the framework of [[Malcev variety|Mal'cev varieties]].

Kashiwara-Schapira, following the terminology of EGA, uses kernel as a synonym of equalizer (and co-kernel of co-equalizer).

## Properties {#Properties}

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

Let $C$ be a category with [[pullback|pullbacks]] and [[zero object]].

In $C$, the kernel of a kernel is 0.

=--

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

By the pasting law for pullbacks we have that the total square

$$
  \array{
     ker ker f &\to& ker f  &\to& 0
     \\
     \downarrow && \downarrow && \downarrow
     \\
     0 &\to& c &\stackrel{f}{\to}& d
  }
$$

is a pullback. Since $0 \to c$ is a [[monomorphism]] and the pullback of a monomorphism along itself is the domain of the monomorphis, we have $ker ker f \simeq 0$.

=--

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

This statement crucially fails to be true in [[higher category theory]]. There, the kernel of a kernel is the based [[loop space object]] of $d$. For this reason where one has [[short exact sequence|short exact sequences]] in 1-category theory, there are instead long [[fiber sequence|fiber sequences]] in higher category theory.

=--

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

In a category $C$ with [[pullback|pullbacks]] and [[pushout|pushouts]] and a [[zero object]], kernel and [[cokernel]] form a pair of [[adjoint functor|adjoint functors]] on the [[arrow category|arrow categories]]

$$
  (coker \dashv ker) : Arr(C) \stackrel{\overset{coker}{\leftarrow}}{\underset{ker}{\to}} Arr(C)
  \,.
$$

=--

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

We check the hom-isomorphism of a pair of [[adjoint functor|adjoint functors]]. An element in the [[hom-set]] $Arr_C(g,ker f)$ is a [[diagram]]

$$
  \array{
     c &\to& ker(f) &\to& 0
     \\
     {}^{\mathllap{g}}\downarrow && \downarrow && \downarrow
     \\
     d &\to& a &\stackrel{f}{\to}& b
  }
  \,.
$$

By the universal property of the [[pullback]], this is the same as a diagram

$$
  \array{
     c  &\to& &\to& 0
     \\
     {}^{\mathllap{g}}\downarrow && && \downarrow
     \\
     d &\to& a &\stackrel{f}{\to}& b
  }
  \,.
$$

By the dual reasoning, an element in $Arr_C(coker g, f)$ is a diagram

$$
  \array{
    c &\stackrel{g}{\to}& d &\to& a
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{f}}
    \\
    0 &\to& coker g &\to& b
  }
  \,.
$$

By the universal property of the [[pushout]] this is equivalently a diagram

$$
  \array{
    c &\stackrel{g}{\to}& d &\to& a
    \\
    \downarrow &&  && \downarrow^{\mathrlap{f}}
    \\
    0 &\to&  &\to& b
  }
  \,.
$$

=--

(This also follows from the general theory of [[generalized kernels]].)

## Examples

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

In the [[category]] [[Ab]] of abelian groups, the kernel of a [[group homomorphism]] $f : A \to B$ is the [[subgroup]] of $A$ on the set $f^{-1}(0)$ of elements of $A$ that are sent to the zero-element of $B$.

=--

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

More generally, for $R$ any [[ring]], this is true in $R$[[Mod]]: the kernel of a morphism of modules is the [[preimage]] of the zero-element at the level of the underlying sets, equipped with the unique sub-module structure on that set.

=--

## Related concepts

* [[generalized kernel]]

* [[rank-nullity theorem]]

* [[fiber]]

* [[homotopy fiber]]

* [[cokernel]]

[[!redirects kernels]]


kernel functor

In noncommutative [[ring]] theory, particularly in the subject of noncommutative [[localization]] of rings, a __kernel functor__ is any left [[exact functor|exact]] additive [[subfunctor]] of the identity functor on the category ${}_R Mod$ of left modules over a ring $R$. There is a bijective correspondence between kernel functors and [[uniform filter]]s of ideals in $R$. A functor $\sigma: {}_R Mod\to {}_R Mod$ is __idempotent__ if $\sigma\sigma = \sigma$ and a preradical if it is additive subfunctor of the identity and $\sigma(M/\sigma(M))=0$ for all $M$ in ${}_R Mod$.
A kernel functor $\sigma: {}_R Mod\to {}_R Mod$ is said to be an __idempotent kernel functor__ if $\sigma(M/\sigma(M))=0$ for all $M$ in ${}_R Mod$; it is idempotent as we see by calculating

$$
\sigma \sigma M = \sigma Ker(M\to M/\sigma M) = Ker (\sigma M\to \sigma(M/\sigma M)) = Ker(\sigma M\to M/\sigma M) = \sigma M
$$

In the last step, we used that $\sigma$ is a subfunctor of the identity, hence the compositions $\sigma M\hookrightarrow M\to M/\sigma M$ and $\sigma M\to \sigma(M/\sigma M)\to M/\sigma M$ coincide.

The basic reference is

* O. Goldman, _Rings and modules of quotients_, J. Algebra __13__, 1969 10--47, [MR245608](http://www.ams.org/mathscinet-getitem?mr=245608), [doi](http://dx.doi.org/10.1016/0021-8693%2869%2990004-0)

which is clearly written from the point of view of a ring theorist. Unfortunately, it just creates another formalism in localization theory of the categories of modules over a ring for basically the same results as [[P. Gabriel]] succeeded by more categorical formulations in his thesis published 7 years earlier. Some of the methods from Goldman, and even more from Gabriel apply for more general [[Grothendieck category|Grothendieck categories]].

* Pascual Jara, Alain Verschoren, Conchi Vidal, _Localization and sheaves: a relative point of view_, Pitman Research Notes in Mathematics Series, 339. Longman, Harlow, 1995. xiv+235 pp.
* J. L. Bueso, P. Jara, A. Verschoren, _Compatibility, stability, and sheaves_, Monographs and Textbooks in Pure and Applied Mathematics, 185. Marcel Dekker, Inc., New York, 1995. xiv+265 pp.
[[!redirects kernel functors]]


kernel method

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### Context
#### Measure and probability theory
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[[!include measure theory - contents]]
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#Contents#
* table of contents
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## Idea

In the context of [[machine learning]], a *kernel method* is effectively the analysis of a data set $D$ through a choice of a [[distance]]/[[metric]]-function $d \;\colon\; D \times D \to \mathbb{R}$ on it. The corresponding [[integral kernels]] $\exp(- \lambda \cdot d(-,-))$ turn out to contain useful information, if chosen correctly.

As a special case, if the data set $D$ is a "choice of rankings", hence the [[set]] of [[permutations]], of $n \in \mathbb{N}$ elements, distance functions come from the [[graph distance]] of [[Cayley graphs]] of the [[symmetric group]], such as the [[Kendall distance]] (or [[Cayley distance]]) with associated kernels such as the [[Mallows kernel]]. With these choices, the kernel method overlaps with [[geometric group theory]].

## Details

A kernel on a set $X$ is a function $K: X \times X \to \mathbb{R}$ (although complex-valued kernels are also used) which is symmetric and positive definite, in the sense that for any $N \geq 1$ and any $x_1,..., x_N \in X$, the matrix $K_{i j} = K(x_i, x_j)$ is positive definite, i.e., $\sum_{i, j} c_i c_j K_{i j} \geq 0$ for all $c_1, ..., c_N \in \mathbb{R}$.

This may be reformulated as a mapping $\phi : X \to H$, where $H$ is a *[[reproducing kernel Hilbert space]]*, a function space in which pointwise evaluation is a continuous linear functional.

The 'kernel' and 'feature map to Hilbert space' stories relate to each other as follows:

$$K(x, .) = \phi (x).$$

The 'reproducing' aspect of $H$ means that

$$\langle f(.), K(x, .) \rangle = f(x),$$

and this is continuous. So we have

$$K(x, y) = \langle \phi (x), \phi(y) \rangle.$$

$H$ is the span of the set of functions $\{K(x, .)| x \in X\}$, and, under certain conditions, when we find the $f \in H$ for which a functional is minimised, it takes the form

$$f(x) = \sum_{i = 1}^m c_i K(x_i, x).$$

Many linear algebra techniques in $H$ just involve the inner product, and so can be conducted as a form of nonlinear algebra back in $X$, using the 'kernel trick'.

Kernels characterise the similarity of the input set. But how do they get chosen? Often a Gaussian radial-basis function (RBF) kernel is selected:

$$K(x, y) = e^{-\|x - y\|^2 / \sigma^2}.$$

Note that there's a [[Bayesianism|Bayesian]] version of kernel methods which uses Gaussian processes.

## Example

Consider binary classification where we look to form a classifier which accurately finds the labels in $\{\pm 1\}$ of a sample from a distribution over $X \times \{\pm 1\}$ on the basis of their $X$ values, given a set of labelled training data. The support vector machine approach to this task looks to find the hyperplane in the associated Hilbert space, $H$, which best separates the images of data points $\phi (x_i)$, so that those with the same label $(y_i)$ fall in the same half space. The control for overfitting comes from finding such a hyperplane that classifies the training data accurately with the largest perpendicular distance to the nearest of them (the support vectors).

## Related concepts

* [[machine learning]]

* [[geometric group theory]]

* [[reproducing kernel Hilbert space]]

* [[Mallows kernel]],

  [[Kendall kernel]]

  [[Cayley distance kernel]]

  [[Markov kernel]]

* [[thermodynamics]]

## References

### General

Survey and introduction:

* Thomas Hofmann, [[Bernhard Schölkopf]], Alexander J. Smola, *Kernel methods in machine learning*, Annals of Statistics 2008, Vol. 36, No. 3, 1171-1220 ([arXiv:math/0701907](https://arxiv.org/abs/math/0701907))

* [[Julien Mairal]], [[Jean-Philippe Vert]], *Machine Learning with Kernel Methods*, lecture notes 2017 ([webpage](http://members.cbio.mines-paristech.fr/~jvert/svn/kernelcourse/course/2021mva/index.html), [pdf](http://members.cbio.mines-paristech.fr/~jvert/svn/kernelcourse/slides/master2017/master2017.pdf), [[MairalVertKernelMethods.pdf:file]])

See also:

* Wikipedia, *[Kernel method](https://en.wikipedia.org/wiki/Kernel_method)*

On positive definiteness via [[Fourier transform]]/[[Bochner's theorem]]:

* [[Kenji Fukumizu]], [[Bharath Sriperumbudur]], [[Arthur Gretton]], [[Bernhard Schölkopf]], *Characteristic Kernels on Groups and Semigroups*, Advances in Neural Information Processing Systems 21 : 22nd Annual Conference on Neural Information Processing Systems 2008 ([NIPS 2008](https://neurips.cc/Conferences/2008/)), 473-480 ([mpg:5466](http://www.is.mpg.de/publications/5466), [pdf](http://www.gatsby.ucl.ac.uk/~gretton/papers/FukSriGreSch09.pdf))

* [[Kenji Fukumizu]], *Theory of Positive Definite Kernel and Reproducing Kernel Hilbert Space* 2008 ([pdf](https://www.ism.ac.jp/~fukumizu/H20_kernel/Kernel_7_theory.pdf), [[FukumizuPositiveDefiniteKernels.pdf:file]])

Proof that the [[Mallows kernel]] and [[Kendall kernel]] are [[positive definite bilinear form|positive definite]]:

* [[Yunlong Jiao]], [[Jean-Philippe Vert]], *The Kendall and Mallows Kernels for Permutations*, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 40, no. 7, pp. 1755-1769, 1 July 2018 ([doi:10.1109/TPAMI.2017.2719680](https://doi.org/10.1109/TPAMI.2017.2719680), [hal:01279273](https://hal.archives-ouvertes.fr/hal-01279273/document))

Discussion of weighted variants:

* [[Yunlong Jiao]], [[Jean-Philippe Vert]], *The Weighted Kendall and High-order Kernels for Permutations*, Proceedings of the 35th International Conference on Machine Learning, PMLR 80:2314-2322, 2018 ([arXiv:1802.08526](https://arxiv.org/abs/1802.08526), [mlr:v80/jiao18a](http://proceedings.mlr.press/v80/jiao18a.html), [hal:hal-01717385](https://hal.inria.fr/hal-01717385))

### In topological machine learning
 {#InTopologicalMachineLearningReferences}

On kernel methods applicable to [[persistence diagrams]]/[[barcodes]] for making [[topological data analysis]] amenable to "[[topological machine learning|topological]]" [[machine learning]]:

* {#RHBK15} Jan Reininghaus, [[Stefan Huber]], Ulrich Bauer, Roland Kwitt, *A Stable Multi-Scale Kernel for Topological Machine Learning*, NIPS'15: Proceedings of the 28th International Conference on Neural Information Processing System, **2** (2015 3070–3078  $[$[arXiv:1412.6821](https://arxiv.org/abs/1412.6821)$]$

* {#KHNL15} Roland Kwitt, [[Stefan Huber]], Marc Niethammer, Weili Lin, [[Ulrich Bauer]], *Statistical Topological Data Analysis -- A Kernel Perspective*, in: *Advances in Neural Information Processing Systems*  (NIPS 2015) $[$[ISBN:9781510825024](https://proceedings.neurips.cc/paper/2015), [doi:10.5555/2969442.2969582](https://dl.acm.org/doi/10.5555/2969442.2969582)$]$

* {#RSL} [[Bastian Rieck]], Filip Sadlo, Heike Leitte, *Topological Machine Learning with Persistence Indicator Functions*,  In: *Topological Methods in Data Analysis and Visualization V* TopoInVis (2017) 87-101 Mathematics and Visualization. Springer, $[$[arXiv:1907.13496](https://arxiv.org/abs/1907.13496), [doi:10.1007/978-3-030-43036-8_6](https://doi.org/10.1007/978-3-030-43036-8_6)$]$

* Raphael Reinauer, Matteo Caorsi, Nicolas Berkouk, *Persformer: A Transformer Architecture for Topological Machine Learning* $[$[arXiv:2112.15210](https://arxiv.org/abs/2112.15210)$]$

[[!redirects kernel methods]]


kernel of integration is the exact differential forms

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## Idea

The [[Stokes theorem]] immediately implies that the [[integration of differential forms|integration]] of an [[exact differential form]] with [[compact support]]  vanishes. But in fact also the converse is true: If the [[integration of differential forms|integration]] of a differential form with [[compact support]] of top degree vanishes, then it is exact (prop. \ref{KernelOfIntegrationOverSmoothManifold} below).

This statement underlies for instance

* [[higher Lie integration]] by the "path method"

* the interpretation of the [[BV-BRST complex]] as computing [[expectation values]] (see [there](BV-BRST+formalism#IntegrationOverManifoldsByBVCohomology)).

## Statement

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

Let $X$ be an [[orientation|oriented]] [[connected topological space|connected]] [[smooth manifold]] of [[finite number|finite]] [[dimension]]. Let $n=\dim X$ and write $\Omega^n_{cp}(X)$ for the [[vector space]] of [[differential n-forms]] with [[compact support]] and

$$
  \int_X \;\colon\; \Omega^n_{cp}(X) \longrightarrow \mathbb{R}
$$

for the [[linear map]] to the [[real numbers]] given by [[integration of differential forms]].

Then the [[kernel]] of this map is precisely the [[exact differential forms]]

$$
  ker\left(\int_X\right) = im(d)
  \,,
$$

hence the [[image]] of the [[de Rham differential]] $d \colon \Omega^{n-1}_{cp} \to \Omega^n_{cp}(X)$.

=--

(e.g. [Lafointaine 15, section 7.3, theorem 7.5](#Lafointaine15))

+-- {: .num_prop #Smooth0TypeIsSheavesOnSmoothMfd}
###### Proposition

At least when $X = \mathbb{R}^n$ is a [[Cartesian space]], then the statement of prop. \ref{KernelOfIntegrationOverSmoothManifold} also holds in smoothly [[indexed sets]] of smooth differential forms.

=--

(e.g. [Lafointaine 15, section 7.3, lemma 7.3](#Lafointaine15))

## Related concepts

* [[de Rham theorem]]

* [[Lie integration]] (proof of [this prop.](Lie+integration#LieIntegrationOfLinenLieAlgebra))

## References

* {#Lafointaine15} Jacques Lafontaine, section 7.3 of _An introduction to differential manifolds_, Springer 2015 ([doi:10.1007/978-3-319-20735-3](https://link.springer.com/book/10.1007/978-3-319-20735-3))


kernel pair

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## Idea

The _kernel pair_ of a [[morphism]] in a [[category]] is the [[fiber product]] of the morphism with itself.

The dual notion is that of [[cokernel pair]].

## Definition

The **kernel pair** of a morphism $f:X\to Y$ in a category $C$ is a pair of morphisms $R\,\rightrightarrows \, X$ which form a [[limit]] of the diagram

$$\array{
X &            &   &            & X \\
  & \searrow^f &   & \swarrow_f \\
  &            & Y \\
}$$

We can think of this as the [[fiber product]] $X \times_Y X$ of $X$ with itself over $Y$, or as the [[pullback]] of $f$ along itself.

## Properties

The kernel pair is always a [[congruence]] on $X$; informally, $R$ is the [[subobject]] of $X \times X$ consisting of pairs of elements which have the same value under $f$ (sometimes called the 'kernel' of a function in $\Set$).

The [[coequalizer]] of the kernel pair, if it exists, is supposed to be the "object of equivalence classes" of the internal [[equivalence relation]] $R$. In other words, it is the [[quotient object]] of $X$ in which [[generalized element]]s are identified if they are mapped by $f$ to equal values in $Y$. In a [[regular category]] (at least), this can be identified with a [[subobject]] of $Y$ called the [[image]] of $f$.

If a morphism has a kernel pair and is a coequalizer, then it is the coequalizer of its kernel pair.  This is a special case of the correspondence of [[generalized kernels]] in enriched categories.

In any category with binary [[pullbacks]], the kernel pair of the [[identity morphism]] $id$ on an object $X$ is the [[diagonal morphism]] $(id,id)$ of $X$ and has a [[coequalizer]] [[isomorphic]] to $X$ itself.

## Related concepts

* [[regular epimorphism]]

* [[regular category]]

* [[exact category]]

[[!redirects kernel pairs]]


kernle

[[!redirects kernel]]


Kerodon

 _[Kerodon](https://kerodon.net/)_ is an online textbook by [[Jacob Lurie]] on  [[homotopy theory]] and [[(∞,1)-category theory]] via [[simplicial homotopy theory]], i.e. on [[Categorical Homotopy Theory|categorical homotopy theory]] following along the lines of Lurie's printed textbook _[[Higher Topos Theory]]_.

The format is modeled on the [[The Stacks Project|Stacks project]].

## See also

* [[Stacks Project]]

* [[bananaspace]]

category: reference


Kerr spacetime

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## Idea ##

The **Kerr spacetime** is the unique stationary axially symmetric [[asymptotically flat]] vacuum (family of) solution(s) to the [[Einstein field equations]]. Informally, it describes an eternal, rotating [[black hole]] inside an otherwise empty universe. It is thought to closely approximate the gravitational field outside of compact rotating objects, to the point that astrophysical black holes are often described as "Kerr".

These black holes are characterized by their [[mass]] and their [[angular momentum]].

## Abstract ##
The definition states the components of the metric tensor in a specific coordinate system, the Boyer-Lindquist coordinates, and compares those to the Minkowski and Schwartzschild metrics. Some properties can be read off directly from the metric tensor, which is done in the properties paragraph.

## Boyer-Lindquist Description ##

Spacetimes in the Kerr family are distinguished by two (real) parameters, $m$ and $a$, respectively interpreted as the hole's [[mass]] and [[angular momentum]] per unit mass. Indeed as $a \to 0$ the geometry approaches that of a mass $m$ "nonrotating" [[Schwartzchild spacetime]].

As in the Schwartzchild spacetime, the observations of distant observers can be described using spherical coordinates $r, \phi, \theta$ on the "space" $\mathbb{R}^3$ and a time coordinate $t \in \mathbb{R}$. These coordinates are called **Boyer-Lindquist coordinates** in the context of the Kerr geometry. They limit to Schwartzchild coordinates with $a \to 0$.

We define two functions mainly as an abbreviation:
$$
\rho^2 = r^2 + a^2 \cos^2(\theta )
$$

$$
\triangle = r^2 - 2 m r + a^2
$$

The following table lists the components of the metric of Minkowski, Schwartzschild and Kerr spacetimes respectivly using the canonical coordinates and the Boyer-Lindquist coordinates for the Kerr spacetime:

 metric
 Minkowski
 Schwartzschild
 Kerr

$g_{tt}$
-1
$-1 + 2 \frac{m}{r}$
$-1 + 2 \frac{mr}{\rho^2}$

$g_{rr}$
+1
$\frac{r}{r - 2m}$
$\frac{\rho^2}{\triangle}$

$g_{\theta\theta}$
$r^2$
$r^2$
$\rho^2$

$g_{\phi\phi}$
$r^2 \sin^2(\theta)$
$r^2 \sin^2(\theta)$
$(r^2 + a^2 + \frac{2 m r a^2 \sin^2(\theta)}{\rho^2}) \sin^2{\theta}$

$g_{ij}$ $i \neq j$
all zero
all zero
all zero except $g_{t \phi} = g_{\phi t} = - \frac{2 m r a \sin^2(\theta)}{\rho^2}$

One gets the [[Kerr-Newman metric]] for an electrically charged source with charge $e$ by replacing the definition of $\triangle$ with

$$
\triangle = r^2 - 2 m r + a^2 + e^2
$$

The family of Kerr spacetimes is classified by the relation of the parameters $a$ and $m$:

* $0 = a$ gives Schwartzschild spacetime

* $0 \lt a^2 \lt m^2$ gives slowly rotating Kerr spacetime (**slow Kerr**)

* $a^2 = m^2$ gives **extreme Kerr spacetime** and

* $m^2 \lt a^2$ gives rapidly rotating Kerr spacetime (**fast Kerr**)

## Properties ##
### Direct Consequences from the Definition ###
There are several coordinate singularities, inlcuding the z-axis (where $\sin(\theta) = 0$) and where $\rho = 0$ and $\triangle = 0$. Points where $\triangle = 0$ define the horizons of Kerr spacetime.

Both $\partial_t$ and $\partial_{\phi}$ are [[Killing vector field]]s, expressing the time invariance and the axial symmetry of the model respectively. Combining the sign changes $t \to -t, \phi \to -\phi$ gives an isometry: Letting time running backwards reverses the rotation.

Kerr spacetime is **asymptotically flat**, that is the Kerr metric approximates the Minkowski metric for large $r$.

### Boyer-Lindquist Blocks ###
The Boyer-Lindquist coordinates are defined on a subset of $\mathbb{R}^2 \times \mathcal{S}^2$ with $t, r$ defined on a copy of $\mathbb{R}$ respectively (actually $r$ is not supposed to take negative values, this definition is for convenience only). There are three subsets where the coordinates fail:

1. The **horizon H** where $\triangle = 0$.

2. The **ring singularity $\Sigma$** where $\rho^2 = 0$

3. The **axis A** where $\sin(\theta) = 0$.

+-- {: .un_def}
###### Definition
The **Boyer-Lindquist blocks I, II, III** are the following open subsets of $\mathbb{R}^2 \times \mathcal{S}^2 - \Sigma$:

1. For slow Kerr, there are two horizons at $r_{\pm}$,
$$
I: r \gt r_+
$$

$$
II: r_- \lt r \lt r_+
$$

$$
III: r \lt r_-
$$

2. For extreme Kerr, there is a single horizon at $r = m$,
$$
I: r \gt m
$$

$$
III: r \lt m
$$
3. For fast Kerr, there is no horizon and $\mathbb{R}^2 \times \mathcal{S}^2 - \Sigma$ can be considered as one block I = III.
=--

Block I is also called the **Kerr exterior** and can be visualized as close to the Newtonian concept of space and time with a central force field.

+-- {: .un_theorem}
###### Theorem
**Causality of I and II**
The Boyer-Lindquist blocks I and II are causal.
=--

For a definition of causality see [[spacetime]].

+-- {: .un_theorem}
###### Theorem
**Noncausality of III**
The Boyer-Lindquist block III is vicious, that is for any two events $p, q$ in III there is a timelike future-pointing curve in III from p to q.
=--

### Extremal spinning black holes

The maximum [[angular momentum]] $J$ of a black hole with [[mass]] $M$ is $J = M^2$.

([[naked singularity]])

One considers the quotient $q \coloneqq J/M^2$. Spinning black holes exist for $q \lt 1$.

### Formation

> the following uses material taken from [this PhysicsSE comment](http://physics.stackexchange.com/a/91284/5603)

High angular momentum presents a barrier preventing collapse to a black hole (at least until this angular momentum is radiated away).

The parameter on which the formation of black hole depends is the ratio $q$ of angular momentum ($J$) to the square of mass ($M$). If $q=J/M^2 \lt 1$ (in relativistic units with $G=1$, $c=1$), then the black hole (non-extremal Kerr black hole, to be precise) can be formed. If $q\gt 1$ then the black hole cannot be formed from all the matter without some mechanism for losing angular momentum (this means of course that some of the mass also has to be lost in the process).

For example, currently for the Sun this parameter is slightly more than 1. (*Of course solar mass is too small to ever form a black hole, angular momentum or not*).

If we are talking stellar collapse (see for instance [HFWLH 02](#HFWLH02)), during the late time evolution of the star it loses considerable portion of its outer shell. Since the outer layers carry most of angular momentum, it is quite possible than as the result of this process the rotation of the actual collapsing object would be slow enough to form the black hole outright.

Alternatively, if the rotation speed of the collapsing star is high enough, during the collapse the part of an angular momentum is retained in the [accretion disk](http://en.wikipedia.org/wiki/Accretion_disk) which is formed around the newly created black hole. The matter from such disk could then fall inside, potentially increasing the ratio $q$, but still it will never reach the limiting value of 1.

Note, that accretion disk (depending on the actual configuration) also can produce the opposite effect [Blandford–Znajek process](http://en.wikipedia.org/wiki/Blandford%E2%80%93Znajek_process) can slow the rotation of a black hole by extracting rotational energy.

Another possibility is that if the rotation is fast enough the black hole is never formed: for instance numerical simulations in ([DSY 04](#DSY04)) found that for $q\gt 1$ the collapsing neutron star forms a torus which then fragments into nonaxisymmetric clumps (without ever producing horizon).

### Higher symmetries

The Kerr spactime admits an extra [[Killing tensor]] and [[Killing-Yano tensor]] (...) See for instance ([JL](#JL)).

## Related concepts

[[!include charged and rotating black holes -- table]]

## References

### General

The original article:

* [[Roy Kerr]], _Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics_, Phys. Rev. Lett. 11, 237 (1963) ([doi:10.1103/PhysRevLett.11.237](https://doi.org/10.1103/PhysRevLett.11.237))

Most textbooks about [[General Relativity]] have chapter about the Kerr spacetime, here is a monograph that specializes on the topic:

* Barrett O'Neill, _The geometry of Kerr black holes._ ([ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0828.53078&format=complete))

A good survey is also in

* Project F, _The spinning black hole_ ([pdf](http://www.eftaylor.com/pub/SpinNEW.pdf))

See also

* Wikipedia, _[Kerr spacetime](http://en.wikipedia.org/wiki/Kerr_spacetime)_

The [[Killing-Yano tensors]] of the Kerr spacetime are discussed in

* {#JL} Jacek Jezierski, Maciej Łukasik, _Conformal Yano-Killing tensor for the Kerr metric and conserved quantities_ ([arXiv:gr-qc/0510058](http://arxiv.org/abs/gr-qc/0510058))

Formation of spinning black holes is discussed in

* {#HFWLH02} Heger, A., Fryer, C. L., Woosley, S. E., Langer, N., & Hartmann, D. H. (2003). *How massive single stars end their life*. The Astrophysical Journal, 591(1), 288. [arXiv:astro-ph/0212469](http://arxiv.org/abs/astro-ph/0212469).

* {#DSY04} Duez, M. D., Shapiro, S. L., & Yo, H. J. (2004). *Relativistic hydrodynamic evolutions with black hole excision*. Physical Review D, 69(10), 104016. [arXiv:gr-qc/040107](http://arxiv.org/abs/gr-qc/0401076).

Kerr-type solutions in dimension $\gt 4$ were first given in

* R. C. Myers and M. J. Perry, Annals Phys. 172, 304 (1986).

In higher dimensions the uniqueness theorem of 4d breaks down, and there are solutions with non-spherical topology, such as [[black rings]]. This plays a role notably in the discussion of [[black holes in string theory]].

On Kerr black holes in the context of the [[holographic principle]] and the [[AdS-CFT correspondence]]:

* [[Monica Guica]], [[Thomas Hartman]], Wei Song, [[Andrew Strominger]], _The Kerr/CFT Correspondence_, Phys. Rev. D **80** (2009) 124008 [[arXiv:0809.4266](http://arxiv.org/abs/0809.4266), [doi:10.1103/PhysRevD.80.124008](https://doi.org/10.1103/PhysRevD.80.124008)]

### Phenomenology
 {#ReferencesObservation}

Observational signature:

* Minyong Guo, Niels A. Obers, Haopeng Yan, _Observational signature of near-extremal Kerr-like black holes in a modified gravity theory at the Event Horizon Telescope_, Phys. Rev. D 98, 084063 (2018) ([arXiv:1806.05249](https://arxiv.org/abs/1806.05249))

Near-extremal spinning black holes are being observed:

* Xinyu Dai, Shaun Steele, Eduardo Guerras, Christopher W. Morgan, Bin Chen, _Constraining Quasar Relativistic Reflection Regions and Spins with Microlensing_ ([arXiv:1901.06007](https://arxiv.org/abs/1901.06007))

  exposition in:

  [[Chandra Observatory]], _[Chandra Views Spinning Black Holes Across Cosmic Sea](https://scitechdaily.com/chandra-views-spinning-black-holes-across-cosmic-sea/)_, July 5 2019

[[!redirects Kerr spacetimes]]

[[!redirects Kerr black hole spacetime]]
[[!redirects Kerr black hole spacetimes]]

[[!redirects Kerr black hole]]
[[!redirects Kerr black holes]]

[[!redirects spinning black hole]]
[[!redirects spinning black holes]]

[[!redirects Kerr solution]]
[[!redirects Kerr metric]]
[[!redirects Kerr black hole]]
[[!redirects Kerr-Newman black hole]]
[[!redirects Boyer-Lindquist coordinates]]
[[!redirects Boyer-Lindquist]]


Kerr-Newman spacetime

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

## Related concepts

[[!include charged and rotating black holes -- table]]

## References

* Wikipedia, _[Kerr-Newman metric](https://en.wikipedia.org/wiki/Kerr–Newman_metric)_

[[!redirects Kerr-Newman spacetimes]]


Kerr-Schild spacetime

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

#Contents#
* table of contents
{:toc}

## Idea

A _Kerr-Schild spacetime_ $(\mathbb{R}^{d,1}, g)$, is a [[spacetime]] which is [[deformation]] of [[Minkowski spacetime]] $(\mathbb{R}^{d,1}, \eta)$ whose [[pseudo-Riemannian metric]] is of the form

$$
  g_{\mu \nu} =
  \eta_{\mu \nu}
  +
  H \cdot k_\my k\nu
$$

where $k$ is a [[lightlike]] [[vector field]] with respect to both metrics, and $H$ is any [[scalar]] [[function]].

## Related concepts

* [[classical double copy]]

## References

The original article is

* [[Roy Kerr]], [[Alfred Schild]], _Some algebraically degenerate solutions of Einstein’s gravitational field equations_, Proc. Symp. Appl. Math 17, 199, 1965

Further developments:

* Tomáš Málek, _Extended Kerr-Schild spacetimes: General properties and some explicit examples_, Class. Quant. Grav. 31 185013 (2014) ([arXiv:1401.1060](https://arxiv.org/abs/1401.1060))

See also

* Wikipedia, _[Kerr-Schild perturbation](https://en.m.wikipedia.org/wiki/Kerr%E2%80%93Schild_perturbations)

[[!redirects Kerr-Schild spacetimes]]

[[!redirects Kerr-Schild metric]]
[[!redirects Kerr-Schild metrics]]


Kervaire invariant

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $X$ a [[framed manifold|framed]] [[smooth manifold]] of [[dimension]] $4k +2$, $k \in \mathbb{N}$,
the **Kervaire invariant** or **Arf-Kervaire invariant**

$$
  Ker(X) \in \mathbb{Z}_2
$$

with values in the [[group of order 2]] is the [[Arf invariant]] of the skew-[[quadratic form]] on the middle dimensional homology group (the [[intersection pairing]]).

## Properties

### Nontrivial Kervaire invariants

Manifolds with non-trivial Kervaire invariant, hence with _Kervaire invariant 1_, exist in [[dimension]]

* $d = 2 = 4\cdot 0 + 2$

* $d = 6 = 4\cdot 1 + 2$

* $d = 14 = 4 \cdot 3 + 2$

* $d =  30 = 4 \cdot 7 + 2$

* $d = 62 = 4 \cdot 15 + 2$

and in no other dimension, except possibly in $d = 126$ (a case that is still open).

This is the statement of (the solution to) the _[[Arf-Kervaire invariant problem]]_.

### For surfaces and Relation to theta characteristic

On a [[surface]] $\Sigma$ a [[framing]] is equivalently a [[spin structure]]. If the surface carries a [[complex manifold]] structure then a spin structure is equivalently a [[theta characteristic]], hence a square root $L$ of the [[canonical bundle]]. Given this, the Kervaire invariant in $\Sigma$ is equal to the [[dimension]] mod 2

$$
  dim(H^0(\Sigma,L)) \; mod \; 2
$$

of the space of [[holomorphic sections]] of $L$:

## Related concepts

[[!include quadratic invariants - table]]

* [[Hopf invariant]], [[Hopf invariant one]]

* [[Arf-Kervaire invariant problem]]

* [[self-dual higher gauge theory]]

## References

* W. Browder, _The Kervaire invariant of framed manifolds and its generalization_, Annals of Mathematics 90 (1969), 157–186.

* John Jones, Elmer Rees, _A note on the Kervaire invariant_ ([pdf](http://www.math.rochester.edu/u/faculty/doug/otherpapers/jonesrees2.pdf))

* Wikipedia, _[Kervaire invariant](http://en.wikipedia.org/wiki/Kervaire_invariant)_

* [[Akhil Mathew]], _[The Kervaire invariant I](https://amathew.wordpress.com/2012/10/01/the-kervaire-invariant-i/)_

On the solution of the [[Arf-Kervaire invariant problem]]:

* [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _On the non-existence of elements of Kervaire invariant one_

On the [[equivariant homotopy theory]] involved:

* [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire problem in algebraic topology: Sketch of the proof_ ([[HHRKervaire.pdf:file]])

More resources are collected at

* [[Douglas Ravenel]], _[A solution to the Arf-Kervaire invariant problem](http://www.math.rochester.edu/people/faculty/doug/kervaire.html)_, web esources 2009

[[!redirects Kervaire invariants]]

[[!redirects Arf-Kervaire invariant]]

[[!redirects Kervaire inavariant one]]


Kesha Hietala

* [personal page](https://khieta.github.io/)

## Selected writings

On [[quantum circuit]] [[certified programming|certification]] with the [[quantum programming language]] *[[SQIR]]*:

* [[Kesha Hietala]], [[Robert Rand]], [[Shih-Han Hung]], Xiaodi Wu, [[Michael Hicks]], *A verified optimizer for Quantum circuits*, Proceedings of the ACM on Programming Languages **5** Issue POPL 37 (2021) 1–29 [[doi:10.1145/3434318](https://doi.org/10.1145/3434318)]

* [[Kesha Hietala]], [[Robert Rand]], [[Shih-Han Hung]], Liyi Li, [[Michael Hicks]], *Proving Quantum Programs Correct*, in *12th International Conference on Interactive Theorem Proving (ITP 2021)*, Leibniz International Proceedings in Informatics (LIPIcs) **193** (2021) [[arXiv:2010.01240](https://arxiv.org/abs/2010.01240)]

* [[Kesha Hietala]], *A verified software toolchain for quantum programming* (2022) [[pdf](https://khieta.github.io/files/drafts/khieta-dissertation.pdf), [blog](https://blog.sigplan.org/2021/06/02/verifying-a-quantum-compiler/)]

category: people


Keshav Dasgupta

* [webpage](http://www.physics.mcgill.ca/~keshav/)

## Selected writings

On the [[BMN matrix model]]:

* {#DSJVR02} [[Keshav Dasgupta]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], _Matrix Perturbation Theory For M-theory On a PP-Wave_, JHEP 0205:056, 2002 ([arXiv:hep-th/0205185](https://arxiv.org/abs/hep-th/0205185))

* [[Keshav Dasgupta]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], _Protected Multiplets of M-Theory on a Plane Wave_, JHEP 0209:021, 2002 ([arXiv:hep-th/0207050](https://arxiv.org/abs/hep-th/0207050))

## Related $n$Lab entries

* [[duality between M-theory on Z2-orbifolds and type IIB string theory on K3-fibrations]]

category: people


Kevin Buzzard

Kevin Buzzard is professor for pure [[mathematics]] at Imperial College London.

* [institute webpage](http://www2.imperial.ac.uk/~buzzard/)

* [personal webpage](http://wwwf.imperial.ac.uk/~buzzard/)

## Selected writings

On [[Xena]], [[formal proof]] and [[proof assistants]] (specifically _[[Lean]]_) in undergaduate mathematics courses:

* _[Xena project](https://xenaproject.wordpress.com/2018/10/07/what-is-the-xena-project/)_

{#OnHoTTApplications} Arguing that real applications of [[homotopy type theory]] have not even been looked into yet (at least as of 2020):

* *Univalent people*, section in: *Where is the fashionable mathematics?* (Feb 2020) [[blog post](https://xenaproject.wordpress.com/2020/02/09/where-is-the-fashionable-mathematics/)]

  > " I don’t want to hear protestations, I want to *see the code*."

* *Is HoTT the way to do mathematics?* (2020) [[pdf](https://www.cl.cam.ac.uk/events/owls/slides/buzzard.pdf)]

  > [[slide 15](https://www.cl.cam.ac.uk/events/owls/slides/buzzard.pdf#page=15):] "Nobody knows because nobody tried."

On formalizing [[perfectoid spaces]] in [[Lean]]:

* [[Kevin Buzzard]], [[Johan Commelin]], [[Patrick Massot]], *Formalising perfectoid spaces*, CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, Pages 299-312 ([doi:10.1145/3372885.3373830](https://doi.org/10.1145/3372885.3373830), [arXiv:1910.12320](https://arxiv.org/abs/1910.12320))

## Related entries

* [[Langlands correspondence]]

* [[formal proof]], [[proof assistant]], [[Lean]]

category: people


Kevin Costello

* [website](https://www.perimeterinstitute.ca/people/kevin-costello)

## Selected writings

On [[perturbative quantum field theory]]:

* Kevin Costello, _[[Renormalization and Effective Field Theory]]_

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

On [[supersymmetry|supersymmetric]] [[field theories]] via [[factorization algebras]]:

* {#Costello11} [[Kevin Costello]], *Notes on supersymmetric and holomorphic field theories in dimension 2 and 4*, talk at *[Geometric and Algebraic Structures in Mathematics](https://www.math.stonybrook.edu/dennisfest/)*, Stony Brook (2011) [[arXiv:1111.4234](https://arxiv.org/abs/1111.4234)]

On [[AdS3/CFT2 duality]]:

* [[Kevin Costello]], [[Natalie Paquette]], _Twisted Supergravity and Koszul Duality: A case study in $AdS_3$_ ([arXiv:2001.02177](https://arxiv.org/abs/2001.02177))

On [[scattering amplitudes]] and [[chiral correlators]] in 4D [[conformal field theories]]

* [[Kevin Costello]], [[Natalie Paquette]], _Celestial holography meets twisted holography: 4d amplitudes from chiral correlators_ ([arXiv:2201.02595](https://arxiv.org/abs/2201.02595))

On [[operator product expansions]] in 2D [[conformal field theories]]:

* [[Kevin Costello]], [[Natalie Paquette]], _On the associativity of one-loop corrections to the celestial OPE_ ([arXiv:2204.05301](https://arxiv.org/abs/2204.05301))

On [[celestial holography]] via [[topological strings]] on [[twistor space]]:

* [[Kevin Costello]], [[Natalie M. Paquette]], [[Atul Sharma]], *Top-down holography in an asymptotically flat spacetime* [[arXiv:2208.14233](https://arxiv.org/abs/2208.14233)]

* [[Kevin Costello]], [[Natalie M. Paquette]], [[Atul Sharma]], *Burns space and holography* [[arXiv:2306.00940](https://arxiv.org/abs/2306.00940)]

## Related entries

* [[TCFT]]

* [[factorization algebra of observables]]

* [[renormalization]], [[effective field theory]]

category: people


Kevin Coulembier

* [personal page](https://www.maths.usyd.edu.au/u/kevinc/)

## Selected writings

On [[Frobenius exact categories]] in view of [[Deligne's theorem on tensor categories]]:

* {#CoulembierEtingofOstrik21} [[Kevin Coulembier]], [[Pavel Etingof]], [[Victor Ostrik]], _On Frobenius exact symmetric tensor categories_ $[$[arXiv:2107.02372](https://arxiv.org/abs/2107.02372)$]$

category: people


Kevin H. Lin

* [MathOverflow page](http://mathoverflow.net/users/83/kevin-h-lin)

category: people


Kevin Iga

* [webpage](http://math.pepperdine.edu/kiga/)

## related $n$Lab entries

* [[adinkra]]

category: people


Kevin K. Ferland

* [personal page](http://facstaff.bloomu.edu/kferland/)

## Selected writings

On cases where [[prime field|$\mathbb{Z}/p$]]-[[equivariant cohomology|equivariant]] [[Bredon cohomology]] groups  are [[free modules]] over the Bredon cohomology of the point:

* [[Kevin K. Ferland]], [[L. Gaunce Lewis, Jr.]], *The $RO(G)$-Graded Equivariant Ordinary Homology of $G$-Cell Complexes with Even-Dimensional Cells for $G=\mathbb{Z}/p$*, Memoirs of the AMS **167** (2004) [[ams:memo-167-794](https://bookstore.ams.org/memo-167-794), 978-ISBN:1-4704-0392-8]

category: people

[[!redirects Kevin Ferland]]


Kevin Lin

I am a grad student at Berkeley, advised by [[Constantin Teleman]].

I am mainly interested in algebraic geometry and mathematical physics. I am particularly interested in mirror symmetry, and so I also dabble in symplectic geometry and category theory.

category: people


Kevin Loo

* [personal page](https://kevin-loo.github.io/about.html)

## Selected writings

On [[conformal blocks]] for [[Liouville theory]]:

* [[Xia Gu]], [[Babak Haghighat]], [[Kevin Loo]], *Irregular Fibonacci Conformal Blocks* [[arXiv:2311.13358](https://arxiv.org/abs/2311.13358)]

category: people


Kevin Sackel

* [webpage](http://math.mit.edu/~ksackel/about.html)

## related $n$Lab entries

* [[H-cohomology]]

category: people


Kevin Walker

* [website](http://canyon23.net/math/)

## Selected writings

On potential [[experiments]] detecting [[uncertainty of fluxes]] in [[quantum electrodynamics|quantum]] [[electromagnetism]]:

* [[Alexei Kitaev]], [[Gregory W. Moore]], [[Kevin Walker]], *Noncommuting Flux Sectors in a Tabletop Experiment* [[arXiv:0706.3410](https://arxiv.org/abs/0706.3410)]

Introducing the [[4d TQFT]] [[Walker-Wang model]]:

* [[Kevin Walker]], [[Zhenghan Wang]], _(3+1)-TQFTs and Topological Insulators_, Frontiers of Physics volume 7, pages 150–159 (2012) ([arXiv:1104.2632](http://arxiv.org/abs/1104.2632), [doi:10.1007/s11467-011-0194-z](https://doi.org/10.1007/s11467-011-0194-z))

## Related $n$Lab entries

* [[blob homology]]

category: people


khaledalomari@gmail.com


Khovanov homology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Khovanov homology_ is a [[knot invariant]] that is a [[categorification]] of the [[Jones polynomial]].

## Interpretation in QFT

Khovanov homology has long been expected to appear as the [[observable]]s in a 4-dimensional [[TQFT]] in higher analogy of how the [[Jones polynomial]] arises as a observables in 3-dimensional [[Chern-Simons theory]]. For instance for $\Sigma : K \to K'$ a [[cobordism]] between two [[knot]]s there is a natural morphism

$$
  \Phi_\Sigma : \mathcal{K}(K) \to \mathcal{K}(K')
$$

between the Khovanov homologies associated to the two knots.

In ([Witten11](#Witten11)) it is argued, following indications in ([Gukov-Schwarz-Vafa 05](#GukovSchwarzVafa04)) that this 4d TQFT is related to the worldvolume theory of the image in [[string theory|type IIA string theory]] of D3-[[brane]]s ending on NS5-[[brane]]s in a type IIB background of the form $\mathbb{R}^9 \times S^1$ with the circle transverse to both kinds of branes, under one [[S-duality]] and one [[T-duality]] operation

$$
  (D3-NS5)
    \stackrel{S}{\mapsto}
  (D3-D5)
    \stackrel{T}{\mapsto}
  (D4-D6)
  \,.
$$

To go from the [[Jones polynomial]] to [[Khovanov homology]], we interpret the circle as Euclidean time. The [[path integral]] with the circle is the [[partition function]] (Witten index), $Tr_{\mathcal{H}}(-1)^F e^{-\beta H}$, of a 5D theory. Khovanov homology is $\mathcal{H}$ itself, rather than the index.

See ([Witten11, p. 14](http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.3216v1.pdf#page=14)).

Earlier indication for this had come from the observation [Witten92](#Witten92) that Chern-Simons theory is the effective background theory for the [[A-model]] 2d [[TCFT]] (see TCFT -- Worldsheet and effective background theories for details).

## Related concepts

* [[categorification in representation theory]]

[[!include table of branes]]

## References

Original sources include

* [[Louis Crane]], [[Igor Frenkel]], _Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases_, J. Math. Phys. __35__ (1994) 5136-5154, [hep-th/9405183](http://arxiv.org/abs/hep-th/9405183)

* [[Igor Frenkel]], [[Mikhail Khovanov]], _Canonical bases in tensor products and graphical calculus for $U_q(\mathfrak{sl}_2)$_, Duke Math. J. __87__ (1997) 409-480, [MR99a:17019](http://www.ams.org/mathscinet-getitem?mr=1446615), [doi](http://dx.doi.org/10.1215/S0012-7094-97-08715-9)

* [[Joseph Bernstein]], [[Igor Frenkel]], [[Mikhail Khovanov]], _A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak{sl}-2)$ by projective and Zuckerman functors, Selecta. Math. __5__ (1999) 199-241, [MR2000i:17009](http://www.ams.org/mathscinet-getitem?mr=1714141), [doi](http://dx.doi.org/10.1007/s000290050047)

* [[Igor Frenkel]], [[Mikhail Khovanov]], [[Catharina Stroppel]], _A categorification of finite-dimensional irreducible representations of quantum $\mathfrak{sl}_2$ and their tensor products_, Selecta Math. (N.S.) __12__ (2006), no. 3-4, 379–431, [MR2008a:17014](http://www.ams.org/mathscinet-getitem?mr=2305608), [doi](http://dx.doi.org/10.1007/s00029-007-0031-y)

* [[Mikhail Khovanov]], *A categorification of the Jones polynomial*, Duke Math. J. __101__ (2000) 359-426 [[arXiv:math/9908171](https://arxiv.org/abs/math/9908171), [doi:10.1215/S0012-7094-00-10131-7](https://projecteuclid.org/journals/duke-mathematical-journal/volume-101/issue-3/A-categorification-of-the-Jones-polynomial/10.1215/S0012-7094-00-10131-7.full), MR1740682 (2002j:57025)]

* M Khovanov, _A functor-valued invariant of tangles_, Algebr. Geom. Topol. __2__ (2002) 665–741 MR1928174 (2004d:57016)

* M Khovanov, _Patterns in knot cohomology. I_, Experiment. Math. __12__ (2003) 365–374
* M Khovanov, An invariant of tangle cobordisms, Trans. Amer. Math. Soc. __358__ (2006), no. 1, 315–327, arXiv:math.QA/0207264, [MR2006g:57046](http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=2171235&loc=fromreflist), [doi](http://dx.doi.org/10.1090/S0002-9947-05-03665-2)
* Raphaël Rouquier, _Khovanov-Rozansky homology and 2-braid groups_, [arxiv/1203.5065](http://arxiv.org/abs/1203.5065)

* [[Carlo Collari]],  _The Functoriality of Khovanov Homology and the Monodromy of Knots_, 2013 ([pdf](https://core.ac.uk/download/pdf/19204099.pdf), [[CollariKhovanovHomology.pdf:file]])

Review and lecture notes:

* {#Bar-Natan} [[Dror Bar-Natan]], _On Khovanov's categorification of the Jones polynomial_, Alg. Geom. Topology __2__ (2002) 337-370, [arXiv:math.GT/0201043](http://arxiv.org/abs/math/0201043)

* [[Mikhail Khovanov]] (notes by [[You Qi]]), *Introduction to categorification*, lecture notes, Columbia University (2010, 2020) [[web](https://www.math.columbia.edu/~khovanov/cat2020/), [web](https://you-qi2121.github.io/mypage/categorificationnotes.html), full:[pdf](https://www.dropbox.com/scl/fi/wdesax1c8il6tgwbi20t4/KhovanovYouQi-Categorification.pdf?rlkey=l5cm3khnzu604ijdnl06o89od&dl=0)]

* Paul Turner, _Five Lectures on Khovanov Homology_, [[math.GT/0606464](http://arxiv.org/abs/math/0606464)]

* [[Mikhail Khovanov]], Robert Lipshitz, _Categorical lifting of the Jones polynomial: a survey_, Bulletin (New Series) of the American Mathematical Society (2022) [[doi:10.1090/bull/1772](https://doi.org/10.1090/bull/1772)]

A proposal for a 4-dimensional [[quantum field theory]] whose [[observable]]s are given by Khovanov homology is discussed in

* {#Witten11} [[Edward Witten]], _Fivebranes and Knots_, Quantum Topology, Volume 3, Issue 1, 2012, pp. 1-137 ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216), [doi:10.4171/QT/26](https://www.ems-ph.org/journals/show_abstract.php?issn=1663-487X&vol=3&iss=1&rank=1))

based on

* [[Sergei Gukov]], [[Albert Schwarz]], [[Cumrun Vafa]], _Khovanov-Rozansky homology and topologicals strings_ , Lett. Math. Phys. __74__ (2005) 53-74, [arXiv:hep-th/0412243](http://arxiv.org/abs/hep-th/0412243), [MR2007a:57014](http://www.ams.org/mathscinet-getitem?mr=2193547), [doi](http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.1007/s11005-005-0008-8)
{#GukovSchwarzVafa04}

and earlier hints in

* [[Edward Witten]], _Chern-Simons gauge theory as a string theory_, The Floer memorial volume, 637–678, Progr. Math. __133__, Birkhäuser 1995, [arXiv/hep-th/9207094](http://arxiv.org/abs/hep-th/9207094), [MR97j:57052](http://www.ams.org/mathscinet-getitem?mr=1362846)
{#Wittem92}

Lecture notes on this and its relation to the [[Jones polynomial]] are in

* [[Edward Witten]], _A New Look At The Jones Polynomial of a Knot_, Clay Conference, Oxford, October 1, 2013 ([[WittenOnJonesPolynomial2013.pdf:file]])

* [[Edward Witten]], _Khovanov Homology And Gauge Theory_, Clay Conference, Oxford, October 1, 2013 ([[WittenOnKhovanovHomology2013.pdf:file]])

See also

* [[Edward Witten]], _Khovanov homology and gauge theory_, [arxiv/1108.3103](http://arxiv.org/abs/1108.3103)

* {#Witten11} [[Edward Witten]], _Fivebranes and Knots_ ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216))

Related $n$Café discussions: [categorification in Glasgow](http://golem.ph.utexas.edu/category/2008/11/categorification_in_glasgow.html), [Kamnitzer on categorifying tangles](http://golem.ph.utexas.edu/category/2009/04/kamnitzer_on_categorifying_tan.html), [link homology in Paris](http://golem.ph.utexas.edu/category/2009/02/link_homology_in_paris.html), [4d QFT and Khovanov homology](http://golem.ph.utexas.edu/category/2011/02/4d_qft_for_khovanov_homology.html)

Parts of the above remarks on the QFT interpretation makes use of comments provided  by [[Jacques Distler]] in this blog discussion.

* [[Dror Bar-Natan]], _Khovanov's homology for tangles and cobordisms_,
Geom. Topol. __9__ (2005), 1443–1499, [MR2006g:57017](http://www.ams.org/mathscinet-getitem?mr=2174270)

See also

* David Clark, [[Scott Morrison]], [[Kevin Walker]], _Fixing the functoriality of Khovanov homology_ ([arXiv:math/0701339](http://arxiv.org/abs/math/0701339) [pdf slides](http://canyon23.net/math/talks/Kh%20200704.pdf))

Discussion via a [[cobordism category]] of "foams":

* {#Khovanov04} [[Mikhail Khovanov]], *$sl(3)$ link homology*, Algebr. Geom. Topol. **4** (2004) 1045-1081 [[arXiv:math/0304375](https://arxiv.org/abs/math/0304375), [doi:10.2140/agt.2004.4.1045](https://doi.org/10.2140/agt.2004.4.1045)]

* [[Marco Mackaay]], [[Pedro Vaz]], *The universal $sl_3$-link homology*, Algebr. Geom. Topol. **7** (2007) 1135-1169 [[doi:10.2140/agt.2007.7.1135](https://doi.org/10.2140/agt.2007.7.1135)]

* [[Marco Mackaay]], [[Pedro Vaz]], *The foam and the matrix factorization $sl_3$ link homologies are equivalent*, Algebr. Geom. Topol. **8** (2008) 309-342 [[arXiv:0710.0771](https://arxiv.org/abs/0710.0771), [doi:10.2140/agt.2008.8.309](https://doi.org/10.2140/agt.2008.8.309)]

* [[Marco Mackaay]], [[Pedro Vaz]], *The diagrammatic Soergel category and $sl(N)$-foams, for $N \gt 3$ [[arXiv:0911.2485](https://arxiv.org/abs/0911.2485)]

* [[Mikhail Khovanov]], [[Louis-Hadrien Robert]], *Foam evaluation and Kronheimer--Mrowka theories*, Advances in Mathematics **376**  (2021) 107433 [[arXiv:1808.09662](https://arxiv.org/abs/1808.09662), [doi:10.1016/j.aim.2020.107433](https://doi.org/10.1016/j.aim.2020.107433)]

* [[Mikhail Khovanov]], [[Louis-Hadrien Robert]], *Conical $SL(3)$ foams*, Journal of Combinatorial Algebra **6** 1/2 (2022) 79-108 [[arXiv:2011.11077](https://arxiv.org/abs/2011.11077), [doi:10.4171/jca/61](https://doi.org/10.4171/jca/61)]

Reviewed in:

* [[Mikhail Khovanov]], *Universal construction, foams and link homology*, lecture series at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#KhovanovMar2023), video 1:[YT](https://www.youtube.com/watch?v=Urh1NNcbCcc), 2:[YT](https://www.youtube.com/watch?v=SPBcAAmGT88)]

More on relation to [[topological quantum field theory]]:

* [[Nitu Kitchloo]], *Symmetry Breaking and Link Homologies I* [[arXiv:1910.07443](https://arxiv.org/abs/1910.07443)]

* [[Nitu Kitchloo]], *Symmetry Breaking and Link Homologies II* [[arXiv:1910.07444](https://arxiv.org/abs/1910.07444)]

* [[Nitu Kitchloo]], *Symmetry Breaking and Link Homologies III* [[arXiv:1910.07516](https://arxiv.org/abs/1910.07516)]

Relation to [[gauge theory]]:

* [[Nitu Kitchloo]], *Symmetry breaking and homotopy types for link homologies*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#KitchlooMar2023), video:[YT](https://www.youtube.com/watch?v=StRyIjpER9I)]

See also:

* Khaled Qazaqzeh, [[Nafaa Chbili]], *On Khovanov Homology of Quasi-Alternating Links*,  Mediterr. J. Math. **19** 104 (2022) [[arXiv:2009.08624](https://arxiv.org/abs/2009.08624), [doi:10.1007/s00009-022-02006-5](https://doi.org/10.1007/s00009-022-02006-5)]

* [[Nafaa Chbili]], *Quasi-alternating links, Examples and obstructions*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#ChbiliMar2023)]

[[!redirects knot homology]]

[[!redirects link homology]]

[[!redirects Khovanov-Rozansky link homology]]
[[!redirects Khovanov-Rozansky homology]]


Kieran O’Grady

* [webpage](http://www1.mat.uniroma1.it/people/ogrady/)

## Selected writings

On exceptional examples of [[compact hyperkähler manifolds]]:

* {#OGrady99} [[Kieran O’Grady]], _Desingularized moduli spaces of sheaves on a K3_, J. Reine Angew. Math. 512 (1999), 49–117 ([arXiv:alg-geom/9708009](https://arxiv.org/abs/alg-geom/9708009), [arXiv:math/9805099](https://arxiv.org/abs/math/9805099))

* {#OGrady03} [[Kieran O’Grady]], _A new six dimensional irreducible symplectic variety_, J. Algebraic Geom. 12 (2003), 435-505 ([arXiv:math/0010187](https://arxiv.org/abs/math/0010187))

category: people


Kiiti Morita

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

## Selected writings

On [[ambidextrous adjunctions]] in [[algebra]] (and introducing that concept, under the name *[[strongly adjoint pairs]]*):

* [[Kiiti Morita]], *Adjoint pairs of functors and Frobenius extensions*, Science Reports of the Tokyo Kyoiku Daigaku, Section A **9** 202/208 (1965) 40-71 [[jstor:43698658](https://www.jstor.org/stable/43698658)]

## Related entries

* [[Morita equivalence]]

category: people

[[!redirects Morita]]


Killing form

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## Idea

The _Killing form_ or _Cartan-Killing form_ is a binary [[invariant polynomial]] that is present on any finite-dimensional [[Lie algebra]].

## Definition

Given a finite-dimensional $k$-[[Lie algebra]] $\mathfrak{g}$ its __Killing form__ $B:\mathfrak{g}\otimes \mathfrak{g}\to k$ is the symmetric bilinear form given by the formula

$$
  B(x,y)
    \,=\,
  tr\big(
    ad(x) \circ ad(y)
  \big)
  \,,
$$

where

$$
  ad(x) \,\coloneqq\,
  [x,-]
  \,:\,
  \mathfrak{g}
    \longrightarrow
  \mathfrak{g}
$$

is the [[linear map]] given by the [[adjoint action]] of $x$, hence the value on $x$ of the [[adjoint representation]] $ad \colon \mathfrak{g} \to Der(\mathfrak{g})$.

If $\{t_a\}$ is a [[linear basis]] for $\mathfrak{g}$ and $\{C^a{}_{b c}\}$ are the structure constants of the Lie algebra in this basis (defined by $[t_a, t_b] = \sum_c C^c_{a b} t_c$), then

$$
  B(t_a, t_b)
    \,=\,
  \sum_{c,d} C^c{}_{a d} C^{d}_{b c}
  \,.
$$

## Properties

The Killing form is an _[[invariant polynomial]]_ in that

$$
  B\big([x,y],z\big)
  \,=\,
  B\big(x, [y,z] \big)
$$

for all $x,y,z \in \mathbb{g}$. This follows from the cyclic invariance of the [[trace]].

For complex Lie algebras $\mathfrak{g}$, nondegeneracy of the Killing form (i.e. being the metric making $\mathfrak{g}$ a [[metric Lie algebra]]) is equivalent to [[semisimple Lie algebra|semisimplicity]] of $\mathfrak{g}$.

{#ForSimple} For [[simple object|simple]] complex Lie algebras, or compact forms of real Lie algebras, any invariant nondegenerate symmetric bilinear form is proportional to the Killing form. Indeed, one often uses a normalisation of the Killing form on a simple Lie algebra that makes it a positive definite [[inner product space|inner product]], such that the algebra elements corresponding to long roots (under the isomorphism $\mathfrak{g}^*\simeq \mathfrak{g}$ induced by the original Killing form) have length $\sqrt{2}$. Both the Killing form and this normalised version give a simple Lie algebra a canonical structure of a [[metric Lie algebra]], one positive definite, and one negative definite. This normalised Killing form is sometimes called, following Pressley and Segal, the **basic inner product** on the Lie algebra.

Here are the basic inner products for the compact forms of the simple real matrix Lie algebras, from ([Wang–Ziller 1985](#Wang-Ziller_85),  page 583):

* $\mathfrak{su}(n)$: $\langle X,Y\rangle = -tr_{\mathbb{C}}(XY)$.
* $\mathfrak{so}(3)$: $\langle X,Y\rangle = -tr_{\mathbb{R}}(XY)/4$.
* $\mathfrak{so}(n)$, $n\geq 5$: $\langle X,Y\rangle = -tr_{\mathbb{R}}(XY)/2$.
* $\mathfrak{sp}(n)$: $\langle X,Y\rangle = -Tr(XY) = -2\Re tr_{\mathbb{H}}(XY)$.

The trace $tr_{k}(\cdot)$ here is the ordinary trace for matrices over the division ring $k$, and the _reduced trace_ $Tr(\cdot)$ for $n\times n$ quaternionic matrices is the composite of the embedding into $2n\times 2n$ complex matrices (thinking of $\mathbb{H}\simeq \mathbb{C}\oplus j\mathbb{C}$), and the ordinary trace of complex matrices.

## Generalizations

Sometimes one considers more generally a Killing form $B_\rho$ for a more general faithful finite-dimensional [[representation]] $\rho$, $B_\rho(x,y)  = tr\big(\rho(x)\rho(y)\big)$. If the Killing form is nondegenerate and $x_1,\ldots,x_n$ is a basis in $L$ with $x_1^*,\ldots,x_n^*$ the dual basis of $\mathfrak{g}^*$, with respect to the Killing form for $\rho$, then the canonical element $r = \sum_i x_i\otimes x_i^*$ defines the __[[Casimir operator]]__ $C(\rho) =(\rho\otimes\rho)(r)$ in the representation $\rho$; regarding that the representation is faithful, if the ground field is $\mathbb{C}$, by [[Schur's lemma]] $C(\rho)$ is a nonzero scalar operator. Instead of Casimir operators in particular faithful representations it is often useful to consider an analogous construction within the [[universal enveloping algebra]], the __[[Casimir element]]__ in $U(\mathfrak{g})$.

## References

For a general discussion see:

* Wikipedia, *[Killing form](https://en.wikipedia.org/wiki/Killing_form)*

Careful discussion around normalisation and related data is in:

* {#Wang-Ziller_85} McKenzie Y. Wang and Wolfgang Ziller, _On normal homogeneous Einstein manifolds_, Annales scientifiques de l’École Normale Supérieure, Série 4 **18** (1985), no. 4, 563–633.

[[!redirects Killing forms]]

[[!redirects Cartan-Killing form]]
[[!redirects Cartan-Killing forms]]


Killing spinor

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## Idea

A _Killing spinor_ on a ([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] $X$ is a _[[spinor]]_  -- a [[section]] of some [[spinor bundle]] $v \in \Gamma(S)$ -- that is taken by the  [[covariant derivative]] of the corresponding [[Levi-Civita connection]] to a multiple of itself

$$
  \nabla_v \psi = \kappa \gamma_v \psi
$$

for some constant $\kappa$.

If that constant is 0, hence if the spinor is _covariant constant_, then one also speaks of a _covariant constant spinor_ or _parallel spinor_ (with respect to the given metric structure).

More generally, a _twistor spinor_ or _conformal Killing spinor_ is a $\psi$ such that

$$
  \nabla_v \psi = \frac{1}{dim(X)} \gamma_v D \psi
  \,,
$$

where $D$ is the given [[Dirac operator]] (e.g. [Baum 00](#Baum00)).

A Killing spinor with non-vanishing $\kappa$ may be understood as a genuine covariantly constant spinor, but with respect to a [[super-Cartan geometry]] modeled not on [[super-Euclidean space]]/[[super-Minkowski spacetime]], but on its spherical/hyperbolic or deSitter/anti-deSitter versions ([Egeileh-Chami 13, p. 60 (8/8)](#EgeilehChami13)).

Similarly a _[[Killing vector]]_ is a covariantly constant [[vector field]].

Pairing two covariant constant spinors to a vector yields a Killing vector.

In [[supergravity]], [[super spacetimes]] which solves the [[equations of motion]] and admit Killing spinors are _[[BPS states]]_ (at least if they are asymptotically flat and of finite mass).

## Related concepts

* [[Killing vector]]

* [[Killing tensor]]

* [[Killing-Yano tensor]]

* [[superisometry]]

* [[supersymmetry and Calabi-Yau manifolds]]

## References

### General

Lecture notes include

* _Parallel and Killing spinor fields_ ([[KillingSpinors.pdf:file]])

* {#Baum00} [[Helga Baum]], _Twistor and Killing spinors in Lorentzian geometry_, Séminaires & Congrès, 4, 2000 ([pdf](http://www.emis.de/journals/SC/2000/4/pdf/smf_sem-cong_4_35-52.pdf))

* [[Helga Baum]], _Conformal Killing spinors and the holonomy problem in Lorentzian geometry_ ([pdf](http://www.mathematik.hu-berlin.de/~baum/publikationen-fr/Publikationen-ps-dvi/IMA06-neu.pdf))

See also

* Özgür Açık, _Field equations from Killing spinors_ ([arXiv:1705.04685](https://arxiv.org/abs/1705.04685))

* [[Ángel Murcia]], [[Carlos S. Shahbazi]], *Parallel spinors on globally hyperbolic Lorentzian four-manifolds*, Ann Glob Anal Geom **61** (2022) 253–292 [[arXiv:2011.02423](https://arxiv.org/abs/2011.02423), [doi:10.1007/s10455-021-09808-y](https://doi.org/10.1007/s10455-021-09808-y)]

  > (on [[globally hyperbolic Lorentzian manifolds]])

Discussion relating to [[Killing vectors]] in [[supergeometry]] ([[superisometries]]) is in

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], pages 1198-1199 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

and later in

* [[Dmitri Alekseevsky]], [[Vicente Cortés]], [[Chandrashekar Devchand]], [[Uwe Semmelmann]], _Killing spinors are Killing vector fields in Riemannian Supergeometry_ ([arXiv:dg-ga/9704002](http://arxiv.org/abs/dg-ga/9704002))

See also

* [[Christian Bär]], _Real Killing spinors and holonomy_, Comm. Math. Phys.
Volume 154, Number 3 (1993), 509-521 ([Euclid](https://projecteuclid.org/euclid.cmp/1104253076))

Discussion regarding the conceptualization of Killing spinors in [[super-Cartan geometry]] is in

* {#EgeilehChami13} Michel Egeileh, Fida El Chami, _Some remarks on the geometry of superspace supergravity_, J.Geom.Phys. 62 (2012) 53-60 ([spire](http://inspirehep.net/record/1333125))

Discussion relating to [[special holonomy]] includes

* Andrei Moroianu, [[Uwe Semmelmann]], _Parallel spinors and holonomy groups_, Journal of Mathematical Physics 41 (2000), 2395-2402 ([arXiv:math/9903062](http://arxiv.org/abs/math/9903062))

Discussion of classification includes

* [[Thomas Friedrich]], _Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten_ Czechoslavakian-GDR-Polish scientific school on differential geometry Boszkowo/ Poland 1978, Sci. Comm., Part 1,2; 104-124 (1979)

* [[Thomas Friedrich]], _Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten_, Colloquium Mathematicum vol. XLIV, Fasc. 2 (1981), 277-290.

### Relation to supersymmetry

General discussion of Killing with an eye towards applications in [[supersymmetry]] is around page 907 in volume II of

* [[Pierre Deligne]], P. Etingof, [[Dan Freed]], L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and [[Edward Witten]], (eds.)  _Quantum Fields and Strings, A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

specifically

in [[heterotic supergravity]]:

* [[Ulf Gran]], [[George Papadopoulos]], Diederik Roest, _Supersymmetric heterotic string backgrounds_, Phys.Lett.B656:119-126, 2007 ([arXiv:0706.4407](https://arxiv.org/abs/0706.4407))

in [[11-dimensional supergravity]]:

* [[Jerome Gauntlett]], Stathis Pakis, _The Geometry of $D=11$ Killing Spinors_, JHEP 0304 (2003) 039 ([arXiv:hep-th/0212008](http://arxiv.org/abs/hep-th/0212008))

for [[G2-structures]] in [[M-theory on G2-manifolds]]:

* [[Thomas Friedrich]], Stefan Ivanov, _Parallel spinors and connections with skew-symmetric torsion in string theory_, AsianJ.Math.6:303-336,2002 ([arXiv:math/0102142](http://arxiv.org/abs/math/0102142))

[[!redirects Killing spinors]]

[[!redirects covariantly constant spinor]]
[[!redirects covariantly constant spinors]]

[[!redirects Killing spinor field]]
[[!redirects Killing spinor fields]]

[[!redirects parallel spinor]]
[[!redirects parallel spinors]]

[[!redirects twistor spinor]]
[[!redirects twistor spinors]]

[[!redirects parallel spinor field]]
[[!redirects parallel spinor fields]]


Killing tensor

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## Definition

For $(X,g)$ a ([[pseudo-Riemannian manifold|pseudo-]])[[Riemannian manifold]] a **Killing [[tensor]]** is a [[section]] of a symmetric power of the [[tangent bundle]]

$$
  K \in Sym^k \Gamma(T X)
$$

which is covariantly constant in that

$$
  \nabla_{(\mu} K_{\alpha_1, \cdots, \alpha_k)} = 0
  \,.
$$

For $k = 1$ this reduces to the notion of [[Killing vector]].

## Properties

For every Killing tensor $K$ on $(X,g)$ the [[dynamics]] of the [[relativistic particle]] on $X$ has a further conserved quantity. In the canonical case $K = g$ this quantity is the [[Hamiltonian]] of the particle (in the case of a relativistic particle its four-velocity normalization).

The analog of this for [[spinning particle]]s and [[superparticle]]s are [[Killing-Yano tensor]]s.

## Related concepts

* [[Killing vector]]

* [[Killing spinor]]

* [[Killing-Yano tensor]].

## References

Named after _[[Wilhelm Killing]]_.

[[!redirects Killing tensors]]


Killing vector > history

> see _[[Killing vector field]]_


Killing vector field

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## Idea

A _Killing vector_ on a ([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] is equivalently

* a _covariantly constant vector field_ : a [[vector field]] $v \in \Gamma(T C)$ that is annihilated by (the symmetrization of) the [[covariant derivative]] of the corresponding [[Levi-Civita connection]];

* an [[infinitesimal space|infinitesimal]] [[isometry]].

Similarly a _Killing spinor_ is a covariantly constant [[spinor]].

## Definition

For $(X,g)$ a [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]]) a [[vector field]] $v \in \Gamma(T X)$ is called a **Killing vector field** if it generates [[isometries]] of the [[metric]] $g$. More precisely, if, equivalently

* the [[Lie derivative]] of $g$ along $v$ vanishes: $\mathcal{L}_v g = 0$;

* the [[flow]] $\exp(v)  : X \times \mathbb{R} \to X$ is a flow by [[isometries]].

## Properties

The [[flows]] of Killing vectors are [[isometries]] of the Riemannian manifold onto itself.

## Related concepts

* [[Killing tensor]]

* [[Killing spinor]]

* [[Killing-Yano tensor]]

[[!redirects Killing vector fields]]

[[!redirects Killing vector]]
[[!redirects Killing vectors]]


Killing-Yano tensor

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## Definition

On a ([[pseudo-Riemannian manifold|pseudo-]])[[Riemannian manifold]] $(X,g)$ a **Killing-Yano [[tensor]]** is a [[differential form]] $\omega$ such that

$$
   \nabla_{(\mu} \omega_{\alpha_0),  \alpha_1, \cdots , \alpha_n}
  = 0
  \,,
$$

where $\nabla$ is the [[covariant derivative]] with respect to the [[Levi-Civita connection]] of $g$.

There is also a variant of **conformal Killing-Yano tensors**

(...)

## Properties

Killing-Yano tensors serve as "square roots" of [[Killing tensor]]. In a [[spacetime]] with a [[Killing tensor]] $H$ the [[relativistic particle]] has an extra conserved quantity. If it refines to a Killing-Yano tensor then also the [[spinning particle]] or [[superparticle]] has an extra odd conserved quantity. If $H = g$ then this is an extra worldline [[supersymmetry]].

## Examples

The [[Kerr spacetime]] admits a conformal Killing-Yano tensor (...)

## Related concepts

* [[Killing vector]]

* [[Killing spinor]]

* [[Killing tensor]]

## References

For instance

* O. P. Santillan, _Killing-Yano tensors and some applications_ ([arXiv:1108.0149](http://arxiv.org/abs/1108.0149))

* Jacek Jezierski, Maciej Łukasik, _Conformal Yano-Killing tensor for the Kerr metric and conserved quantities_ ([arXiv:gr-qc/0510058](http://arxiv.org/abs/gr-qc/0510058))

* W. Dietz and R. Rüdiger, _Space-Times Admitting Killing-Yano Tensors. I_ Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 375, No. 1762 (Mar. 31, 1981), pp. 361 ([JSTOR](http://www.jstor.org/stable/2990311))

[[!redirects Killing-Yano tensors]]


kilogram

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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

The _kilogram_ (kilogramme, $kg$) is a [[unit]] of [[mass]].  It is $1000$ grams (grammes, $g$).

The kilogram is defined by setting the [[Planck constant]] to be exactly
$$
6.62607015\times 10^{-34} kg\cdot m^2\cdot s^{-1}
$$
and using the previously given definitions of the [[second]] and the [[metre]].

The kilogram used to be defined as the mass of a certain physical object, _le Grand K_, kept by the BIPM in Paris.

## References

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

[[!redirects kilogram]]
[[!redirects kilogramme]]
[[!redirects kg]]

[[!redirects gram]]
[[!redirects gramme]]


Kim Lin Chew

__Kim Lin Chew__ is an [[algebra|algebraist]] with main interests in [[ring]] theory. Among his main works, he studied quotient/localization constructions, including in an approach via modular closure operators (including analogues of [[Lawvere-Tierney topologies]] in the context of the abelian categories of modules).

PhD advised by David Carrithers Murdoch and James Victor Whittaker:

* math [genealogy](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=17723) page

## Selected writings

* _On extensions of rings and modules_, PhD thesis, University of British Columbia (1965) [[pdf](https://open.library.ubc.ca/media/download/pdf/831/1.0080549/1)]
* _Closure operations in the study of rings of quotients_, Bull. Math. Soc. Nanyang Univ. (1965) 1-20.

category: people


Kimball Martin

* [webpage](http://www2.math.ou.edu/~kmartin/)

## related $n$Lab entries

* [[automorphic form]]

category: people


Kimmo Rosenthal

[homepage](http://www.math.union.edu/people/faculty/rosenthk.html)

[list of publications](http://www.math.union.edu/people/faculty/publications/rosenthk.html)

[[!redirects Kimmo I. Rosenthal]]
[[!redirects K. I. Rosenthal]]

category: people


kinematic tangent space

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###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#Contents#
* table of contents
{: toc}

+-- {: .num_section #sectiona }
=--

## Introduction ## {: .num_section}

A _kinematic tangent space_ of a [[generalized smooth space]] at some point is an equivalence class of smooth [[curves]] through that point, regraded as equivalent if their first [[derivatives]] coincide. This generalizes the notion of [[tangent space]] of a [[differentiable manifold]]. The alternative notion is that of _[[operational tangent space]]_ which for differentiable manifolds coincides, but more generally need not.

[[generalized smooth space|Generalised smooth spaces]] are, as the name suggests, generalisations of [[smooth manifolds]]. They share many common properties and constructions with manifolds. One of the most basic constructions applied to manifolds is that of the [[tangent space]]. There are, indeed, several different constructions of tangent spaces, but as can be shown they are all equivalent for smooth manifolds (see [Spivak 79](#Spivak79), for example). This is no longer true for more general smooth spaces, indeed even for [[Fréchet manifolds]] the various notions are no longer equivalent.

One of the more intuitive definitions of the tangent space of a manifold is as the space of derivatives of smooth curves. Roughly speaking, in this view the tangent space of a manifold at a point is the home of all derivatives of smooth curves through that point. To differentiate this notion from the other definitions, it is sometimes called the **kinematic tangent space**.

Once we go beyond ordinary smooth manifolds, the kinematic tangent space of a smooth space $X$ may not have some or all of its "usual" properties. In particular, it may not even be a [[vector space]]. (If it is then $X$ is also called a _[[microlinear space]]_, at least in the context of [[synthetic differential geometry]] and possibly under some more conditions. ) However, it will always be a kind of *partial* vector space.

It is also possible to define "higher" kinematic tangent spaces. This is not done by iterating the construction, but rather by considering a slightly different construction: instead of [[derivatives]] of [[curves]], we can take the second derivative of curves whose first derivative vanishes. For an ordinary manifold, this yields nothing extra as it is merely observing that the tangent space of the zero vector is again the tangent space. However, for more general spaces it can provide more tangent vectors. For example, the kinematic tangent space at $0$ of the space $[0,1)$ is just $\{ 0\} $ because any curve $\alpha $ with $\alpha (0) = 0$ must also have $\alpha '(0) = 0$. But there are curves $\alpha $ with $\alpha (0) = 0$ and $\alpha '(0) = 0$ but $\alpha''(0) \ne 0$. These curves define a tangent vector that "sees" the inward direction from $0$.

+-- {: .num_section #sectionb }
=--
## Definition ## {: .num_section}

Unless otherwise specified, we shall assume that we have fixed an arbitrary category of [[generalised smooth space]]. The general construction does not rely on a particular choice, though of course one should be consistent.

+-- {: .num_defn #defna }
###### Definition ######
Let $X$ be a smooth space. Let $x \in X$ be a point. The *kinematic tangent space* at $x$, written $T_{x} X$, is defined as the set of equivalence classes of smooth curves $\alpha \colon \mathbb{R} \to X$ with the property that $\alpha (0) = x$ under the relation $\alpha \simeq \beta $ if $(f \circ \alpha )'(0) = (f \circ \beta )'(0)$ for all smooth functions $f \colon X \to \mathbb{R} $.
=--

As mentioned in the introduction, this need not be a [[vector space]]. However, it will always have some of the structure of a vector space. There is a special vector, which we write as $0$ (or $0_{x}$) represented by the constant curve at $x$. And for $v \in T_{x} X$ and $\lambda \in \mathbb{R} $ we can define $\lambda v$ by taking a representative for $v$, say $\alpha $, and letting $\lambda v$ be the equivalence class of $t \mapsto \alpha (\lambda t)$. What fails is the existence of a globally defined addition. It may be defined for a given pair of vectors: for $u, v \in T_{x} X$ we say that the sum $u + v$ exists if, for representatives $\alpha $ and $\beta $ of $u$ and $v$ respectively, there is a smooth curve $\gamma $ with $\gamma (0) = x$ and
$$
(f \circ \gamma )'(0) = (f \circ \alpha )'(0) + (f \circ \beta )'(0)
$$
for all smooth $f \colon X \to \mathbb{R} $. It follows from the definition that if $u + v$ exists, it is unique. Also, all the identities one would expect from addition hold providing all the terms are defined.

One nice property of the kinematic tangent spaces is that the construction of the analogous tangent *bundle* is straightforward. This is because all of our categories are cocomplete and cartesian closed.

+-- {: .num_defn #defnb }
###### Definition ######
Let $X$ be a smooth space. The *tangent bundle* of $X$ is defined to be the quotient of the smooth space $C^{\infty } (\mathbb{R},X)$ by the relation $\alpha \simeq \beta $ if $\alpha (0) = \beta (0)$ and for all smooth $f \colon X \to \mathbb{R} $, $(f \circ \alpha )'(0) = (f \circ \beta )'(0)$.
=--

If $X$ is *functionally Hausdorff* - that is, smooth functions separate points - then the equivalence class can be written more elegantly as: $\alpha \simeq \beta $ if for all $f \colon X \to \mathbb{R} $ and $i \in \{ 0,1\} $, $(f \circ \alpha )^{(i)}(0) = (f \circ \beta )^{(i)}(0)$.

It should be pointed out at this early stage that when considering the space of tangent vectors at a particular point, say $x \in X$, then there are potentially two smooth structures on this set coming from the above definitions. We have the space $T_{x} X$ defined as the quotient of curves based at $x$ and the space $(T X)_{x}$ defined as the fibre of the full tangent bundle. The inclusion of based curves in all curves descends to a smooth map $T_{x} X \to (T X)_{x}$ which is a bijection on the underlying sets, but this need not be a diffeomorphism.

The definition of the kinematic tangent spaces and bundle are closely related to the alternative definition of the tangent space in terms of derivations (sometimes called the [[operational tangent space]]). One could say that the kinematic tangent space is the space of curves seen through the lens of derivations.

Now if the first derivative of a curve vanishes, then its second derivative acts as a derivation since:
$$
\begin{aligned}
((f g) \circ \alpha )''(0) &= (f \circ \alpha )''(0) (g \circ \alpha )(0) + 2 (f \circ \alpha )'(0) (g \circ \alpha )'(0) + (f \circ \alpha )(0) (g \circ \alpha )''(0) \\    &= (f \circ \alpha )''(0) (g \circ \alpha )(0) + (f \circ \alpha )(0) (g \circ \alpha )''(0).
\end{aligned}
$$
With this in mind, we can define higher level kinematic tangent spaces.

+-- {: .num_defn #defnc }
###### Definition ######
Let $X$ be a smooth space. Let $x \in X$ be a point. Let $k \in \mathbb{N} $. The *$k$th kinematic tangent space* at $x$, written $T_{x,k} X$, is defined as the set of equivalence classes of smooth curves $\alpha \colon \mathbb{R} to X$ with the property that $\alpha (0) = x$ and $(f \circ \alpha )^{(j)}(0) = 0$ for $1 \le j \le k - 1$ and smooth $f \colon X \to \mathbb{R} $. The relation is $\alpha \simeq \beta $ if $(f \circ \alpha )^{(k)}(0) = (f \circ \beta )^{(k)}(0)$ for all smooth functions $f \colon X \to \mathbb{R} $.

The *$k$th kinematic tangent bundle* is defined similarly.
=--

There is an obvious map $T_{x,k} X \to T_{x,m k} X$ for $m \in \mathbb{N} $ given by precomposition with $t^{m}$. This defines a diagram indexed by the [[poset]] $\mathbb{N}$ with divisibility as the order. We can define the *full* kinematic tangent space at $x$ as the colimit of this diagram.

The structure of the higher kinematic tangent spaces is somewhat complicated. For example, in the second kinematic tangent space then $(-1) \cdot v = v$.

In the following we shall concentrate on the first kinematic tangent space.

+-- {: .num_section #sectionc }
=--
## Properties ## {: .num_section}

In this section, we shall gather various results on kinematic tangent spaces that will be of use elsewhere.

Because the kinematic tangent space is defined using curves, it has nice properties with respect to limits and mapping spaces. However, as it also uses quotients, these properties are not necessarily as nice as they could be.

+-- {: .num_prop #propositiona }
###### Proposition ######
Let $X,Y$ be smooth spaces. There is a natural isomorphism $T(X \times Y) \to T X \times T Y$.
=--

+--   {: .proof }
The   map in the given direction is obvious: the projection $X \times Y \to X$ is smooth and so induces a map $T(X \times Y) \to T X$. Together with the other projection, we get a map:
$$
T(X   \times Y) \to T X \times T Y.
$$
For   the other direction, we need to consider curves. The result will hinge on the fact that when testing the equivalence relation for curves in $X \times Y$, it is enough to consider functions $X \times Y \to \mathbb{R} $ that factor through one of the projections.

To see this, consider curves $\alpha \colon \mathbb{R} \to X$ and $\beta \colon \mathbb{R} \to Y$ and let $\gamma \colon \mathbb{R} \to X \times Y$ be their product. Let $f \colon X \times Y \to \mathbb{R} $ be a smooth function. The composition $f \circ \gamma $ factors as:
$$
\mathbb{R} \xrightarrow{\Delta } \mathbb{R} \times \mathbb{R} \xrightarrow{\alpha \times \beta } X \times Y \xrightarrow{f} \mathbb{R}.
$$
Thus the derivative of $f \circ \gamma $, at $0$, factors through the derivative of $(s,t) \mapsto f(\alpha (s),\beta (t))$ at $(0,0)$. As this map is smooth, its derivative agrees with the vector of partial derivatives. As every beginner in calculus knows, to compute the, say, $s$-derivative we "hold $t$ fixed and vary $s$". That is to say, we compute the derivative of:
$$
s \mapsto f(\alpha (s),\beta (0)).
$$
Let $y_{0} = \beta (0)$ and define $g \colon X \to \mathbb{R} $ as $g(x) = f(x,y_{0})$. Then the above function agrees with $g \circ \alpha $. For the other side, let $x_{0} = \alpha (0)$ and define $h \colon Y \to \mathbb{R} $ as $h(y) = f(x_{0},y)$.

Hence
$$
(f \circ \gamma )'(0) = \begin{bmatrix} (g \circ \alpha )'(0) & (h \circ \beta )'(0) \end{bmatrix} D \Delta _{0}.
$$
From this we see that the map $T (X \times Y) \to T X \times T Y$ is injective. It is clearly surjective, and thus it remains to show that it is a diffeomorphism.

This follows from the fact that they can both be written as quotients of the same space. For this, we use the natural isomorphism $C^{\infty } (\mathbb{R}, X) \times C^{\infty } (\mathbb{R},Y) \cong C^{\infty } (\mathbb{R}, X \times Y$.
=--

For mapping spaces, we have an obvious map in one direction. A tangent vector at a map is an infinitesimal deformation of that map. Evaluation yields deformations at each point which fit together to give a family of deformations. A simple diagram chase shows that everything involved is smooth.

+-- {: .num_prop #propositionb }
###### Proposition ######
Let $S, X$ be smooth spaces. There is a natural transformation $TC^{\infty } (S, X) \to C^{\infty } (S,T X)$.
=--

+-- {: .proof }
The natural transformation is build up as follows. We start with the evaluation map, $S \times C^{\infty } (S,X) \to X$. Applying the tangent bundle functors, we obtain:
$$
T (S \times C^{\infty } (S,X)) \to T X.
$$
We can compose that with the natural isomorphism from Proposition [1](#propositiona) to obtain:
$$
T S \times T C^{\infty } (S,X) \to T X.
$$
Composing this with the zero section on $T S$ yields:
$$
S \times T C^{\infty } (S,X) \to T X.
$$
Whereupon the adjunction comes in to play to produce:
$$
T C^{\infty } (S,X) \to C^{\infty } (S, T X)
$$
as required.
=--

## Related concepts

* [[operational tangent bundle]]

* [[synthetic tangent bundle]]

## References ##

* {: #citeMR53283} **Spi79** [[Michael Spivak]], _A comprehensive introduction to differential geometry_ Vol. I. Wilmington, Del., 1979.
 {#Spivak79}

[[!redirects kinematic tangent spaces]]
[[!redirects kinematic tangent bundle]]
[[!redirects kinematic tangent bundles]]


kinematics and dynamics

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

## Idea

In the context of [[mechanics]] (broadly construed), one distinguishes between *kinematics* and *dynamics*:

1. __Kinematics__ concerns (only) the [[physical fields]], [[states]] and [[observables]], including the [[spaces]] and [[algebras]] (such as a [[phase space]] or [[Hilbert space]] [[space of states|of (pure) states]] and an appropriate [[algebra of observables]]) into which they are organised;

2. __Dynamics__ additionally treats the evolution of the system in [[time]] or even [[spacetime]]: as given by a [[Lagrangian]] and [[action functional]] and as given by the [[action]] of [[Hamiltonian]] [[quantum observables]] on physical states.

   In the [[Schrödinger picture]], we think of the [[states]] as evolving, while the [[observables]] evolve in the [[Heisenberg picture]]. In the [[interaction picture]] we think of the states as evolving with respect to a given time evolution and the observables to evolve, too, with respect to a [[perturbation theory|perturbation]] of this time evolution.

## Formalization

The notions of kinematics and dynamics may be formally defined in the two formalizations of [[quantum field theory]]: [[FQFT]] and [[AQFT]].

### In FQFT

Consider a [[quantum field theory]] as given by a [[strong monoidal functor]]

$$
  Z : Bord_n^S \to \mathcal{C}
$$

from a [[category of cobordisms]] with $S$-[[structure]] (for instance [[conformal structure]] or [[Riemannian structure]]) to some [[symmetric monoidal category]] $\mathcal{C}$.

Then:

* the _kinematics_ of $Z$ is $Z_0 : Ob(Bord_n^S) \to Ob(\mathcal{C})$, the action of the [[functor]] $Z$ on [[object]]s.

* with that given, the _dynamics_ of $Z$ is $Z_1 : Mor(Bord_n^S) \to Mor(\mathcal{C})$, the action of the functor $Z$ on [[morphism]]s.

This means that as we regard an $n$-dimensional [[QFT]] as an [[FQFT|extended QFT]] given by an [[n-functor]]

$$
  Z : Bord_n^S \to \mathcal{C}
$$

to from a [[(infinity,n)-category of cobordisms]] with $S$-structure to some [[symmetric monoidal (infinity,n)-category]] $\mathcal{C}$ the dichotomy between kinematics and dynamics may be regarded as being blurred a bit: we can regard the action $Z_0$ on [[object]]s as the genuine kinematics and the action $Z_n$ on [[n-morphism]]s as the genuine dynamics, and then the actions $Z_{1 \leq k \leq n-1}$ as interpolating between these two notions.

### In AQFT

(...)

## Examples

### In $\sigma$-models

We discuss the notions of kinematics and dynamics for [[sigma-model]] [[QFT]]s.

#### Charged particle

Given a  [[Riemannian manifold|Riemannian]] [[target space]] $(X,g)$ and a  [[background gauge field]]s given by a [[circle n-bundle with connection|circle bundle with connection]] $(P \to X, \nabla)$, the corresponding [[sigma-model]] quantum field theory is, as an [[FQFT]], a [[functor]]

$$
  Z : Bord_1^{Riemn} \to Hilb
  \,.
$$

It sends

* each [[object]]  -- a [[point]] -- to the space $\Gamma(P \times_{U(1)} \mathbb{C})$ of square integrable [[section]]s of the [[line bundle]] [[associated bundle|associated]] with the background gauge field;

* each [[morphism]] $(\bullet \stackrel{t}{\to} \bullet)$ to the [[operator]] $\exp(i t \nabla^* \nabla)$, where $\nabla^* \nabla$ is the [[Laplace-Beltrami operator]] of the [[covariant derivative]] $\nabla$.

So in terms of the background field data we have:

* the kinematics is encoded in a smooth [[principal bundle]] -- hence a [[cocycle]] in [[smooth infinity-groupoid -- structures|smooth cohomology]];

* the dynamics is encoded in a [[connection on a bundle|connection]] on this bundle -- hence a refinement of the above cocycle to [[ordinary differential cohomology]].

[[!redirects kinematics and dynamics]]
[[!redirects dynamics and kinematics]]

[[!redirects kinematics]]
[[!redirects dynamics]]


kinetic action

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

## Idea

In [[physics]] the part of the [[action functional]] of a [[theory (physics)|theory]] which describes the [[dynamics]] of  [[free fields]] without [[interaction]] is called the _kinetic action_.

The [[Legendre transform]] of the kinetic action is the _[[kinetic energy]]_ ([[Dirichlet energy]]).

## Related concepts

[[!include action (physics) - table]]

[[!redirects kinetic action functional]]
[[!redirects kinetic action functionals]]


kinetic energy

## Idea

The [[energy]] associated with (via [[Legendre transform]]) a [[kinetic action functional]] ([[Dirichlet energy]]).

[[!redirects kinetic energies]]


kinetic Lagrangian

#Contents#
* table of contents
{:toc}

## Idea

The summand of the [[Lagrangian]] corresponding to the [[kinetic action]].

## Related concepts

* [[kinetic energy]]

[[!redirects kinetic Lagrangians]]


kinetic theory

# Related concepts

* [[thermodynamics]]

## References

* Wikipedia, _[Kinetic theory](http://en.wikipedia.org/wiki/Kinetic_theory)_