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vanishing ideal | https://ncatlab.org/nlab/source/vanishing+ideal |
## Idea
Let $X$ be some kind of [[space]] with $\mathcal{O}(X)$ some kind of [[ring]] of [[functions]] on $X$. Then for $S \subset X$ a subspace, of kind, its _vanishing ideal_ is the [[ideal]] $I(S) \subset \mathcal{O}(X)$ of all those functions that vanish when restricted to $S$.
## Related concepts
* [[Zariski topology]]
[[!redirects vanishing ideals]]
|
Vardan Oganesyan | https://ncatlab.org/nlab/source/Vardan+Oganesyan |
* [personal page](https://www.vardanoganesyan.com/)
## Selected writings
A proposal for [[homotopy theory]] of [[symplectic manifolds]]:
* [[Vardan Oganesyan]], *The first step towards symplectic homotopy theory* [[arXiv:2304.10529] (https://arxiv.org/abs/2304.10529)]
category: people |
variable | https://ncatlab.org/nlab/source/variable |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
In [[logic]] and [[type theory]], a variable is a symbol that stands for an arbitrary instantiation of some given [[type]] (although in single-typed theories the type is trivial). Thus, every variable of type $T$ is a [[term]] of type $T$ (although typically there are other terms).
Logicians have developed several ways to precisely specify what variables are and how to keep track of them, and the only thing more annoyingly pedantic than learning one of these is translating between different ones.
## Free and bound variables
If we keep track of [[context]], every [[context extension|introduction of a variable]] changes the context. Thus, whatever [[terms]] of [[type]] $T$ there may (or may not) be in a given context $\Gamma$, in the context $(\Gamma, x\colon T)$ ---which is $\Gamma$ extended by a __free variable__ $x$ of type $T$--- there is (at least) one more term of type $T$, and that term is $x$ itself.
Conversely, we may be able to take a term $a$ in the context $(\Gamma, x\colon T)$ and apply some operation $f$ to it to create a term $f(a)$ (possibly of a different type) in the context $\Gamma$; then any appearances of the variable $x$ in the term $a$ have become __bound variables__ in the term $f(a)$. That is, the appearances of $x$ are bound by the operator $f$, and $x$ has no meaning outside of its scope.
So for example, if (say in something simple like [[Peano arithmetic]], supplemented with the abbreviations used in practice) I write $2x$, this is a term (for a natural number) in a context with a free variable $x$, whereas if I write $3 \sum_{x=0}^4 2x + 5$, this is a term (also for a natural number) in a context with no free variables; here, $x$ is bound. The variable $x$ has a meaning after the operator $\sum_{x=0}^4$ (whose notation also specifies the variable that will have a meaning inside it) and before the plus sign (which, following the standard order of operations, marks the end of the scope of $\sum_{x=0}^4$) but has no meaning outside of that (where the $3$ and $5$ are, or even where the $0$ and $4$ are). And indeed, the ultimate meaning of $2x$ depends on what $x$ is, but the ultimate meaning of $3 \sum_{x=0}^4 2x + 5$ is simply $65$.
## In categorical semantics
In the [[internal language]] of a [[category]] $\mathcal{C}$ a [[morphism]]
$$
f\colon B \to A
$$
is a _[[term]]_ $f(x)$ of _[[type]]_ $A$ where $x$ is
a **free variable** of _type_ $B$, which in symbols is given by
$$
x\colon B \vdash f(x)\colon A
\,.
$$
We may think of the _free variables_ here as being placeholders for all the [[generalized elements]]
$U \stackrel{x}{\to} B$ of $B$.
Then the assertion
$$
x\colon B \vdash f(x)\colon A
$$
indicates that with $B \stackrel{f}{\to} A$ given we may send $U \stackrel{x}{\to} B$ to the [[composition]]
$$
(U \stackrel{x}{\to} B \stackrel{f}{\to}A ) = (U \stackrel{f(x)}{\to} A)
\,.
$$
A free variable becomes a **bound variable** after application of a [[quantifier]]: for instance the image of $f\colon P \to X$ under [[base change]]
$$
\mathcal{C}_{/X} \stackrel{\to}{\stackrel{\leftarrow}{\to}}
\mathcal{C}
$$
represents, for the [[left adjoint]] the [[type]]
$$
\exists x\colon X , P(x)
$$
([[existential quantifier]]), and the the [[right adjoint]] the type
$$
\forall x\colon X , P(x)
$$
([[universal quantifier]]) in which now $x$ is a _bound variable_.
## Related concepts
* [[quantification]]
* [[variable rule]]
* [[context extension]]
* [[proposition]]
## References
{#References}
A textbook account with an eye towards applications in [[computer science]] is in section 1.2 of
* {#Harper} [[Robert Harper]], _Practical Foundations for Programming Languages_ ([web](http://www.cs.cmu.edu/~rwh/plbook/book.pdf))
An exposition on the relation between free variables and [[universal quantification]] is in
* [[Andrej Bauer]], _Free variables are not "implicitly universally quantified"!_ ([web](http://math.andrej.com/2012/12/25/free-variables-are-not-implicitly-universally-quantified/))
[[!redirects variable]]
[[!redirects variables]]
[[!redirects free variable]]
[[!redirects free variables]]
[[!redirects bound variable]]
[[!redirects bound variables]]
[[!redirects dependent variable]]
[[!redirects dependent variables]]
[[!redirects metavariable]]
[[!redirects metavariables]]
|
variable rule | https://ncatlab.org/nlab/source/variable+rule |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[dependent type theory]], what is known as the *variable rule* (or *projection rule*) is the [[structural rule|structural]] [[inference rule]] which, informally speaking, asserts for any [[dependent type]] $A$ the apparent [[tautology]] that "given a variable $a \colon A$" (as a [[hypothesis]]/[[antecedent]]) "then we have that variable $a \colon A$" (as a [[succedent]]).
$$
VAR
\frac{
\Gamma
\;\;\;\;
\vdash
\;\;\;\;
A \colon Type
}{
\Gamma
,\;
a \colon A
\;\;\;\;
\vdash
\;\;\;\;
a \colon A
}
$$
As usual in [[dependent type theory]], the meaning of this rule is a little less trivial than it may superficially seem by the generic [[dependent type|type dependency]] involved: In the [[antecedent]] the variable may be picked out of a [[context]] [[type telescope|telescope]], and in the [[succedent]] it is implicitly [[weakening rule|weakened]] to depend on that full context, -- which includes $a \colon A$ itself!
Accordingly, the [[categorical semantics]] [[categorical semantics of dependent types|as dependent types]] of the variable rule is given by the [[diagonal]] morphism on the [[object]] interpreting $A$, regarded as a [[section]] of the [[pullback bundle|pullback]] of the [[bundle]]/[[display map]] $A \to \Gamma$ to its own total space:
<center>
<img src="/nlab/files/StructuralTypeInferenceRules-230326b.jpg" width="650">
</center>
## Related concepts
* [[variable]]
* [[structural rule]]
[[weakening rule]]
[[substitution rule]]
[[exchange rule]]
## References
* {#Jacobs98} [[Bart Jacobs]], p. 121 in: *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998) [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]
* {#UFP13} [[Univalent Foundations Project]], p. 422 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]
> (in the context of [[homotopy type theory]])
* {#Rijke18} [[Egbert Rijke]], p. 4 of: *Dependent type theory* [[pdf](https://www.andrew.cmu.edu/user/erijke/hott/dtt.pdf)], Lecture 1 in: *[[Introduction to Homotopy Type Theory]]*, lecture notes, CMU (2018) [[pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [[Rijke-IntroductionHoTT-2018.pdf:file]], [webpage](https://www.andrew.cmu.edu/user/erijke/hott/)]
> (in the context of [[homotopy type theory]])
[[!redirects identity rule]]
|
variational bicomplex | https://ncatlab.org/nlab/source/variational+bicomplex |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
For $X$ a ([[spacetime]]) [[manifold]] and $E \to X$ a [[bundle]] (in [[physics]] called the _[[field bundle]]_) with [[jet bundle]] $Jet(E) \to X$, the **variational bicomplex** is essentially the [[de Rham complex]] $(\Omega^\bullet(Jet( E)),\mathbf{d})$ of $Jet(E)$ with [[differential forms]] $\Omega^n(Jet(E)) = \bigoplus_{h+v=n} \Omega^{h,v}(E)$ bigraded by [[horizontal differential form|horizontal]] degree $h$ (with respect to $X$) and vertical degree $v$ (along the [[fiber]]s of $j_\infty E$)). Accordingly the [[differential]] decomposes as
$$
\mathbf{d} = d + \delta
\,,
$$
where $\mathbf{d}$ is the [[de Rham differential]] on $Jet(E)$, $d$ is called the **horizontal differential** and $\delta$ is called the **vertical differential**.
With $E \to X$ thought of as a [[field bundle]] over [[spacetime]]/[[worldvolume]], then $d$ is a measure for how quantities change over spacetime, while $\delta$ is the [[variational calculus|variational]] differential that measures how quantities change as the field configurations are varied.
Accordingly, much of [[classical mechanics]] and [[classical field theory]] on $X$ is formalized in terms of the variational bicomplex. For instance
* a field configuration is a [[section]] of $E$;
* a [[Lagrangian]] is an element $L \in \Omega^{n,0}(E)$;
* a [[local action functional]] is a map
$$
S : \Gamma(E) \to \mathbb{R}
$$
of the form
$$
S(\phi) = \int_X L(j^\infty \phi)
\,,
$$
* the [[Euler-Lagrange equation]] is
$$
E(L) := \delta L \mod im d = 0
$$
* the [[covariant phase space]] is the locus
$$
\{
\phi \in \Gamma(E) | E(L)(j^\infty \phi) = 0
\}
$$
* a [[conserved current]] is an element $\eta\in \Omega^{n-1,0}(E)$ that is horizontally closed on the covariant phase space
$$
d \eta = 0 \mod E(L)
$$
* a [[symmetry]] is an evolutionary vector field $v$ such that
$$
v(L) = 0 \mod im d
$$
* [[Noether's theorem]] asserts that every [[symmetry]] induces a [[conserved current]].
## Definition
Let $X$ be a [[smooth manifold]] and $p : E \to X$ some smooth [[bundle]] over $X$. Write $Jet(E) \to X$ for the corresponding [[jet bundle]].
### The bicomplex
The spaces of [[sections]] $\Gamma(E)$ and $\Gamma(Jet(E))$ canonically inherit a generalized smooth structure that makes them [[diffeological spaces]]: we have a [[pullback]] [[diagram]] of diffeological spaces
$$
\array{
\Gamma(E) & \longrightarrow & *
\\
\downarrow && \downarrow^{id}
\\
[X,E] &\stackrel{p_*}{\longrightarrow}& [X,X]
}
\,.
$$
This induces the [[evaluation map]]
$$
X \times \Gamma(E) \to E
\,.
$$
and composed with the jet prolongation
$$
j^\infty : \Gamma(E) \to \Gamma(Jet(E))
$$
it yields a smooth map ([[homomorphism]] of [[diffeological spaces]])
\[
\label{EProlongation}
e_\infty : X \times \Gamma(E) \stackrel{(id,j^\infty)}{\to}
X \times \Gamma(Jet(E))
\stackrel{ev}{\to}
Jet(E)
\,.
\]
Write
$$
\Omega^{\bullet, \bullet}(X \times \Gamma(E))
$$
for the [[cochain complex]] of smooth differential forms on the [[product]] $X \times \Gamma(E)$, bigraded with respect to the differentials on the two factors
$$
\mathbf{d} \coloneqq d + \delta
\,,
$$
where the $\mathbf{d}$, $d$ and $\delta$, are the [[de Rham differentials]] of $X\times\Gamma(E)$, $X$ and $\Gamma(E)$, respectively.
+-- {: .num_defn }
###### Definition
The **variational bicomplex** of $E \to X$ is the sub--bi-complex of $\Omega^{\bullet, \bullet}(X \times \Gamma(E))$ that is the [[image]] of the pullback of forms along the map $e_\infty$ (eq:EProlongation):
$$
e_\infty^* : \Omega^{\bullet}(Jet(E)) \to
\Omega^\bullet(X \times \Gamma(E))
\,.
$$
We write
$$
\Omega^{\bullet, \bullet}_{loc}
\coloneqq
im (e_\infty^*)
$$
and speak of the bicomplex of **local forms** on sections on $E$.
=--
The bicomplex structure on $\Omega^{\bullet, \bullet}_{loc}$ is attributed in ([Olver 86](#Olver86)) to ([Takens 79](#Takens79)). The above formulation as a sub-bicomplex of the evident bicomplex of forms on $X \times \Gamma(E)$ is due to [[Zuckerman 87, p. 5](#Zuckerman87)].
<img src="https://ncatlab.org/nlab/files/VariationalBicomplex.jpg" width="550">
### More on the horizontal differential complex
{#MoreOntheHorizontalComplex}
+-- {: .num_remark #CharacterizationOfHorizontalDifferential}
###### Remark
In terms of a [[coordinate chart]] $(x^i, u^\alpha,u^\alpha_i,u^\alpha_{i j},\cdots)$ of $E$ covering a coordinate chart $(X^i)$ of $X$, the action of the horizontal differential on functions $f \in C^\infty(Jet(E))$ is given by the formula for the [[total derivative]] operation, but with concrete differentials substituted by the respective jet coordinates:
$$
d_h f
\;\coloneqq\;
\sum_i
\left(
\frac{\partial f}{\partial x^i}
+
\frac{\partial f}{\partial u^\alpha}u^\alpha_i
+
\sum_j \frac{\partial f}{\partial u^\alpha_j} u^\alpha_{j i}
+
\cdots
\right)
d x^i
\,.
$$
([Anderson 89, p. 10](#Anderson89)).
=--
More abstractly, the horizontal differential is characterized as follows:
+-- {: .num_prop #HorizontalDifferentialCompatibleWithPullbackAlongJetProlongations}
###### Proposition
The horizontal differential takes horizontal forms to horizontal forms, and for all sections $\phi \in \Gamma(E)$ it respects [[pullback of differential forms]] along the jet prolongation $j_\infty \phi \in \Gamma(Jet(E))$
$$
(j^\infty \phi)^\ast \circ d_h = d \circ (j^\infty \phi)^\ast
$$
(where on the right we have the ordinary de Rham differential on the base space).
=--
Yet more abstractly, the horizontal complex may be understood in terms of [[differential operators]] and the [[jet comonad]] as follows.
+-- {: .num_remark #HorizontalComplexViaDifferentialOperators}
###### Remark
A horizontal differential $n$-form $\alpha$ on $Jet(E) \to X$ is equivalently a homomorphism of [[bundles]] over $X$
$$
\alpha \colon Jet(E) \longrightarrow \wedge^n T^\ast X
$$
from the [[jet bundle]] $Jet(E)$ to the [[exterior bundle]] $\wedge^n T^\ast X$. This in turn is, by the discussion there, equivalently a [[differential operator]] $\alpha \colon E \to \wedge^n T^\ast X$.
Now of course also the [[de Rham differential]] $d_X$ on $X$ is a differential operator $\wedge^n T^\ast X \to \wedge^n T^\ast X$. In view of this, the horizontal differential of the variational bicomplex is just the [[composition]] operation of differential operators, with horizontal forms regarded as differential operators as above.
By the fact that differential operators are the [[co-Kleisli morphisms]] of the [[Jet comonad]], this means that the horizontal differential is
$$
d_H \alpha
\colon
Jet(F)
\longrightarrow
Jet(Jet(F))
\stackrel{Jet(\alpha)}{\longrightarrow}
Jet(\wedge^n T^\ast X)
\stackrel{\tilde d_X}{\longrightarrow}
\wedge^n T^\ast X
\,.
$$
=--
(e.g. [Krasil'shchik-Verbovertsky 98, around def. 3.27](#KrasilshchikVerbovertsky98), [Krasil'shchik-Vinogradov 99, ch 4, around def. 1.8](#KrasilshchikVinogradov99))
### Evolutionary vector fields
Vector fields on $J^\infty E$ also split into a direct sum of **vertical** and **horizontal** ones, respectively being annihilated by contraction with any horizontal $1$-forms or with any vertical $1$-forms, $\mathfrak{X}(J^\infty E) = \mathfrak{X}_H(J^\infty E) \oplus \mathfrak{X}_V(J^\infty E)$. A special kind of vertical vector field $v \in \mathfrak{X}_V(J^\infty E)$ is called an **[[evolutionary vector field]]** provided it satisfies $\mathcal{L}_v d = d \mathcal{L}_v$ and $\mathcal{L}_v = \iota_v \delta + \delta \iota_v$, we denote the subspace of evolutionary vector fields as $\mathfrak{X}_{ev}(J^\infty E) \subset \mathfrak{X}_V(J^\infty E)$.
## Properties
### Horizontal, vertical, and total cohomology
Let $E \to X$ be a smooth [[fiber bundle]] over a base [[smooth manifold]] $X$ of [[dimension]] $n.$ Write $J^\infty E \to X$ for the [[jet bundle]] of $E\to X$.
Write
$$
\mathcal{F}^s(J^\infty E) \coloneqq I (\Omega^{n,s}(J^\infty E))
$$
for the projection of $(n,s)$-forms to the image of the "interior Euler operator" ([Anderson 89, p. 21 (50/318)](#Anderson89)).
+-- {: .num_prop #ELComplexHasSameCohomologyAsDeRhamComplex}
###### Proposition
**(Takens acyclicity theorem)**
The [[cochain cohomology]] of the [[Euler-Lagrange complex]]
$$
0 \to \mathbb{R}
\to
\Omega^{0,0}(J^\infty E)
\stackrel{d_H}{\to}
\Omega^{1,0}(J^\infty E)
\stackrel{d_H}{\to}
\cdots
\stackrel{d_H}{\to}
\Omega^{n,0}(J^\infty E)
\stackrel{E}{\to}
\mathcal{F}^1(J^\infty E)
\stackrel{\delta_V}{\to}
\mathcal{F}^2(J^\infty E)
\stackrel{\delta_V}{\to}
\cdots
$$
is [[isomorphism|isomorphic]] to the [[de Rham cohomology]] of the total space $E$ of the given fiber bundle.
=--
For smooth functions of locally bounded jet order this is due to ([Takens 79](#Takens79)). A proof is also in ([Anderson 89, theorem 5.9](#Anderson89)).
For smooth functions of globally bounded order and going up to the Euler-Lagrange operator $E$, this is also shown in ([Deligne 99, vol 1, p.188](#Deligne99)).
### The fundamental variational formula
+-- {: .num_defn }
###### Definition {#BicomplexDefinition}
A **source form** is an element $\alpha$ in $\Omega^{n,1}_{loc}$ such that
$$
\alpha_\phi(\delta \phi)
$$
depends only on the 0-jet of $\delta \phi$.
=--
+-- {: .num_prop #VariationOfLagrangian}
###### Proposition
Let $L \in \Omega^{n,0}_{loc}$.
Then there is a unique source form $E(L) $ such that
$$
\delta L = E(L) - d \Theta
\,.
$$
Moreover
* $E(L)$ is independent of changes of $L$ by $d$-exact terms:
$$
E(L) = E(L + d K)
\,.
$$
* $\Theta$ is unique up to $d$-exact terms.
=--
This is [[Zuckerman 87, theorem 3](#Zuckerman87)].
Here $E$ is the _[[Euler-Lagrange equations|Euler-Lagrange operator]]_ .
+-- {: .num_defn }
###### Definition
Write
$$
\Omega = \delta \Theta
\,.
$$
=--
+-- {: .num_remark #HorizontalDOfOmega}
###### Remark
By prop. \ref{VariationOfLagrangian} have
$$
d \Omega = -\delta E(L)
\,.
$$
=--
+-- {: .num_prop #SecondOrderVariationVanishesOnShell}
###### Proposition
$\delta E$ vanishes when restricted to vertical tangent vectors based in [[covariant phase space]] (but not necessarily tangential to it).
$$
\delta E(L) |_{T_{E(L) = 0} \Gamma(E)} = 0
\,.
$$
=--
This is [[Zuckerman 87, lemma 8](#Zuckerman87)].
### Presymplectic covariant phase space
+-- {: .num_cor }
###### Corollary
The form $\Omega$ is a [[conserved current]].
=--
+-- {: .proof}
###### Proof
By remark \ref{HorizontalDOfOmega} and
prop. \ref{SecondOrderVariationVanishesOnShell}.
=--
+-- {: .num_defn #ThePresymplecticForm}
###### Definition
For $\Sigma \subset X$ a [[compact space|compact]] [[closed manifold|closed]] submanifold of [[dimension]] $n-1$, one says that
$$
\omega := \int_\Sigma \Omega \in \Omega^{0,2}_{loc}
$$
is the [[presymplectic structure]] on [[covariant phase space]] relative to $\Sigma$.
=--
+-- {: .num_prop }
###### Proposition
The 2-form $\omega$ is indeed closed
$$
\delta \omega = 0
$$
and in fact exact:
$$
\theta := \int_\Sigma \Theta
$$
is its _presymplectic potential_ .
$$
\delta \theta = \omega
\,.
$$
=--
### Symmetries
Let $L \in \Omega^{n,0}_{loc}$.
+-- {: .num_defn }
###### Definition
An evolutionary vertical vector field $v \in \mathfrak{X}_{ev}(J^\infty E)$ is a **[[symmetry]]** if
$$
v(L) = 0 \mod im d
\,.
$$
=--
+-- {: .num_prop }
###### Proposition
The presymplectic form $\omega$ from def. \ref{ThePresymplecticForm} is annihilated by the Lie derivative of the vector field on $\Gamma(E)$ induced by a symmetry.
=--
This appears as [[Zuckerman 87, theorem 13](#Zuckerman87)].
## Elementary formalization in differential cohesion
{#FormalizationInDifferentialCohesion}
We discuss aspects of an [[elementary (infinity,1)-topos|elementary]]
formalization in [[differential cohesion]] of the concept of the variational bicomplex .
> under construction
Let $\mathbf{H}$ be a context of [[cohesion]] and [[differential cohesion]], with
* [[flat modality]] denoted $\flat$;
* [[infinitesimal shape modality]] denoted $\Im$.
Choose
1. an [[object]] $\Sigma \in \mathbf{H}$, the _base space_ (or _[[spacetime]]_ or _[[worldvolume]]_);
1. an object $E \in \mathbf{H}_{/\Sigma}$, the _[[field bundle]]_,
1. an object $\mathbf{A} \in Stab(\mathbf{H}_{/\Sigma}) \stackrel{\Omega}{\to} \mathbf{H}_{/\Sigma}$, the _[[differential cohomology|differential]] [[coefficients]]_.
Write
* $\mathbf{H}_{/\Sigma}
\stackrel{\overset{\sum_\Sigma}{\longrightarrow}}{\stackrel{\overset{\Sigma^\ast}{\longleftarrow}}{\underset{\prod_\Sigma}{\longrightarrow}}}
\mathbf{H}$
for the [[base change]] [[adjoint triple]] over $\Sigma$, the [[étale geometric morphism]] of the [[slice (infinity,1)-topos]] $\mathbf{H}_{/\Sigma}$;
* $\Gamma_X \coloneqq \flat \circ \prod_\Sigma \colon \mathbf{H}_{/\Sigma} \to \mathbf{H}$ for the external [[space of sections]] functor;
* $i \colon \Sigma \longrightarrow \Im(\Sigma)$ for the $\Sigma$-component of the [[unit of a monad|unit]] of $\Im$;
* $Jet_\Sigma \coloneqq i^\ast i_\ast$ for the induced [[jet comonad]];
* $\mathbf{H}_{/\Sigma} \stackrel{\overset{}{\longleftarrow}}{\underset{\iota}{\longrightarrow}} PDE(\mathbf{H})_{\Sigma}$ for the [[Eilenberg-Moore category]] of $Jet_\Sigma$-[[algebras over a monad|coalgebras]] (the objects are [[differential equations]] with [[variables]] in $\Sigma$, the morphisms are [[differential operators]] between these, preserving solution spaces), manifested as a [[topos of coalgebras]] over $\mathbf{H}$;
the (non-full) [[direct image]] of this [[geometric morphism]] is the [[co-Kleisli category]] of the [[jet comonad]] and so for $\phi \colon free(E) \to free(F)$ a morphism in $PDE(\mathbf{H})_\Sigma$, we write $\tilde f \colon Jet(E) \to F$ for the corresponding co-Kleisli morphism in $\mathbf{H}_{/\Sigma}$;
We record the following simple fact,
which holds generally since the [[jet comonad]] $Jet_\Sigma$ is a [[right adjoint]]
(to the [[infinitesimal disk bundle]] functor), hence preserves [[terminal objects]], and $\Sigma \in \mathbf{H}_{/\Sigma}$ is the [[terminal object]]:
+-- {: .num_prop #JetSigmaPreservesSigma}
###### Proposition
The essentially unique morphism
$$
Jet(\Sigma) \stackrel{\simeq}{\longrightarrow} \Sigma
$$
in $\mathbf{H}_{/\Sigma}$ in an [[equivalence]].
=--
+-- {: .num_defn #JetProlongationInDifferentialCohesion}
###### Definition
The _jet prolongation_ map
$$
j \colon \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(Jet(E))
$$
is the the Jet functor itself, regarded,
in view of prop. \ref{JetSigmaPreservesSigma},
as taking [[sections]] to sections via
$$
(\Sigma \stackrel{\sigma}{\to} E)
\;\;\mapsto
\;\;
\left(
\Sigma \stackrel{\simeq}{\to} Jet(\Sigma)
\stackrel{Jet(\sigma)}{\longrightarrow}
Jet\left(E\right)
\right)
\,.
$$
=--
+-- {: .num_defn #HorizontalFormInDifferentialCohesion}
###### Definition
For $E \in \mathbf{H}_{\Sigma}$ a [[bundle]] over $\Sigma$,
then a _horizontal $\mathbf{A}$-form_ on the [[jet bundle]]
$Jet(E)$ is a morphism in $PDE(\mathbf{H})_{\Sigma}$ of the form
$$
\alpha \colon \iota E \to \iota \mathbf{A}
\,.
$$
For $d \colon \iota Et_\Sigma\Sigma^\ast \mathbf{A}\to \iota Et_\Sigma\Sigma^\ast \mathbf{A}'$ a morphism in $\mathbf{PDE}(\mathbf{H})_{\Sigma}$, then
the induced _horizontal differential_ is the operation of horizontal forms sending $\alpha$ to the composite
$$
d \alpha \colon \iota E
\stackrel{\alpha}{\longrightarrow}
\iota \mathbf{A}
\stackrel{d}{\longrightarrow} \iota \mathbf{A}'
\,.
$$
=--
+-- {: .num_remark }
###### Remark
Since all objects in def. \ref{HorizontalFormInDifferentialCohesion}
are in the [[co-Kleisli category]] of the [[jet comonad]], the morphism $\alpha$ there is equivalently a morphism in $\mathbf{H}_{/\Sigma}$ of the form
$$
\tilde \alpha \colon Jet(E) \longrightarrow \mathbf{A}
\,.
$$
For the special case that $E = \Sigma$ in def. \ref{HorizontalFormInDifferentialCohesion}, then $Jet_{\Sigma}(\Sigma)\simeq\Sigma$ and so a horizontal $\mathbf{A}$-form on $\Sigma$ we call just a an $\mathbf{A}$-form.
=--
+-- {: .num_prop}
###### Proposition
The horizontal differential of def. \ref{HorizontalFormInDifferentialCohesion} commutes with pullback of horizontal differential forms $\alpha$ along the jet prolongation, def. \ref{JetProlongationInDifferentialCohesion}, of any field section $\sigma \in \Gamma_X(E)$.
In detail: for
* $d \colon \iota Et_\Sigma\Sigma^\ast \mathbf{A} \longrightarrow \iota Et_\Sigma\Sigma^\ast \mathbf{A}'$ a morphism,
* $\alpha \colon \iota E \to \iota Et_\Sigma\Sigma^\ast \mathbf{A}$ a horizontal $\mathbf{A}$-form on $Jet(E)$, def. \ref{HorizontalFormInDifferentialCohesion};
* $\sigma \in \Gamma_\Sigma(E)$ a [[field (physics)|field]] [[section]],
then there is a [[natural equivalence]]
$$
j(\sigma)^\ast (d \alpha)
\simeq
d (j(\sigma)^\ast \alpha)
\,.
$$
=--
+-- {: .proof}
###### Proof
Since all objects are in the [[direct image]] $free\colon \mathbf{H} \to PDE(\mathbf{H})_\Sigma$, this is an equivalence of morphisms in the [[co-Kleisli category]] of the [[jet comonad]], hence is equivalently an equivalence of co-Kleisli composites of morphisms in $\mathbf{H}$.
As such, the left hand side of the equality is given in $\mathbf{H}$ by the composite morphism
$$
\Sigma
\stackrel{\simeq}{\to}
Jet(\Sigma)
\stackrel{Jet(\sigma)}{\longrightarrow}
Jet(E)
\stackrel{}{\to}
Jet(Jet(E))
\stackrel{Jet(\tilde \alpha)}{\longrightarrow}
Jet(\mathbf{A})
\stackrel{\tilde d}{\longrightarrow}
\mathbf{A}'
\,,
$$
thought of as bracketed to the right. By [[natural transformation|naturality]] of the Jet-counit this is equivalently
$$
Jet(\Sigma) \stackrel{\simeq}{\to} Jet(Jet(\Sigma))
\stackrel{Jet(Jet(\sigma))}{\longrightarrow}
Jet(Jet(E))
\stackrel{Jet(\tilde \alpha)}{\longrightarrow}
Jet(\mathbf{A})
\stackrel{\tilde d}{\longrightarrow}
\mathbf{A}'
\,,
$$
By functorality of $Jet(-)$ this is equivalent to
$$
Jet(\Sigma)
\stackrel{\simeq}{\to}
Jet(Jet(\Sigma)) \stackrel{Jet ( \tilde \alpha \circ Jet(\sigma) )}{\longrightarrow}
Jet(\mathbf{A})
\stackrel{\tilde d}{\to}
\mathbf{A}'
$$
which is the right hand side of the equivalence to be proven.
=--
## Related concepts
* [[local Lagrangian]], [[Euler-Lagrange form]], [[source form]], [[Lepage form]]
* [[variational calculus]]
* [[secondary differential calculus]]
* [[variational sequence]]
* [[BV-BRST complex]]
* [[BV-BRST variational bicomplex]]
* [[presymplectic current]], [[covariant phase space]]
## References
The variational bicomplex was introduced independently in
* [[Alexandre Vinogradov]], _A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints_ , Sov. Math. Dokl. 19 (1978) 144–148.
* W. M. Tulczyjew, _The Euler-Lagrange resolution_ , in Lecture Notes in Mathematics 836 22–48 (Springer-Verlag, New York 1980).
* T. Tsujishita, _On variation bicomplexes associated to differential equations_, Osaka J. Math. 19 (1982), 311–363.
See also
* {#Takens79} [[Floris Takens]], _A global version of the inverse problem of the calculus of variations_ J. Diff. Geom. 14 (1979) 543-562
* {#Deligne99} [[Pierre Deligne]] et al. _[[Quantum Fields and Strings]]_
An introduction is in
* [[Ian Anderson]], _Introduction to the variational bicomplex_, in _Mathematical aspects of classical field theory_, Contemp. Math. 132 (1992) 51–73, [gBooks](http://books.google.com/books?id=NuiJ0c72_gQC&lpg=PA51&ots=-zAVQLViUn&dq=Zuckerman%20variational%20problems)
A careful discussion that compares the two versions (one over smooth functions globally of finite jet order, one over smooth functions locally of finite jet order) is in
* G. Giachetta, L. Mangiarotti, [[Gennadi Sardanashvily]], _Cohomology of the variational bicomplex on the infinite order jet space_, Journal of
Mathematical Physics 42, 4272-4282 (2001) ([arXiv:math/0006074](http://arxiv.org/abs/math/0006074))
Textbook accounts include
* {#Olver86} [[Peter Olver]], section 5.4 of _Applications of Lie groups to differential equations_, Springer Graduate Texts in Mathematics 107 (1986)
* {#Anderson89} [[Ian Anderson]], _The variational bicomplex_, Utah State University 1989 ([[AndersonVariationalBicomplex.pdf:file]])
* {#KrasilshchikVerbovetsky98} [[Joseph Krasil'shchik]], [[Alexander Verbovetsky]], _Homological Methods in Equations of Mathematical Physics_ ([arXiv:math/9808130](http://arxiv.org/abs/math/9808130))
* {#KrasilshchikVinogradov99} [[Joseph Krasil'shchik]], [[Alexandre Vinogradov]] et al. (eds.) _Symmetries and Conservation Laws for Differential Equations of Mathematical Physics_, AMS 1999
Other surveys include
* Juha Pohjanpelto, _Symmetries, Conservation Laws, and Variational Principles for Differential Equations_ (2014) ([pdf slides](http://math.usask.ca/~cheviakov/bluman2014/talks/Pohjanpelto.pdf))
An early discussion with application to [[covariant phase spaces]] and their [[presymplectic structure]] is in
* {#Zuckerman87} [[Gregg Zuckerman]], *Action principles and global geometry*, in: [[Shing-Tung Yau]] (ed.) *Mathematical Aspects of String Theory*, World Scientific (1987) 259-284 [[[ZuckermanVariation.pdf:file]], [doi:10.1142/0383](https://doi.org/10.1142/0383)]
An invariant version (under group action) is in
* [[Irina Kogan]], [[Peter Olver]], _The invariant variational bicomplex_, [pdf](http://www4.ncsu.edu/~iakogan/papersPDF/sivKoganOlver.pdf)
A more detailed version of this is in
* [[Irina Kogan]], [[Peter Olver]], _Invariant Euler-Lagrange Equations and the Invariant Variational Bicomplex_, [pdf](http://www4.ncsu.edu/~iakogan/papersPDF/ivbKoganOlver-cor.pdf)
See also
* [[Victor Kac]], _An explicit construction of the complex of variational calculus and Lie conformal algebra cohomology_, talk at Algebraic Lie Theory, Newton Institute 2009, [video](http://sms.cam.ac.uk/media/538761)
An application to [[multisymplectic geometry]] is discussed in
* Thomas Bridges, Peter Hydon, Jeffrey Lawson, _Multisymplectic structures and the variational bicomplex_ ([pdf](http://personal.maths.surrey.ac.uk/st/T.Bridges/PAPERS/MPCPS-Paper-09024.pdf))
Discussion in the context of [[supergeometry]] is in
* [[Gennadi Sardanashvily]], _Grassmann-graded Lagrangian theory of even and odd variables_ ([arXiv:1206.2508](https://arxiv.org/abs/1206.2508))
Discussion in the convenient context of [[smooth sets]]:
* {#GiotopoulosSati23} [[Grigorios Giotopoulos]], [[Hisham Sati]], §5.1 & §7.1 in: *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]
[[!redirects horizontal derivative]]
[[!redirects horizontal derivatives]]
[[!redirects vertical derivative]]
[[!redirects vertical derivatives]]
[[!redirects variational differential form]]
[[!redirects variational differential forms]]
[[!redirects total spacetime derivative]]
[[!redirects total spacetime derivatives]]
[[!redirects horizontal variational complex]]
[[!redirects horizontal variational complexes]]
|
variational calculus | https://ncatlab.org/nlab/source/variational+calculus |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
**Variational calculus** -- sometimes called **secondary calculus** -- is a version of [[differential calculus]] that deals with local extremization of [[nonlinear functionals]]: extremization of differentiable functions on non-finite dimensional spaces such as [[mapping spaces]], [[spaces of sections]] and hence [[spaces of histories]] of [[field (physics)|fields]] in [[field theory]].
Specifically, it studies the _[[critical points]]_ , i.e. the points where the first variational derivative of a functional vanishes, for functionals on spaces of [[sections]] of [[jet bundles]]. The kinds of equations specifying these critical points are [[Euler-Lagrange equations]].
This applies to, and is largely motivated from, the study of [[action functionals]] in [[physics]]. In [[classical physics]] the critical points of a specified action functional on the space of field configurations encode the physically observable configurations.
There are strong [[cohomology|cohomological]] tools for studying variational calculus, such as the [[variational bicomplex]] and [[BV-BRST formalism]].
## In terms of smooth spaces
{#InTermsOfSmoothSpaces}
We discuss some basics of variational calculus of functional in terms of [[smooth spaces]] and in particular in terms of [[diffeological spaces]].
### Smooth functionals
{#SmoothFunctionals}
Let $X$ be a [[smooth manifold]].
Let $\Sigma$ be a [[smooth manifold|smooth]] [[manifold with boundary]] $\partial \Sigma \hookrightarrow \Sigma$.
Write
$$
[\Sigma, X] \in Smooth0Type
$$
for the [[smooth space]] (a [[diffeological space]]) which is the [[mapping space]] from $\Sigma$ to $X$, hence such that for each $U \in $ [[CartSp]] we have
$$
[\Sigma, X](U) = C^\infty(U \times \Sigma, X)
\,.
$$
+-- {: .num_defn #MappingSpaceWithNonVaryingBoundary}
###### Definition
Write
$$
[\Sigma, X]_{\partial \Sigma}
\coloneqq
[\Sigma, X] \times_{[\partial \Sigma,X]} \flat [\partial \Sigma,X]
$$
for the [[pullback]] in smooth spaces
$$
\array{
[\Sigma,X]_{\partial \Sigma}
&\to&
\flat [\partial \Sigma, X]
\\
\downarrow && \downarrow
\\
[\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X]
}
\,,
$$
where
* the bottom morphism is the restriction $[\partial \Sigma \hookrightarrow \Sigma, X]$ of configurations to the boundary;
* the right vertical morphism is the [[counit of an adjunction|counit]] of the $(Disc \dashv \Gamma)$-[[adjunction]] on smooth spaces.
=--
+-- {: .num_prop #PlotsOfMappingSpaceWithNonVaryingBoundary}
###### Proposition
The [[smooth space]] $[\Sigma, X]_{\partial \Sigma}$ is a [[diffeological space]] whose underlying set is $C^\infty(\Sigma,X)$ and whose $U$-plots for $U \in $ [[CartSp]] are smooth functions
$\phi \colon U \times \Sigma \to X$ such that $\phi(-,s) \colon U \to X$ is the constant function for all $s \in \partial \Sigma \hookrightarrow \Sigma$.
=--
+-- {: .num_defn #SmoothFunctional}
###### Definition
A **functional** on the mapping space $[\Sigma, X]$ is a [[homomorphism]] of smooth spaces
$$
S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R}
\,.
$$
=--
### Functional derivative
{#FunctionalDerivative}
Write
$$
\mathbf{d} \colon \mathbb{R} \to \Omega^1
$$
for the [[de Rham differential]] incarnated as a [[homomorphism]] of [[smooth spaces]] from the [[real line]] to the smooth [[moduli space]] of [[differential 1-forms]].
+-- {: .num_defn}
###### Definition
The **[[functional derivative]]**
$$
\mathbf{d}S
\in
\Omega^1([\Sigma,X]_{\partial \Sigma})
$$
of a functional $S$, def. \ref{SmoothFunctional}, is simply its [[de Rham differential]] as a homomorphism of smooth spaces, hence the composite
$$
\mathbf{d} S \colon [
\Sigma, X]_{\partial \Sigma}
\stackrel{S}{\to}
\mathbb{R}
\stackrel{\mathbf{d}}{\to}
\Omega^1
\,.
$$
=--
+-- {: .num_defn}
###### Definition
This means that for each smooth parameter space $U \in $ [[CartSp]] and for each smooth $U$-parameterized collection
$$
\phi \colon U \times \Sigma \to X
$$
of smooth functions $\Sigma \to X$, constant in the parameter $U$ in some neighbourhood of the boundary of $\Sigma$,
$$
S_\phi \colon [\Sigma,X]_{\partial \Sigma}(U) \to C^\infty(U,\mathbb{R})
$$
is the function that sends each such $U$-collection of configurations to the $U$-collection of their values under $S$. Then
$$
(\mathbf{d}S)_\phi \in \Omega^1(U)
$$
is the smooth [[differential 1-form]]
$$
(\mathbf{d}S)_\phi = \mathbf{d}(S(\phi))
\,.
$$
=--
+-- {: .num_example}
###### Example
Let $\Sigma = [0,1] \hookrightarrow \mathbb{R}$ be the standard [[interval]]. Let
$$
L(-,-) \mathbf{d}t \in \Omega^1([0,1], C^\infty(\mathbb{R}^2))
$$
and consider the functional
$$
S
\colon
([0,1] \stackrel{\gamma}{\to} X)
\mapsto
\int_{0}^1 L(\gamma(t), \dot \gamma(t)) d t
\,.
$$
Then for $U = \mathbb{R}$ and any smooth $U$-parameterized collection $\{\gamma_{u} \colon \Sigma \to X\}_{u \in I}$ the functional derivative takes the value
$$
\begin{aligned}
\mathbf{d}S_{\gamma_{(-)}}
& =
\left(
\frac{d}{d u} \int_0^1 L(\gamma_u(t), \dot \gamma_u(t)) dt
\right)
\mathbf{d}u
\\
& =
\int_{0}^1
\left(
\frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
\frac{d \gamma_u(t)}{d u}
+
\frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
\frac{\partial \dot \gamma_u(t)}{\partial u}
\right)
\mathbf{d} u
\\
& =
\int_{0}^1
\left(
\frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
\frac{d \gamma_u(t)}{d u}
+
\frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
\frac{\partial }{\partial t}\frac{\partial \gamma_u(t)}{\partial u}
\right)
\mathbf{d} u
\\
& =
\int_{0}^1
\left(
\frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
-
\frac{\partial}{\partial t}\frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
\right)
\frac{\partial \gamma_u(s)}{\partial u}
\mathbf{d}u
\end{aligned}
\,.
$$
Here we used [[integration by parts]] and used that the boundary term vanishes
$$
\begin{aligned}
\int_{0}^1 \frac{\partial}{\partial t}
\left(
\frac{\partial}{\partial \dot\gamma} L(\gamma_u(s), \dot \gamma_u(s))
\frac{\partial \gamma_u(s)}{\partial u}
\right)
d s
& =
\left(
\frac{\partial}{\partial \dot\gamma} L(\gamma_u(1), \dot \gamma_u(1))
\frac{\partial \gamma_u(1)}{\partial u}
\right)
-
\left(
\frac{\partial}{\partial \dot\gamma} L(\gamma_u(0), \dot \gamma_u(0))
\frac{\partial \gamma_u(0)}{\partial u}
\right)
\\
& = 0
\end{aligned}
$$
since by prop. \ref{PlotsOfMappingSpaceWithNonVaryingBoundary} $\gamma_{(-)}(1)$ and $\gamma_{(-)}(0)$ are constant.
=--
## In terms of the variational bicomplex
In the special case that the functional to be varied comes from a [[Lagrangian density]], then its variational derivative is the image under [[transgression of variational differential forms|transgression]] of the [[vertical derivative]] in the [[variational bicomplex]] of differential forms on the given [[jet bundle]].
(...)
## Related concepts
* [[Euler-Lagrange equations]], [[shell]]
* [[source form]], [[evolutionary vector field]], [[evolutionary derivative]]
* [[de Donder-Weyl formalism]]
* [[phase space]]
* [[variational bicomplex]], [[secondary differential calculus]]
* [[Lagrange multiplier]]
* [[variational sequence]]
## References
Exposition of variational calculus in terms of [[jet bundles]] and [[Lepage forms]] and aimed at examples from [[physics]] is in
* {#MusilovaHronek16} [[Jana Musilová]], [[Stanislav Hronek]], _The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories_, Communications in Mathematics, Volume 24, Issue 2 (Dec 2016) ([doi.org/10.1515/cm-2016-0012]( https://doi.org/10.1515/cm-2016-0012))
Fundamental texts on variational calculus include
* [[Ian Anderson]], _The variational bicomplex_, ([[AndersonVariationalBicomplex.pdf:file]])
* Hubert Goldschmidt, [[Shlomo Sternberg]], _The Hamilton-Cartan formalism in the calculus of variations_, Annales de l'institut Fourier 23 no. 1 (1973), p. 203-267 [numdam](http://www.numdam.org/item?id=AIF_1973__23_1_203_0)
* [[Peter Olver]], _Applications of Lie groups to differential equations_, Springer; _Equivalence, invariants, and symmetry_, Cambridge Univ. Press 1995.
* [[Demeter Krupka]], _Introduction to global variational geometry_, 2015
* Olga Krupková, _The geometry of ordinary variational equations_, Springer 1997, 251 p.
* Robert Hermann, _Some differential-geometric aspects of the Lagrange variational problem_, Illinois J. Math. __6__, 1962, 634–673 [MR145457](http://www.ams.org/mathscinet-getitem?mr=145457) [euclid](http://projecteuclid.org/euclid.ijm/1255632711);
_Differential geometry and the calculus of variations_, Acad. Press 1968
* J. Jost, X. Li-Jost, _Calculus of variations_, CUP 1998
* {#Zuckerman} [[Gregg Zuckerman|G. J. Zuckerman]], _Action Principles and Global Geometry_ , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259€284. ([[ZuckermanVariation.pdf:file]])
Zuckerman's ideas are used in
* [[Marco Benini]], [[Alexander Schenkel]], _Poisson algebras for non-linear field theories in the Cahiers topos_, [arxiv/1602.00708](http://arxiv.org/abs/1602.00708)
Examples: [[Jürgen Jost]], _Variational problems from physics and geometry_, [pdf](http://www.mis.mpg.de/fileadmin/jjost/variational_problems_from_physics_and_geometry.pdf)
* J. J. Duistermaat, _On the Morse index in variational calculus_, Adv. Math. __21__ (1976), 2, 173--195 [pdf](http://www.maths.ed.ac.uk/~aar/papers/duistermaat.pdf).
Some new results are in
* [[E. Getzler]], _A Darboux theorem for Hamiltonian operators in the formal calculus of variations_, Duke Math. J. __111__, n. 3 (2002), 535-560, [MR2003e:32026](http://www.ams.org/mathscinet-getitem?mr=1885831), [doi](http://dx.doi.org/10.1215/S0012-7094-02-11136-3)
* Alberto De Sole, Victor G. Kac, _The variational Poisson cohomology_, [arxiv/1106.0082](http://arxiv.org/abs/1106.0082)
Geometric extremization problems (e.g. minimal surfaces), see also [[geometric measure theory]]:
* [[Jürgen Jost]], _The geometric calculus of variations: a short survey and a list of open problems_, Exposition. Math. __6__ (1988), no. 2, 111–143, [MR89h:58036](http://www.ams.org/mathscinet-getitem?mr=938179)
* H. Federer, _Geometric measure theory_, Springer 1969(especially appendices to Russian transl.)
* Frederick J., Jr. Almgren, Almgren's big regularity paper (book form of a 1970s preprint)
Discussion in the context of [[BV formalism]]:
* Arthemy V. Kiselev, _The geometry of variations in Batalin-Vilkovisky formalism_, [http://arxiv.org/abs/1312.1262](http://arxiv.org/abs/1312.1262)
Other references
* J. C. P. Bus, _The Lagrange multiplier rule on manifolds and optimal control of nonlinear systems_, SIAM J. Control and Optimization __22__, 5, 1984, 740-757 [pdf](http://oai.cwi.nl/oai/asset/2552/2552A.pdf)
In the [[covariant phase space]]-perspective:
* L. Vitagliano, _Secondary calculus and the covariant phase space_, [pdf](http://diffiety.ac.ru/preprint/2008/02-08.pdf)
{#Vitagliano}
On [[smooth sets]] as a [[convenient category of spaces|convenient category]] for [[variational calculus]] of [[Lagrangian quantum field theory|Lagrangian]] [[classical field theory]]:
* [[Grigorios Giotopoulos]], [[Hisham Sati]], *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]
### By functorial analysis and $\mathcal{D}$-geometry
See also references at [[diffiety]].
A formalism for variational calculus based on [[functorial analysis]] (with a precise relation with functional analytic methods and jet formalism) and a long list of examples of variational problems arising in [[classical mechanics]] and [[quantum field theory]] are collected in
* [[Frédéric Paugam]], _Towards the mathematics of quantum field theory_ ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics.pdf))
The formulation of variational calculus in terms of [[diffeological spaces]] is mentioned for instance in section 1.65 of
* [[Patrick Iglesias-Zemmour]], _Diffeology_ ([pdf](http://math.huji.ac.il/~piz/documents/Diffeology.pdf#page=64))
{#PIZ}
* [[Frédéric Paugam]], _Histories and observables in covariant field theory_ ([arXiv:1010.3210](http://arxiv.org/abs/1010.3210)), sec. 2.4
{#Paugam}
following section 2.3.20 of
* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _[[Chiral Algebras]]_
For variational calculus in [[nonstandard analysis]] see survey
* Vítor Neves, _Nonstandard calculus of variations, a survey_, [pdf](http://www2.mat.ua.pt/vneves/nsa/CalcVar-vitor6.pdf)
category: analysis, physics
[[!redirects calculus of variations]]
[[!redirects variational derivative]]
[[!redirects variational derivatives]]
|
variational calculus - contents | https://ncatlab.org/nlab/source/variational+calculus+-+contents |
**[[variational calculus]]**
## Differential geometric version
* [[variational bicomplex]]
* [[local Lagrangian]], [[local action functional]]
* [[Euler-Lagrange form]], [[source form]]
* [[Lepage form]]
* [[principle of extremal action]], [[Euler-Lagrange equations]], [[de Donder-Weyl formalism]]
* [[covariant phase space]]
* [[Noether's theorem]]
* [[conserved current]], [[charge]]
* [[symmetry]]
## Derived differential geometric version
* [[BV-BRST complex]]
* [[D-module]]
<div markdown="1">[Edit this sidebar](/nlab/edit/variational+calculus+-+contents)</div>
|
variational sequence | https://ncatlab.org/nlab/source/variational+sequence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Given a [[jet bundle]] $J^r E$ of jets of finite order $r \in \mathbb{Z}$, its _variational sequence_ ([Krupka 90](#Krupka90)) is a [[resolution]] of its [[constant sheaf]] of [[locally constant functions]] with values in the [[real numbers]] by a [[chain complex]] of quotient of sheaves of [[differential forms]] by combinations of [[contact forms]] and their [[de Rham differentials]]. (e.g. [PRWM 15, section 2.2](#PRWM15)).
The variational sequence is a sub-sequence of the [[Euler complex]] as obtained from the theory of the [[variational bicomplex]] ([Krupka 90](#Krupka90), see e.g. [PRWM 15,theorem 1](#PRWM15)).
## Related concepts
* [[Poincaré lemma]]
* [[Lepage form]]
## References
The concept was introduced in
* {#Krupka90} [[Demeter Krupka]], _Variational Sequences on Finite Order Jet Spaces_, Proc. Diff. Geom. Appl.; J. Janyška, D. Krupka eds., World Sci. (Singapore, 1990) 236–254.
Review includes
* [[Demeter Krupka]], _Introduction to Global Variational Geometry_
* {#PRWM15} M. Palese, O. Rossi, E. Winterroth, J. Musilova, _Variational sequences, representation sequences and applications in physics_ ([arXiv:1508.01752](http://arxiv.org/abs/1508.01752))
[[!redirects variational sequences]] |
variety | https://ncatlab.org/nlab/source/variety |
## Idea
A *variety* may denote
* [[algebraic variety]]
* [[analytic variety]]
* [[variety of algebras]] in [[universal algebra]]
In the first two cases, at least locally, varieties are given by satisfaction of a family of equations, given by polynomials (algebraic variety), analytic functions (analytic variety). The variety of algebras consists to some extent similarly of algebras satisfying some list of equations for algebraic operations true for any elements in the place of free variables.
## Related concepts
* [[complex variety]]
* [[locus]]
[[!redirects varieties]]
|
variety of algebras | https://ncatlab.org/nlab/source/variety+of+algebras |
* tic
{: toc}
## Idea
The notion of *variety of algebras* is a classical notion from [[universal algebra]] that subsumes nearly all of the usual kinds of [[algebra|algebraic]] objects, such as [[groups]], [[rings]], [[vector spaces]] (over a fixed [[field]]), and so on. Those algebraic objects that do not form a variety of algebras, such as [[fields]], can still usually be easily described as a subcategory of some variety of algebras (in this case, [[commutative rings]]).
A variety of algebras is not to be confused with an [[algebraic variety]], although in fact there is a connection: both concepts are (roughly speaking) given by a set of equations, but an algebraic variety is the set of elements (of a fixed generic structure) that satisfy the equations, while a variety of algebras is the class of structures in which *every* element satisfies the equations.
## Definitions
A __signature__ (in this context) is a [[set]], whose elements are called __operations__, to each of which is assigned a [[cardinal number]] ($0, 1, 2, \ldots$) called its [[arity]]. A signature is __finitary__ if its set of operations is [[finite set|finite]] (in the strictest sense for the purposes of [[constructive mathematics]]) and every arity is finite (so a [[natural number]]); similar definitions apply to other [[arity classes]]. Given a signature $\sigma$ and a set $V$, whose elements are called __variables__[^1], a __word__ (in $\sigma$ using $V$) is a (well-founded directed rooted) [[tree]] in which each node is labelled by either a variable or an operation, such that every node labelled by a variable has no branches away from the root and every node labelled by an operation $o$ has as many branches away from the root as the arity of $o$. An __axiom__ (in $\sigma$ using $V$) is a pair of such words; we write the axiom consisting of the words $v$ and $w$ as $v = w$ or $\vdash v = w$. A __theory__ (in our sense) consists of a signature $\sigma$ and a set of axioms in $\sigma$. An algebraic theory is __finitary__ if its signature is finitary and its set of equational axioms is finite ([[Kuratowski-finite]] in constructive mathematics); analogous remarks apply to other arity classes.
[^1]: For a finitary signature, we may without loss of generality use the set $V \coloneqq \{x_0, x_1, x_2, \ldots\}$, effectively the set of [[natural numbers]], as the only set of variables; similar remarks apply to other arity classes.
A __variety of algebras__ is given precisely by a theory in this sense, but it is more usual to think of the variety as being the class (or [[category]]) of models, so we continue with the definitions.
Given a theory $T$, a __model__ or __algebra__ of $T$ consists of a [[set]] $A$ together with, for each operation of $T$ with arity $\kappa$, a [[function]] to $A$ from its $\kappa$th [[cartesian power]] $A^\kappa$ such that, for each axiom $\vdash v = w$ and each assignment of elements of $A$ to the variables in that axiom, the equation holds that is given by applying the operations to the elements of $A$ as indicated by the trees defining $v$ and $w$ (the meaning of which we hope is obvious). We say that $A$ __carries__ the operations and __satisfies__ the axioms.
Given a algebras $A$ and $B$ of the same (quasi)-algebraic theory, a __homomorphism__ from $A$ to $B$ consists of a [[function]] $f\colon A \to B$ such that, for each operation, the diagram
$$ \array {
A^\kappa & \stackrel{f^\kappa}\rightarrow & B^n \\
\downarrow & & \downarrow \\
A & \stackrel{f}\rightarrow & B
} $$
commutes. We say that $f$ __preserves__ the operations.
It is fairly straightforward to define a model (and homomorphisms) in any [[cartesian monoidal category]] $C$, replacing [[Set]] in the two paragraphs above. But note that the operations and axioms still form sets, not objects of $C$; the theory $T$ is independent of the modelling category. In any case, given any theory $T$ and any cartesian monoidal category $C$, the models of $T$ in $C$ (as [[objects]]) and the homomorphisms between them (as [[morphisms]]) form a [[category]], which is the __variety of $T$-algebras in $C$__.
## Generalisations
A __sorted variety__ or __many-sorted variety__ is a generalisation. A __sorted signature__ or __many-sorted signature__ consists of a set, whose elements are called __sorts__, together with operations and arities as before, except that now an arity is a [[list]] of __input__ sorts instead of merely a natural number (the length of the list) together with one __output__ sort. Each variable in an axiom comes equipped with a sort, which we may also classify as an output sort; then each branch in (the tree reprsenting) a word must link output sort to input sort. The root of a word gives the entire word a sort; the two words in an equational axiom must have the same sort. Then a model of a sorted variety consists of one set for each sort, and the rest should be obvious. A sorted variety is __finitary__ if it satisfies the finitarity conditions of a variety and its set of sorts is also finite. An ordinary variety as above may be called __unsorted__ or __single-sorted__ to avoid invoking the [[red herring principle]]; it is the same thing as a sorted variety with exactly one sort.
Another generalisation is a __quasivariety__. While an axiom in a variety gives an equation, an axiom in a quasivariety is a [[Horn clause]], stating that one equation (the __consequent__) holds whenever each equation in some set of equations (the __premises__) holds. A Horn clause is __finitary__ if this set of premises is (Kuratowski)-finite; a quasivariety is __finitary__ if it satisfies the fintarity conditions of a variety and additionally each Horn clause is finitary. An ordinary variety is a quasivariety in which each Horn clause has an [[empty set]] of premises.
## Examples
For each definition of an appropriate algebraic object, there is a corresponding variety of algebras, a few of which are given below:
* The variety of _[[groups]]_ is given by a theory with three operations $m,e,i$ (traditionally called multiplication, identity, and inverse), with respective arities $2,0,1$ (so multiplication is a binary operation, identity is a nullary operation, and inverse is a unary operation). There are five axioms, given by the following pairs of trees (with their traditional names):
* associative law:
\[ \label{ass} \array {
& & & & m \\
& & & \nearrow & & \nwarrow \\
& & m & & & & z \\
& \nearrow & & \nwarrow \\
x & & & y
} \;\;\;=\;\;\; \array {
& & m \\
& \nearrow & & \nwarrow \\
x & & & & m \\
& & & \nearrow & & \nwarrow \\
& & y & & & & z
} \]
* right unit law:
\[ \label{runit} \array {
& & m \\
& \nearrow & & \nwarrow \\
x & & & & e
} \;\;\;=\;\;\; \array {
x
} \]
* left unit law:
\[ \label{lunit} \array {
& & m \\
& \nearrow & & \nwarrow \\
e & & & & x
} \;\;\;=\;\;\; \array {
x
} \]
* right inverse law:
\[ \label{rinv} \array {
& & m \\
& \nearrow & & \nwarrow \\
x & & & & i \\
& & & & \uparrow \\
& & & & x
} \;\;\;=\;\;\; \array {
e
} \]
* left inverse law:
\[ \label{linv} \array {
& & m \\
& \nearrow & & \nwarrow \\
i & & & & x \\
\uparrow \\
x
} \;\;\;=\;\;\; \array {
e
} \]
Then a model of this variety is a set $A$ together with functions $m\colon A^2 \to A$, $e\colon 1 \to A$ (which may be identified with an element of $A$), and $i\colon A \to A$ such that, for all $x,y,z$ in $A$, $m(m(x,y),z) = m(x,m(y,z))$, $m(e,x) = x$, $m(x,e) = x$, $m(x,i(x)) = e$, and $m(i(x),x) = e$. In other words, an algebra in this variety is simply a [[group]]. Similarly, a homomorphism of such algebras is simply a [[group homomorphism]]. Also, an algebra in this variety in the category $C$ is a [[group object]] in $C$.
Note that it is not necessary to draw explicit trees; these can be recovered from the succinct equations like $m(m(x,y),z) = m(x,m(y,z))$; we will do this below.
* The variety of _[[monoids]]_ is given by a subtheory of the theory of groups. It has only the operations $m,e$ and the axioms (eq:ass),(eq:runit),(eq:lunit). An algebra in this theory is a monoid, and a homomorphism of such algebras is a monoid homomorphism (including the condition that it preserve the identity).
* There is no variety of _[[cancellative monoid]]s_, but there is a quasivariety of them. In addition to the axioms for a monoid, we have an axiom stating that the equation $x = y$ follows from the equation $m(x,z) = m(y,z)$, which we write succinctly as $m(x,z) = m(y,z) \vdash x = y$.
* The variety of _[[abelian groups]]_ is given by a supertheory of the theory of groups. In addition to the operations and axioms of the variety of groups, it has an additional axiom, the commutative law: $m(x,y) = m(y,x)$.
* The variety of (associative unital) _[[rings]]_ has five operations $a,z,n,m,e$, with respective arities $2,0,2,0,1$, and ten axioms. Five of these axioms are those of groups, applied to $a,z,n$ in place of $m,e,i$; three of these axioms are those of monoids (applied to $m,e$ as before). The remaining two axioms are:
* left distributive law: $m(x,a(y,z)) = a(m(x,y),m(x,z))$,
* right distributive law: $m(a(x,y),z) = a(m(x,z),m(y,z))$.
* There is no variety or quasivariety of [[fields]]; the elements with multiplicative inverses are not identified by an equation. One can, however, define fields as certain rings and then get the correct notion of field homomorphism as a ring homomorphism between rings that happen to be fields.
* The variety of (real) _[[vector spaces]]_ has infinitely many operations and axioms (so is not finitary). First, there are operations $a,z,n$ with the arities and axioms of an abelian group (as with rings). In addition, there is one operation $m_r$ of arity $1$ for each [[real number]] $r$, satisfying these axioms:
* left distributive law: for each real number $r$, $m_r(a(x,y)) = a(m_r(x),m_r(y))$,
* right distributive law: for each real number $r$ and real number $s$, $m_{r+s}(x) = a(m_r(x),m_r(y))$ (where $r + s$ is the sum of $r$ and $s$ as real numbers),
* associative law: for each real number $r$ and real number $s$, $m_{r s}(x) = m_r(m_s(x))$ (where $r s$ is the product of $r$ and $s$ as real numbers),
* unit law: $m_1(x) = x$ (where $1$ the real number [[1]]).
* There is a similar variety of vector spaces over any given [[ground field]], but there is no variety of vector spaces over arbitrary fields, since we would have to specify the field as well as the vector space. The problem is not really that there is no variety of fields; there is no variety of (left) [[modules]] over arbitrary [[commutative rings]] either. However, there is a _sorted_ variety of modules over arbitrary rings, with two types, one for the ring and one for the module. Even though this variety includes the non-finitary variety of real vector spaces, it is itself a finitary sorted variety.
## As algebraic theories
A variety of algebras is one way to look at an [[algebraic theory]].
As a [[logic]]al theory, everything that we wish to say can be said in (typed or untyped, to match the variety) [[cartesian logic]] (although one might want to use something stronger, such as full classical or intuitionistic predicate logic with equality). We use one function symbol (of matching arity) for each operation, and each axiom in the variety becomes an axiom in the logic, a universally quantified equation. Then the [[models]] of this logical theory are the same as the algebras in the variety.
As a [[Lawvere theory]], an untyped finitary variety of algebras defines a category $L$ whose objects are of the form $A^n$ for $n$ a natural number. The morphisms of the category are those which are necessary to make $A^n$ the $n$-fold [[product]] of $A$, along with one morphism for each operation in the variety. Each axiom of the variety corresponds to a commutative diagram, and $L$ is freely generated by these as a [[sketch]]. Then the models of the variety in a cartesian monoidal category $C$ are the same as the [[product-preserving functor|product-preserving]] functors from $L$ to $C$. Conversely, any [[locally small category|locally small]] (and hence [[small category|small]]) cartesian monoidal category whose objects are all of the form $A^n$ may be thought of as a Lawvere theory and defines an untyped variety of algebras, with operation for each morphism to $A$ and one axiom for each commutative diagram. (This can all be generalised to sorted varieties as well.)
A variety of algebras can be seen as a cartesian version of a (typed or untyped) [[operad]]; whereas an operad has models in any [[symmetric monoidal category]], a variety of algebras has models only in a cartesian monoidal category.
A variety of algebras is traditionally identified with its category $M$ of models in $Set$ (or even with simply the [[class]] of objects of this category), but it then becomes unclear what an algebra in the variety would be in some other category $C$. However, it is worth noting that $M$ is always an [[algebraic category]]; see below for the construction of free objects.
## Free algebras
Given a variety $V$ and a [[cartesian monoidal category]] $C$ (such as $Set$), let $C^V$ be the category of algebras in $V$ in $C$, with homomorphisms as morphisms. There is an obvious [[forgetful functor]] $U\colon C^V \to C$; $C$ is a [[cocomplete category|cocomplete]] $W$-[[W-pretopos|pretopos]] (or if $C$ is just a $W$-pretopos if $V$ is finitary), then $U$ has a [[left adjoint]], the [[free functor]] $F\colon C \to C^V$. In fact, this adjunction is [[monadic adjunction|monadic]]; we can identify the models of $V$ in $C$ with the [[module of a monad|modules/algebras]] of the [[monad]] $U \circ F$.
Given an object $A$ of $C$, we construct the free algebra $F(A)$ as follows. First, let $W(A)$ be the [[inductive type]] in $C$ given by the signature of $V$, that is the [[initial algebra]] of the [[endofunctor]] $A \mapsto \sum_o A^{\#o}$, where $o$ ranges over the operations of $V$ and $\#o$ is the arity of $o$. Then each axiom specifies an internal binary [[relation]] on $W(A)$; the [[colimit]] of these is $F(A)$. Note that, if $C$ is $Set$ and $A$ is a set, then $W(A)$ consists precisely of the words (defined as trees above) with variables from $A$.
The __word problem__ for a variety $V$ consists of deciding, whenever $A$ is a [[finite set]], whether two elements of $W(A)$ are equal in $F(A)$. As a problem in algorithmic decidability, the word problem can be very difficult; for example, it is:
* trivial for [[Boolean algebras]],
* impossible for [[Heyting algebras]],
* nontrivial but possible for [[lattices]],
even though each of these is a supertheory of the next.
## Literature
* [[Jiří Adámek]], [[Lawvere|F.W. Lawvere]], J. Rosický, _On the duality between varieties and algebraic theories_, Algebra universalis 2003, __49__:1, pp 35–49 [doi](https://doi.org/10.1007/s000120300002)
[[!redirects variety of algebras]]
[[!redirects varieties of algebras]]
[[!redirects quasivariety of algebras]]
[[!redirects quasivarieties of algebras]]
[[!redirects word problem]]
[[!redirects word problems]]
|
Varmo Vene | https://ncatlab.org/nlab/source/Varmo+Vene |
* [personal page](https://kodu.ut.ee/~varmo/)
## Selected writings
On [[comonads in computer science]]:
* {#UustaluVene08} [[Tarmo Uustalu]], [[Varmo Vene]], *Comonadic Notions of Computation*, Electronic Notes in Theoretical Computer Science **203** 5 (2008) 263-284 [[doi:10.1016/j.entcs.2008.05.029](https://doi.org/10.1016/j.entcs.2008.05.029)]
category: people |
Varun Gupta | https://ncatlab.org/nlab/source/Varun+Gupta |
## Selected writings
On [[codimension]]=2 [[defect brane]] [[brane intersection|intersections]] of [[M5-brane|M5]]-[[probe branes]] in an [AdS7-CFT6](AdS-CFT+correspondence#AdS7CFT6) [[background field|background]] (i.e. in the [[near horizon limit]] of [[black brane|black]] [[M5-branes]], cf. the [[super 3-brane in 6d]]):
* [[Varun Gupta]], *Holographic M5 branes in $AdS_7 \times S^4$*, J. High Energ. Phys. **2021** 32 (2021) [[arXiv:2109.08551](https://arxiv.org/abs/2109.08551), <a href="https://doi.org/10.1007/JHEP12(2021)032">doi:10.1007/JHEP12(2021)032</a>]
* [[Varun Gupta]], *More Holographic M5 branes in $AdS_7 \times S^4$* [[arXiv:2301.02528](https://arxiv.org/abs/2301.02528)]
category: people
|
Varvara Karpova | https://ncatlab.org/nlab/source/Varvara+Karpova |
* [webpage](http://hessbellwald-lab.epfl.ch/page-77160-en.html)
## Selected writings
Reviewing [[topological K-theory]]:
* {#Karpova09} [[Varvara Karpova]], _Complex topological K-theory_, 2009 ([pdf](http://infoscience.epfl.ch/record/162450/files/karpova.semestre.hess2.pdf), [[KarpovaTopologicalKTheory.pdf:file]])
On left-[[transferred model structures]] and [[model structures on functors]]:
* [[Marzieh Bayeh]], [[Kathryn Hess]], [[Varvara Karpova]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *Left-induced model structures and diagram categories*, in: *Women in Topology: Collaborations in Homotopy Theory*, Contemporary Mathematics **641** American Mathematical Society (2015) [[arXiv:1401.3651](http://arxiv.org/abs/1401.3651), [ISBN:978-1-4704-2495-4](https://bookstore.ams.org/conm-641)]
## Related $n$Lab entries
* [[topological K-theory]]
category: people
[[!redirects Karpova Varvara]] |
Vasilis Niarchos | https://ncatlab.org/nlab/source/Vasilis+Niarchos |
* [institute page](https://www.physics.uoc.gr/en/faculty/v.niarchos)
* [InSpire page](https://inspirehep.net/authors/995738)
## Selected writings
On the *[[blackfold]]*-approach to describing [[black branes]]:
* [[Roberto Emparan]], [[Troels Harmark]], [[Vasilis Niarchos]], [[Niels A. Obers]], *Essentials of Blackfold Dynamics*, JHEP 1003 063 (2010) [[arXiv:0910.1601](https://arxiv.org/abs/0910.1601), <a href="https://doi.org/10.1007/JHEP03(2010)063">doi:10.1007/JHEP03(2010)063</a>]
* [[Roberto Emparan]], [[Troels Harmark]], [[Vasilis Niarchos]], [[Niels A. Obers]], *Blackfolds in Supergravity and String Theory*, JHEP 1108 (2011) 154 [[arXiv:1106.4428](https://arxiv.org/abs/1106.4428)<a href="https://doi.org/10.1007/JHEP08(2011)154">doi:10.1007/JHEP08(2011)154</a>]
On [[black brane]] [[M2-M5 brane bound states]]:
* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *M2-M5 blackfold funnels*, Journal of High Energy Physics **2012** 175 (2012) [[arXiv:1205.1535](https://arxiv.org/abs/1205.1535), <a href="https://doi.org/10.1007/JHEP06(2012)175">doi:10.1007/JHEP06(2012)175</a>]
* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *Entropy of the self-dual string soliton*, Journal of High Energy Physics **2012** 134 (2012) [[arXiv:1206.2935](https://arxiv.org/abs/1206.2935), <a href="https://doi.org/10.1007/JHEP07(2012)134">doi:10.1007/JHEP07(2012)134</a>]
* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *The black M2-M5 ring intersection spins* [[arXiv:1302.0854](https://arxiv.org/abs/1302.0854)]
* [[Vasilis Niarchos]], *Localizing the black M2-M5 intersection* (2012) [[pdf](https://hep.physics.uoc.gr/Slides/Fall_2012/VasilisNiarchos.pdf), [[Niarchos-BlackM2M5Branes.pdf:file]]]
* Jay Armas, [[Vasilis Niarchos]], [[Niels A. Obers]], Section 2 of: _Thermal transitions of metastable M-branes_, J. High Energ. Phys. (2019) 2019: 128 [[arXiv:1904.13283](https://arxiv.org/abs/1904.13283)]
category: people
|
Vasily Dolgushev | https://ncatlab.org/nlab/source/Vasily+Dolgushev |
* [webpage](https://www.math.temple.edu/~vald/)
## Selected writings
On [[orbifold cohomology]] of [[symplectic orbifolds]]:
* [[Vasily Dolgushev]], [[Pavel Etingof]], _Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology_, Int. Math. Res. Not. 2005, no. 27, 1657-1688 ([arXiv:math/0410562](https://arxiv.org/abs/math/0410562))
On [[graph complexes]] and the [[Lie algebra]] of the [[Grothendieck-Teichmüller group]]:
* {#DolgushevRogers12} [[Vasily Dolgushev]], [[Christopher Rogers]], _Notes on Algebraic Operads, Graph Complexes, and Willwacher's Construction_, In: Mathematical aspects of quantization 583 (2012): 25-145. ([arXiv:1202.2937](https://arxiv.org/abs/1202.2937))
* [[Vasily Dolgushev]], _A manifestation of the Grothendieck-Teichmuellergroup in geometry_ ([slides pdf](https://math.temple.edu/~vald/AMSslides.pdf))
## Related entries
* [[deformation quantization]], [[Grothendieck-Teichmüller group]]
category: people
|
Vasily Pestun | https://ncatlab.org/nlab/source/Vasily+Pestun |
* [personal webpage](https://pestun.ihes.fr/)
* arXiv [hep-th author page](https://arxiv.org/search/hep-th?searchtype=author&query=Pestun%2C+V), [cs author page](https://arxiv.org/search/cs?searchtype=author&query=Pestun%2C+V)
## Selected writings
On [[equivariant de Rham cohomology]] and [[equivariant localization]]:
* [[Vasily Pestun]], _Review of localization in geometry_ ([arXiv:1608.02954](https://arxiv.org/abs/1608.02954)), chapter in: [[Vasily Pestun]], [[Maxim Zabzine]] (eds.) _Localization techniques in quantum field theories_, J. Phys. A: Math. Theor. 50 440301 ([doi:10.1088/1751-8121/aa63c1](https://iopscience.iop.org/article/10.1088/1751-8121/aa63c1), [arXiv:1608.02952](https://arxiv.org/abs/1608.02952), [full pdf](http://pestun.ihes.fr/pages/LocalizationReview/LocQFT.pdf))
On [[tensor networks]] for [[machine learning]] [[natural language]]:
* [[Vasily Pestun]], [[Yiannis Vlassopoulos]], *Tensor network language model*, [[arXiv:1710.10248](https://arxiv.org/abs/1710.10248)]
category: people |
Vassili Gorbounov | https://ncatlab.org/nlab/source/Vassili+Gorbounov |
* [webpage](http://www.abdn.ac.uk/ncs/profiles/v.gorbunov)
category: people
|
Vassiliev invariant | https://ncatlab.org/nlab/source/Vassiliev+invariant |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Knot theory
+-- {: .hide}
[[!include knot theory - contents]]
=--
=--
=--
# Vassiliev knot invariants
* table of contents
{: toc}
## Idea ##
The space of [[knots]] in the [[Euclidean space]] $\mathbb{R}^3$ (or in the [[3-sphere]] $S^3$) is an [[open subset|open]] [[submanifold]] of the [[smooth loop space]]. [[knot invariant|Knot invariants]] are [[locally constant functions]] on this manifold. The [[complement]] of the space of knots is called the [[discriminant]] and consists of all [[singular knots]].
If we consider those singular knots with only a finite number of double points, we can build a [[cubical complex]] from this data. The vertices in the complex are labelled by the isotopy classes of knots, and more generally the $n$-cubes by the isotopy classes of singular knots with $n$ double points (and a few other technical pieces of information). The boundary operator resolves a double crossing either upwards or downwards according to the orientation at the crossing.
A **Vassiliev invariant** is simply a cubical morphism from this complex to an abelian group that vanishes above a certain degree.
## Definition ##
One does not need the language of cubical complexes to _define_ Vassiliev invariants. Rather, there is a general method whereby a [[knot invariant]] can be extended to all [[singular knots]] with only finitely many double points (and no other singularities) using the [[Vassiliev skein relations]].
+-- {: .num_defn #vinv}
###### Definition
A **Vassiliev invariant** of degree (or order) $\le n$ is a knot invariant whose extension to singular knots (with double points) vanishes on all singular knots with more than $n$ double points.
=--
As is standard, it is of degree $n$ if it is of degree $\le n$ but not $\le n - 1$. Vassiliev invariants are also called **finite type invariants**.
+-- {: .num_remark}
###### Remark
The degree of Vassiliev invariants defines a filtration on the space of knots (and more particularly, on the [[algebra of knots]]). Two knots are $n$-equivalent if all the Vassiliev invariants of degree $\le n$ agree on them. In particular, a knot that is $n$-equivalent to the unknot is said to be $n$-trivial.
=--
## Properties
### Relation to Chord diagrams and weight systems
A function which is constant on nonsingular knots may be extended to a Vassiliev invariant of degree 0 by applying the [[Vassiliev skein relations]], and conversely, any Vassiliev invariant of degree 0 must be constant on nonsingular knots. Likewise, any Vassiliev invariant of degree 1 must be constant on nonsingular knots.
Any [[singular knot]] $f : S^1 \to \mathbb{R}^3$ with $n$ distinct double points $x_1,\dots,x_n \in \mathbb{R}^3$ gives rise to a [[chord diagram]] of order $n$, consisting of the circle $S^1$ with a chord connecting each pair of points $f^{-1}(x_1), \dots, f^{-1}(x_n)$.
The importance of this construction for singular knots comes from the fact that any finite type invariant determines a function on chord diagrams:
+-- {: .un_theorem}
######Theorem
Let $v$ be a Vassiliev invariant of degree $\le n$. Then the value of $v$ on a singular knot with $n$ distinct double points depends only on the [[chord diagram]] of the singular knot, and not on the knot itself.
=--
Conversely, one can ask which functions on chord diagrams come from finite type invariants. The answer is that Vassiliev invariants (of degree $\le n$) can essentially be identified with _[[weight systems]]_ (of order $n$), which are functions on chord diagrams (of order $n$) satisfying two properties called the "[[1T relation]]" (or "framing independence") and the "[[4T relation]]": see Theorem 1 of [Bar-Natan 95](#BarNatan95) (or Theorem 6.2.13 of [Lando & Zvonkin](#LandoZvonkin)):
+-- {: .num_prop #WeightSystemsAreAssociatedGradedOfVassilievInvariants}
###### Proposition
**([[weight systems are associated graded of Vassiliev invariants]])**
For [[ground field]] $k = \mathbb{R}, \mathbb{C}$ the [[real numbers]] or [[complex numbers]], there is for each [[natural number]] $n \in \mathbb{N}$ a canonical [[linear isomorphism]]
$$
\mathcal{V}_n/\mathcal{V}_{n-1}
\underoverset{\simeq}{\phantom{AAAA}}{\longrightarrow}
\big(
\mathcal{A}_n^u
\big)^\ast
$$
from
1. the [[quotient vector space]] of order-$n$ [[Vassiliev invariants]] of [[knots]] by those of order $n-1$
1. to the space of [[unframed weight systems]] of order $n$.
In other words, in [[characteristic zero]], the [[graded vector space]] of [[unframed weight systems]] is the [[associated graded vector space]] of the [[filtered vector space]] of [[Vassiliev invariants]].
=--
([Bar-Natan 95, Theorem 1](#BarNatan95), following [Kontsevich 93](#Kontsevich93))
### Relation to homology of loop spaces of configuration spaces of points
{#RelationToHomologOfLoopSpacesOfConfigurationSpaces}
We discuss the relation between Vassiliev invariants and the [[Euler characteristic]] of the [[ordinary homology]] of [[loop spaces]] of [[configuration spaces of points]]:
For $n, q \in \mathbb{N}$ and $q \geq 1$, write
1. $\underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big)$ for the [[configuration space of points|configuration space of n ordered points]] in [[Euclidean space]] $\mathbb{R}^{q+2}$;
1. $\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big)$ for the corresponding [[based loop space]] (for any choice of base point);
1. $H_\bullet\Big(\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big)$ for the [[ordinary homology]] of this loop space, with [[coefficients]] in the [[complex numbers]];
1. $\chi H_\bullet\Big(\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big)$ for the [[Euler characteristic]]-series of the homology
Write also
1. $V^n_k$ for the [[complex vector space]] of [[Vassiliev invariants]] of order $k$ for [[pure braids]] with $n$ strands;;
1. $A^n_k$ for the [[complex vector space]] spanned by the horizontal [[chord diagrams]] with $n$ vertical strands modulo the "horizontal 4T relation"
such that there is an [[linear isomorphism]]
$$
V^n_k/V^n_{k-1}
\simeq
(A^n_k)^\ast
$$
between the [[quotient vector space]] of [[Vassiliev invariants]] and the [[dual vector space]] of [[chord diagrams]].
Then:
The [[Euler characteristic]]-series (...) of the homology of the loop spaces of configuration spaces
$$
\chi H_\bullet\Big(\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big)
\;=\;
\Big[
\big(
1 - t^q
\big)
\cdot
\big(
1 - 2 t^q
\big)
\cdots
\big(
1 - (n-1) t^q
\big)
\Big]^{-1}
$$
and is related to the complex [[dimensions]] of spaces of Vassiliev invariants according to
\[
\label{HomotopyOfOmegaConfR3IsBraidVassiliev}
\chi
H_\bullet
\Big(
\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{3} \big), \mathbb{C}
\Big)
\;=\;
\underset{k \in \mathbb{N}}{\sum}
dim_{\mathbb{C}}\big( A^n_k \big)
t^k
\]
([Cohen-Gitler 01, Prop. 9.1](#CohenGitler01), based on [Cohen 76](#Cohen76) and [Kohno 94](#Kohno94))
\linebreak
Alternatively, we have the combination of the following two facts, via [[weight systems]]:
+-- {: .num_prop #WeightSystemsAreCohomologyOfLoopSpaceOfConfigurationSpace}
###### Proposition
**([[weight systems are cohomology of loop space of configuration space]])**
For [[ground field]] $k = \mathbb{R}$ the [[real numbers]],
there is a canonical [[injection]] of the [[real vector space]] $\mathcal{W}$ of [[framed weight systems]] (eq:SpaceOfWeightSystemsAsLinearDual) into the [[real cohomology]] of the [[based loop spaces]] of the ordered [[configuration spaces of points]] in 3-[[dimension|dimensional]] [[Euclidean space]]:
$$
\mathcal{W}
\;\overset{\;\;\;\;}{\hookrightarrow}\;
H^\bullet
\Big(
\Omega
\underset{{}^{\{1,\cdots,n\}}}{Conf}
\big(
\mathbb{R}^3
\big)
\Big)
$$
=--
This is stated as [Kohno 02, Theorem 4.2](#Kohno02).
+-- {: .num_prop #WeightSystemsAreAssociatedGradedOfVassilievInvariants}
###### Proposition
**([[weight systems are associated graded of Vassiliev invariants]])**
For [[ground field]] $k = \mathbb{R}, \mathbb{C}$ the [[real numbers]] or [[complex numbers]], there is for each [[natural number]] $n \in \mathbb{N}$ a canonical [[linear isomorphism]]
$$
\mathcal{V}_n/\mathcal{V}_{n-1}
\underoverset{\simeq}{\phantom{AAAA}}{\longrightarrow}
\big(
\mathcal{A}_n^u
\big)^\ast
$$
from
1. the [[quotient vector space]] of order-$n$ [[Vassiliev invariants]] of [[knots]] by those of order $n-1$
1. to the space of [[unframed weight systems]] of order $n$.
In other words, in [[characteristic zero]], the [[graded vector space]] of [[unframed weight systems]] is the [[associated graded vector space]] of the [[filtered vector space]] of [[Vassiliev invariants]].
=--
([Bar-Natan 95, Theorem 1](#BarNatan95), following [Kontsevich 93](#Kontsevich93))
## Examples
1. The $n$th coefficient of the [[Conway polynomial]] is a Vassiliev invariant of order $\le n$.
(...)
## Related concepts ##
[[!include chord diagrams and weight systems -- table]]
* [[Kontsevich integral]]
* [[universal Vassiliev invariant]]
* [[knot theory]]
* [[Jones polynomial]]
* [[singularity]]
## References
### General
The original articles are
* [[Viktor Vassiliev]], _Complements of discriminants of smooth maps: topology and applications_, Amer. Math. Soc. 1992 ([ams:mmono-98](https://bookstore.ams.org/mmono-98))
* {#Kontsevich93} [[Maxim Kontsevich]], _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf), [[KontsevichVassilievInvariants93.pdf:file]])
* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (<a href="https://doi.org/10.1016/0040-9383(95)93237-2">doi:10.1016/0040-9383(95)93237-2</a>, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))
Lecture notes:
* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_, in: *Geometry and Physics*, Lecture Notes in Pure and Applied Mathematics, volume 184, Marcel Dekker Inc. 1996 ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009), [ISBN:0-8247-9791-4](https://users-math.au.dk/swann/Aarhus-proceedings-1995/index.html))
Textbook accounts:
* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))
* [[David Jackson]], [[Iain Moffat]], _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))
* {#Lescop20} [[Christine Lescop]], _Invariants of links and 3-manifolds from graph configurations_ ([arXiv:2001.09929](https://arxiv.org/abs/2001.09929))
Further review:
* [[Simon Willerton]], _On the Vassiliev invariants for knots and for pure braids_, 1997 ([hdl:1842/11581](http://hdl.handle.net/1842/11581), [ethos.663801](https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.663801), [[WillertonVasdilievInvariants1997.pdf:file]])
* [[Dror Bar-Natan]], _Finite Type Invariants_, in: J.-P. Francoise, G.L. Naber and Tsou S.T. (eds.) Encyclopedia of Mathematical Physics, Oxford: Elsevier, 2006, volume 2 page 340
([arXiv:math/0408182](https://arxiv.org/abs/math/0408182))
* {#LandoZvonkin} Sergei K. Lando and Alexander K. Zvonkin, Chapter 6 of: _Graphs on Surfaces and Their Applications_, Springer, 2004.
More literature is listed at
* [[Dror Bar-Natan]], [[Sergei Duzhin]], _[Bibliography of Vassiliev Invariants](http://www.pdmi.ras.ru/~duzhin/VasBib/Long)_
See also
* Mathworld, _[Vassiliev invariant](http://mathworld.wolfram.com/VassilievInvariant.html)_
Concrete computations:
* [[Jan Kneissler]], _On spaces of connected graphs I: Properties of Ladders_, Proc. Internat. Conf. "Knots in Hellas '98", Series on Knots and Everything, vol. 24 (2000), 252-273 ([arXiv:math/0301018](https://arxiv.org/abs/math/0301018))
* [[Jan Kneissler]], _On spaces of connected graphs II: Relations in the algebra Lambda_, Jour. of Knot Theory and its Ramif. vol. 10, no. 5 (2001), 667-674 ([arXiv:math/0301019](https://arxiv.org/abs/math/0301019))
* [[Jan Kneissler]], _On spaces of connected graphs III: The Ladder Filtration_, Jour. of Knot Theory and its Ramif. vol. 10, no. 5 (2001), 675-686 ([arXiv:math/0301020](https://arxiv.org/abs/math/0301020))
* [[Pierre Vogel]], _Algebraic structures on modules of diagrams_, Journal of Pure and Applied Algebra Volume 215, Issue 6, June 2011, Pages 1292-1339 ([pdf](https://webusers.imj-prg.fr/~pierre.vogel/diagrams.pdf))
### Relation to Jones polynomial:
Relation to the [[Jones polynomial]]:
* Joan S. Birman; Xiao-Song Lin, _Knot polynomials and Vassiliev's invariants_, Inventiones mathematicae (1993) Volume: 111, Issue: 2, page 225-270 ([https://dml:144077](https://eudml.org/doc/144077))
Relation to other polynomial [[knot invariants]]:
* Myeong-Ju Jeong, Chan-Young Park, _Polynomial invariants and Vassiliev invariants_, Geom. Topol. Monogr. 4 (2002) 89-101 ([arxiv:math/0211045](https://arxiv.org/abs/math/0211045))
### Relation to homology of loop spaces of configuration spaces
Relation to the [[Euler characteristic]] of the [[ordinary homology]] of [[loop spaces]] of [[configuration spaces of points]]
* {#CohenGitler01} [[Fred Cohen]], [[Samuel Gitler]], _Loop spaces of configuration spaces, braid-like groups, and knots_, In: Jaume Aguadé, Carles Broto, [[Carles Casacuberta]] (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel 2001 ([doi:10.1007/978-3-0348-8312-2_7](https://doi.org/10.1007/978-3-0348-8312-2_7))
based on
* {#Kohno94} [[Toshitake Kohno]], _Vassiliev invariants and de Rham complex on the space of knots_, In: Yoshiaki Maeda, Hideki Omori and [[Alan Weinstein]] (eds.), _Symplectic Geometry and Quantization_, Contemporary Mathematics 179 (1994): 123-123 ([doi:10.1090/conm/179](http://dx.doi.org/10.1090/conm/179))
* {#Cohen76} [[Fred Cohen]], _The homology of $\mathcal{C}_{n+1}$-Spaces, $n \geq 0$. In: The Homology of Iterated Loop Spaces. Lecture Notes in Mathematics, vol 533. Springer, Berlin, Heidelberg 1976 ([doi:10.1007/BFb0080467](https://doi.org/10.1007/BFb0080467))
See also
* {#Kohno02} [[Toshitake Kohno]], _Loop spaces of configuration spaces and finite type invariants_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arXiv:math/0211056](https://arxiv.org/abs/math/0211056))
* [[Toshitake Kohno]], Section 3.1 in: *Conformal field theory and topology*, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs __210__. Iwanami Series in Modern Mathematics. Amer. Math. Soc. 2002 $[$[AMS:mmono-210](https://bookstore.ams.org/mmono-210)$]$
### As Chern-Simons amplitudes
Discussion of higher order Vassiliev invariants as [[Chern-Simons theory]]-[[correlators]], hence as [[configuration space of points|configuration space]]-[[integrals]] of [[wedge products]] of [[Chern-Simons propagators]] assigned to [[edges]] of [[Feynman diagrams]] in the [[graph complex]]:
* [Kontsevich 93, Section 5](#Kontsevich93)
* {#AF96} [[Daniel Altschuler]], [[Laurent Freidel]], _Vassiliev knot invariants and Chern-Simons perturbation theory to all orders_, Commun. Math. Phys. **187** (1997) 261-287 [[arXiv:q-alg/9603010](https://arxiv.org/abs/q-alg/9603010), [doi:10.1007/s002200050136](https://doi.org/10.1007/s002200050136)]
* [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Configuration spaces and Vassiliev classes in any dimension_, Algebr. Geom. Topol. 2 (2002) 949-1000 ([arXiv:math/9910139](https://arxiv.org/abs/math/9910139))
* [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Algebraic structures on graph cohomology_, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 ([arXiv:math/0307218](https://arxiv.org/abs/math/0307218))
Reviewed in:
* {#Volic13} [[Ismar Volić]], Section 4 of: _Configuration space integrals and the topology of knot and link spaces_, [Morfismos, Vol 17, no 2, 2013](https://fdocuments.co/amp/document/morfismos-vol-17-no-2-2013.html) ([arxiv:1310.7224](https://arxiv.org/abs/1310.7224))
See also
* J. de-la-Cruz-Moreno, H. García-Compeán, E. López, _Vassiliev Invariants for Flows Via Chern-Simons Perturbation Theory_ ([arXiv:2004.13893](https://arxiv.org/abs/2004.13893))
### As observables on fuzzy spheres
Relation of [[Dp-D(p+2)-brane bound states]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]) to [[Vassiliev braid invariants]] via [[chord diagrams]] computing [[radii]] of [[fuzzy spheres]]:
* [[Sanyaje Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: _Resolving brane collapse with $1/N$ corrections in non-Abelian DBI_, Nucl. Phys. B703 (2004) 236-276 ([arxiv:hep-th/0405256](https://arxiv.org/abs/hep-th/0405256))
* [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))
* [[Simon McNamara]], Section 4 of: _Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels_, 2006 ([spire:1351861](http://inspirehep.net/record/1351861), [pdf](https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf), [[McNamara06.pdf:file]])
* [[Constantinos Papageorgakis]], p. 161-162 of: _On matrix D-brane dynamics and fuzzy spheres_, 2006 ([[Papageorgakis06.pdf:file]])
### Vassiliev invariants of braids
Vassiliev invariants of [[braid group|braids]] via [[horizontal chord diagrams]]:
* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))
* Louis Funar, _Vassiliev invariants I: Braid groups and rational homotopy theory_ ([arXiv:q-alg/9510008](https://arxiv.org/abs/q-alg/9510008))
category: geometry, topology
[[!redirects Vassiliev knot invariant]]
[[!redirects Vassiliev knot invariants]]
[[!redirects Vassiliev invariants]]
[[!redirects Vassiliev finite type invariants]]
[[!redirects Vassiliev finite type invariant]]
[[!redirects Vassiliev finite-type invariants]]
[[!redirects Vassiliev finite-type invariant]]
[[!redirects finite type invariants]]
[[!redirects finite type invariant]]
[[!redirects finite-type invariants]]
[[!redirects finite-type invariant]]
[[!redirects Vassiliev link invariant]]
[[!redirects Vassiliev link invariants]]
[[!redirects Vassiliev invariants of braids]]
[[!redirects Vassiliev braid invariant]]
[[!redirects Vassiliev braid invariants]]
|
Vassiliev skein relation | https://ncatlab.org/nlab/source/Vassiliev+skein+relation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **Vassiliev skein relation** is a way to extend [[knot invariants]] to [[singular knots]] (at least, to singular knots where the only singularities are double points). If $v$ is a knot invariant that takes values in an abelian group, then it is extended to singular knots using the relation
$$
v(L_d) = v(L_+) - v(L_-)
$$
where $L_d$ is a singular knot with a double point and $L_+$, respectively $L_-$, are formed from $L_d$ by replacing the double point by a positively oriented, respectively negatively oriented, crossing.
\begin{tikzcd}[sep={between origins, 30pt}]
&
{} && {}
&[+30pt] &
{} \ar[-Latex, from=ddrr, line width=1.5pt] && {}
&[+30pt] &
{} && {}
\\
L_d =
&
& \raisebox{-2.5pt}{\scalebox{1.8}{$\bullet$}}
&
&
L_+ =
&
&
&
&
L_- =
\\
&
{} \ar[-Latex, uurr, line width=1.5pt]
&&
{} \ar[-Latex, uull, line width=1.5pt]
&&
{} \ar[-Latex, uurr, line width=1.5pt, crossing over]
&&
{}
&&
{} \ar[-Latex, uurr, line width=1.5pt]
&&
{} \ar[-Latex, uull, line width=1.5pt, crossing over]
\end{tikzcd}
## References
General discussion:
* Wikipedia, _[Skein relation](https://en.wikipedia.org/wiki/Skein_relation)_
Discussion in the context of [[quantization of 3d Chern-Simons theory]]:
* {#GelcaUribe10a} [[Razvan Gelca]], [[Alejandro Uribe]], _From classical theta functions to topological quantum field theory_ ([arXiv:1006.3252](http://arxiv.org/abs/1006.3252), [slides pdf](http://www.math.ttu.edu/~rgelca/berk.pdf))
* {#GelcaUribe10b} [[Razvan Gelca]], [[Alejandro Uribe]], _Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$_ ([arXiv:1007.2010](http://arxiv.org/abs/1007.2010))
category: knot theory
[[!redirects Vassiliev skein relations]]
[[!redirects skein relation]]
[[!redirects skein relations]]
|
Vassily Gorbounov | https://ncatlab.org/nlab/source/Vassily+Gorbounov |
**Vassily Gorbounov** is a professor of mathematics at Aberdeen. Much of his mathematical work is inspired by [[quantum field theory]], especially [[topological field theory|topological]] and [[conformal field theory]] as well as with [[superstring theory]].
* [institute page](https://www.hse.ru/en/org/persons/216359816)
## Selected writings
* [[Imma Gálvez]], Vassily Gorbounov, [[Andrew Tonks]], _Homotopy Gerstenhaber structures and vertex algebras_, [math/0611231.QA](https://arxiv.org/abs/math/0611231)
On the [[quaternionic cobordism ring]]:
* {#GorbunovRay92} [[Vassily Gorbounov]], [[Nigel Ray]], _Orientations of $Spin$ Bundles and Symplectic Cobordism_, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 ([[GorbunovRaySpinBundles.pdf:file]], [doi: 10.2977/prims/1195168855](https://www.ems-ph.org/journals/show_abstract.php?issn=0034-5318&vol=28&iss=1&rank=4))
* [[Vassily Gorbounov]], [[Nigel Ray]], [[Paul Turner]], _On the Hopf Ring for a Symplectic Oriented Spectrum_, American Journal of Mathematics
Vol. 117, No. 4 (Aug., 1995), pp. 1063-1088 ([doi:10.2307/2374960](https://doi.org/10.2307/2374960))
On [[chiral differential operators]]:
* [[Vassily Gorbounov]], [[Fyodor Malikov]], [[Vadim Schechtman]], _Gerbes of chiral differential operators_ Math. Res. Lett. __7__ (2000), no. 1, 55--66, [MR2002c:17040](https://mathscinet.ams.org/mathscinet-getitem?mr=2002c:17040), [math.AG/9906117](https://arxiv.org/abs/math/9906117);
* _Gerbes of chiral differential operators. II. Vertex algebroids_, Invent. Math. __155__ (2004), no. 3, 605--680, [MR2005e:17047](http://www.ams.org/mathscinet-getitem?mr=2005e:17047), [math.AG/0003170](http://arxiv.org/abs/math/0003170), [doi](https://doi.org/10.1007/s00222-003-0333-4);
* _Gerbes of chiral differential operators. III_, in: The orbit method in geometry and physics (Marseille, 2000), 73--100, Progr. Math. __213__, Birkhäuser 2003, [MR2005a:17028](https://mathscinet.ams.org/mathscinet-getitem?mr=2005a:17028), [math.AG/0005201](https://arxiv.org/abs/math/0005201),
* _On chiral differential operators over homogeneous spaces_, Int. J. Math. Math. Sci. __26__ (2001), no. 2, 83--106, [MR2002g:14020](https://mathscinet.ams.org/mathscinet-getitem?mr=2002g:14020), [math.AG/0008154](https://arxiv.org/abs/math/0008154), [doi](http://doi.org/10.1155/S0161171201020051)
* Vassily Gorbounov, [[Owen Gwilliam]], Brian Williams, _Chiral differential operators via Batalin-Vilkovisky quantization_, [pdf](http://people.mpim-bonn.mpg.de/gwilliam/cdo.pdf)
category: people
[[!redirects Vassili Gorbounov]] |
Vassily Gorbunov > history | https://ncatlab.org/nlab/source/Vassily+Gorbunov+%3E+history | |
Vaughan Jones | https://ncatlab.org/nlab/source/Vaughan+Jones |
* [website](http://math.berkeley.edu/~vfr/)
* [Wikipedia entry](http://en.wikipedia.org/wiki/Vaughan_Jones)
## related $n$Lab entries
* [[knot invariant]]
* [[Jones polynomial]]
* [[planar algebra]]
category: people
|
Vaughan Pratt | https://ncatlab.org/nlab/source/Vaughan+Pratt | Vaughan Pratt is Professor Emeritus of Computer Science at Stanford University. Prior to his retirement in 2000 his research focused on theory and technical applications of computers. After retirement he started Tiqit Computers to promote what are now called netbooks. His current interests are in [[universal algebra]] and [[category theory]], signal processing, principles of written and spoken communication, thin film optics, and philosophy of mind. His most recent four Ph.D. students are in acoustic signal processing, including Parham Aarabi, named to MIT's TR35 for 2005, and Keyvan Mohajer, CEO of Melodis Corp. He is responsible for 38 of Sophus Lie's 990 descendants.
He has written extensively on [[Chu spaces]], taking simple examples to show the potential of the idea, and encouraging the use of generalisations of it.
* [website](http://boole.stanford.edu/pratt.html)
* [Chu Space Web Site (HTML)](http://chu.stanford.edu/)
## Selected writings
On [[quantum programming]] via [[quantum logic]] understood as [[linear type theory]] [[categorical semantics|interpreted in]] [[symmetric monoidal categories]]:
* {#Pratt92} [[Vaughan Pratt]], *Linear logic for generalized quantum mechanics*, in Proceedings of *[Workshop on Physics and Computation (PhysComp'92)](https://www.computer.org/csdl/proceedings/phycmp/1992/12OmNx19jVS)* (1992) 166-180 [[doi:10.1109/PHYCMP.1992.615518](https://doi.ieeecomputersociety.org/10.1109/PHYCMP.1992.615518), [pdf](http://boole.stanford.edu/pub/ql.pdf), [[Pratt-LinearLogicForQuantum.pdf:file]]]
On [[linear logic]] and the [[Chu construction]]:
* {#Pratt93} [[Vaughan Pratt]], *The second calculus of binary relations*, Mathematical Foundations of Computer Science 1993. MFCS 1993, Lecture Notes in Computer Science **711**, Springer (1993) [[doi:10.1007/3-540-57182-5_9](https://doi.org/10.1007/3-540-57182-5_9)]
> "Linear logic is seen in its best light as the
realization of the [[Curry-Howard isomorphism]] for [[linear algebra]]"
On [[Chu spaces]]:
* [[Vaughan Pratt]], *Chu Spaces* (1999) [[pdf](http://boole.stanford.edu/pub/coimbra.pdf), [[Pratt-ChuSpaces.pdf:file]]]
* [[Vaughan Pratt]], *Linear process algebra*, in: *Distributed Computing and Internet Technology ICDCIT 2011*, Lecture Notes in Computer Science **6536**, Springer (2011) [[doi:10.1007/978-3-642-19056-8_6](https://doi.org/10.1007/978-3-642-19056-8_6), [pdf](http://boole.stanford.edu/pub/bhub.pdf)]
On the relation between [[toposes]] and [[abelian categories]] (cf. *[[AT-categories]]*):
* [[Vaughan Pratt]] in conversation with [[Peter Freyd]], *Topos-abelian categories that are neither*, mailing list comments (1999) [[catlist:1999/atcat](http://www.mta.ca/~cat-dist/catlist/1999/atcat), [[Freyd_AT-categories.txt:file]]]
category: people |
vDbl | https://ncatlab.org/nlab/source/vDbl |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
By $vDbl$ is denoted the [[2-category]] of [[virtual double categories]].
## References
* [[Geoff Cruttwell]] and [[Mike Shulman]], _A unified framework for generalized multicategories_, [arXiv:0907.2460](http://arxiv.org/abs/0907.2460)
{#CruttwellShulman}
category: category |
Vect | https://ncatlab.org/nlab/source/Vect |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Given a [[field]] $k$, the _category of $k$-vector spaces_ **$Vect_k$** is the [[category]] whose
1. [[object|objects]] are [[vector space|vector spaces]],
1. [[morphisms]] are [[linear maps]].
If the field $k$ is understood, one often just writes $Vect$.
Via [[direct sum]] and [[tensor product of vector spaces]] $\otimes_k$, this becomes a [[symmetric monoidal category]] in two compatible ways, making it a [[distributive monoidal category]], in particular a [[rig category]].
The study of $Vect$ is called _[[linear algebra]]_.
## Properties
For a field $k$, the category $Vect_k$ is [[ cocomplete category | cocomplete ]].
### Splitting lemma
Assuming the [[axiom of choice]] (and essentially by the [[basis theorem]]):
\begin{proposition}\label{ShortExactSequencesSplit}
In $Vect$ every [[short exact sequence]] [[split short exact sequence|splits]].
\end{proposition}
(See [there](split+exact+sequence#SESOfVectorSpacesSplits).)
\begin{remark}
On [[FinDimVect]] this is a [[categorification]] of the [[rank-nullity theorem]].
\end{remark}
## Related categories
### Finite-dimensional vector spaces
The [[full subcategory]] of Vect consisting of [[finite-dimensional vector spaces]] may be denoted **[[FinDimVect]]**.
This is a [[compact closed category]] (see [here](finite-dimensional+vector+space#CompactClosure)).
$FinDimVect$ is where most of ordinary [[linear algebra]] lives, although much of it makes sense in all of $Vect$. See also at _[[quantum information theory in terms of dagger-compact categories]]_.
On the other hand, anything involving transposes or [[inner products]] really takes place in $Fin$ [[Hilb]].
### Modules
More generally, for $R$ any [[ring]] (not necessarily a [[field]]) then the analog of $Vect$ is the category $R$[[Mod]] of $R$-[[modules]] and module homomorphisms between them.
### Vector bundles
For $X$ a suitable [[space]] of sorts, there is the category [[Vect(X)]] of [[vector bundles]] over $X$. Specifically for $X$ a [[topological space]], there is the category of [[topological vector bundles]] over $X$. For $X = \ast$ the [[point space]], then this is [[equivalence of categories|equivalently]] the category of plain vector spaces:
$$
Vect(\ast) \simeq Vect
\,.
$$
category: category
[[!redirects category of vector spaces]]
[[!redirects categories of vector spaces]]
[[!redirects category of vector bundles]]
[[!redirects categories of vector bundles]]
[[!redirects VectorSpaces]]
|
Vect(X) | https://ncatlab.org/nlab/source/Vect%28X%29 |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $X$ a suitable [[space]] of sorts, the _category of vector bundles over $X$_ is the [[category]] denoted $Vect(X)$ whose
1. [[objects]] are [[vector bundles]] over $X$,
1. [[morphisms]] are vector bundle [[homomorphisms]] over $X$.
Specifically for $X$ a [[topological space]], there is the category of [[topological vector bundles]] over $X$.
Via [[direct sum of vector bundles]] and [[tensor product of vector bundles]] this becomes a [[symmetric monoidal category]] in two compatible ways, making it a [[distributive monoidal category]], in particular a [[rig category]].
For $X$ a [[compact Hausdorff space]] then the [[Grothendieck group]] of $Vect(X)$ is the [[topological K-theory]] group $K(X)$.
## Relation to other categories
### Vector spaces
For $X = \ast$ the [[point space]], then this is [[equivalence of categories|equivalently]] the category [[Vect]] of plain vector spaces:
$$
Vect(\ast) \simeq Vect
\,.
$$
### Vector bundles over general bases
See *[[VectBund]]*
### Higher vector bundles
An analog in [[homotopy theory]]/[[higher category theory]] is the [[(infinity,1)-category]] of [[(infinity,1)-module bundles]].
[[!redirects VectBund(X)]]
[[!redirects VectBund(B)]]
|
VectBund | https://ncatlab.org/nlab/source/VectBund |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
This entry is concerned with [[categories]] $VectBund$ of [[vector bundles]] whose [[morphisms]] are allowed to cover non-trivial [[maps]] between their base spaces.
This means that after choosing a [[ground field]] and an ambient category of [[spaces]] (e.g. [[Sets]] [[TopologicalSpaces]], [[SmoothManifolds]], etc.):
* the [[objects]] of $VectBund$ are [[vector bundles]] over any base space $B$ in the ambient category, which we shall denote like this:
$$
\left[
\array{
E
\\
\big\downarrow
\\
B
}
\right]
$$
* the [[morphisms]] of $VectBund$ are [[commuting squares]] (in the ambient category of spaces) of the form
$$
\array{
E &\overset{\phi}{\longrightarrow}& E'
\\
\big\downarrow && \big\downarrow
\\
B &\underset{f}{\longrightarrow}& B'
}
$$
which are [[fiber]]-wise [[linear maps]].
Notice here that given the base map $f$, such a diagram is equivalent to a homomorphism of vector bundles over $B$ from $E$ into the [[pullback vector bundle]] $f^\ast E'$, which we may denote by the same symbol $\phi$:
$$
\array{
E && \overset{\phi}{\longrightarrow} && f^\ast E'
\\
& \searrow && \swarrow
\\
&& B
}
$$
Accordingly, for each base space $B$ there is a [[full subcategory]]-inclusion
$$
VectBund(B)
\xhookrightarrow{\phantom{--}}
VectBund
$$
of the category [[VectBund(X)|VectBund(B)]] of vector bundles over $B$,
## Properties
### Bireflective inclusion of zero-bundles
{#BireflectiveInclusionOfZeroBundles}
The construction which to any base space $B$ assigns the [[zero-vector bundle]] over $B$ constitutes a [[bireflective subcategory]]-inclusion (see [there](bireflective+subcategory#ZeroVectorBundlesAmongVectorBundles)) of the category of base spaces into that over vector bundles over these base spaces.
### Closed monoidal structures
{#ClosedMonoidalStructures}
For a fixed base space $B$, the [[monoidal category]]-[[structures]] on [[VectBund(B)|$VectBund_B$]] are well known, given [[fiber]]-wise by the respective structures on [[VectorSpaces]]:
* the [[cartesian monoidal structure]] on $VectBund(B)$ is given by [[direct sum of vector bundles]], which in turn is [[fiber]]-wise the [[direct sum]] of [[vector spaces]];
* the other [[symmetric monoidal structure]] on $VectBund(B)$ is given by [[tensor product of vector bundles]], which in turn is [[fiber]]-wise the [[tensor product of vector spaces]].
But analogous -- if subtly different -- monoidal structures exist on the total category $VectBund$, discussed in the following:
* [Cartesian monoidal structure on VectBund](#TensorMonoidalStructur)
* [Tensor monoidal structure on VectBund](#TensorMonoidalStructur).
In discussing this now, we (have to and want to) assume that the ambient [[category]] of [[TopologicalSpaces]] is itself [[cartesian closed category|cartesian closed]] (a "[[convenient category of topological spaces]]"), such as that of [[compactly generated topological spaces]] or of [[D-topological spaces]].
#### Cartesian monoidal structure
{#CartesianMonoidalStructure}
The [[cartesian product]] in $VectBund$ is readily checked to be the "external direct sum", namely the result of [[pullback of vector bundles|pulling back]] the two vector bundles to the [[product space]] of their base spaces along the [[projection maps]]
$$
\array{
&&
B \times B'
\\
& \mathllap{ {}^{pr_{B}} }\swarrow
&&
\searrow \mathrlap{ {}^{ pr_{B'} } }
\\
B && && B'
}
$$
and then forming the [[direct sum of vector bundles]] there:
\[
\label{CartesianProduct}
\left[
\begin{array}{c}
E
\\
\big\downarrow
\\
B
\end{array}
\right]
\times
\left[
\begin{array}{c}
E'
\\
\big\downarrow
\\
B'
\end{array}
\right]
\;\;\simeq\;\;
\left[
\begin{array}{c}
(pr_{B})^\ast E
\oplus
(pr_{B'})^\ast {E'}
\\
\big\downarrow
\\
B \times B'
\end{array}
\right]
\]
From this one finds that the cartesian [[mapping space]] ([[internal hom]])
$$
Maps(-,-)
\;\colon\;
VectBund^{op}
\times
VectBund
\longrightarrow
VectBund
$$
is given by forming the vector bundle whose base is the space of vector bundle homomorphisms $\phi$ covering maps of base spaces $f$, and whose fiber over $(f,\phi)$ is (independent of $\phi$ and) given by the [[vector space|vector]]-[[space of sections]] $\Gamma_B(-)$ of the [[pullback vector bundle]] $f^\ast E'$ of the [[codomain]] bundle $E'$ along $f$:
$$
Maps
\left(
\left[
\begin{array}{c}
E
\\
\big\downarrow
\\
B
\end{array}
\right]
\;,\;\;
\left[
\begin{array}{c}
E'
\\
\big\downarrow
\\
B'
\end{array}
\right]
\right)
\;\;\simeq\;\;
\left[
\begin{array}{c}
(f,\phi)
\mapsto
\Gamma_B\big(f^\ast E'\big)
\\
\big\downarrow
\\
\left\{
\array{
f \colon B \to B',
\\
\phi
\colon E \underset{lin/B}{\to} f^\ast E'
}
\;\;
\right\}
\end{array}
\right]
$$
Namely, one checks the defining [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism)
$$
Hom
\big(
-
,\,
Map(-,-)
\big)
\;\;\simeq\;\;
Hom
\big(
- \times -
,\,
-
\big)
$$
by testing it on vector bundles over the [[point space]] (ignoring the topology for the moment) and using (eq:CartesianProduct) on the right:
$$
Hom
\left(
\left[
\begin{array}{c}
V
\\
\big\downarrow
\\
\ast
\end{array}
\right]
\;,\;\;
Maps
\left(
\left[
\begin{array}{c}
E
\\
\big\downarrow
\\
B
\end{array}
\right]
\;,\;\;
\left[
\begin{array}{c}
E'
\\
\big\downarrow
\\
B'
\end{array}
\right]
\right)
\right)
\;\;
\underoverset
{\sim}
{
\Big(
v
\mapsto
\big(
(f,\phi)
,
\sigma_v
\big)
\Big)
\;\mapsto\;
\big(f, (\phi + \sigma_{(-)}) \big)
}{\longrightarrow}
\;\;
Hom
\left(
\left[
\begin{array}{c}
E \oplus V
\\
\big\downarrow
\\
B
\end{array}
\right]
\;,\;\;
\left[
\begin{array}{c}
E'
\\
\big\downarrow
\\
B'
\end{array}
\right]
\right)
$$
#### Tensor monoidal structure
{#TensorMonoidalStructure}
Consider now the [[symmetric monoidal structure]] on $VectBund$ given by the [[external tensor product]] [[external tensor product of vector bundles|of vector bundles]]
$$
\otimes
\;\colon\;
VectBund \times VectBund
\longrightarrow
VectBund
$$
which itself is constructed by first [[pullback vector bundle|pulling back]] both bundles to the [[product space]] of their base spaces, and then forming their [[tensor product of vector bundles]] there:
\[
\label{TensorProduct}
\left[
\begin{array}{c}
E
\\
\big\downarrow
\\
B
\end{array}
\right]
\otimes
\left[
\begin{array}{c}
E'
\\
\big\downarrow
\\
B'
\end{array}
\right]
\;\;\coloneqq\;\;
\left[
\begin{array}{c}
(pr_{B})^\ast E
\otimes
(pr_{B'})^\ast {E'}
\\
\big\downarrow
\\
B \times B'
\end{array}
\right]
\]
The corresponding [[internal hom]]
$$
LMaps
\;\colon\;
VectBund^{op} \times VectBund
\longrightarrow
VectBund
$$
forms the vector bundle whose base space is the [[compact-open topology|mapping space]] of base spaces, and whose [[fiber]] over $f \colon B \to B'$ is the [[vector space]] $[E, f^\ast E']_B$ of vector bundle homomorphisms $E \to f^\ast E'$ over $B$:
$$
LMaps
\left(
\left[
\begin{array}{c}
E
\\
\big\downarrow
\\
B
\end{array}
\right]
\;,\;\;
\left[
\begin{array}{c}
E'
\\
\big\downarrow
\\
B'
\end{array}
\right]
\right)
\;\;\coloneqq\;\;
\left[
\begin{array}{c}
f
\mapsto
\big[E, f^\ast E'\big]_B
\\
\big\downarrow
\\
\left\{
\array{
f \colon B \to B'
}
\;\;
\right\}
\end{array}
\right]
$$
To see this, one checks the required hom-isomorphism
$$
Hom
\big(
-
,\,
LMap(-,-)
\big)
\;\;\simeq\;\;
Hom
\big(
- \otimes -
,\,
-
\big)
$$
over the point, now using (eq:TensorProduct) on the right:
$$
Hom
\left(
\left[
\begin{array}{c}
V
\\
\big\downarrow
\\
\ast
\end{array}
\right]
\;,\;\;
LMaps
\left(
\left[
\begin{array}{c}
E
\\
\big\downarrow
\\
B
\end{array}
\right]
\;,\;\;
\left[
\begin{array}{c}
E'
\\
\big\downarrow
\\
B'
\end{array}
\right]
\right)
\right)
\;\;
\underoverset
{\sim}
{
\Big(
v
\mapsto
\big(f, \sigma_v\big)
\Big)
\;\mapsto\;
\big(f, \sigma_{(-)} \big)
}{\longrightarrow}
\;\;
Hom
\left(
\left[
\begin{array}{c}
E \otimes V
\\
\big\downarrow
\\
B
\end{array}
\right]
\;,\;\;
\left[
\begin{array}{c}
E'
\\
\big\downarrow
\\
B'
\end{array}
\right]
\right)
$$
#### Distributive monoidal structure
{#DistributiveMonoidalStructure}
With respect to the [[coproduct]] and the [[external tensor product]] [[external tensor product of vector bundles|of vector bundles]], $VectBund$ is a [[distributive monoidal category]]. We spell this out in detail for the case of [[discrete topological space|discrete]] base spaces (sets):
\begin{definition}
For any [[ground field]], write [[VectBund|$Vect_{Set}$]] for the [[category]] of [[indexed sets]] of [[vector spaces]].
\end{definition}
\begin{remark}
We may and will present [[objects]] $\mathcal{V}$ of [[VectBund|$Vect_{Set}$]] as [[pairs]] consisting of a [[set]] $S$ and a [[function]] $\mathcal{V}_{(-)}$ (really a [[functor]] on the [[discrete category]] on $S$) to [[Vect]]:
$$
\left(
\array{
S &\longrightarrow& Vect
\\
s &\mapsto& \mathcal{V}_s
}
\right)
\;\;
\in
\;\;
Vect_{Set}
\mathrlap{\,.}
$$
\end{remark}
\begin{definition}
\label{ExternalTensorProductInVectSet}
The "[[external tensor product|external]]" [[tensor product]] on [[VectBund|$Vect_{Set}$]] is the [[functor]]
$$
\boxtimes
\,\colon\,
Vect_{Set} \times Vect_{Set}
\longrightarrow
Vect_{Set}
$$
given by
$$
\big(
\mathcal{V}_{(-)} \,\colon\, S \to \Vect
\big)
\,\boxtimes\,
\big(
\mathcal{V}'_{(-)} \,\colon\, S' \to \Vect
\big)
\;\;
\coloneqq
\;\;
\left(
\array{
S \times S' &\longrightarrow& \Vect
\\
(s, s')
&\mapsto&
\mathcal{V}_s \otimes \mathcal{V}'_{s'}
}
\right)
\mathrlap{\,.}
$$
\end{definition}
\begin{proposition}
\label{CoproductInVectSet}
The [[coproduct]] in [[VectBund|$Vect_{Set}$]] is given by [[disjoint union]] of [[bundles]]:
$$
\big(
\mathcal{V}^{(1)}_{(-)} \,\colon\, S_1 \to \Vect
\big)
\;
\sqcup
\;
\big(
\mathcal{V}^{(1)}_{(-)} \,\colon\, S_2 \to \Vect
\big)
\;\;
\simeq
\;\;
\left(
\array{
S_1 \sqcup S_2 &\longrightarrow& \Vect
\\
s_i &\mapsto& \mathcal{V}^{(i)}_{s_i}
}
\right)
$$
\end{proposition}
\begin{proof}
It is immediate to check the [[universal property]] characterizing the coproduct.
\end{proof}
\begin{proposition}
The [[external tensor product]] $\boxtimes$
(Def. \ref{ExternalTensorProductInVectSet})
[[distributive monoidal category|distributes]] over the coproduct (Prop. \ref{CoproductInVectSet}):
$$
\mathcal{E} \boxtimes ( \mathcal{V}^{(1)} \sqcup \mathcal{V}^{(2)})
\;\simeq\;
\big(
\mathcal{E} \boxtimes \mathcal{V}^{(1)}
\big)
\,\sqcup\,
\big(
\mathcal{E} \boxtimes \mathcal{V}^{(2)}
\big)
$$
and hence gives a [[distributive monoidal category]]:
$$
\big(
Vect_{Set}
,
\sqcup
,
\boxtimes
\big)
\;\in\;
DistMonCat
\mathrlap{\,.}
$$
\end{proposition}
\begin{proof}
Unwinding the above definitions and using that [[Set]] is a [[distributive category]], we have the following sequence of [[natural isomorphisms]]:
$$
\begin{array}{l}
\mathcal{E}
\,\boxtimes\,
\big(
\mathcal{V}^{(1)}
\,\sqcup\,
\mathcal{V}^{(2)}
\big)
\\
\;\equiv\;
\big(
\mathcal{E}_{(-)} \,\colon\, S_E \to Vect
\big)
\,\boxtimes\,
\Big(
\big(
\mathcal{V}^{(1)}_{(-)} \,\colon\, S_1 \to Vect
\big)
\,\sqcup\,
\big(
\mathcal{V}^{(2)}_{(-)} \,\colon\, S_2 \to Vect
\big)
\Big)
\\
\;\simeq\;
\big(
\mathcal{E}_{(-)} \,\colon\, S_E \to Vect
\big)
\,\boxtimes\,
\left(
\array{
S_1 \sqcup S_2 &\longrightarrow& Vect
\\
s_i &\mapsto& \mathcal{V}^{(i)}_{s_i}
}
\right)
\\
\;\simeq\;
\left(
\array{
S_E \times (S_1 \sqcup S_2)
&\longrightarrow& Vect
\\
(s_E , s_i) &\mapsto&
\mathcal{E}_{s_E} \otimes \mathcal{V}^{(i)}_{s_i}
}
\right)
\\
\;\simeq\;
\left(
\array{
(S_E \times S_1) \,\sqcup\, (S_2 \times S_2)
&\longrightarrow& Vect
\\
(s_E , s_i) &\mapsto&
\mathcal{E}_{s_E} \otimes \mathcal{V}^{(i)}_{s_i}
}
\right)
\\
\;\simeq\;
\left(
\array{
S_E \times S_1
&\longrightarrow& Vect
\\
(s_E , s_1)
&\mapsto&
\mathcal{E}_{s_E} \otimes \mathcal{V}^{(1)}_{s_1}
}
\right)
\,\sqcup\,
\left(
\array{
S_E \times S_2
&\longrightarrow& Vect
\\
(s_E , s_2)
&\mapsto&
\mathcal{E}_{s_E} \otimes \mathcal{V}^{(2)}_{s_2}
}
\right)
\\
\;\equiv\;
\mathcal{E} \boxtimes \mathcal{V}^{(1)}
\,\sqcup\,
\mathcal{E} \boxtimes \mathcal{V}^{(2)}
\end{array}
$$
\end{proof}
#### Amalgamation of monoidal and parameter structures
{#AmalgamationOfMonoidalAndParameterStructures}
> under construction
We want to point out a way in which $VectBund$ with its [[external tensor product]]-structure is the canonical "amalgamation" of
1. vector spaces equipped with their tensor product,
1. vector spaces equipped with parametrization, namely vector bundles,
at least in the case that the base space is discrete.
The argument generalizes immediately to [[flat vector bundles]] over arbitrary base spaces (by enhancing the following parameter [[sets]] to [[fundamental groupoids]]), and in fact to [[(infinity,1)-vector bundle|$(\infty,1)$-vector bundles]] with [[flat infinity-connection|flat $\infty$-connections]] (by enhancing the [[fundamental groupoids]] further to [[fundamental infinity-groupoids|fundamental $\infty$-groupoids]] and the [[category of vector spaces]] to the [[(infinity,1)-category of chain complexes]]).
In order to formalize this idea, we need to work inside a [[2-category]] which suitably subsumes all of the following
[[very large category|very large]] [[(2,1)-categories]]:
* $Cat$ of [[categories]]
* $MonCat$ of [[monoidal categories]];
* $CoCartCart$ of [[cocartesian monoidal categories]];
* $DistMonCat$ of [[distributive monoidal categories]] (with [[underlying]] [[cocartesian monoidal categories]]).
Here we understand that
* the [[1-morphisms]] in these [[2-categories]] are [[functors]] which respect all given [[tensor products]] up to [[isomorphism]], i.e. [[strong monoidal functors]]
* the [[2-morphisms]] are [[natural isomorphisms]] between these functors.
The system of [[forgetful functor|forgetful]] [[2-functors]] between these [[(2,1)-categories]] forms a square [[diagram]] which we may regard as the [[image]] of a corresponding [[contravariant functor|contravariant]] [[pseudofunctor]] from the [[commuting square]]-[[diagram category]] to [[2Cat]]:
$$
\mathbf{C}
\;\;
\colon
\;\;
\array{
(1,0) &\longrightarrow& (1,1)
\\
\big\uparrow && \big\uparrow
\\
(0,0) &\longrightarrow& (1,0)
}
\;\;\;\;
\mapsto
\;\;\;\;
\array{
MonCat &\longleftarrow& DistMonCat
\\
\big\downarrow && \big\downarrow
\\
Cat &\longleftarrow& CoCartCat
}
$$
The 2-categorical context which we are after is then the [[(infinity,1)-Grothendieck construction|(2,1)-]][[Grothendieck construction]] $\int \mathbf{C}$ on this [[pseudofunctor]]: Here [[1-morphisms]] are functors which strictly preserve the [[tensor products]] present on their [[domain]] category, but whose [[codomain]] may be equipped with further [[tensor products]] (though not with fewer tensor products).
**Claim.** In $\int \mathbf{C}$ the following [[diagram]] is a ([[homotopy pushout|homotopy]]) [[pushout]]
\begin{tikzcd}
\big(
\mathrm{Vect}
,
\otimes
\big)
\ar[rr]
&&
\big(
\mathrm{Vect}_{\mathrm{Set}}
,
\sqcup
,
\boxtimes
\big)
\\
\mathrm{Vect}
\ar[u]
\ar[
rr
]
&&
\big(
\mathrm{Vect}_{\mathrm{Set}}
,
\sqcup
\big)
\ar[u]
\end{tikzcd}
\begin{proof}
We check the [[universal property]]. To this end, observe that any [[cocone]] under the diagram must have as tip a [[distributive monoidal category]], since only these can receive morphisms in $\int \mathbf{C}$ both from a monoidal and from a cocartesian category.
This means that a general [[cocone]] looks like the following solid diagram:
\begin{tikzcd}
&&
&[-10pt]
\big(
\mathcal{C}
,
\sqcup
,
\otimes
\big)
\\[-10pt]
\big(
\mathrm{Vect}
,
\otimes
\big)
\ar[rr]
\ar[urrr]
&&
\big(
\mathrm{Vect}_{\mathrm{Set}}
,
\sqcup
,
\boxtimes
\big)
\ar[
ur,
dashed
]
\\
\mathrm{Vect}
\ar[u]
\ar[
rr
]
&&
\big(
\mathrm{Vect}_{\mathrm{Set}}
,
\sqcup
\big)
\ar[u]
\ar[uur]
\end{tikzcd}
We need to see that this uniquely admits the dashed arrow. Unwinding the definitions, the existence of the dashed arrow means that any functor $F \,\colon\, Vect_{Set} \to \mathcal{C}$ which preserves both the coproduct of vector bundles as well as the tensor product on fibers already respects the [[external tensor product]]. But this follows by the [[distributive monoidal category|distributivity]] and using that any [[set]] is the [[coproduct]] of its elements:
$$
\begin{array}{l}
F\big( \mathcal{E} \boxtimes \mathcal{E}' \big)
\\
\;\simeq\;
F
\Bigg(
\underset{s \in S}{\coprod}
\left(
\array{
\{s\} &\longrightarrow& Vect
\\
s &\mapsto& \mathcal{E}_s
}
\right)
\;\;
\boxtimes
\;\;
\underset{s' \in S'}{\coprod}
\left(
\array{
\{s'\} &\longrightarrow& Vect
\\
s' &\mapsto& \mathcal{E}'_{s'}
}
\right)
\Bigg)
\\
\;\simeq\;
F
\Bigg(
\underset{(s, s') \in S \times S'}{\coprod}
\Big(
\big(
(s \colon \{s\}) \mapsto \mathcal{E}_s
\big)
\boxtimes
\big(
(s' \colon \{s'\}) \mapsto \mathcal{E}_{s'}
\big)
\Big)
\Bigg)
\\
\;\simeq\;
\underset{(s, s') \in S \times S'}{\coprod}
\bigg(
\Big(
F
\big(
(s \colon \{s\}) \mapsto \mathcal{E}_s
\big)
\Big)
\otimes
\Big(
F
\big(
(s' \colon \{s'\}) \mapsto \mathcal{E}_{s'}
\big)
\Big)
\bigg)
\\
\;\simeq\;
\Big(
F
\big(
\underset{s \in S}{\coprod}
(s \colon \{s\}) \mapsto \mathcal{E}_s
\big)
\Big)
\otimes
\Big(
F
\big(
\underset{s' \in S'}{\coprod}
(s' \colon \{s'\}) \mapsto \mathcal{E}_{s'}
\big)
\Big)
\\
\;\simeq\;
F(\mathcal{E}) \otimes F(\mathcal{E}')
\,.
\end{array}
$$
\end{proof}
## Related concepts
* [[Mod]]
* [[VectBund(B)]]
* [[tangent (infinity,1)-category|tangent $\infty$-category]]
* [[parameterized spectra]]
* [[doubly closed monoidal category]]
[[!redirects VectorBundles]]
|
vector | https://ncatlab.org/nlab/source/vector |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A **vector** is an element in a [[vector space]].
The archetypical examples are [[tangent vectors]].
## Related concepts
* [[wave vector]]
* [[direction of a vector]]
[[!redirects vectors]] |
vector boson | https://ncatlab.org/nlab/source/vector+boson |
#Contents#
* table of contents
{:toc}
## Idea
A [[boson]] with [[spin]] 1.
## Examples
In the [[standard model of particle physics]]: the [[photon]], the [[W-boson]], the [[Z-boson]] and the [[gluon]].
## References
* Wikipedia, _[Vecor boson](http://en.wikipedia.org/wiki/Vector_boson)_
[[!redirects vector bosons]] |
vector bundle | https://ncatlab.org/nlab/source/vector+bundle |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
# Vector bundles
* table of contents
{: toc}
## Idea
Given some context of [[geometry]], then a _vector bundle_ is a collection of [[vector spaces]] that varies in a geometric way over a given base [[space]] $X$: over each [[generalized element|element]] $x \in X$ there is a [[vector space]] $V_x$, called the _[[fiber]]_ over $x$, and as $x$ varies in $X$, the fibers vary along in a geometric way. One also says that vector bundles are _[[fiber bundles]]_ whose fiber carries [[vector space]]-structure. Hence the theory of vector bundle is _parameterized_ [[linear algebra]]. The vector bundles over a fixed base $X$ form a category [[Vect(X)|$VectBund_X$]], and as the base space is allowed to vary these fit into a global category [[VectBund]].
For example
* in [[topology]] a _[[topological vector bundle]]_ is a collection of [[vector spaces]] which "vary continuously" over a [[topological space]] $X$,
* in [[differential geometry]] a _[[differentiable vector bundle]]_ is a collection of vector space which "varies differentiably" over a [[differentiable manifold]],
* in [[algebraic geometry]] an _[[algebraic vector bundle]]_ is a collection of vector spaces which vary algebraically over a [[scheme]].
and so on.
One requires that "locally", on small enough patches of the base space $X$, the variation of the fibers is constant up to isomorphism (one says the vector bundle is "locally trivial"), but the key point of vector bundles is that there may be non-trivial structure in how the collection of vector spaces "globally glues together".
For example if $X = S^1$ is the [[circle]] regarded as a [[topological space]] in the standard way, and if we consider [[real vector spaces]], then there are up to [[isomorphism]] two different $\mathbb{R}$-vector bundles over $S^1$ whose [[fibers]] look like the 1-dimensional real vector space $\mathbb{R}$ itself, namely
1. the [[cylinder]]
<img src="https://ncatlab.org/nlab/files/cylinder.jpg" width="190">
1. the [[Möbius strip]]:
<img src="https://ncatlab.org/nlab/files/moebiusstrip.jpg" width="200">
(In these pictures each vertical interval is to be thought of as a stand-in for a copy of the [[real line]] $\mathbb{R}$.)
Clearly for the cylinder nothing special happens to the fibers as one moves around the circle (one says this is a _trivial vector bundle_) while the Möbius strip is "locally trivial" but globally has a twist: as one moves once around the circle the original fiber comes back identified with its reflection at the origin.
<div style="float:right;margin:0 10px 10px 0;">
<img src="https://ncatlab.org/nlab/files/TangentSpaceToSphere.png" width="250">
<blockquote>
graphics grabbed from <a href="Hatcher">Hatcher</a>
</blockquote>
</div>
An important class of examples of vector bundles are [[tangent bundles]] of [[differentiable manifolds]] $X$. Here the [[vector space]] at each point of $X$ is the [[tangent space]] of that point, the space of all [[tangent vectors]] based at that point. The graphics on the right shows one of the tangent space of the [[2-sphere]].
Dually, given an [[embedding of differentiable manifolds]] into a [[Euclidean space]], then the [[normal vectors]] to the tangent bundle span a vector bundle called the _[[normal bundle]]_ of the embedding.
All the usual operations on [[finite dimensional vector spaces]] in [[linear algebra]] generalize to vector bundles by applying them [[fiber]]-wise. For instance there is [[direct sum of vector bundles]] and the [[tensor product of vector bundles]] over the same base space.
To the extent that the base [[space]] $X$ is encoded in its [[algebra of functions]] (tautologically in [[algebraic geometry]] or via [[Gelfand duality]] in [[topology]]), the [[Serre-Swan theorem]] asserts that vector bundles over $X$ are equivalently encoded in the [[projective modules]] over these algebras constituted by their [[sections]].
Vector bundles have various applications and uses:
1. their [[Grothendieck group]] under [[direct sum of vector bundles]] yields [[topological K-theory]], an interesting [[generalized (Eilenberg-Steenrod) cohomology theory]];
1. a [[reduction of the structure group]] of vector bundles encodes actual [[geometry]] on the base space; when applied to [[tangent bundles]] such _[[G-structures]]_ on vector bundles encode for instance [[orthogonal structure]], [[Riemannian geometry]], [[complex geometry]], [[symplectic geometry]], [[conformal geometry]] etc. (in general: [[Cartan geometry]]); when applied to [[normal bundles]] these [[G-structures]] give rise, via [[Thom's theorem]], to [[Thom spectra]] and [[cobordism theory]];
1. equipping differentiable vector bundles with [[connection on a vector bundle]] is the basis for [[Chern-Weil theory]] and for the application of vector bundles in [[physics]], where they model [[gauge fields]] and [[instanton sectors]]; see also at _[[fiber bundles in physics]]_.
## Definition
### Standard
See at _[[topological vector bundle]]_
### Sheaf-theoretic version
Vector bundles can also be defined via [[sheaf and topos theory|sheaf theory]], which permits easy transport to general [[Grothendieck toposes]]. Let $Sh(X)$ be the [[category]] of ([[set]]-valued) [[sheaf|sheaves]] on $X$. The sheaf of continuous local sections of the product projection
$$X \times \mathbb{R} \to X$$
forms a [[local ring]] object $R$; when interpreted in the [[internal logic]] of $Sh(X)$, it is the Dedekind [[real numbers object]]. Then, according to a [[Serre-Swan theorem|theorem of Richard Swan]], in its sheaf-theoretic incarnation a vector bundle is the same thing as a [[projective module|projective R-module]].
* A theorem of Kaplansky states "every [[projective module]] over a [[local ring]] is [[free module|free]]". When interpreted in [[sheaf semantics]] ([[Kripke-Joyal semantics]]), the [[existential quantifier]] implicit in "free" is interpreted _locally_, so we can consider a vector bundle as a locally free module over the Dedekind reals.
### Virtual vector bundles
In one class of models for [[K-theory]] -- [[generalized (Eilenberg-Steenrod) cohomology]] theory -- cocycles are represented by $\mathbb{Z}_2$-graded vector bundles (pairs of vector bundles, essentially) modulo a certain equivalence relation. In that context it is sometimes useful to consider a certain variant of infinite-dimensional $\mathbb{Z}_2$-graded vector bundles called [[vectorial bundle | vectorial bundles]].
Much else to be discussed...
## Examples
* [[canonical bundle]]
* [[valence bundle]]
* ...
## Related concepts
* [[principal bundle]], [[associated bundle]]
* **vector bundle**:
* [[VectBund(X)]], [[VectBund]]
* [[real vector bundle]]
* [[complex vector bundle]]
* [[holomorphic vector bundle]], [[pseudoholomorphic vector bundle]]
* [[universal vector bundle]]
* [[rank]] of a vector bundle
* [[dual vector bundle]]
* [[module bundle]]
* [[direct sum of vector bundles]]
* [[short exact sequence of vector bundles]]
* [[connection on a vector bundle]]
* [[flat vector bundle]]
* [[real vector bundle]], [[complex vector bundle]]
* [[super vector bundle]]
* [[measurable field of Hilbert spaces]]
* [[2-vector bundle]]
* [[(∞,1)-vector bundle]] / [[(∞,n)-vector bundle]]
## Literature
{#Literature}
* [[Glenys Luke]], [[Alexandr S. Mishchenko]], *Vector bundles and their applications*, Math. and its Appl. **447** Kluwer (1998) [[doi:10.1007/978-1-4757-6923-4](https://doi.org/10.1007/978-1-4757-6923-4), [MR99m:55019](http://www.ams.org/mathscinet-getitem?mr=99m:55019)]
* А. С. Мищенко, _Векторные расслоения и их применения_ (Russian; A. S. Mishchenko, Vector bundles and their applications) Nauka, Moscow, 1984. 208 pp.
* Howard Osborn, _Vector bundles. Vol. 1. Foundations and Stiefel-Whitney classes_, Pure and Appl. Math. __101__, Academic Press 1982. xii+371 pp. [MR85e:55001](http://www.ams.org/mathscinet-getitem?mr=85e:55001)
* [[Dale Husemöller]], _Fibre bundles_, McGraw-Hill 1966 (300 p.); Springer GTM 1975 (327 p.), 1994 (353 p.).
* [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], _[[Basic Bundle Theory and K-Cohomology Invariants]]_,
Lecture Notes in Physics, Springer 2008 ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf))
An exposition with an eye towards [[gauge theory]] is in section 16.1 of
* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_
* [[Raoul Bott]], [[Loring Tu]], _Differential forms in algebraic topology_, Graduate Texts in Mathematics __82__, Springer 1982. xiv+331 pp.
Discussion with an eye towards [[K-theory]] is in
* [[Max Karoubi]], _K-theory. An introduction_, Grundlehren der Mathematischen Wissenschaften __226__, Springer 1978. xviii+308 pp.
* {#Hatcher} [[Allen Hatcher]], _Vector bundles and K-Theory_, (partly finished book) [web](https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html)
[[!redirects vector bundles]]
|
vector field | https://ncatlab.org/nlab/source/vector+field |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A __vector field__ is a [[section]] of a [[vector bundle]].
More specifically, a vector field is a _[[tangent vector field]]_ which is a [[section]] of a [[tangent bundle]]. Hence this is a function which picks a [[tangent vector]] at each point of a manifold, such that this assignment is suitably [[differentiable function|differentiable]].
Equivalently this is a rank $(1,0)$-[[tensor field]] on $X$
## Properties
### As derivations of smooth functions
Vector fields may be identified with [[derivations]]
on the [[algebra of functions|algebra of]] [[smooth functions]].
See the article *[[derivations of smooth functions are vector fields]]*.
## Related concepts
* [[flow of a vector field]]
* [[differential form]]
* [[derivations of smooth functions are vector fields]]
* [[frame field]]
* [[Hamiltonian vector field]]
* [[Reeb vector field]]
* [[evolutionary vector field]]
* [[field (physics)]]
## References
For instance section 1.3 and 1.4 of
* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_
[[!redirects vector fields]]
|
vector G-space | https://ncatlab.org/nlab/source/vector+G-space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Let $G$ be a [[finite group]]. We write
\[
\label{OrbitCategory}
G Orbits \;\subset\; Categories
\]
for the [[orbit category]] of $G$.
+-- {: .num_defn #FiniteDimensionalRationalVectorGSpaces}
###### Definition
**([[rational vector space|rational]] [[vector G-spaces]])**
We say that the _category of finite-dimensional vector G-spaces_ is
the [[functor category|category of functors]] from the [[opposite category|opposite]] of the [[orbit category]] to the [[category]] of [[finite-dimensional vector space|finite-dimensional]] [[rational vector spaces]]:
\[
\begin{aligned}
Vector G Spaces^{fin}_{\mathbb{Q}}
&
\coloneqq
\;
PSh
\Big(
G Orbits
\,,\,
VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}}
\Big)
\\
&
\coloneqq
\;
Functors
\Big(
G Orbits^{op}
\,,\,
VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}}
\Big)
\end{aligned}
\]
Its [[opposite category]] we call the
_category of finite-dimensional dual vector G-spaces_:
\[
\label{FunctorCategoryFromOrbitCategoryToOppositeRationalVectorSpaces}
\begin{aligned}
\big(
Vector G Spaces^{fin}_{\mathbb{Q}}
\big)^{op}
&
\simeq
\;
PSh
\Big(
G Orbits^{op}
\,,\,
VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}}
\Big)
\\
&
=
\;
Functors
\Big(
G Orbits
\,,\,
VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}}
\Big)
\end{aligned}
\]
(using in the first line that forming [[dual linear maps]] is an [[equivalence of categories]] from [[finite dimensional vector spaces]] to their [[opposite category]].)
In generalization of (eq:FunctorCategoryFromOrbitCategoryToOppositeRationalVectorSpaces), dropping the finiteness condition, we write
\[
\label{DualVectorGSpaces}
DualVector G Spaces
\;\coloneqq\;
Functors
\Big(
G Orbits
\,,\,
VectorSpaces_{\mathbb{Q}}
\Big)
\]
=--
The category (eq:DualVectorGSpaces) is denoted $Vec_G^\ast$ in ([Triantafillou 82](#Triantafillou82)).
## Properties
### Projective objects
{#ProjectiveObjects}
> Beware that this section uses different notational conventions than the rest of the entry. None of the rest of the entry is necessary for reading this section here.
Notation:
* $G$ a [[finite group]],
* $N(H)/H$ the "[Weyl group](Weyl+group#InEquivariantHomotopyTheory)" of a [[subgroup]] $H \subset G$,
* $Orb_G$ the [[orbit category]] of $G$, equivalently thought of as the [[full subcategory]] of [[G-sets]] $G/H$ of [[cosets]] for [[subgroups]] $H \subset G$,
* $\mathbb{Q}$ the [[ground field]] of [[rational numbers]],
* $Mod_{\mathbb{Q}}^{Orb_G^{op}}$ the [[presheaf category]] on $Orb_G$ with [[coefficients]] in [[rational vector spaces]],
* $\mathbb{Q}[K]$ the [[group algebra]] over the [[rational numbers]], of a [[finite group]] $K$.
\linebreak
\begin{definition}
\label{TheProjectiveGenerators}
([T82, Def. 3.1](#Triantafillou82))
\linebreak
For
* $H \subset G$,
* $V_H \,\in\, Mod_{ \mathbb{Q}[N(H)/H] }$
define
$$
\begin{array}{l}
\underline{V}_H
\;\in\;
Mod_{\mathbb{Q}}^{Orb_G^{op}}
\\
\underline{V}_H
\;\equiv\;
\mathbb{Q}\big[
Orb_G(-,G/H)
\big]
\underset{
\mathbb{Q}[N(H)/H]
}{\otimes}
V_H
\mathrlap{\,.}
\end{array}
$$
\end{definition}
\begin{remark}\label{UnderlineVHatStageH}
Since the [[Weyl group]] of $H$ is (see [there](Weyl+group#eq:AsAutomorphismGroupInOrbitCategory)) the [[endomorphism monoid]] of $G/H$ in the $G$-[[orbit category]]
$$
N(H)/H
\;\simeq\;
{Orb_G}(G/H,G/H)
\;\;
$$
we have with Def. \ref{TheProjectiveGenerators} canonical isomorphism:
\[
\begin{array}{ccl}
\underline{V}_H(G/H)
&\equiv&
\mathbb{Q}\big[
Orb_G(G/H,G/H)
\big]
\underset{
\mathbb{Q}[N(H)/H]
}{\otimes}
V_H
\\
&\simeq&
V_H
\mathrlap{\,.}
\end{array}
\]
\end{remark}
\begin{proposition}
\label{ProjectiveGeneratorsAreIndeedProjective}
([T82, Prop. 3.2](#Triantafillou82))
\linebreak
The objects $\underline{V}_H$ from Def. \ref{TheProjectiveGenerators} are [[projective object|projective]] in $Mod_{\mathbb{Q}}^{Orb_G^{op}}$.
\end{proposition}
\begin{proof}
We need to show that dashed [[lifts]] in the following [[diagrams]] exist, where $p$ is an [[epimorphism]]:
\begin{tikzcd}[sep=20pt]
&& M
\ar[
dd,
->>,
"{ p }"
]
\\
\\
\underline{V}_H
\ar[
rr,
"{ f }"
]
\ar[
uurr,
dashed,
"{ \widehat{f} }"
]
&&
N
\end{tikzcd}
Since plain [[rational vector spaces]] are [[free modules]] (by the [[basis theorem]]), hence [[projective modules]], hence [[projective objects]] in [[Vect]], each $Orb_G$-component of this diagram separately has such a lift, in particular we may choose a lift $\widehat{f_{G/H}}$ at stage $G/H$:
\begin{tikzcd}[sep=20pt]
&&
M(G/H)
\ar[
dd,
->>,
"{ p_{G/H} }"
]
\\
\\
V_H
\ar[
rr,
"{ f_{G/H} }"{swap}
]
\ar[
uurr,
dashed,
"{ \widehat{f_{G/H}} }"
]
&&
N(G/H)
\mathrlap{\,,}
\end{tikzcd}
where we have identified $V_H$ in the bottom left via Rem. \ref{UnderlineVHatStageH}.
With this local lift in hand, we obtain a global lift by setting:
\begin{tikzcd}[row sep=0pt]
&
\underline{V}_H
\ar[rr, "{ \widehat{f} }"]
&&
M
\\
G/K
&
\mathbb{Q}\big[
\mathrm{Orb}_G(G/K,G/H)
\big]
\underset{
\mathclap{
\raisebox{-2pt}{
\scalebox{.6}{$
\mathbb{Q}[N(H)/H]
$}
}
}
}{\otimes}
V_H
\ar[
rr,
"{ \widehat{f}_{G/K} }"
]
&&
M(G/H)
\\
&
\big(
G/K \xrightarrow{\phi} G/H
\big)
\otimes
v_H
\ar[
rr,
phantom,
"{ \longmapsto }"
]
&&
M(\phi)\big(
\widehat{f_{G/H}}(v_H)
\big)
\mathrlap{\,.}
\end{tikzcd}
This is clearly a [[natural transformation]] (by the [[contravariant functor|contravariant functoriality]]) of $M$), and it is a lift by [[natural transformation|naturality]] of $p$ and $f$:
$$
\begin{array}{l}
p_{G/K}
\circ
\widehat{f}_{G/K}
\big(
(G/K \xrightarrow{\phi} G/H)
\otimes
v_H
\big)
\\
\;\equiv\;
p_{G/K}
\circ
M(\phi)
\circ
\widehat{f_{G/H}}
\big(
(G/H \xrightarrow{id} G/H)
\otimes
v_H
\big)
\\
\;=\;
N(\phi)
\circ
p_{G/H}
\circ
\widehat{f_{G/H}}
\big(
(G/H \xrightarrow{id} G/H)
\otimes
v_H
\big)
\\
\;=\;
N(\phi)
\circ
f_{G/H}
\big(
(G/H \xrightarrow{id} G/H)
\otimes
v_H
\big)
\\
\;=\;
f_{G/K}
\circ
\underline{V}_H(\phi)
\big(
(G/H \xrightarrow{id} G/H)
\otimes
v_H
\big)
\\
\;=\;
f_{G/K}
\big(
(G/K \xrightarrow{\phi} G/H)
\otimes
v_H
\big)
\mathrlap{\,.}
\end{array}
$$
\end{proof}
\begin{proposition}\label{CharacterizationOfProjectives}
([T82, Prop. 3.4](#Triantafillou82))
\linebreak
Every [[projective object]] $P \,\in\, Mod_{\mathbb{Q}}^{Orb_G^{op}}$ is a [[direct sum]] of projective generators as in Prop. \ref{TheProjectiveGenerators}.
\end{proposition}
\begin{proof}
We make a bunch of choices:
First, in each [[conjugacy class]] $[H]$ of [[subgroups]] $G$ choose one representative $H \subset G$.
For that $H \hookrightarrow G$, consider the joint [[span]] of the [[images]] of
$$
P(G/H \twoheadrightarrow G/H')
\;\colon\;
P(G/H') \to P(G/H)
$$
for all intermediate [[subgroup]]-inclusions $H \hookrightarrow H' \hookrightarrow G$:
$$
0
\to
\underset{ H' \supset H }{
\textstyle{\sum}
}
im\big(
P(G/H \twoheadrightarrow G/H')
\big)
\hookrightarrow
P(G/H)
\twoheadrightarrow
P(G/H)
\big/
\underset{ H' \supset H }{
\textstyle{\sum}
}
im\big(
P(G/H \twoheadrightarrow G/H')
\big)
\to
0
\,.
$$
On the right we have exhibited the [[quotient vector space]] of the inclusion of the joint images on the left (hence the joint [[cokernel]]) making a [[short exact sequence]] of [[rational vector spaces]].
Since [every short exact sequence of vector spaces splits](split+exact+sequence#OfVectorSpaces), we may next *choose* a [[split exact sequence|splitting]]:
\[
\label{TheSplitting}
\sigma_H
\;\;\colon\;\;
P(G/H)
\big/
\underset{ H' \supset H }{
\textstyle{\sum}
}
im\big(
P(G/H \twoheadrightarrow G/H')
\big)
\xhookrightarrow{\phantom{---}}
P(G/H)
\,,
\]
whose [[image]] we denote by
\[
\label{VHInProjectiveCover}
V_H
\;\;\;\coloneqq\;\;\;
\sigma
\Big(
P(G/H)
\big/
\underset{ H' \supset H }{
\textstyle{\sum}
}
im\big(
P(G/H \twoheadrightarrow G/H')
\big)
\Big)
\xhookrightarrow{\phantom{---}}
P(G/H)
\,.
\]
(In fact, we need this splitting $N(H)/H$-equivariantly: Since we are in [[characteristic zero]] this follows by the fact that every $N(H)/H$-[[linear representation|representation]] splits as a [[direct sum]] of [[irreducible representations]], and by the first part of [[Schur's lemma]], which says that there are no non-zero maps between distinct such direct summands.)
Via these (images of) chosen splittings (eq:VHInProjectiveCover), we may define a morphism in $Mod_{\mathbb{Q}}^{Orb_G^{op}}$ as follows, out of the direct sum of their underlined versions from \ref{TheProjectiveGenerators}:
\[
\label{ComparisonMorphismForProjectiveDecomposition}
\array{
p
&\colon&
\underset{
[H]
}{\oplus}
\underline{V}_H
&\longrightarrow&
P
\\
p_{G/K}
&\colon&
\underset{
[H]
}{\oplus}
\mathbb{Q}
\big[
Orb_G(G/K, G/H)
\underset{
\mathbb{Q}[N(H)/H]
}{\otimes}
V_H
\big]
&\longrightarrow&
P(G/K)
\\
&&
\big(
G/K \xrightarrow{ f } G/H
\big)
\,\otimes\,
v_H
&\mapsto&
P(f)(v_H)
\,,
}
\]
which is manifestly [[natural transformation|natural]] in $G/K$ (via [[contravariant functor|contravariant functoriality]] of $P$) and hence well-defined:
\begin{tikzcd}[
row sep=0pt,
column sep=10pt
]
&
\bigoplus_{[H]}
\underline{V}_H
\ar[
rrrr,
"{ p }"
]
&[-30pt]
&&
&[-20pt]
P
\\
G/K
&
\underset{[H]}{\bigoplus}
\,
\mathbb{Q}
\big[
\mathrm{Orb}_G\big(
G/K
,\,
G/H
\big)
\underset{
\mathclap{
\raisebox{-2pt}{
\scalebox{.6}{$
\mathbb{Q}[N(H)/H]
$}
}
}
}{\otimes}
V_H
\big]
\ar[
rrrr,
"{ p_{G/K} }"
]
\ar[
dddd,
"{
\underset{[H]}{\bigoplus}
\,
\mathbb{Q}
[
\mathrm{Orb}_G(
\phi
,\,
G/H
)
\underset{
\mathclap{
\raisebox{-2pt}{
\scalebox{.6}{$
\mathbb{Q}[N(H)/H]
$}
}
}
}{\otimes}
V_H
]
}"{description}
]
&&&&
P(G/K)
\ar[
dddd,
"{ P(\phi) }"
]
\\[-10pt]
&
&
\big(
G/K \xrightarrow{f} G/H
\big)
\otimes
v_H
\ar[
rr,
phantom,
"{ \longmapsto }"
]
\ar[
dd,
phantom,
"{ \longmapsto }"{rotate=-90}
]
&&
P(f)(v_H)
\ar[
dd,
phantom,
"{ \longmapsto }"{rotate=-90}
]
\\
\phantom{A}
\\
&
&
\big(
G/K' \xrightarrow{ \phi }
G/K \xrightarrow{f} G/H
\big)
\otimes
v_H
\ar[
rr,
phantom,
"{ \longmapsto }"
]
&&
P(\phi)\big(P(f)(v_H)\big)
\\
G/K'
\ar[
uuuu,
"{ \phi }"
]
&
\underset{[H]}{\bigoplus}
\,
\mathbb{Q}
\big[
\mathrm{Orb}_G\big(
G/K'
,\,
G/H
\big)
\underset{
\mathclap{
\raisebox{-2pt}{
\scalebox{.6}{$
\mathbb{Q}[N(H)/H]
$}
}
}
}{\otimes}
V_H
\big]
\ar[
rrrr,
"{ p_{G/K'} }"{swap}
]
&&&&
P(G/K')
\end{tikzcd}
Since all the direct summands on the left are projective by Prop. \ref{ProjectiveGeneratorsAreIndeedProjective}, it is now sufficient to prove that (eq:ComparisonMorphismForProjectiveDecomposition) is an [[isomorphism]]. Since isomorphisms in [[functor categories]] are detected objectwise and since [[rational vector spaces]] form a [[balanced category]] (see [there](balanced+category#AbelianCategoriesAreBalanced)) for this it is sufficient to show that for all $K \subset G$ the morphism $p_{G/K}$ (eq:ComparisonMorphismForProjectiveDecomposition) is both an [[epimorphism]] and a [[monomorphism]].
First to see that that $p_{G/K}$ is an epimorphism: To start with, it is clearly surjective onto the summand $V_K$. Hence it is next sufficient to show that given $v_K \in P(G/K)$ which is in the image under $P(G/K \twoheadrightarrow G/H)$ of some $\widehat{v}_H \in P(G/H)$ then it is also in the image of $p_{G/K}$. As before, this is clear for those $\widehat{v}_H \in V_H$. Hence next, as before, it is sufficient to show this for those $\widehat{v}_H$ which are in the image under some $P(G/H \twoheadrightarrow G/H')$ of some $v_{H'} \in P(G/H')$... And so on. Since $G$ is a [[finite group]], this recursive argument eventually terminates with $V_G = P(G/G)$.
Finally, to see that $p_{G/K}$ (eq:ComparisonMorphismForProjectiveDecomposition) is a monomorphism. It is here (only) that we use the assumption that $P$ is [[projective object|projective]]. With the previous point, this implies a [[lift]] $p'$ in the following diagram in $Mod_{\mathbb{Q}}^{Orb_G^{op}}$:
\begin{tikzcd}[sep=25pt]
&
\underset{[H]}{\oplus} \underline{V}_H
\ar[
d,
->>,
"{ p }"
]
\\
P
\ar[r, equals]
\ar[ur, dashed, "{ p' }"]
&
P
\end{tikzcd}
Hence if $v,w \,\in\, P(G/K)$ such that $p'_{G/K}(v) = p'_{G/K}(w)$ then $p_{G/K} \circ p'_{G/K}(v) = p_{G/k} \circ p'_{G/K}(w)$ hence $v = w$, whence each $p'_{G/K}$ is injective.
\end{proof}
\begin{corollary}([T82, Prop. 3.6](#Triantafillou82))
Every object $N \,\in\, Mod_{\mathbb{Q}}^{Orb_G}$ admits a [[projective cover]] in the sense of a [[projective object]] $\underset{[H]}{\oplus} \underline{V}_H$ and an [[epimorphism]] $p \,\colon\,\underset{[H]}{\oplus} \underline{V}_H \twoheadrightarrow N$.
\end{corollary}
\begin{proof}
The construction and verification is verbatim as in the proof of Prop. \ref{CharacterizationOfProjectives}, omitting only the proof of injectivity in the last step.
\end{proof}
\linebreak
### Injective objects (1)
{#InectiveObjects}
We spell out aspects of the discussion of injective objects in the copresheaf category dual to the [above](#ProjectiveObjects) discussion of projective objects in the presheaf category but left implicit in [Triantafillou 1982, p. 517](#Triantafillou82).
Notation:
* $G$ a [[finite group]],
* $N(H)/H$ the "[Weyl group](Weyl+group#InEquivariantHomotopyTheory)" of a [[subgroup]] $H \subset G$,
* $Orb_G$ the [[orbit category]] of $G$, equivalently thought of as the [[full subcategory]] of [[G-sets]] $G/H$ of [[cosets]] for [[subgroups]] $H \subset G$,
* $\mathbb{Q}$ the [[ground field]] of [[rational numbers]],
* $Mod_{\mathbb{Q}}^{Orb_G}$ the [[copresheaf]] [[functor category]] on $Orb_G$ with [[coefficients]] in [[rational vector spaces]],
* $\mathbb{Q}[K]$ the [[group algebra]] over the [[rational numbers]], of a [[finite group]] $K$.
\linebreak
\begin{definition}
\label{TheInjectiveGenerators}
(dual to [T82, Def. 3.1](#Triantafillou82))
\linebreak
For
* $H \subset G$,
* $V^H \,\in\, Mod_{ \mathbb{Q}[N(H)/H] }$
define
$$
\begin{array}{l}
\underline{V}^H
\;\in\;
Mod_{\mathbb{Q}}^{Orb_G}
\\
\underline{V}^H
\;\equiv\;
Mod_{\mathbb{Q}[N(H)/H]}
\Big(
\mathbb{Q}\big[
Orb_G(-,G/H)
\big]
,\,
V^H
\Big)
\mathrlap{\,.}
\end{array}
$$
\end{definition}
\begin{proposition}\label{CharacterizationOfInjectives}
(dual to [T82, Prop. 3.4](#Triantafillou82))
\linebreak
Every [[injective object]] $I \,\in\, Mod_{\mathbb{Q}}^{Orb_G}$ is a [[direct sum]] of injective generators as in Prop. \ref{TheInjectiveGenerators}.
\end{proposition}
\begin{proof}
We make a bunch of choices:
First, in each [[conjugacy class]] $[H]$ of [[subgroups]] $G$ choose one representative $H \subset G$.
For that $H \hookrightarrow G$, consider the [[intersection]] of the [[kernels]] of
$$
I(G/H \twoheadrightarrow G/H')
\;\colon\;
I(G/H) \to I(G/H')
$$
for all intermediate [[subgroup]]-inclusions $H \hookrightarrow H' \hookrightarrow G$,
and an $N(H)/H$-equivariant splitting
\[
\label{TheSplittingForInjectiveEnvelopes}
\tau_H
\;\;\colon\;\;
I(G/H)
\twoheadrightarrow
\underset{ H' \supset H }{
\textstyle{\bigcap}
}
ker\big(
I(G/H \twoheadrightarrow G/H')
\big)
\,\equiv\,
V^H
\,.
\]
With this, we may define a morphism in $Mod_{\mathbb{Q}}^{Orb_G}$ as follows,
\[
\label{ComparisonMorphismForInjectiveDecomposition}
\array{
i
&\colon&
I
&\longrightarrow&
\underset{
[H]
}{\oplus}
\underline{V}^H
\\
i_{G/K}
&\colon&
I(G/K)
&\longrightarrow&
\underset{
[H]
}{\oplus}
Mod_{\mathbb{Q}[N(H)/H]}
\Big(
\mathbb{Q}
\big[
Orb_G(G/K, G/H)
,\,
V^H
\big]
\Big)
\\
&&
i_K
&\mapsto&
\underset{[H]}{\oplus}
\Big(
\big(
G/K \xrightarrow{f} G/H
\big)
\mapsto
\tau_{H} \circ I(f)(i_K)
\Big)
\,,
}
\]
where it is the [[functor|functoriality]] of $I$ which implies both that the maps on the right are $N(H)/H$-equivariant and that this transformation is [[natural transformation|natural]] in $G/K$:
\begin{tikzcd}[
row sep=0pt,
column sep=10pt
]
&
I
\ar[rrrr, "{ i }"]
&[-10pt]
&&
&[-50pt]
\bigoplus_{[H]}
\underline{V}^H
\\
G/K
\ar[
dddd,
"{ \phi }"
]
&
I(G/K)
\ar[
rrrr,
"{ i_{G/K} }"
]
\ar[
dddd,
"{ I(\phi) }"
]
&&&&
\underset{[H]}{\bigoplus}
\,
\mathrm{Mod}_{N(H)/H}
\Big(
\mathrm{Orb}_G
\big(G/K ,\, G/H \big)
,\,
V^H
\Big)
\ar[
dddd,
"{
\mathrm{Mod}_{N(H)/H}
\Big(
\mathrm{Orb}_G
\big(\phi ,\, G/H \big)
,\,
V^H
\Big)
}"{description}
]
\\
& &
i_K
\ar[
rr,
phantom,
"{ \longmapsto }"
]
\ar[dd, phantom, "{ \longmapsto }"{rotate=-90}]
&&
\oplus_{[H]}
\Big(
\big( G/K \xrightarrow{f} G/H \big)
\mapsto
\tau_H \circ I(f)(i_K)
\Big)
\ar[dd, phantom, "{ \longmapsto }"{rotate=-90}]
\\
\phantom{A}
\\
&&
I(\phi)(i_K)
\ar[
rr,
phantom,
"{ \longmapsto }"
]
&&
\oplus_{[H]}
\Big(
\big( G/K' \xrightarrow{f'} G/H \big)
\mapsto
\tau_H \circ I(f')\circ I(\phi) (i_K)
\Big)
\\
G/K'
&
I(G/K')
\ar[
rrrr,
"{ i_{G/K'} }"{swap}
]
&&&&
\underset{[H]}{\bigoplus}
\,
\mathrm{Mod}_{N(H)/H}
\Big(
\mathrm{Orb}_G
\big(G/K' ,\, G/H \big)
,\,
V^H
\Big)
\end{tikzcd}
Now to check that this map $i$ is in fact an isomorphism if $I$ is injective (...)
\end{proof}
(...)
### Injective objects (2)
> The following is another survey of aspects of the injective objects from [Triantafillou 1982](#Triantafillou82). This is from a different edit using different notational conventions than the previous subsection. Eventually both subsections should be harmonized and merged.
+-- {: .num_example #RestrictionOfVectorGSpacesToWeyGroupRepresentations}
###### Example
(restriction of vector $G$-spaces to Weyl group representations)
Let $H \subset G$ any [[subgroup]]. Notice that its [[Weyl group]] is the [[automorphism group]] of its [[coset space]] in the [[orbit category]]:
\[
\label{WeylGroup}
G Orbits
\big(
G/H
\,,\,
G/H
\big)
\;\;
\simeq
\;\;
Aut_{G Orbits}
\big(
G/H
\big)
\;\;
\simeq
\;\;
W_G(H)
\;\;
\coloneqq
\;\;
N_G(H)/H
\]
This gives a [[full subcategory]]-inclusion
$$
\mathbf{B}
W_G(H)
\;
\overset{\;\;i_H\;\;}{\hookrightarrow}
\;
G Orbits
$$
of the [[delooping]] category of they Weyl group into the [[orbit category]] of $G$ (eq:OrbitCategory), and hence a restriction functor
\[
\label{RestrictionFromOrbitCategoryRepresentations}
W_G(H) Representations^{fin}_{l,\mathbb{Q}}
\overset{
\;\;\;
(-)^\ast
\,\circ\,
i_H^\ast
\;\;\;
}{\longleftarrow}
\big(
Vector G Spaces_{\mathbb{Q}}^{fin}
\big)^{op}
\]
or more generally
\[
\label{NonFiniteRestrictionFromOrbitCategoryRepresentations}
W_G(H) Representations^{fin}_{r,\mathbb{Q}}
\overset{
\;\;\;
i_H^\ast
\;\;\;
}{\longleftarrow}
DualVector G Spaces
\]
=--
By the general [[end]]-formula for [[right Kan extension]] ([here](Kan+extension#eq:RightKanExtensionViaEndFormula)), this restriction functor has a [[right adjoint]], given as follows:
+-- {: .num_defn #InjectiveAtomsOfGEquivariantDualVectorSpaces}
###### Definition
**(injective atoms of dual vector $G$-spaces)**
For $H \subset G$ a [[subgroup]] and
$$
V \;\in\; W_G(H) Representations^{fin}_{l,\mathbb{Q}}
$$
a [[rational vector space|rational]] [[finite number|finite]] [[dimension|dimensional]] left [[representation]] of the [[Weyl group]] of $H$ in $G$, write
$$
\array{
G Orbits
&
\overset{
I_H(V)
}{\longrightarrow}
&
\mathbb{Q}VectorSpaces
\\
G/K
&\mapsto&
W_G(H) Representations
\Big(
\mathbb{Q}
\big[
G Orbits
(
G/K, G/H
)
\big]
\,,\,
V^\ast
\Big)
}
$$
for the [[functor]] from the $G$-[[orbit category]] to [[rational vector spaces]] which assigns to a [[coset space]] $G/K$ the vector space of [[homomorphisms]] of right [[actions]] by the [[Weyl group]] (eq:WeylGroup) from the [[hom-set]] $G Orbits\big(G/K, G/H \big)$ to the [[dual vector space]] equipped with its [[dual linear map|dual action]].
More generally, for
$$
V^\ast \;\in\; W_G(H) Representations_{r,\mathbb{Q}}
$$
set
$$
\array{
G Orbits
&
\overset{
I_H(V^\ast)
}{\longrightarrow}
&
\mathbb{Q}VectorSpaces
\\
G/K
&\mapsto&
W_G(H) Representations
\Big(
\mathbb{Q}
\big[
G Orbits
(
G/K, G/H
)
\big]
\,,\,
V^\ast
\Big)
}
$$
This construction extends to a [[functor]] [[right adjoint]] to the restriction (eq:RestrictionFromOrbitCategoryRepresentations):
$$
W_G(H) Representations_{\mathbb{Q}}
\underoverset{
\underset{
\;\;\;
I_H
\;\;\;
}{
\longrightarrow
}
}{
\overset{
\;\;\;
(-)^\ast
\,\circ\,
i_H^\ast
\;\;\;
}{\longleftarrow}
}
{\bot}
DualVector G Spaces
$$
=--
([Triantafillou 82, (4.1)](#Triantafillou82), [Golasinski 97a, Lemma 1.1](#Golasinski97a), [Scull 08, Def. 2.2, Lemma 2.3](#Scull08))
+-- {: .num_prop #InjectiveAtomsOfGEquivariantDualVectorSpacesAreIndeedInjectiveAtoms}
###### Proposition
$\,$
**(i)**
The objects of the form $I_H(V^\ast)$ (Def. \ref{InjectiveAtomsOfGEquivariantDualVectorSpaces}) are [[injective objects]] in dual [[vector G-spaces]] (Def. \ref{FiniteDimensionalRationalVectorGSpaces}).
**(ii)** Every [[injective]] dual vector $G$-space is a [[direct sum]] of objects of this form, specifically (see Def. \ref{InjectiveEnvelope} below):
$$
\underline{V}
\;\in\;
DualVector G Spaces
\;\;\;
\text{is injective}
\;\;\;\;\;\;\;
\Leftrightarrow
\;\;\;\;\;\;\;
\underline{V}
\;\simeq\;
\underset{
\big[
H \subsetneqq G
\big]_{conj}
}{\bigoplus}
\,
I_H
\left(
\underset{
K \supset H
}{\bigcap}
ker
\big(
\underline{V}(G/H)
\overset{
\underline{V}(G/H \to G/K)
}{\longrightarrow}
\underline{V}(G/K)
\big)
\right)
$$
=--
([Triantafillou 82, Section 3 and p. 10](#Triantafillou82), [Scull 08, Lemma 2.4, Prop. 2.5](#Scull08))
+-- {: .num_example #EquivariantPLDeRhamComplex}
###### Example
**(equivariant PL de Rham complex in injective dual vector $G$-space)**
Let $S \in G SimplicialSets$ a [[simplicial set]] equipped with $G$-[[action]], say that the _[[equivariant PL de Rham complex]]_ is the [[functor]] on the [[orbit category]]
$$
\array{
G Orbits
&
\overset{
\Omega^\bullet_{PLdR}
\big(
Maps(-,X)^G
\big)
}{\longrightarrow}
&
dgcAlgebras
\\
G/H
&\mapsto&
\Omega^\bullet_{PLdR}
\big(
X^H
\big)
}
$$
which to a [[coset space]] $G/H$ assigns the [[PL de Rham complex]] of the $H$-[[fixed locus]] $X^H \subset X$.
Then the underlying dual [[vector G-space]]
$$
\array{
G Orbits
&
\overset{
\Omega^\bullet_{PLdR}
\big(
Maps(-,X)^G
\big)
}{\longrightarrow}
&
dgcAlgebras
&\overset{}{\longrightarrow}&
VectorSpaces_{\mathbb{Q}}
}
$$
is an [[injective object]] (degreewise, in fact).
=--
([Triantafillou 82, Prop. 4.3](#Triantafillou82))
+-- {: .num_cor }
###### Corollary
Any [[equivariant PL de Rham complex]] (Def. \ref{EquivariantPLDeRhamComplex}) is a [[fibrant object]] in the [[model structure on equivariant connective dgc-algebras]].
=--
(also [Scull 08, Lemma 5.2](#Scull08))
+-- {: .num_defn #InjectiveEnvelope}
###### Definition
**([[injective envelope]] of [[dual vector G-spaces]])**
For $\underline{V} \in DualVector G Spaces$ (eq:DualVectorGSpaces),
its _[[injective envelope]]_ is
$$
\underset{
\big[
H \subset G
\big]_{conj}
}{\bigoplus}
\,
I_H
\left(
\underset{
K \supsetneqq H
}{\bigcap}
ker
\big(
\underline{V}(G/H)
\overset{
\underline{V}(G/H \to G/K)
}{\longrightarrow}
\underline{V}(G/K)
\big)
\right)
\,,
$$
where
1. the [[direct sum]] is over [[conjugacy classes]] of [[subgroups]], with $H \subset G$ on the right any one representative of its conjugacy class,
1. for $H = G$ the argument of $I_H$ is taken to be all of $\underline{V}(G/G)$,
1. $I_H(-)$ is the injective atom construction from Def. \ref{InjectiveAtomsOfGEquivariantDualVectorSpaces}.
=--
([Triantafillou 82, p. 10](#Triantafillou82), [Scull 01, Prop. 7.34](#Scull01), [Scull 08, Def. 2.6](#Scull08))
+-- {: .num_defn #TensorProductOfInjectiveDualVectorSpaces}
###### Lemma
**(tensor product of injective dual vector $G$-spaces)**
The object-wise [[tensor product]] of two finite-dimensional [[injective object|injective]] [[dual vector G-spaces]] (Def. \ref{FiniteDimensionalRationalVectorGSpaces}) is again injective.
=--
This is proven as [Golasinski 97b, Lemma 3.6](#Golasinski97b) (use [Golasinski 97b, Remark 1.2](#Golasinski97b) to see that the Lemma does apply to the ordinary tensor product of finite-dimensional vector spaces).
Beware that incorrect versions of this statement had been circulating; for discussion of the literature see [Golasinski 97b, p. 3](#Golasinski97b) and [Scull 01, Prop. 7.36](#Scull01)
## Examples
### Over $G = \mathbb{Z}_2$
{#ExamplesOverCyclicGroupOfOrder2}
+-- {: .num_example #OrbitCategoryOfCyclicGroupOfOrder2}
###### Example
**([[orbit category]] of [[cyclic group of order 2|Z/2Z]])**
For equivariance group the [[cyclic group of order 2]]:
$$
G \;\coloneqq\; \mathbb{Z}_2 \;\coloneqq\; \mathbb{Z}/2\mathbb{Z}
\,.
$$
the [[orbit category]] looks like this:
\[
\label{OrbitCategoryOfZMod2}
\mathbb{Z}_2
Orbits
\;=\;
\left\{
\array{
\mathbb{Z}_2/1
&
\overset{
\phantom{AAAAA}
}{
\longrightarrow
}
&
\mathbb{Z}_2/\mathbb{Z}_2
\\
Aut = \mathbb{Z}_2
&&
Aut = 1
}
\right\}
\]
i.e.:
$$
\begin{aligned}
\mathbb{Z}_2 Orbits
\big(
\mathbb{Z}_2/\mathbb{Z}_2
\,,\,
\mathbb{Z}_2/\mathbb{Z}_2
\big)
\;\simeq\;
1
\\
\mathbb{Z}_2 Orbits
\big(
\mathbb{Z}_2/1
\,,\,
\mathbb{Z}_2/\mathbb{Z}_2
\big)
\;\simeq\;
\ast
\\
\mathbb{Z}_2 Orbits
\big(
\mathbb{Z}_2/\mathbb{Z}_2
\,,\,
\mathbb{Z}_2/1
\big)
\;\simeq\;
\varnothing
\\
\mathbb{Z}_2 Orbits
\big(
\mathbb{Z}_2/1
\,,\,
\mathbb{Z}_2/1
\big)
\;\simeq\;
\mathbb{Z}_2
\end{aligned}
$$
=--
Write
$$
\mathbf{1}, \mathbf{1}_{sgn}
\;\in\;
\mathbb{Z}_2 Representations
$$
for the two [[irreducible representations]] (the [[trivial representation]] and the [[sign representation]], respectively) of the [[Weyl group]] $W_{\mathbb{Z}_2}(1) = \mathbb{Z}_2$.
Their induced injective dual vector $\mathbb{Z}_2$-spaces, according to Def. \ref{InjectiveAtomsOfGEquivariantDualVectorSpaces}, are:
$$
I_1(\mathbf{1})
\;\;
\colon
\;\;
\;\;\;\;\;
\array{
\mathbb{Z}_2/1
&\mapsto&
\mathbb{Z}_2 Reps
\Big(
\underset{
\simeq \, \mathbf{1} \oplus \mathbf{1}_{sgn}
}{
\underbrace{
\mathbb{Q}
\big[
\mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/1 )
\big]
}
}
\,,\,
\mathbf{1}
\Big)
&
\simeq
&
\mathbf{1}
\\
\big\downarrow
&&
\\
\mathbb{Z}_2/\mathbb{Z}_2
&\mapsto&
\mathbb{Z}_2 Reps
\Big(
\underset{
\simeq \, 0
}{
\underbrace{
\mathbb{Q}
\big[
\mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/1 )
\big]
}
}
\,,\,
\mathbf{1}
\Big)
& \simeq
&
0
}
$$
and
$$
I_1(\mathbf{1}_{sgn})
\;\;
\colon
\;\;
\;\;\;\;\;
\array{
\mathbb{Z}_2/1
&\mapsto&
\mathbb{Z}_2 Reps
\Big(
\underset{
\simeq \, \mathbf{1} \oplus \mathbf{1}_{sgn}
}{
\underbrace{
\mathbb{Q}
\big[
\mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/1 )
\big]
}
}
\,,\,
\mathbf{1}_{sgn}
\Big)
&
\simeq
&
\mathbf{1}_{sgn}
\\
\big\downarrow
&&
\\
\mathbb{Z}_2/\mathbb{Z}_2
&\mapsto&
\mathbb{Z}_2 Reps
\Big(
\underset{
\simeq \, 0
}{
\underbrace{
\mathbb{Q}
\big[
\mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/1 )
\big]
}
}
\,,\,
\mathbf{1}_{sgn}
\Big)
& \simeq
&
0
}
$$
Similarly, write
$$
\mathbf{1} \;\in\; 1 Representations
$$
for the unique [[irrep]] of the [[Weyl group]] $W_{\mathbb{Z}_2}(\mathbb{Z}_2) = 1$.
Its induced injective dual vector $\mathbb{Z}_2$-spaces, according to Def. \ref{InjectiveAtomsOfGEquivariantDualVectorSpaces}, is:
$$
I_{\mathbb{Z}_2}(\mathbf{1})
\;\;
\colon
\;\;
\;\;\;\;\;
\array{
\mathbb{Z}_2/1
&\mapsto&
1 Reps
\Big(
\underset{
\simeq \, \mathbf{1}
}{
\underbrace{
\mathbb{Q}
\big[
\mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/\mathbb{Z}_2 )
\big]
}
}
\,,\,
\mathbf{1}
\Big)
&
\simeq
&
\mathbf{1}
\\
\big\downarrow
&&
&&
\big\downarrow{}^{\mathrlap{\mathrm{id}}}
\\
\mathbb{Z}_2/\mathbb{Z}_2
&\mapsto&
1 Reps
\Big(
\underset{
\simeq \, \mathbf{1}
}{
\underbrace{
\mathbb{Q}
\big[
\mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/\mathbb{Z}_2 )
\big]
}
}
\,,\,
\mathbf{1}
\Big)
& \simeq
&
\mathbf{1}
}
$$
## Related concepts
* [[equivariant chain complex]]
* [[equivariant dgc-algebra]], [[model structure on equivariant dgc-algebras]]
* [[equivariant rational homotopy theory]]
## References
* {#Triantafillou82} [[Georgia Triantafillou]], _Equivariant minimal models_, Trans. Amer. Math. Soc. **274** (1982) 509-532 [[jstor:1999119](http://www.jstor.org/stable/1999119)]
* {#Golasinski97a} [[Marek Golasiński]], _Componentwise injective models of functors to DGAs_, Colloquium Mathematicum, Vol. 73, No. 1 (1997) ([dml:21048](https://eudml.org/doc/210480), [[GolasinskiInjectiveModels.pdf:file]])
* {#Golasinski97b} [[Marek Golasiński]], _Injective models of $G$-disconnected simplicial sets_, Annales de l'Institut Fourier, Volume 47 (1997) no. 5, p. 1491-1522 ([numdam:AIF_1997__47_5_1491_0](http://www.numdam.org/item/?id=AIF_1997__47_5_1491_0))
* {#Scull01} [[Laura Scull]], _Rational $S^1$-equivariant homotopy theory_, Transactions of the AMS, Volume 354, Number 1, Pages 1-45 2001 ([pdf](http://www.ams.org/journals/tran/2002-354-01/S0002-9947-01-02790-8/S0002-9947-01-02790-8.pdf), [doi:10.1090/S0002-9947-01-02790-8](https://doi.org/10.1090/S0002-9947-01-02790-8))
* {#Scull08} [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2))
[[!redirects vector G-spaces]]
[[!redirects dual vector G-space]]
[[!redirects dual vector G-spaces]]
[[!redirects equivariant vector space]]
[[!redirects equivariant vector spaces]]
|
vector meson | https://ncatlab.org/nlab/source/vector+meson |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _vector meson_ is a [[meson]] which (in [[Wigner classification]]) is in the [[vector representation]] or [[pseudo-vector representation]] of the [[Lorentz group]].
This is in contrast to [[scalar mesons]].
## Examples
Examples of [[vector mesons]]:
* [[omega meson]]
* [[rho meson]]
* [[K* meson]]
Examples of [[pseudo-vector mesons]]:
* [[a1-meson]]
* [[f1-meson]]
\begin{imagefromfile}
"file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
"width": 600,
"unit": "px",
"margin": {
"top": 0,
"right": 10,
"bottom": 0,
"left": 20
}
\end{imagefromfile}
## Properties
* [[hidden local symmetry]]
* [[vector meson dominance]]
## Related concepts
* [[scalar meson]], [[tensor meson]]
## References
(See also the references at _[[meson]]_.)
* F. Nichitiu, _An Introduction to the vector meson_, Swansea Hadron Spect. 1995:0219-240 ([spire:405666](https://inspirehep.net/literature/405666))
* Eef van Beveren, George Rupp, _Scalar and axial-vector mesons_, Eur. Phys. J. A31:468-473, 2007 ([arXiv:hep-ph/0610199](https://arxiv.org/abs/hep-ph/0610199))
See also:
* Wikipedia, _[Vector meson](https://en.wikipedia.org/wiki/Vector_meson)_
[[!redirects vector mesons]]
[[!redirects pseudovector meson]]
[[!redirects pseudovector mesons]]
[[!redirects pseudo-vector meson]]
[[!redirects pseudo-vector mesons]]
[[!redirects vector meson field]]
[[!redirects vector meson fields]]
|
vector meson dominance | https://ncatlab.org/nlab/source/vector+meson+dominance |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The phenomenon or principle called _vector meson dominance_ (VMD) is a tight relation between [[quantum hadrodynamics]] and [[quantum electrodynamics]], where the electrically neutral [[light meson|light]] [[vector meson]] [[field (physics)|fields]] $V_\mu$ (the neutral [[rho-meson]] $\rho^0_\mu$, the [[omega-meson]] $\omega_\mu$ and the [[phi-meson]]) are seen to be on par with, or even identified with, the [[electromagnetism|electromagnetic]] [[hadronic current]] $J^{hadr}$, at least to some approximation.
### Phenomenology
{#Phenomenology}
In [[particle physics]] [[phenomenology]], _vector meson dominance_ is the observation that the [[interaction]] of [[hadrons]] with [[photons]] is dominated by [[interactions]] that procceed via excange of [[vector mesons]].
Specifically, the [[decay]] of [[electron]]/[[positron]]-[[pairs]] into [[hadrons]]
$$
e^+ + e^- \to H
$$
(reverse to the [[purely leptonic decay]] $H \to e^+ e^-$ of [[hadrons]]) is dominated by the [[light meson|light]] [[vector mesons]], in that the [[scattering cross section]] $\sigma(e^+ e^- \to H)$ is peaked at those [[energies]] corresponding to the [[rest masses]] of the vector mesons:
\begin{imagefromfile}
"file_name": "VectorMesonDominanceInDileptonCrossSectionI.jpg",
"width": 420,
"caption": "[Piller-Weise 90, Fig. 1](#PillerWeise90)",
"float": "right",
"margin": {
"top": 0,
"right": 10,
"bottom": 10,
"left": 0,
"unit": "px"
}
\end{imagefromfile}
The first graphics shows the measured cross section (in [[physical unit|units]] of that of the decay $e^+ e^- \to \mu^+ \mu_-$ of [[electron]] [[dileptons]] into [[muon]] [[dileptons]]): The [[light vector mesons]] [[rho-meson|ρ]]$^0$, [[omega-meson|ω]] and [[phi-meson|ϕ]] correspond to the the spikes below 1 [[GeV]], the further spikes correspond to [[heavy vector mesons]], with [[charmonium]] $J/\psi$ around 4 [[GeV]].
\begin{imagefromfile}
"file_name": "VectorMesonDominanceInDileptonCrossSectionII.jpg",
"width": 420,
"caption": "[Piller-Weise 90, Fig. 2](#PillerWeise90)",
"float": "left",
"margin": {
"top": 0,
"right": 10,
"bottom": 10,
"left": 0,
"unit": "px"
}
\end{imagefromfile}
The second graphics shows the same data, now zoomed into the region of the [[light vector mesons]]. One sees clearly that in this region the [[graph of a function|graph]] of the [[scattering cross section]] is completely dominated first of all by the broad [[rho-meson|ρ]]$^0$ peak at a [[mass]] of $\sim 770$ [[MeV]] (see also [Murphy-Yount 71, Fig. 4](#MurphyYount71)), accompanied just by two sharp spikes, corresponding to the [[omega-meson|ω]] at $\sim 783$ [[MeV]] and the [[phi-meson|ϕ]] at $\sim 1020$ [[MeV]] (see also [Schildknecht 72, Table 1](#Schildknecht72)).
### Theory
In the [[effective field theory]] of [[quantum hadrodynamics]], _vector meson dominance_ refers to a proposal for how to formulate the [[theory (physics)|theory]] guided by the above [[phenomenology]]:
Here the dominance is encoded by the _field-current identity_ ([Gell-Mann & Zachariasen 61](#GellMannZachariasen61), [Kroll, Lee & Zumino 76, (1.3)](#KrollLeeZumino76), [Sakurai 69, p. 54 onwards](#Sakurai69), review in [Piller-Weise 90, (4)](#PillerWeise90)):
$$
J^{hadr}_\mu
\;\sim\;
V_\mu
\,,
$$
essentially identifying the electromagnetic [[hadron current]] with the joint neutral [[light meson|light]] [[vector meson]] [[field (physics)|field]].
This implies in particular that all [[coupling constants]] of [[interactions]] with an [[omega-meson]]/neutral [[rho-meson]] are proportional, by the same factor, to the corresponding electromagnetic coupling (reviewed in [Schildknecht 05, p. 3](#Schildknecht05)).
In terms of a [[Lagrangian density]], this is encoded by meson/photon _mixed terms_ of the following form ([Kroll, Lee & Zumino 76, (2.7)](#KrollLeeZumino76),[Sakurai 69, p. 67](#Sakurai69)):
\[
\label{VMD1Lagrangian}
\mathbf{L}_{VMD1}
\;\sim\;
d V \wedge \star_4 d A
\;+\;
V \wedge \star_4 J^{hadr}
\]
obtained from the Lagrangian density $\mathbf{L}_{EM} \;\sim\; d A \wedge \star_{4} d A + A \wedge \star_4 J_{hadr}$ of [[Maxwell theory]] by exchanging a [[photon]] field variable $A$ with a [[vector meson]] [[field (physics)|field]] $V$ (reviewed in [OCPTW 95, p. 10](#OCPTW95), [Schildknecht 05, p. 4](#Schildknecht05)).
There is also an alternative, supposedly equivalent, Lagrangian formulation, less elegant but now in more widespread use. To distinguish the two one speaks of "VMD1" for the formulation (eq:VMD1Lagrangian) and of "VMD2" for the alternative formulation (reviewed in [OCPTW 95. (26)](#OCPTW95), [Schildknecht 05, (8)](#Schildknecht05)).
## Properties
### Derivation from holography
From [OCPTW 95](#OCPTW95):
> No direct translation between the [[standard model of particle physics|Standard Model]] and [[vector meson dominance|VMD]] has yet been made.
\linebreak
From [Rho et al. 16](AdS-QCD#RhoEtAl16):
> One can make $[$[[chiral perturbation theory]]$]$ consistent with [[QCD]] by suitably matching the [[correlators]] of the [[effective field theory|effective theory]] to those of [[QCD]] at a [[scale]] near $\Lambda$. Clearly this procedure is not limited to only one set of [[vector mesons]]; in fact, one can readily generalize it to an infinite number of hidden [[gauge fields]] in an [[effective field theory|effective]] [[Lagrangian density|Lagrangian]]. In so doing, it turns out that a fifth dimension is "deconstructed" in a (4+1)-dimensional (or 5D) [[Yang-Mills theory|Yang–Mills]] type form. We will see in Part III that such a structure arises, [[top-down model building|top-down]], in [[string theory]].
> $[...]$
> $[$this [[holographic QCD]]$]$ model comes out to describe — unexpectedly well — low-energy properties of both [[mesons]] and [[baryons]], in particular those properties reliably described in quenched [[lattice QCD]] simulations.
> $[...]$
> One of the most noticeable results of this [[AdS/QCD|holographic model]] is the first derivation of vector dominance (VD) that holds both for [[mesons]] and for [[baryons]]. It has been
somewhat of an oddity and a puzzle that [[Jun John Sakurai|Sakurai's]] [[vector meson dominance|vector dominance]] — with the lowest [[vector mesons]] [[rho meson|ρ]] and [[omega meson|ω]] — which held very well for [[pion|pionic]] [[form factors]] at low momentum transfers famously failed for [[nucleon]] [[form factors]]. In this [[AdS/QCD|holographic model]], the [[vector meson dominance|VD]] comes out automatically for both the [[pion]] and the [[nucleon]] provided that the infinite $[$[[Kaluza-Klein mechanism|KK-]]$]$tower is included. While the [[vector meson dominance|VD]] for the [[pion]] with the infinite tower is not surprising given the successful Sakurai VD, that the [[vector meson dominance|VD]] holds also for the [[nucleons]] is highly nontrivial. $[...]$ It turns out to be a consequence of a [[AdS/QCD|holographic]] [[Cheshire cat principle|Cheshire Cat phenomenon]]
## Related concepts
* [[vector meson]]
* [[form factor]]
* [[hadron current]]
[[!include effective field theories of nuclear physics -- contents]]
## References
### General
The original articles:
* {#GellMannZachariasen61} [[Murray Gell-Mann]], Fredrik Zachariasen, _Form Factors and Vector Mesons_, Phys. Rev. 124, 953 (1961) ([doi:10.1103/PhysRev.124.953](https://doi.org/10.1103/PhysRev.124.953))
* {#KrollLeeZumino76} Norman M. Kroll, T. D. Lee, [[Bruno Zumino]], _Neutral Vector Mesons and the Hadronic Electromagnetic Current_, Phys. Rev. 157, 1376 (1967) ([arXiv:10.1103/PhysRev.157.1376](https://doi.org/10.1103/PhysRev.157.1376))
* {#Sakurai69} [[Jun John Sakurai]], Chapter III of: _Currents and Mesons_, Chicago Lectures in Physics, based on notes by George Barry, University of Chicago Press (1969) ([ISBN: 9780226733838](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3622598.html))
Review:
* {#MurphyYount71} Fred V. Murphy, David E. Yount, _Photons as hadrons_, Sci.Am. 225N1 (1971) 1, 94-104 ([spire:41546](https://inspirehep.net/literature/41546), [doi:10.1038/scientificamerican0771-94](https://doi.org/10.1038/scientificamerican0771-94))
* {#Schildknecht72} [[Dieter Schildknecht]], _Vector meson dominance, photo- and electroproduction from nucleons_, in: _Photon-Hadron Interactions II_ Springer Tracts in Modern Physics, vol 63. Springer 1972 ([doi:10.1007/BFb0041507](https://doi.org/10.1007/BFb0041507))
* {#PillerWeise90} G. Piller, [[Wolfram Weise]], _Vector meson dominance: Selected topics_ 1990 ([spire310958](https://inspirehep.net/literature/310958), [[PillerWeiseVMD.pdf:file]])
* {#OCPTW95} H. B. O'Connell, B. C. Pearce, A. W. Thomas, A. G. Williams, _Rho-omega mixing, vector meson dominance and the pion form-factor_, Prog. Part. Nucl. Phys. 39:201-252, 1997 ([arXiv:hep-ph/9501251](https://arxiv.org/abs/hep-ph/9501251))
* {#Schildknecht05} [[Dieter Schildknecht]], _Vector Meson Dominance_, Acta Phys. Polon. B37:595-608, 2006 ([arXiv:hep-ph/0511090](https://arxiv.org/abs/hep-ph/0511090))
See also
* Wikipedia, _[Vector meson dominance](https://en.wikipedia.org/wiki/Vector_meson_dominance)_
More:
* H. B. O'Connell, A. G. Williams, M. Bracco, G. Krein, _Vector Meson Mixing and Charge Symmetry Violation_, Phys. Lett. B370:12-16, 1996 ([arXiv:hep-ph/9510425](https://arxiv.org/abs/hep-ph/9510425))
* A. G. Williams, _New results in vector meson dominance and rho meson physics_, Proceedings of the APCTP Workshop on Astro-Hadron Physics, "Properties of Hadrons in Matter", Seoul, 25-31 October, 1997 ([arXiv:hep-ph/9712405](https://arxiv.org/abs/hep-ph/9712405))
* {#BDDL09a} M. Benayoun, P. David, L. DelBuono, O. Leitner, _A Global Treatment Of VMD Physics Up To The $\phi$: I. $e^+ e^-$ Annihilations, Anomalies And Vector Meson Partial Widths_, Eur. Phys. J. C65:211-245, 2010 ([arXiv:0907.4047](https://arxiv.org/abs/0907.4047))
* {#BDDL09a} M. Benayoun, P. David, L. DelBuono, O. Leitner, _A Global Treatment Of VMD Physics Up To The $\phi$: II. $\tau$ Decay and Hadronic Contributions To $g-2$_, Eur. Phys. J. C68:355-379, 2010 ([arXiv:0907.5603](https://arxiv.org/abs/0907.5603))
* Avner Karasik, _Vector dominance, one flavored baryons, and QCD domain walls from the "hidden" Wess-Zumino term_ ([arXiv:2010.10544](https://arxiv.org/abs/2010.10544))
### Via holographic QCD
{#ReferencesViaHolographicQCD}
Derivation of vector meson dominance via [[holographic QCD]]:
* D.T. Son, M.A. Stephanov, _QCD and dimensional deconstruction_, Phys. Rev. D69 (2004) 065020 ([arXiv:hep-ph/0304182](https://arxiv.org/abs/hep-ph/0304182))
(from the point of view of [[hidden local symmetry]])
* Sungho Hong, Sukjin Yoon, [[Matthew Strassler]], _On the Couplings of Vector Mesons in AdS/QCD_, JHEP 0604 (2006) 003 ([arXiv:hep-th/0409118](https://arxiv.org/abs/hep-th/0409118))
* Sungho Hong, Sukjin Yoon, [[Matthew Strassler]], _On the Couplings of the Rho Meson in AdS/QCD_ ([cds:816440](https://cds.cern.ch/record/816440), [arXiv:hep-ph/0501197](https://arxiv.org/abs/hep-ph/0501197))
* Leandro Da Rold, Alex Pomarol, _Chiral symmetry breaking from five dimensional spaces_, Nucl. Phys. B721:79-97, 2005 ([arXiv:hep-ph/0501218](https://arxiv.org/abs/hep-ph/0501218))
and specifically in the [[Witten-Sakai-Sugimoto model]]:
* {#SakaiSugimoto05} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], p. 18 and Section 5 of: _More on a holographic dual of QCD_, Progr. Theor. Phys. 114: 1083-1118, 2005 ([arXiv:hep-th/0507073](https://arxiv.org/abs/hep-th/0507073))
[[!redirects current-field identity]]
[[!redirects current-field identities]]
[[!redirects field-current identity]]
[[!redirects field-current identities]]
[[!redirects VMD]]
[[!redirects VMD1]]
[[!redirects VMD2]]
|
vector representation | https://ncatlab.org/nlab/source/vector+representation |
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### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Let $G \coloneqq Spin(V) \overset{\pi}{\to} SO(V)$ be a [[spin group]] [[group extension|extension]] of a [[special orthogonal group]], or more generally a [[Pin group]]-extension of an [[orthogonal group]] (or [[Lorentz group]], ...).
Then a [[spin representation]] of $Spin(V)$ is called the _vector representation_ if it comes via $\pi$ from the defining [[linear representation]] of $SO(V)$ on the [[vector space]] underlying the given [[inner product space]] $V$.
## Related concepts
* [[real spin representation]]
* [[pseudovector representation]]
* [[vector meson]]
[[!redirects vector representations]] |
vector space | https://ncatlab.org/nlab/source/vector+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Linear algebra
+-- {: .hide}
[[!include higher linear algebra - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Definition
For $k$ a [[field]] or a [[division ring]], a __vector space__ over $k$
(or a $k$-vector space) is a [[module]] over the [[ring]] $k$. When the vector space is fixed, its elements are called _vectors_, the field $k$ is referred to as the base field of the ground field of the vector space, and the elements of $k$ are called _scalars_.
Sometimes a vector space over $k$ is called a __$k$-linear space__. (Compare '$k$-[[linear map]]'.) If $k$ is only a division ring then we
carefully distinguish the left $k$-vector spaces and right $k$-vector spaces.
The [[category]] of vector spaces is typically denoted [[Vect]], or
$Vect_k$ if we wish to make the field $k$ (the *ground field*) explicit. So
$$
Vect_k \coloneqq k Mod
\,.
$$
This category has vector spaces over $k$ as objects, and $k$-linear maps between these as morphisms.
### Multisorted notion
Alternatively, one sometimes defines "vector space" as a two-sorted notion; taking the field $k$ as one of the sorts and a module over $k$ as the other. More generally, the notion of "module" can also be considered as two-sorted, involving a ring and a module over that ring.
This is occasionally convenient; for example, one may define the notion of [[topological vector space]] or topological module as an [[internalization]] in $Top$ of the multisorted notion. This procedure is entirely straightforward for topological modules, as the notion of module can be given by a two-sorted Lawvere theory $T$, whence a topological module (for instance) is just a product-preserving functor $T \to Top$. One may then define a topological vector space as a topological module whose underlying (discretized) ring sort is a field.
## Properties
Every [[free object|free]] vector space admits a [[basis of a vector space|basis]].
The _[[basis theorem]]_, which is equivalent to the [[axiom of choice]], states that every vector space is a free vector space.
## Related concepts
* **vector space**, [[dual vector space]],
* [[finite-dimensional vector space]]
* [[real vector space]], [[complex vector space]]
* [[topological vector space]], [[convenient vector space]]
* [[virtual vector space]]
* [[tensor product of vector spaces]]
* [[real structure]], [[complex structure]], [[quaternionic structure]]
* [[vector bundle]],
* [[lattice in a vector space]]
* [[vector G-space]]
* [[2-vector space]], [[n-vector space]]
* [[inner product space]]
* [[linear operator]], [[matrix]], [[determinant]], [[eigenvalue]], [[eigenvector]]
## References
The concept of vector spaces seem to have been first introduced in:
* [[Giuseppe Peano]], *Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva*, Fratelli Bocca Editori, Torino, 1888, pp. XI, 171 $[$[Mathematica Italiana opere 138](http://mathematica.sns.it/opere/138), [pdf](http://mathematica.sns.it/media/volumi/138/Calcolo%20geometrico%20secondo%20l'Ausdehnungslhere%20di%20H.%20Grassmann_bw.pdf)$]$
> (referring to [[Hermann Grassmann]]'s *[[Ausdehnungslehre]]*)
An early expository account is in §14 of
* [[Bartel Leendert van der Waerden]], _[[Moderne Algebra]]_, Volume I, Springer, 1930.
The literature on vector spaces is now extremely large, including lots of elementary linear algebra textbooks.
* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §3 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]
Classics include:
* [[Michael Artin]], Algebra
* [[Israel M. Gelfand]], Lectures on linear algebra
* P. R. Halmos, Finite dimensional vector spaces
* [[M M Postnikov]], Lectures on geometry, semester 2: Linear algebra
Affine spaces are sets which are torsors over the abelian group of vectors of a vector space. Thus vector spaces may serve as a basis for the affine and for the Euclidean geometry. This approach has been invented by [[Hermann Weyl]] in 1918. Dieudonné wrote an influential book on such an approach to 2d and 3d Euclidian geometry, in which the basics of vector spaces in low dimension is introduced along the way (the book is intended for high school teachers):
* Jean Alexandre Dieudonné, Linear algebra and geometry
Discussion of vector space in [[univalent foundations of mathematics]] ([[homotopy type theory]]) is announced to appear in:
* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Chapter 8 of: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/)]
[[!redirects vector space]]
[[!redirects vector spaces]]
[[!redirects linear space]]
[[!redirects linear spaces]]
[[!redirects Grassmannian]]
|
vectorial bundle | https://ncatlab.org/nlab/source/vectorial+bundle |
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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _vectorial bundle_ ([Gomi 08](#Gomi08)) is a $\mathbb{Z}_2$-[[graded vector space|graded]] [[vector bundle]] $E$ of [[finite number|finite]] [[rank of a vector bundle|rank]], equipped with an odd [[endomorphism]] $h \;\colon\; E \to E$. The [[homomorphisms]] of vectorial bundles are such that the [[endomorphism]] $h$ acts like canceling parts of the even and odd degree of $E$ against each other.
This way vectorial bundles lend themselves to the description of [[topological K-theory]]. In particular, they allow a geometric model for [[twisted K-theory]].
## Definition
For $X$ a [[topological space]], the [[category]] $VectrBund(X)$ of **vectorial bundles** on $X$ has
* as [[objects]] $(E \stackrel{h}{\to} E)$ [[finite number|finite]] [[rank of a vector bundle|rank]] Hermitean $\mathbb{Z}_2$-graded [[vector bundle | vector bundles]] $E\to X$ equipped with a self-adjoint [[endomorphism]] $h$ of odd degree. In [[matrix calculus]]
$$
E = \left(
\array{
E_{0}
\\
E_{1}
}
\right)
$$
$$
h =
\left(
\array{
0 & h_{10}
\\
h_{01} & 0
}
\right)
$$
* as [[morphisms]] $\phi : (E,h) \to (E',h)$ [[equivalence classes]] of morphisms $\phi \colon E \to E'$ of vector bundles such that
$$
\array{
E &\stackrel{\phi}{\longrightarrow}& E
\\
{}^{\mathllap{h}}\big\downarrow && \big\downarrow^{\mathrlap{h'}}
\\
E &\stackrel{\phi}{\longrightarrow}& E
}
\,,
$$
where two such maps are regarded as equivalent, $\phi \sim \phi'$, already if they coincide on the kernel of $h^2_x$ for each point $x$.
In particular, we have the following two important special cases:
* the case that $h = 0$ -- in this case all eigenvalues of all $h_x^2$ are zero. and hence maps $\phi, \phi' : (E,0) \to (E',0)$ represent the same morphism precisely if they are actually equal as morphisms $\phi, \phi' : E \to E'$ of vector bundles.
(Notice that there is only the 0-morphism $(E,0) \to (E',h')$ for $h' \neq 0$.)
This yields a canonical inclusion
$$
VectBund(X) \hookrightarrow VectrBund(X)
$$
by sending $E \mapsto (E \stackrel{0}{\to} E)$.
* the case that $E = \left( \array{V \\ V}\right)$
and $h = \left(
\array{
0 & Id \\
Id & 0
}
\right)
$
Here $E_x|_{\lt \mu \lt 1} = 0$ and hence two morphisms
$\phi, \phi' : (E,h) \to (E',h')$ are identified already if they agree on the 0-vector. In other words, _all_ morphisms out of such $(E,h)$ are identified. In particular they are all equal to the 0-morphism to $(0,0)$. Therefore the bundles of this form represent the 0-element.
**Definition**
Say two vectorial bundles $(E,h)$, $(E',h')$ on $X$ are **[[concordance|concordant]]** if there is a vectorial bundle on $X \times [0,1]$ which restricts to them at either end, respectively.
Let
$
(E,h)^{\vee}
=
$ be the degree-reversed bundle to $(E,h)$.
**Lemma**
There is a concordance
$$
E \oplus E^\vee \to 0 .
$$
## References
The definition of vectorial bundles is due to Furuta. It is recalled and applied to the study of [[K-theory]] and [[twisted K-theory]] in
* {#Gomi08} [[Kiyonori Gomi]], _Twisted K-theory and finite-dimensional approximation_ ([arXiv:0803.2327](http://arxiv.org/abs/0803.2327))
[[!redirects vectorial bundles]] |
Veeravalli Varadarajan | https://ncatlab.org/nlab/source/Veeravalli+Varadarajan |
* [webpage](http://www.math.ucla.edu/~vsv/)
## selected writings
* {#Varadarajan04} [[Veeravalli Varadarajan]], _[[Supersymmetry for mathematicians]]: An introduction_, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I 2004
(on [[supersymmetry]])
* [[Riccardo D'Auria]], [[Sergio Ferrara]], M. A. Lledó, [[Veeravalli Varadarajan]], _Spinor Algebras_, J.Geom.Phys. 40 (2001) 101-128 ([arXiv:hep-th/0010124](http://arxiv.org/abs/hep-th/0010124))
(on [[spin groups]] and [[spin representations]])
* {#Varadarajan01} [[Veeravalli Varadarajan]], _Spin(7)-subgroups of SO(8) and Spin(8)_, Expositiones Mathematicae, 19 (2001): 163-177 (<a href="https://doi.org/10.1016/S0723-0869(01)80027-X">doi:10.1016/S0723-0869(01)80027-X</a>, [pdf](https://core.ac.uk/download/pdf/81114499.pdf))
(on [[Spin(7)]]-[[subgroups]] of [[Spin(8)]])
## related $n$Lab entries
* [[spin representation]]
* [[Spin(7)]]
* [[supersymmetry]], [[geometry of physics -- supersymmetry]]
* [[conformal compactification]]
category: people
|
Velayudhan Parameswaran Nair | https://ncatlab.org/nlab/source/Velayudhan+Parameswaran+Nair |
* [spire page](https://inspirehep.net/authors/996257)
## Selected writings
Precursor to [[twistor string theory]]:
* [[Velayudhan Parameswaran Nair]], *A current algebra for some gauge theory amplitudes*, Physics Letters B Volume 214, Issue 2, 17 November 1988, Pages 215-218 (<a href="https://doi.org/10.1016/0370-2693(88)91471-2">doi:10.1016/0370-2693(88)91471-2</a>, [spire:24212](https://inspirehep.net/literature/24212))
category: people
[[!redirects V. Parameswaran Nair]]
[[!redirects V. P. Nair]] |
Velo-Zwanziger problem | https://ncatlab.org/nlab/source/Velo-Zwanziger+problem |
+-- {: .rightHandSide}
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Velo-Zwanziger problem_ is is a phenomenon in theoretical [[physics]]: the "evident" [[action functionals]] for [[field (physics)|fields]] which
1. have [[spin]] higher than 1
1. are in an irreducible [[mass|massive]] representation of the Lorentz group
1. have [[minimal coupling]] to [[electromagnetism]]
tend to have [[equations of motion]] which are not [[hyperbolic differential operator|hyperbolic]] and hence may violate the usual [[causal locality|causality]] relation of [[general relativity]].
## References
The original references are
* G. Velo, D. Zwanziger, Phys. Rev. 186 (1969) 1337,
* G. Velo, D. Zwanziger, Phys. Rev. 188 (1969) 2218;
* G. Velo, Nucl. Phys. B 43 (1972) 389
* M. Hortacsu, Phys. Rev. D 9, 928 (1974).
Adjustments of the [[action functional]] for [[spin]] [[mass|massive]] $3/2$-particles ([[Rarita-Schwinger field]]) by non-[[minimal coupling]] that evades the Velo-Zwanziger problem is discussed in
* [[Massimo Porrati]], Rakibur Rahman, _Causal Propagation of a Charged Spin 3/2 Field in an External Electromagnetic Background_, Phys. Rev. D80:025009, 2009 ([arXiv:0906.1432](http://arxiv.org/abs/0906.1432))
Discussion for the higher [[string]] excitations that appear in [[string theory]] is for instance in
* [[Massimo Porrati]], Rakibur Rahman, [[Augusto Sagnotti]], _String Theory and The Velo-Zwanziger Problem_, Nucl. Phys. B846:250-282, 2011 ([arXiv:1011.6411](http://arxiv.org/abs/1011.6411))
|
velocity | https://ncatlab.org/nlab/source/velocity |
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###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
Given a [[Cartesian space]] $V$ and a [[smooth curve]] $r:\mathbb{R} \to V$, the **velocity** of $r$ is given by the [[derivative]] of $r$: $v \coloneqq \partial_t r$.
## Related concepts
* [[curve]]
* [[acceleration]]
* [[jerk]]
In Euclidean spaces:
* [[speed]]
* [[rapidity]]
* [[areal velocity]]
* [[torque]]
## References
See the references at *[[differential geometry of curves and surfaces]]*.
See also
+ Wikipedia, _[Velocity](https://en.wikipedia.org/wiki/Velocity)_ |
Veneziano amplitude | https://ncatlab.org/nlab/source/Veneziano+amplitude |
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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Veneziano amplitude_ is the tree-level [[string scattering amplitude]] for the scattering of four [[open string|open]] [[bosonic strings]] in their [[tachyon]] [[quantum state|states]]. Hence equivalently it is a certain [[n-point function|4-point function]] in a certain [[boundary field theory|boundary]] [[2d CFT]].
Historically, this amplitude first arose in [[dual resonanace models]] and the realization that it has a natural interpretation as the scattering of [[strings]] was one of the main original motivations for the development of [[string theory]].
## Related concepts
* [[Virasoro amplitude]]
* In [[p-adic string theory]] (see there) one considers [[adelic integral]]-variants of the Veneziano amplitude.
* [[vacuum amplitude]]
## References
* [[Michael Green]], [[John Schwarz]], [[Edward Witten]], section 1.1 of _Superstring theory_ Volume 1, Cambridge Monographs on Mathematical Physics, 1987
* [[Gerard 't Hooft]], section 1.2 and section 6 of _Introduction to string theory_ 2004 ([pdf](http://www.phys.uu.nl/~thooft/lectures/stringnotes.pdf))
* _The Veneziano amplitude_ [pdf](http://www.thphys.uni-heidelberg.de/~palti/Stringcourse/problemset8.pdf)
Discussion in the context of [[Riemann hypothesis and physics]] is in
* {#HJM15} [[Yang-Hui He]], Vishnu Jejjala, Djordje Minic, _From Veneziano to Riemann: A String Theory Statement of the Riemann Hypothesis_ ([arXiv:1501.01975](http://arxiv.org/abs/1501.01975))
Textbook review includes (16.49) in
* [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], _String Scattering Amplitudes and Low Energy Effective Field Theory_, chapter 16 in _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013
See also:
* Clifford Cheung, Grant N. Remmen, *Veneziano Variations: How Unique are String Amplitudes?* [[arXiv:2210.12163](https://arxiv.org/abs/2210.12163)]
* Nicholas Geiser, Lukas W. Lindwasser, *Generalized Veneziano and Virasoro amplitudes* [[arXiv:2210.14920](https://arxiv.org/abs/2210.14920)]
[[!redirects Veneziano amplitudes]] |
Venkata S. R. Redrouthu | https://ncatlab.org/nlab/source/Venkata+S.+R.+Redrouthu |
* [institute page](https://www.asifequbal.com/team/venkata-subbarao-redrouthu)
* [GoogleScholar page](https://scholar.google.com/citations?user=vFG-qJYAAAAJ)
category: people
[[!redirects Venkata Redrouthu]] |
Venkatesa Chandrasekaran | https://ncatlab.org/nlab/source/Venkatesa+Chandrasekaran |
* [InSpire page](https://inspirehep.net/authors/1812047)
## Selected writings
On [[von Neumann algebra factors]] [[algebra of observables|of observables]] for [[quantum field theory on curved spacetime|quantum field theory on]] [[de Sitter spacetime]]:
* [[Venkatesa Chandrasekaran]], [[Roberto Longo]], [[Geoff Penington]], [[Edward Witten]], *An Algebra of Observables for de Sitter Space*, Journal of High Energy Physics **2023** 82 (2023) [[arXiv:2206.10780](https://arxiv.org/abs/2206.10780), <a href="https://doi.org/10.1007/JHEP02(2023)082">doi:10.1007/JHEP02(2023)082</a>]
category: people |
Vera Serganova | https://ncatlab.org/nlab/source/Vera+Serganova |
* [personal page](https://math.berkeley.edu/people/faculty/vera-serganova)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Vera_Serganova)
## Selected writings
On [[Lie groups]] and [[quantum groups]]:
* [[Richard Borcherds]], [[Mark Haiman]], [[Theo Johnson-Freyd]], [[Nicolai Reshetikhin]], [[Vera Serganova]], *Berkeley Lectures on Lie Groups and Quantum Groups*, 2020 ([pdf](http://categorified.net/LieQuantumGroups.pdf))
category: people
|
Verdier duality | https://ncatlab.org/nlab/source/Verdier+duality |
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### Context
#### Duality
+--{: .hide}
[[!include duality - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What is called _Verdier duality_ is the refinement of [[Poincaré duality]] from [[ordinary cohomology]] to [[abelian sheaf cohomology]].
The Grothendieck [[six operations]] formalism is a formalization of aspects of Verdier duality.
In general abstract formulation one says that a locally compact [[site]] $X$ (e.g. a [[locally compact topological space]]) _satisfies Verdier duality_ if there exists a [[derived functor|derived]] [[right adjoint]] $\mathbb{L}p^!$ ([[exceptional inverse image]]) to the operation $\mathbb{R}p_!$ of [[direct image with compact support]] for the terminal map $X \to \ast$. More generally, in the relative situation, for $f \colon X \to Y$ a map, Verdier duality of $f$ means that $\mathbb{R}f_!$ has a derived right adjoint. If here $X$ is a [[scheme]] and $f_! \simeq f_\ast$ coincides with the [[direct image]] ("[[Grothendieck context]]") then this is called _[[Grothendieck duality]]_. Therefore one often speaks of _Verdier-Grothendieck duality_.
If these adjoints exist, then one has the _[[dualizing object in a closed category]]_ ([[dualizing module]]) $\omega \coloneqq \mathbb{L}p^! k$ (the image under the extra right adjoint of the [[unit object]]) and for every [[abelian sheaf]] $V$ on $X$ can define the [[dual object in a closed category]]
$$
\mathbb{D}V \coloneqq [V,\omega]
\,.
$$
The analog of [[Poincaré duality]] is then the statement that the [[abelian sheaf cohomology]] with [[coefficients]] in $V$ is dual to the abelian sheaf cohomology with [[compact support]] with coefficients in $\mathbb{D}V$.
## Implementations
A fairly general class of implementations of Verdier-Grothendieck duality is discussed in ([Joshua, theorem 1.3](#Joshua)).
With assumption ([Joshua, 4.3](#Joshua)) this derives that dualization intertwines the two pushforwards as $\mathbb{D} \circ f_\ast \simeq f_! \circ \mathbb{D}$ ([Joshua, corollary 5.4](#Joshua)).
## Related concepts
* [[Verdier-Grothendieck context]]
* [[Grothendieck duality]]
* [[Poincaré duality]]
## References
The duality is named after [[Jean-Louis Verdier]].
Accounts of the traditional notion include
* [[Amnon Neeman]], _The Grothendieck duality theorem via Bousfield's techniques and Brown representability_, Jour. Amer. Math. Soc. 9 (1996), 205–236. ([JSTOR](http://www.jstor.org/stable/2152846))
{#Neeman96}
* Wikipedia, _[Verdier duality](http://en.wikipedia.org/wiki/Verdier_duality)_
* Liviu Niculaescu, _The derived category of sheaves and the Poincaré-Verdier duality_ ([pdf](http://www3.nd.edu/~lnicolae/Verdier-ams.pdf))
* [[Jacob Lurie]], lecture notes on _Verdier duality_ ([2011 pdf](http://www.math.harvard.edu/~lurie/287xnotes/Lecture19.pdf), [2013 pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.348.5739&rep=rep1&type=pdf), [2014 pdf](http://people.math.harvard.edu/~lurie/282ynotes/LectureXXI-Verdier.pdf))
A general abstract formalization is in
* [[Roy Joshua]], _Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories_ ([[JoshuaDuality.pdf:file]])
{#Joshua}
Discussion in the context of [[higher algebra]]/[[higher geometry]] is in
* [[Roy Joshua]], _Generalised Verdier duality for presheaves of spectra—I_, Journal of Pure and Applied Algebra Volume 70, Issue 3, 29 March 1991, Pages 273–289
* [[Jonathan Block]] and [[Andrey Lazarev]], _Homotopy Theory and Generalized Duality for Spectral Sheaves_, IMRN International Mathematics Research Notices
1996, No. 20 ([pdf](http://www.math.upenn.edu/~blockj/papers/BlockLazarevSpectralSheaves.pdf))
* [[Vladimir Drinfeld]], [[Dennis Gaitsgory]], section 5.3 _On some finiteness questions for algebraic stacks_ ([arXiv:1108.5351](http://arxiv.org/abs/1108.5351))
* [[Jacob Lurie]], section 4.2 of _[[Representability theorems]]_
Formulation in terms of an equivalence between sheaves and cosheaves is in
* [[Peter Schneider]], _Verdier duality on the building_, Journal reine angew. Math. 494, 1998
* [[Justin Curry]], _Sheaves, Cosheaves and Applications_ ([arXiv:1303.3255](http://arxiv.org/abs/1303.3255))
* [[Jacob Lurie]], section 5.5.5 of _[[Higher Algebra]]_
Discussion in the context of prestacks and diagrams of schemes
* [[Dennis Gaitsgory]], _The Atiyah-Bott formula for the cohomology of the moduli space of bundles on a curve_ ([arXiv:1505.02331](http://arxiv.org/abs/1505.02331))
* Alexey Kalugin, _On categorical approach to Verdier Duality_ ([arXiv:1505.06922](http://arxiv.org/abs/1505.06922))
[[!redirects Verdier dualities]] |
Verdier product | https://ncatlab.org/nlab/source/Verdier+product | ## Definition
Verdier product is an analogue of the [[Gabriel product]] in triangulated context.
## Literature
* [[Wendy Lowen]], [[Michel Van den Bergh]], _On compact generation of deformed schemes_, Adv. Math. __244__ (2013) 441--464 [doi](https://doi.org/10.1016/j.aim.2013.04.024)
* Gustavo Jasso, [[Bernhard Keller]], [[Fernando Muro]], _The triangulated Auslander--Iyama correspondence_, [arXiv:2208.14413](https://arxiv.org/abs/2208.14413)
|
Verdier site | https://ncatlab.org/nlab/source/Verdier+site |
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### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _Verdier site_ (named after [[Jean-Louis Verdier]]) is a [[site]] with an extra condition on the [[coverage]] that ensures that one can refine every [[hypercover]] in the context of [[simplicial presheaf|simplicial presheaves]] on that site in a particularly nice.
For a discussion of the use of Verdier sites see [[descent]].
## Definition
A **Verdier site** is a [[site]] such that for every morphism $U \to X$ which is part of a covering [[sieve]] on $X$, also
$$
U \stackrel{(Id_U,Id_U)}{\to} U \times_X U
$$
is in a covering [[sieve]] of $U \times_X U$.
## References
section 9 of
* [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], _Hypercovers and simplicial presheaves_ ([arXiv](http://arxiv.org/abs/math/0205027))
[[!redirects Verdier sites]]
|
Verdier's abelianization functor | https://ncatlab.org/nlab/source/Verdier%27s+abelianization+functor | Given a [[triangulated category]] $T$ there is an [[abelian category]] $A$ and an [[additive functor]] $T\to A$ satisfying certain universal property; this functor is due to [[Verdier]] and is called the Verdier's abelianization functor. A readable and comprehensive introduction is in the monograph
* A. Neeman, _Triangulated categories_, Annals of Mathematics Studies __148__, Princeton University Press 2001, viii+449 pp.
[[!redirects Verdier abelianization functor]] |
Verdier-Grothendieck context | https://ncatlab.org/nlab/source/Verdier-Grothendieck+context |
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###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The specialization of a context of [[six operations]] $(f^\ast \dashv f_\ast)$, $(f_! \dashv f^!)$ to the case that the "[[projection formula]]" $Y \otimes f_! X \simeq f_!(f^\ast Y \otimes X)$ holds naturally in $X,Y$. ([May 05, def. 2.12](#May05))
## Properties
In a Verdier-Grothendieck context, [[dual object in a closed category|duality]] intertwines $f_!$ with $f_\ast$ and $f^!$ with $f^\ast$.
([May 05, section 3](#May05), [Joshua, corollary 5.4](#Joshua))
## Related concepts
* [[Verdier duality]], [[Grothendieck duality]]
* [[Wirthmüller context]], [[Grothendieck context]]
## References
A general abstract discussion of the axioms and their consequences is in
* {#May05} [[Halvard Fausk]], [[Po Hu]], [[Peter May]], *Isomorphisms between left and right adjoints*, Theory and Applications of Categories, **11** 4 (2003) 107-131 [[tac:11-04](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), [pdf](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf)]
A fairly general class of implementations is in
* [[Roy Joshua]], _Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories_ ([[JoshuaDuality.pdf:file]])
{#Joshua}
[[!redirects Verdier-Grothendieck contexts]]
[[!redirects Grothendieck-Verdier context]]
[[!redirects Grothendieck-Verdier contexts]] |
Vergleichende Betrachtungen über neuere geometrische Forschungen | https://ncatlab.org/nlab/source/Vergleichende+Betrachtungen+%C3%BCber+neuere+geometrische+Forschungen |
This page collect material related to the article
* {#Klein1872} [[Felix Klein]], _[[Vergleichende Betrachtungen über neuere geometrische Forschungen]]_ (1872) Mathematische Annalen volume 43, pages 63–100 1893 ([doi:10.1007/BF01446615](https://doi.org/10.1007/BF01446615))
English translation by M. W. Haskell:
_A comparative review of recent researches in geometry_, Bull. New York Math. Soc. 2, (1892-1893), 215-249. ([euclid:1183407629](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-2/issue-10/A-comparative-review-of-recent-researches-in-geometry/bams/1183407629.full), LaTeX version retyped by Nitin C. Rughoonauth: [arXiv:0807.3161](https://arxiv.org/abs/0807.3161))
on foundations of [[differential geometry]] in the notion of [[group actions]].
For more see at _[[Erlangen program]]_.
category: people |
Verity on descent for strict omega-groupoid valued presheaves | https://ncatlab.org/nlab/source/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves |
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###Context###
#### Locality and descent
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[[!include descent and locality - contents]]
=--
=--
=--
This entry is about a theorem by [[Dominic Verity]] that characterizes the [[descent]] condition for [[(∞,1)-sheaves]]/[[∞-stack]]s that take values not in arbitrary [[∞-groupoid]]s, but in [[strict ∞-groupoid]]s.
#Contents#
* table of contents
{:toc}
## Verity's theorem
The details are here:
* [[Dominic Verity]], _[[VerityDescent.pdf:file]]_ (pdf)
Here is an abstract that served as an abstract for a talk on this at the _Australian Category Seminar_ at Macquarie University on Wednesday 27th of May 2009.
<strong>Abstract</strong>
>In the literature one can find a number of different [[limit]] notions which one might refer to as a "[[descent]] construction". Generally speaking, these may all be regarded as a kind of [[weak limit|lax, pseudo]] or [[homotopy limit]] of a [[simplicial object|co-simplicial diagram]] of objects in some theory of "spatially-enriched" categories. While each of these notions certainly deserves to bear the [[descent]] name, it is not necessarily immediately clear how they may be related in any more specific mathematical sense.
>Recently I was asked by [[Urs Schreiber]] if I knew how a couple of these descent notions might be related formally, and so spent a little time contemplating this problem. My hope is that this talk might achieve two things, firstly I hope to provide a little of the intuition which leads us to define and study such descent constructions. Then I would like to discuss a specific answer to Urs' question, which gives a precise relationship between [[Ross Street]]'s [descent construction](http://arxiv.org/abs/math.CT/0303175) for [[strict omega-category|strict ∞-categories]] (or more precisely [[strict ∞-groupoid]]s in this case) and the [[descent for simplicial presheaves|simplicial descent construction]] used to characterise the fibrant objects in [[model structure on simplicial presheaves|model categories of simplicial sheaves]].
## Idea
Generally, [[models for ∞-stack (∞,1)-toposes]] are provided by a [[model structure on simplicial presheaves|model structure on presheaves with values in simplicial sets]].
As for all [[combinatorial simplicial model category|combinatorial simplicial model categories]] the $(\infty,1)$-topos [[presentable (∞,1)-category|presented]] by this model structure is the full [[SSet]]-[[enriched category|enriched subcategory]] on fibrant and cofibrant objects.
By a theorem by Dugger-Isaksen-Hollander on the projective [[local model structure on simplicial presheaves]] the _fibrant_ simplicial presheaves are those that
* take values in [[∞-groupoid]]s (i.e. the [[simplicial set]]s assigned by them are [[Kan complex]]es)
* and satisfy [[descent]].
While general [[∞-groupoid]]s are useful due to their generality and conceptual simplicity, for many concrete computations it is useful to get a more concrete [[algebraic definition of higher category|algebraic model]] and consider just [[strict ∞-groupoid]]s. Under the [[oriental]]-[[nerve]]
$$
Str \omega Grpd \stackrel{N}{\hookrightarrow} \infty Grpd
$$
the strict $\omega$-groupoids form a [[subcategory]] of all [[∞-groupoid]]s. This is to be regarded as a refinement of the [[Dold-Kan correspondence]] which embeds strict $\omega$-groupoids _with abelian group structure_ equivalently modeled as [[chain complex]]es into all $\infty$-groupoids
$$
Ch_+
\stackrel{\simeq}{\to} StrAb \omega Grpd
\hookrightarrow
Str \omega Grpd \stackrel{N}{\hookrightarrow} \infty Grpd
\,.
$$
It is a familiar process to restrict general [[∞-stack]]s to those that factor through the entire inclusion: this is the topic of [[homological algebra]] and restricts the general notion of [[cohomology]] to that of [[abelian sheaf cohomology]].
What we are interested in here is a notion in between the fully strictly abelian context and the fully general context: that of strict $\omega$-groupoid valued $\infty$-stacks inside all $\infty$-stacks. This may be thought of as [[nonabelian algebraic topology|nonabelian homological algebra]] that uses not [[chain complex]]es of sheaves but [[crossed complex]]es.
In his work
* [[Ross Street]], _Categorical and combinatorial aspects of descent theory_ ([arXiv](http://arxiv.org/abs/math.CT/0303175))
Ross Street had proposed a formulation of the [[descent]] condition for such $Str \omega Grpd$-valued $\infty$-stacks (see the corresponding section at [[descent]] for the details).
The question remained open how that definition of descent on $Str \omega Grpd$-valued presheaves relates to the general one of $\infty Grpd$-valued presheaves under the above inclusion.
It is this question that Dominic Verity's theorem answers.
In words, Verity's theorem says that Ross Street's descent conditon on a $Str \omega Grpd$-valued presheaf $A$ is the correct one if the [[hypercover]] $\pi : Y \to X$ along which one checks descent is sufficiently well behaved -- in that the [[simplicial object|cosimplicial]] $\infty$-groupoid $A(Y)$ is [[Reedy category|Reedy fibrant]].
[[!redirects Verity on descent for strict omega-groupoid valued presheave]] |
Verity-Gray tensor product | https://ncatlab.org/nlab/source/Verity-Gray+tensor+product | #Idea#
The _Verity-Gray tensor product_ or _lax Gray tensor product of [[stratified simplicial set]]s_ is a [[tensor product]] on the [[category]] $Strat$ of [[stratified simplicial set]]s which when restricted to [[complicial set]]s, i.e. [[oriental|omega-nerve]]s of [[strict omega-category|strict omega-categories]] reproduces the [[Crans-Gray tensor product]] on strict $\omega$-categories.
#Definition#
Let $(X, t X)$ and $(Y, t Y)$ be [[stratified simplicial set]]s. Then their **Verity-Gray tensor product** $(X, t X) \otimes (Y, t Y)$ is given by
$$
(X, t X) \otimes (Y, t Y)
:=
(X \times Y, q(t X, t Y))
\,,
$$
where $X \times Y$ is the [[cartesian product]] of [[simplicial set]]s (hence the standard monoidal structure on [[SSet]]), while $q(t X, t Y)$, the set of thin cells, is $tX\times tY$ for the Gray product, and for the lax-Gray product is enlarged as described in the paper.
#References#
definition 128 of
* Dominic Verity, _Complicial sets_ ([arXiv](http://arxiv.org/abs/math/0410412))
definition 59, page 32 of
* Dominic Verity, _Weak complicial sets I_ ([arXiv](http://arxiv.org/abs/math/0604414))
slide 60 of
* Dominic Verity, _Weak complicial sets and internal quasi-categories_ ([arXiv](http://www.mat.uc.pt/~categ/ct2007/slides/verity.pdf)) |
Verlinde formula | https://ncatlab.org/nlab/source/Verlinde+formula |
#Contents#
* table of contents
{:toc}
## Idea
Generally, a _Verlinde formula_ gives the [[dimension]] of a [[space of states]] of [[Chern-Simons theory]] for a given [[gauge group]] $G$. Depending on which one of various different algebraic means to expresses these spaces is used, the Verlinde formula equivalently computes the dimension of spaces of non-abelian [[theta functions]], the dimension of objects in a [[modular tensor category]] and so forth.
There are also Verlinde formulas in algebraic geometry (proved by Faltings) and a related one in the theory of [[vertex operator algebra]]s (proved only in very special cases).
## Related concepts
* [[fusion ring]].
## References
Original article:
* [[Erik Verlinde]], *Fusion rules and modular transformations in 2D conformal field theory*, Nuclear Physics B **300** (1988) 360-376 $[$<a href="https://doi.org/10.1016/0550-3213(88)90603-7">doi:10.1016/0550-3213(88)90603-7</a>$]$
Proofs of the Verlinde formula:
* {#TUY89} A. Tsuchiya, [[Kenji Ueno]], and Y. Yamada, *Conformal field theory on the universal family of stable curves with gauge symmetry* In: *Conformal field theory and solvable lattice models* Adv. Stud. Pure Math. **16** (1989), 297–372.
* {#Faltings94} [[Gerd Faltings]], *A proof for the Verlinde formula*, J. Alg. Geom. **3** (1994) 347–374
* {#Huang} [[Yi-Zhi Huang]], *Vertex operator algebras, the Verlinde conjecture and modular tensor categories*, Proc. Nat. Acad. Sci. **102** (2005) 5352-5356 $[$[arXiv:math/0412261](http://arxiv.org/abs/math/0412261), [doi:10.1073/pnas.0409901102](https://doi.org/10.1073/pnas.0409901102)$]$
Review in:
* [[Martin Schottenloher]], *Mathematical Aspects of the Verlinde Formula*, in: *A Mathematical Introduction to Conformal Field Theory*, Lecture Notes in Physics **759**, Springer (2008) $[$[doi:10.1007/978-3-540-68628-6](https://link.springer.com/book/10.1007/978-3-540-68628-6), [web](https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/), [pdf](https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/11_978-3-540-68625-5_Ch11_23-08-08.pdf)$]$
where it says (p. 213):
> Several mathematicians have worked on the problem of proving the Verlinde formula, starting with $[$[TUY89](#TUY89)$]$ and coming to a certain end with $[$[Fal94](#Faltings94)$]$. These proofs are all quite difficult to understand.
* [[Ralph Blumenhagen]], [[Erik Plauschinn]], §4.3 of: *Introduction to Conformal Field Theory -- With Applications to String Theory*, Lecture Notes in Physics **779**, Springer (2009) [[doi:10.1007/978-3-642-00450-6](https://doi.org/10.1007/978-3-642-00450-6)]
Introduction:
* [[Shigeru Mukai]], _An introduction to invariants and moduli_, Cambridge Univ. Press 2003
See also:
* Dowker, _On Verlinde's formula for the dimensions of vector bundles on moduli spaces_, [iopscience](http://iopscience.iop.org/0305-4470/25/9/033)
* Juergen Fuchs, Christoph Schweigert, _A representation theoretic approach to the WZW Verlinde formula_, 1997
A generalization to [[logarithmic 2d CFTs]] has been suggested in:
* [[Thomas Creutzig]], [[Terry Gannon]], _Logarithmic conformal field theory, log-modular tensor categories and modular forms_, J. Phys. A __50__, 404004 (2017) ([doi:10.1088/1751-8121/aa8538](https://doi.org/10.1088/1751-8121/aa8538), [arXiv:1605.04630](https://arxiv.org/abs/1605.04630))
[[!redirects Verlinde formulas]]
|
Verlinde ring | https://ncatlab.org/nlab/source/Verlinde+ring |
#Contents#
* table of contents
{:toc}
## Idea
For $G$ a suitable [[Lie group]], the _Verlinde ring_ is the collection of [[isomorphism classes]] of [[positive energy representations]] of the corresponding [[loop group]], equipped with the "fusion" [[tensor product]].
The Verlinde ring is also understood as being the ring of [[equivariant cohomology|equivariant]] [[twisted K-theory]] classes on $G$ ([FHT](#FHT)) and, essentially equivalently, of [[Chan-Paton gauge fields]] over [[D-branes]] in the [[WZW model]].
## Related concepts
* [[fusion ring]]
* [[twisted ad-equivariant K-theory]]
* [[twisted ad-equivariant Tate K-theory]]
## References
### General
Due to
* [[Erik Verlinde]], _Fusion rules and modular transformations in 2D conformal field theory_, Nuclear Physics B __300__ (3): 360–376, (1988) (<a href="https://doi.org/10.1016/0550-3213(88)90603-7">doi:10.1016/0550-3213(88)90603-7</a>)
> We study conformal field theories with a finite number of primary fields with respect to some chiral algebra. It is shown that the fusion rules are completely determined by the behavior of the characters under the modular group. We illustrate with some examples that conversely the modular properties of the characters can be derived from the fusion rules. We propose how these results can be used to find restrictions on the values of the central charge and conformal dimensions.
See also
* [[Domenico Fiorenza]], [[Alessandro Valentino]], _$(3,2,1)$-TQFTs and Verlinde algebras_ ([MO question](http://mathoverflow.net/q/74593/381), [MO answer](http://mathoverflow.net/a/263829/381))
* Wikipedia, _[Verlinde algebra](https://en.wikipedia.org/wiki/Verlinde_algebra)_
### Relation to twisted ad-equivariant K-theory
On [[twisted ad-equivariant K-theory]] of [[compact Lie groups]] and the identification with the [[Verlinde ring]] of [[positive energy representations]] of their [[loop group]]:
* [[Daniel S. Freed]], _The Verlinde algebra is twisted equivariant K-theory_, Turk. J. Math 25 (2001) 159-167 ([arXiv:math/0101038](https://arxiv.org/abs/math/0101038))
{#FHT}
* [[Daniel S. Freed]], [[Michael Hopkins]], [[Constantin Teleman]],
1. _[[Loop Groups and Twisted K-Theory]] I_,
J. Topology, 4 (2011), 737-789
([arXiv:0711.1906](http://arxiv.org/abs/0711.1906), [doi:10.1112/jtopol/jtr019](https://doi.org/10.1112/jtopol/jtr019))
1. _[[Loop Groups and Twisted K-Theory]] II_,
J. Amer. Math. Soc. 26 (2013), 595-644
([arXiv:0511232](http://arxiv.org/abs/math/0511232), [doi:10.1090/S0894-0347-2013-00761-4](https://doi.org/10.1090/S0894-0347-2013-00761-4))
1. _[[Loop Groups and Twisted K-Theory]] III_,
Annals of Mathematics, Volume 174 (2011) 947-1007
([arXiv:math/0312155](http://arxiv.org/abs/math/0312155), [doi:10.4007/annals.2011.174.2.5](http://dx.doi.org/10.4007/annals.2011.174.2.5))
* [[Daniel S. Freed]], [[Constantin Teleman]],
_Dirac families for loop groups as matrix factorizations_,
Comptes Rendus Mathematique, Volume 353, Issue 5, May 2015, Pages 415-419
([arxiv/1409.6051](http://arxiv.org/abs/1409.6051), [doi:10.1016/j.crma.2015.02.011](https://doi.org/10.1016/j.crma.2015.02.011))
Review:
* [[Valentin Zakharevich]], _K-Theoretic Computation of the Verlinde Ring_, thesis 2018 ([hdl:2152/67663](http://hdl.handle.net/2152/67663), [pdf](http://www.math.jhu.edu/~vzakharevich/research/Dissertation.pdf))
[[!redirects Verlinde rings]]
[[!redirects Verlinde algebra]]
[[!redirects Verlinde algebras]] |
Verma module | https://ncatlab.org/nlab/source/Verma+module |
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###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Given a [[semisimple Lie algebra]] $\mathfrak{g}$ over a field $F$ with [[Borel subalgebra]] $\mathfrak{b}\subset\mathfrak{g}$ a _Verma module_ is the [[induced module]]
$$
U(\mathfrak{g})\otimes_{U(\mathfrak{b})} F_\lambda
$$
where $F_\lambda$ is one dimensional [[representation]] of $U(\mathfrak{b})$ corresponding to the [[character]] $\lambda$ of $\mathfrak{b}$.
This construction is used also for many other Lie algebras with triangular decomposition, for quantized enveloping algebras and many other generalizations.
## Related concepts
* [[BGG resolution]]
[[!redirects Verma modules]]
|
Veronika Hubeny | https://ncatlab.org/nlab/source/Veronika+Hubeny |
* [webpage](https://hubeny.physics.ucdavis.edu/)
## Selected writings
Introducing the [[fluid/gravity correspondence]]:
* [[Sayantani Bhattacharyya]], [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _Nonlinear Fluid Dynamics from Gravity_, JHEP 0802:045, 2008 ([arXiv:0712.2456](https://arxiv.org/abs/0712.2456))
More on the [[fluid/gravity correspondence]]:
* [[Veronika Hubeny]], _The Fluid/Gravity Correspondence: a new perspective on the Membrane Paradigm_, Class. Quant. Grav. 28:114007, 2011 ([arXiv:1011.4948](https://arxiv.org/abs/1011.4948))
* [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _The fluid/gravity correspondence_ ([arXiv:1107.5780](https://arxiv.org/abs/1107.5780))
category: people |
Veronique Godin | https://ncatlab.org/nlab/source/Veronique+Godin |
* [website](http://math.ucalgary.ca/profiles/veronique-godin)
## Related entries
* [[string topology]], [[HCFT]]
* [[ribbon graph]]
category: people |
Verschiebung morphism | https://ncatlab.org/nlab/source/Verschiebung+morphism | (under construction)
##Idea
The *Verschiebung morphism* (German for 'shift') is an endomorphism, related to the [[Frobenius morphism]].
##Witt vectors
For a commutative ring $A$, let $\mathbb{W}(A)$ denote the set $1+T A[[T]]$
of power series in $A$ with constant term one. Given a positive integer $n$, the Verschiebung homomorphism is the additive homomorphism $V_n : \mathbb{W}(A) \to \mathbb{W}(A)$ defined by the formula $V_n(f(T)) = f(T^n)$.
[[!redirects Verschiebung morphisms]]
|
vertex | https://ncatlab.org/nlab/source/vertex |
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### Context
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
A __vertex__ is a _[[point]]_ in a [[graph]] or [[simplicial set]] or similar. In the context of [[simplicial sets]] it is a 0-[[dimension|dimensional]] [[simplex]].
## Terminological remarks
Vertices are also sometimes called **points** (somewhat old-fashioned, e.g. [Harary, Chapter 2](#HararyGraphTheory)) or **nodes** (more frequently in [[computer science]]). In specialized contexts other terms are used. For instance, the two classes of vertices of a [[Petri net]] are called "places" or "species" and "transitions" or "reactions", respectively;
in an [[automaton]] the vertices are called "states";
and for [[quasicategories]] (particular simplicial sets) the vertices are also called [[objects]].
Historically, the use of "vertex" for a point of an abstract graph probably derives from the natural connections between [[graph theory]] and the theory of [[polyhedra]], which have "vertices". One such connection is [[Ernst Steinitz|Steinitz's]] [[Steinitz's theorem|1916 theorem]] on the one-dimensional skeleta of [[polyhedra]] in three-dimensional [[Euclidean space]].
## Related concepts
* [[graph]], [[directed graph]], [[quiver]], [[digraph]]
* [[simplicial set]]
* [[edge]]
* **vertex**, [[simplex]], [[dendrex]]
## References
* Frank Harary: Graph Theory. Addison-Wesley. 1969
{#HararyGraphTheory}
[[!redirects vertex]]
[[!redirects vertices]]
[[!redirects node]]
[[!redirects nodes]]
|
vertex coloring | https://ncatlab.org/nlab/source/vertex+coloring | A (proper) **vertex coloring** of a [[graph]] is a labelling of its vertices with the property that no two vertices sharing an edge have the same label, while an **$n$-coloring** is a vertex coloring using at most $n$ distinct labels. The **chromatic number** $\chi(G)$ of a graph $G$ is the least $n$ such that $G$ has an $n$-coloring.
Vertex coloring may also be expressed in terms of graph homomorphisms: a $n$-coloring of $G$ is the same thing as a graph homomorphism $G \to K_n$ into the complete graph on $n$ vertices.
This formulation makes clear the duality between colorings and _cliques_: an $n$-[[clique]] in $G$ is the same thing as a homomorphism $K_n \to G$.
## Related concepts
* [[knot coloring]]
## References
* Chris Godsil and Gordon Royle (2001), _Algebraic Graph Theory_, Springer.
* Pavol Hell and Jaroslav Nešetřil (2001), _Graphs and Homomorphisms_, Oxford University Press.
Two posts by [[Tom Leinster]] at the n-Category Café on vertex-coloring and a categorical formulation of _Hedetniemi's conjecture_:
* [[Tom Leinster]], ["Colouring a Graph"](https://golem.ph.utexas.edu/category/2013/04/colouring_a_graph.html), n-Category Café post of April 12, 2013.
* [[Tom Leinster]], ["Graph Colouring and Cartesian Closed Categories"](https://golem.ph.utexas.edu/category/2014/12/graph_colouring_and_cartesian.html), n-Category Café post of December 5, 2014.
Other references:
* Wikipedia: [Graph coloring](https://en.wikipedia.org/wiki/Graph_coloring) |
vertex operator algebra | https://ncatlab.org/nlab/source/vertex+operator+algebra |
+-- {: .rightHandSide}
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Vertex operator algebras (or "vertex algebras", for short) are [[algebras]] with a product-operation parameterized by points in the [[complex plane]].
Vertex operator algebras equipped with an [[action]] of the [[Virasoro algebra]] encode the local ([[genus of a surface|genus-0]] behaviour) of [[2-dimensional conformal field theories]]. Here one may think of the [[complex plane]] as the [[Riemann sphere]] and of the $z$-parameterized product operation in the vertex algebras as being the [[n-point function|3-point function]] of the [[2d CFT]] with field insertions at the points 0, $z$ and $\infty$. In fact one vertex operator algebra encodes (only) one chiral/holomorphic half of such a genus-0 2d CFT; a full genus-0 2d CFT is given by the combination of two vertex operator algebras called a _[[full field algebra]]_.
The traditional definition of _vertex operator algebra_ (VOA) is long and tends to be somewhat unenlightening. But due to ([Huang 91](#Huang)) it is now known that vertex operator algebras have equivalently an [[FQFT]]-type characterization which manifestly captures this relation to [[n-point functions]] in the [[2d CFT]]:
There is a [[monoidal category]] or [[operad]] whose [[morphism|morphisms]] are conformal spheres with $n$-punctures marked as incoming and one puncture marked as outgoing (each puncture equipped with a conformally parametrized annular neighborhood). Composition of such spheres is by gluing along punctures. This can be regarded as a category $2Cob_{conf}^0$ of 2-dimensional genus-0 conformal [[cobordism|cobordisms]].
As shown by theorems by [[Yi-Zhi Huang]] and [[Liang Kong]], a vertex operator algebra is precisely a _[[holomorphic representation]]_ of this category, or [[algebra over an operad]] for this [[operad]] i.e. an genus-0 [[conformal field theory|conformal]] [[FQFT]], hence a [[monoidal functor]]
$$
V : 2Cob_{conf}^0 \to Vect
$$
such that its component $V_1$ is a holomorphic function on the moduli space of conformal punctured spheres.
## Standard definition
(under construction) A __vertex algebra__ consists of the following data:
* nonnegatively graded complex vector space $V = \oplus_{n =0}^\infty V_n$
* vacuum vector $|0\rangle\in V_0$
* a shift operator $T: V\to V$
* operation $Y = Y(-,z) : V\to End V [ [z,z^{-1}] ]$
This data satisfy three axioms:
(vacuum)
(translation axiom)
(locality)
In practice, the most important case is when the shift $T$ is related to a conformal vector "the energy momentum tensor" $\omega$ whose Fourier components are related to the generators $L_i$ of the [[Virasoro Lie algebra]] $Vir$. In that case we talk about a __conformal vertex algebra__.
There is a notion of a module over a vertex algebra. A conformal vertex algebra $A$ is said to be __rational__ if every $A$-module is completely reducible. Then it follows that there are only finitely many nonisomorphic simple $A$-modules.
## Properties
{#Properties}
### Category of vertex operator algebras
{#CategoryOfVOAs}
Vertex operator algebras naturally form a [[category]] (see section 2.4 of ([FrenkenHuangLepowsky](#FrenkenHuangLepowsky)). This is naturally a [[monoidal category]] with respect to [[tensor product]] of VOAs (section 2.5).
This is [[equivalence of categories|equivalent]] to
* the category of algebras over the holomorphic punctured sphere operad ([Huang](#HuangCFT));
* the category of vertex operator coalgebras ([Hubbard](#Hubbard)).
### Modular category of modules over a VOA
{#ModularTensorCategories}
The [[category]] of [[modules]]/[[representations]] over a given vertex operator algebra is a [[modular tensor category]], ([Huang](#Huang))
### Goddard-Thorn theorem
* [[Goddard-Thorn theorem]]
### Relation to $A_\infty$-algebras and RG fixed points
{#RelationToAInfinityAlgebras}
A [[functor]] from the category of [[BRST complex|BRST]]-VOAs to that of [[A-infinity algebra]]s is described in
* [[Anton Zeitlin]], _Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory_ ([arXiv](http://arxiv.org/abs/0910.3652))
and argued to algebraically encode the [[effective QFT|effective]] [[string theory]] background encoded by the [[CFT]] given by the VOA.
### Relation to conformal nets
Subject to some conditions, from a vertex operator algebra one may induce a [[conformal net]] and conversely ([Carpi-Kawahigahshi-Longo-Weiner 15](#CarpiKawahigahshiLongoWeiner15), [Carpi 16](#Carpi16)) also ([Gui 18](#Gui18)).
## Examples
A class of examples are _[[current algebras]]_ .
A database of examples is given by ([Gannon-Höhn](#GannonHoehn)).
The [[Moonshine]] module over the [[Griess algebra]] admits the structure of a vertex operator algebra, which has
* rank 24;
* is a self-[[dual object]] in the category of VOAs;
* has trivial degree-1 subspaces.
A conjecture due to
* [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, _Vertex operator algebras and the monster_, Pure and Applied Mathematics __134__, Academic Press, New York 1998.
says that the Moonshine VOA is, up to isomorphism the unique VOA with these properties. See at _[[monster vertex algebra]]_.
## Variants
* [[Poisson vertex algebra]]
* [[rational vertex operator algebra]]
* [[super vertex operator algebra]]
* [[sheaf of vertex operator algebras]], [[vertex operator algebroid]]
## Related concepts
* [[operator product expansion]], [[conformal bootstrap]]
* [[affine Lie algebra]]
* [[conformal field theory]],
* [[factorization algebra]],
* [[automorphism of a vertex operator algebra]]
## References
### General
* [[Victor Kac]], *Vertex algebras for beginners*, Amer. Math. Soc., Univ. Lecture Series **10** (1998) [[ams:ulect-10-r](https://bookstore.ams.org/ulect-10-r), [ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0924.17023&format=complete)]
* [[Edward Frenkel]], [[David Ben-Zvi]]: _Vertex algebras and algebraic curves_, Math. Surveys and Monographs __88__, AMS (2001) xii+348 pp. [[review](http://www.ams.org/journals/bull/2002-39-04/S0273-0979-02-00955-2/S0273-0979-02-00955-2.pdf), [ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1106.17035&format=complete)]
* {#FrenkenHuangLepowsky} [[Igor Frenkel]], [[Yi-Zhi Huang]], [[James Lepowsky]], _On Axiomatic approaches to Vertex Operator Algebras and Modules_, Memoirs of the AMS Vol 104, No 494 (1993)
Brief review:
* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], pp. 4 of: *Algebraic structures in two-dimensional conformal field theory*, Encyclopedia of Mathematical Physics [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]
See also:
* [what-is-the-motivation-for-a-vertex-algebra](http://mathoverflow.net/questions/53988/what-is-the-motivation-for-a-vertex-algebra) - a MathOverflow discussion on the motivation
* [The online database of Vertex Operator Algebras and Modular Categories](http://www.math.ksu.edu/~gerald/voas)
* [[Keith Hubbard]], _The duality between vertex operator algebras and coalgebras, modules and comodules_, [[pdf](http://www.faculty.sfasu.edu/hubbardke/duality.pdf)]
Relation to [[sporadic groups]]:
* [[John Duncan]], _Vertex operator algebras and sporadic groups_, [pdf](http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.1306v1.pdf); _Moonshine for Rudvalis's sporadic group I_, [pdf](http://arxiv.org/PS_cache/math/pdf/0609/0609449v2.pdf)
Vertex operator algebras over [[non-archimedean fields]]:
* [[Victor Kac]], _Non-Archimedean vertex algebras_ [[arXiv:2304.09651](https://arxiv.org/abs/2304.09651)]
Classification of strongly rational holomorphic vertex operator algebras of [[central charge]] $\leq 24$ (of relevance in [[heterotic string theory]] and [[monstrous moonshine]]):
* [[Adrian Norbert Schellekens]], *Meromorphic $c = 24$ Conformal Field Theories*, Commun. Math. Phys. **153** (1993) 159-186 [[doi:10.1007/BF02099044](https://doi.org/10.1007/BF02099044)]
* [[Chongying Dong]], [[Geoffrey Mason]], *Holomorphic vertex operator algebras of small central charges*, Pacific Journal of Mathematics **213** 2 (2004) 253--266 [[arXiv:math/0203005](https://arxiv.org/abs/math/0203005), [doi:10.2140/pjm.2004.213.253](http://dx.doi.org/10.2140/pjm.2004.213.253)]
* Philip Boyle Smith, Ying-Hsuan Lin, Yuji Tachikawa, Yunqin Zheng, *Classification of chiral fermionic CFTs of central charge $\leq 16$* [[arXiv:2303.16917](https://arxiv.org/abs/2303.16917)]
* Brandon C. Rayhaun, *Bosonic rational conformal field theories in small genera, chiral fermionization, and symmetry/subalgebra duality* [[arXiv:2303.16921](https://arxiv.org/abs/2303.16921)]
* [[Gerald Höhn]], [[Sven Möller]], *Classification of self-dual vertex operator superalgebras of central charge at most 24* [[arXiv:2303.17190](https://arxiv.org/abs/2303.17190)]
### As algebras over the holomorphic sphere operad
{#AsOperadAlgebras}
The original article with the interpretation of vertex operator algebras as holomorphic algebras over the holomorphic punctured sphere operad is
* {#Huang91} [[Yi-Zhi Huang]], _Geometric interpretation of vertex operator algebras_, Proc. Natl. Acad. Sci. USA __88__ (1991) pp. 9964-9968 ([doi:10.1073/pnas.88.22.9964](https://doi.org/10.1073/pnas.88.22.9964))
A standard textbook summarizing these results is
* {#HuangCFT} [[Yi-Zhi Huang]], _Two-dimensional conformal geometry and vertex operator algebras_, Progr. in Math. Birkhauser (1997) [gbooks](http://books.google.hr/books?isbn=0817638296)
As mentioned in the [acknowledgements](http://books.google.com/books?id=SUUdknTpYjEC&pg=PR11&lpg=PR11&dq=%22vertex+operator+algebras%22+%22trimble%22&source=bl&ots=v3GHx_ra2M&sig=TW5MzAtDi4n_gJjvUHb6eELoAXQ&hl=en&ei=9jdnSoXqKJiCtge6xqDuDw&sa=X&oi=book_result&ct=result&resnum=4) there, [[Todd Trimble]] and [[Jim Stasheff]] had a hand in making the operadic picture manifest itself here.
See also:
* [[Yi-Zhi Huang]], _Meromorphic open-string vertex algebras_, J. Math. Phys. __54__, 051702 (2013) [[doi:10.1063/1.4806686](http://dx.doi.org/10.1063/1.4806686)]
Other operadic approaches are known, e.g. an earlier one in:
* Bojko Bakalov, Alessandro D'Andrea, [[Victor Kac]], _Theory of finite pseudoalgebras_, Adv. Math. __162__ (2001), no. 1, 1--140, [MR2003c:17020](http://www.ams.org/mathscinet-getitem?mr=2003c:17020)
and even earlier treatments via coloured [[operads]] and [[generalized multicategories]], for references (Snydal, Soibelman, Beilinson-Drinfeld) see [[relaxed multicategory]].
More recently Huang's student [[Liang Kong]] has been developing these results further, generalizing them to open-closed CFTs and to non-chiral, i.e. completely general CFTs. See
* {#Kong08} [[Liang Kong]], _Open-closed field algebras_ Commun. Math. Physics. 280, 207-261 (2008), [math.QA/0610293](http://arxiv.org/abs/math/0610293).
### As factorization algebras
Discussion of vertex operator algebras as [[factorization algebras of observables]] is in section 6.3 and section 6.5 of
* [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis ([pdf](http://math.berkeley.edu/~gwilliam/thesis.pdf))
{#Gwilliam}
* Emily Cliff, _Universal factorization spaces and algebras_, [arxiv:1608.08122](https://arxiv.org/abs/1608.08122)
> We introduce categories of weak factorization algebras and factorization spaces, and prove that they are equivalent to the categories of ordinary factorization algebras and spaces, respectively. This allows us to define the pullback of a factorization algebra or space by an \'etale morphism of schemes, and hence to define the notion of a universal factorization space or algebra. This provides a generalization to higher dimensions and to non-linear settings of the notion of a vertex algebra.
### As coalgebras over a comonad
{#ReferencesAsCoalgebrasOverAComonad}
On vertex operator algebras as [[coalgebras over a comonad]]:
* [[Jethro van Ekeren]], *A quantum field comonad* [[arXiv:2305.18223](https://arxiv.org/abs/2305.18223)]
### Relation to modular tensor categories
The [[representation categories]] of (rational) vertex operator algebras ([[modular tensor categories]]) are discussed in
* {#Huang} [[Yi-Zhi Huang]], *Vertex operator algebras, the Verlinde conjecture and modular tensor categories*, Proc. Nat. Acad. Sci. **102** (2005) 5352-5356 $[$[arXiv:math/0412261](http://arxiv.org/abs/math/0412261), [doi:10.1073/pnas.0409901102](https://doi.org/10.1073/pnas.0409901102)$]$
* {#GannonHoehn} [[Terry Gannon]], [[Gerald Höhn]], Hiroshi Yamauchi, _[The online database of Vertex Operator Algebras and Modular Categories](http://www.math.ksu.edu/~gerald/voas/)_
### As chiral algebras
An algebrogeometric version is due Beilinson and Drinfel'd and called the [[chiral algebra]].
* E. Frenkel, N. Reshetikhin, _Towards deformed chiral algebras_, [q-alg/9706023](http://arXiv.org/abs/q-alg/9706023)
* Ruthi Hortsch, [[Igor Kriz]], Ales Pultr, _A universal approach to vertex algebras_, [arxiv/1006.0027](http://arxiv.org/abs/1006.0027)
Much algebraic insight to algebaric structures in CFT is in unfinished notes
* [[A. Beilinson]], [[B. Feigin]], B. Mazur, _Notes on conformal field theory_, <http://www.math.sunysb.edu/~kirillov/manuscripts.html>
### Relation to 2d conformal field theory
Relation to [[2d CFT]]:
* [[Yi-Zhi Huang]], _Two-dimensional conformal geometry and vertex operator algebras_ Birkhäuser (1997) ([doi:10.1007/978-1-4612-4276-5](https://link.springer.com/book/10.1007/978-1-4612-4276-5))
For [[orbifolds]]:
* [[Yi-Zhi Huang]], _Representation theory of vertex operator algebras and orbifold conformal field theory_ ([arXiv:2004.01172](https://arxiv.org/abs/2004.01172))
Relation specifically to [[conformal nets]]:
* {#CarpiKawahigahshiLongoWeiner15} [[Sebastiano Carpi]], [[Yasuyuki Kawahigashi]], [[Roberto Longo]], Mihály Weiner, _From vertex operator algebras to conformal nets and back_, Memoirs of the American Mathematical Society **254** 1213 (2018) [[doi:10.1090/memo/1213](https://doi.org/10.1090/memo/1213), [arXiv:1503.01260](http://arxiv.org/abs/1503.01260)]
* {#Carpi16} [[Sebastiano Carpi]], _Operator algebras and vertex operator algebras_, Contribution to the Proceedings of the 14th Marcel Grossmann Meeting - MG14 (Rome, 2015) [[arXiv:1603.06742](http://arxiv.org/abs/1603.06742), [doi:10.1142/9789813226609_0508](https://doi.org/10.1142/9789813226609_0508)]
and for [[super vertex operator algebras]]:
* [[Sebastiano Carpi]], [[Tiziano Gaudio]], [[Robin Hillier]], *From vertex operator superalgebras to graded-local conformal nets and back* [[arXiv:2304.14263](https://arxiv.org/abs/2304.14263)]
See also
* James E. Tener, _Representation theory in chiral conformal field theory: from fields to observables_ ([arXiv:1810.08168](https://arxiv.org/abs/1810.08168))
Relation of the corresponding [[ribbon categories]]:
* {#Gui18} Bin Gui, _Categorical extensions of conformal nets_ ([arXiv:1812.04470](https://arxiv.org/abs/1812.04470))
### Deformations
There is an interesting theory of deformation quantization of VOAs from
* [[Pavel Etingof]], [[David Kazhdan]], _Quantization of Lie bialgebras. V. Quantum vertex operator algebras_,
Selecta Math. (N.S.) 6 (2000), no. 1, 105--130, [MR2002i:17022](http://www.ams.org/mathscinet-getitem?mr=2002i:17022)
Deformation quantization of chiral algebras are studied by
* [[Dmitry Tamarkin]], _Deformations of chiral algebras_, Proceedings of the ICM, Beijing 2002, vol. 2, 105--118
A class of "free" vertex algebras are also quantized using [[Batalin-Vilkovisky formalism]], with results important for understanding [[mirror symmetry]], in
* [[Si Li]], _Vertex algebras and quantum master equation_ ([arxiv/1612.01292](https://arxiv.org/abs/1612.01292))
[[!redirects vertex algebra]]
[[!redirects vertex algebras]]
[[!redirects Vertex operator algebra]]
[[!redirects vertex operator algebras]]
[[!redirects VOA]]
[[!redirects VOAs]]
|
vertex operator algebroid | https://ncatlab.org/nlab/source/vertex+operator+algebroid |
+-- {: .rightHandSide}
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###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[sheaf of vertex operator algebras]] which in degree-0 coincides with the [[structure sheaf]] of the base space is a kind of "[[horizontal categorification]]" of [[vertex operator algebra]], hence also called a _vertex operator algebroid_ or _vertex algebroid_ for short.
## References
* [[Vassily Gorbounov]], [[Fyodor Malikov]], [[Vadim Schechtman]], _Gerbes of chiral differential operators. II. Vertex algebroids_, Invent. Math. __155__ (2004), no. 3, 605--680, [MR2005e:17047](http://www.ams.org/mathscinet-getitem?mr=2005e:17047), [math.AG/0003170](http://arxiv.org/abs/math/0003170), [doi](http://dx.doi.org/10.1007/s00222-003-0333-4);
* [[Paul Bressler]], _Vertex algebroids I_ ([arXiv:math/0202185](http://arxiv.org/abs/math/0202185))
[[!redirects vertex operator algebroids]]
[[!redirects vertex algebroid]]
[[!redirects vertex algebroids]] |
vertical categorification | https://ncatlab.org/nlab/source/vertical+categorification | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
#### Higher Category Theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
Roughly speaking, _vertical categorification_ is a procedure in which structures are generalized from the context of [[set theory]] to [[category theory]] or from category theory to [[higher category theory]].
What precisely that means may depend on circumstances and authors, to some extent. The following lists some common procedures that are known as categorification. They are in general different but may in cases lead to the same categorified notions, as discussed in the examples.
See also [[categorification in representation theory]].
## Variants
### As a section of decategorification
What has a more specific definition is the process of [[decategorification]]: this concretely quotients out [[k-morphism]]s in a [[n-category]] to produce a $k$-category for some $k \lt n$, in particular a (possibly large) [[set]] (the set of isomorphism classes or equivalence classes of objects) when $k = 0$.
One may understand _vertical categorification_ as any operation that is a [[section]] of this [[decategorification]] operation, and many examples in the literature are of this kind.
#### Examples
* The maybe archetypical example of categorification as something that is taken by decategorification to the identity operation is the categorification of the [[set]] $\mathbb{N}$ of [[natural numbers]] to the [[category]] [[FinSet]] of [[finite sets]]:
The set $\mathbb{N}$ is a [[0-category]], while $FinSet$ is a [[1-category]]. The [[decategorification|isomorphism classes]] of $FinSet$ however are in canonical bijection with the elements of $\mathbb{N}$:
$$ \mathbb{N} \simeq FinSet_\sim .$$
So $\mathbb{N}$ is the [[decategorification]] of $FinSet$ and accordingly $FinSet$ is a (vertical) categorification of $\mathbb{N}$.
From the point of view of [[category theory]], there is some justification to the converse of this statement: the study of the natural numbers is nothing but the study of the isomorphism classes in $FinSet$. In a way, the very notion of **counting** is about this. This point is made nicely in [BaDo98](http://arxiv.org/abs/math.QA/9802029).
But there are other categorifications of the natural numbers, for instance the category [[FinDimVect]] of [[finite dimensional vector spaces]] (over any given [[field]]). This is not equivalent to [[FinSet]], but still, its set of [[isomorphism classes]] is also canonically in [[bijection]] with the [[natural numbers]]:
$$ \mathbb{N} \;\simeq\; FinDimVect_{/\sim} \,. $$
### As internalization in $n Cat$
Common structures such as [[group]]s, [[ring]]s, etc. may be defined entirely diagrammatically as a collection of objects, morphisms and [[2-morphisms]] in a [[category]] like [[Set]] (the 2-morphisms then are the identities required to hold between the given morphisms).
For instance a [[group]] is an object $G$ in [[Set]] equipped with a morphism $\mu : G \times G \to G$ and so on, and equipped with a 2-morphism from $\mu \circ Id_G \times \mu $ to $\mu \circ \mu \times Id_G$.
In such diagrammatic incarnation, these definitions may be [[internalization|internalized]] into other categories. For instance a group internal to [[Diff]] is a [[Lie group]].
But similarly one can also internalize in categories of higher categories. Let [[Cat]] be the category of (small) categories, regarded as an ordinary category. Then a group internal to [[Cat]] is a [[strict 2-group]]. This is thought of as a notion of a categorified group.
And under the decategorification functor $Cat \to Set$ every categorified group $\mathbf{G}$ in this sense maps to an ordinary group $G$ and one could speak of $\mathbf{G}$ being "a categorification" of $G$. But notice again how highly non-unique such categorification is. In particular, every ordinary group may trivially be regarded as a strict 2-group. As such every ordinary group is at the same time one of its own categorifications, in this sense.
Not every [[2-group]] is a [[strict 2-group]] that is a group object internal to the 1-category [[Cat]]. In general, a [[2-group]] is a _weak_ group object in [[Cat]]: where in [[Set]] all 2-morphisms in the diagrammatic description of the concept of group necessarily had to be identities, in [[Cat]] they could be taken to be non-trivial. But if they are, one will usually want 3-morphisms (which now are necessarily identities) to relate various combinations of these 2-morphisms.
One speaks of this process of categorification using weak internalization of diagrammatic descriptions as categorification of structures _up to coherent higher equivalences_ or _up to coherent higher homotopies_ . One way to make this systematic is discussed below.
#### Examples
In the sense of "weakly internalizing in a higher categorical context" we have the following examples of categorification:
* the categorification of [[sheaf]] is [[stack]], 2-stack, 3-stack, ... [[∞-stack]].
* accordingly, the categorification of ([[sheaf]]-)[[topos]] is ([[stack]]-)2-topos, ... [[∞-stack]] [[(∞,1)-topos]]
* using these [[∞-stack]] [[(∞,1)-topos]] notions one can then speak of [[∞-space]]s [[Lie ∞-groupoid]]s and the like.
* Since [[simplicial object]]s in any category usually [[model category|model]] [[higher category theory|higher categorial]] structures, many simplicial objects may be regarded as secretly being objects in [[higher category theory]]. For instance [[simplicial group]]s, being [[Kan complex]]es, may be thought of as [[∞-groupoid]]s equipped with a strict group structure, i.e. as _strict $\infty$-groups_. By the [[Dold-Kan correspondence]] it follows then that many objects in [[homological algebra]] such as [[chain complex]]es are equivalent incarnations of these simplicial (abelian) groups. This way large parts of [[homological algebra]] may be regarded as being secretly about "categorified linear algebra" in some sense. For instance the [[dg-category]] of [[chain complex]]es in many contexts plays the role of a [[stable (∞,1)-category]] of "$(\infty,1)$-vector spaces". Notably [[L-∞-algebra]]s may be thought of as [[Lie algebra]]s weakly internalized in $(\infty,1)$-vector spaces understood in this sense.
* When understanding higher linear algebra in this sense, there are important $\infty$-categorifications of notions like integral transforms, [[Fourier transform]]s etc. Examples of such categorified linear algebra are [[Fourier-Mukai transform]]s or more generally [[integral transform]]s. A general theory of this is described at [[geometric ∞-function theory]]. Also [[geometric Langlands]] duality fits into this context.
* [categorified Gram-Schmidt process](Gram-Schmidt+process#CategorifiedGramSchmidtProcess)
* [[(∞,1)-category theory|(∞,1)-]]categorification of [[differential calculus]] is [[Goodwillie calculus]]
[[(∞,1)-category theory|(∞,1)-]]categorification of [[derivative]] is [[Goodwillie derivative]]
[[(∞,1)-category theory|(∞,1)-]]categorification of [[chain rule]] is [[Goodwillie chain rule]]
etc.
### As homotopy coherent resolution
The above process of categorification by coherently weak internalization into higher categorical contexts can be made systematic at least in some cases.
If the structures being defined are [[algebra over an operad|algebras over]] an [[operad]] $T$ one may think of regarding $T$ as an [[(∞,1)-operad]]s and then consider the structures of its algebras as algebras over an $(\infty,1)$-operad. These are infinity-categorified versions of the original structures.
#### Examples
For instance this way
* the notion of ordinary associative [[algebra]] is sent to that of $A_\infty$-[[A-∞-algebra|algebra]]
* the notion of ordinary [[Lie algebra]] to $L_\infty$-[[L-∞-algebra|algebra]].
## Contrast to horizontal categorification
Some people also speak of [[horizontal categorification]] as categorification. This is to be distinguished from vertical categorification.
Some people just say 'oidification' for horizontal categorification, in which case it is consistent to speak of vertical categorificaton as just _categorification_ .
## Homotopification versus laxification
(Vertical) categorification can often be usefully decomposed into two operations.
* In **groupoidal categorification**, which may also be called **homotopification** or **groupoidification** (although the latter term also has a [[groupoidification|different meaning]], we allow objects to come with automorphisms, those automorphisms to come with automorphisms, and so on. In the limit, this involves replacing sets by [[∞-groupoids]] or [[homotopy types]] (hence the name "homotopification").
* In **directed categorification**, which may also be called **directification** or **laxification**, we allow morphisms that were previously required to be invertible to instead be noninvertible (i.e. "directed").
If you like [[negative thinking]], then instead of saying that categorification 'replaces [[sets]] by [[categories]]' (to quote Wikipedia), you can say that we replace [[truth values]] by sets, especially the truth values of [[equality|equations]]. That is, we acknowledge that there may be many different ways in which something may be true, and in particular many different ways in which two things may be the same. And then it is meaningful to ask whether two ways in which these things are the same are the same way (and if so, whether two ways that *they* are the same are the same way, etc).
However, when we apply "replace truth values by sets" to the truth values of the equality relation of a set, we end up with a groupoid, since the equality of a set is symmetric. Thus, while two [[elements]] of a set simply may (or may not) be equal, two [[objects]] of a [[groupoid]] may be [[isomorphic]] in many different ways. And while two [[parallel morphisms|parallel isomorphisms]] in a groupoid may be equal, two parallel [[equivalences]] in a $2$-groupoid may be isomorphic in many different ways. Thus, this gives us groupoidal categorification, or homotopification.
To get from groupoids to categories, we need to also allow things which were previously invertible to be noninvertible, i.e. perform "directification." We could also do this first starting from a set, obtaining a poset (in which the symmetric relation "is equal to" has been replaced by the non-symmetric one "is less than or equal to"). Then when we homotopify a poset, we get a category: while one element $x$ of a [[poset]] may precede an element $y$, there may be many different [[morphisms]] from one object $x$ of a category to an object $y$.
This can also be understood naturally in the language of [[(n,r)-categories]]. Recall that an $(n,r)$-category can be defined as an [[∞-category]] in which all cells above dimension $r$ are invertible, and all cells above dimension $n$ are trivial. Thus, groupoidal categorification can be understood as increasing $n$ but keeping $r$ constant, while directification can be understood as increasing $r$ but keeping $n$ constant.
## Related entries
* [[(∞,1)-categorification]]
* [[higher structures]]
* [[categorification in representation theory]]
* [[categorified Dold-Kan correspondence]]
## References
The terminology goes back to
* {#CraneYetter96} [[Louis Crane]], [[David Yetter]],
_Examples of categorification_ ([arXiv:q-alg/9607028](http://arxiv.org/abs/q-alg/9607028))
and was further amplified in
* {#BaezDolan98} [[John Baez]], [[James Dolan|James Dolan]], _Categorification_, in [[Ezra Getzler]], [[Mikhail Kapranov]] (eds.) _Higher Category Theory_, Contemp. Math. 230, American Mathematical Society, Providence, Rhode Island, 1998, pp. 1-36 ([arXiv:math.QA/9802029](http://arxiv.org/abs/math.QA/9802029))
Exposition:
* [[John Baez]], _[What is categorification?](http://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html)_
Lecture notes:
* [[Alistair Savage]], *Introduction to categorification* [[arXiv:1401.6037](https://arxiv.org/abs/1401.6037), slides:[pdf](https://alistairsavage.ca/talks/2014-savage-intro-to-categorification.pdf)]
* Dylan Madden, *Introductory Categorification* (2015) [[pdf](https://warwick.ac.uk/fac/sci/maths/people/staff/madden/1102171m.pdf)]
> (elementary introduction to [[category theory]] leading up to the example of the categorified [[Weyl algebra]])
* [[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)]
> (focus on [[link homology]])
* [[Yuri Ximenes Martins]], part II of _Categorical and Geometrical Methods in Physics_, Master’s thesis, UFMG, 2018. ([pdf](https://repositorio.ufmg.br/bitstream/1843/32053/1/dissertacao_yuri_FINAL.pdf), [slides](https://math-phys.group/wp-content/uploads/2020/08/slides_yuri.pdf))
* [[Yuri Ximenes Martins]], Chapter 4 of _Introdução à Teoria da Homotopia Abstrata_, NEA, 2018. ([arXiv:2008.05302](https://arxiv.org/abs/2008.05302), [hal-02908896v1](https://hal.archives-ouvertes.fr/hal-02908896v1))
A general notion of categorification for structures defined by [[cartesian monads]], which specializes to produce weak [[n-categories]] in the sense of Leinster, can be found here:
* [[D. Bourn]], J. Penon, _Catégorification de structures définies par monade cartésienne_, Cahiers de Topologie et Géométrie Différentielle Catégoriques __46__, no. 1 (2005), p. 2-52, [numdam](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_2005__46_1_2_0).
[[!redirects vertical categorification]]
[[!redirects vertical categorifications]]
[[!redirects categorification]]
[[!redirects categorifications]]
[[!redirects categorify]]
[[!redirects categorified]]
[[!redirects ∞-categorification]]
[[!redirects ∞-categorifications]]
[[!redirects groupoidal categorification]]
[[!redirects groupoidal categorifications]]
|
vertical composition | https://ncatlab.org/nlab/source/vertical+composition |
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#### 2-Category theory
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#### Higher category theory
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[[!include higher category theory - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
## Definition
In a [[2-category]] (and more generally in [[higher category theory]]) [[2-morphism]]s have a composition along the 1-morphisms that they go between.
$$
\array{
& \nearrow &\Downarrow& \searrow
\\
x &&\to&& y
\\
& \searrow &\Downarrow& \nearrow
}
\,.
$$
This is in contrast to the other composition operation, along objects, which is called [[horizontal composition]].
## Examples
* In the 2-category [[Cat]], 2-morphisms are [[natural transformation]]s and vertical composition is the composition of these, the composition in the corresponding [[functor category]].
* For $A$ an [[abelian group]] and $\mathbf{B}^2 A$ its double [[delooping]] [[2-groupoid]], vertical composition (as well as horizontal composition) is given by the group product operation in $A$.
## References
For the notion of vertical composition of [[natural transformations]] (ie. composition in [[functor categories]]):
* [[Saunders MacLane]], §II.4 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]
[[!redirects vertical compositions]]
|
vertical differential form | https://ncatlab.org/nlab/source/vertical+differential+form |
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
=--
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=--
#Contents#
* table of contents
{:toc}
## Definition
### In differential geometry
Let $\pi : P \to X$ be a [[bundle]] in the category [[Diff]] of [[smooth manifold]]s.
The [[dg-algebra]] $\Omega^\bullet_{vert}(P)$ of _vertical differential forms_ on $P$ is the quotient of the [[de Rham complex]] dg-algebra $\Omega^\bullet(P)$ of all forms on $P$, by the dg-ideal of [[horizontal differential forms]], hence of all those forms that vanish when any one vector in their arguments is a [[vertical vector field]] in that it is in the [[kernel]] of the [[differential]] $d \pi : T P \to T X$.
For a trivial bundle $P = X \times F$ the underlying complex of $\Omega^\bullet_{vert}(P)$ is $\wedge^\bullet_{C^\infty(X \times F)} \Gamma(T^* F)$.
## Related concepts
* [[horizontal differential form]]
* [[Lie integration]]
## References
* Philippe Bonnet, Alexandru Dimca, _Relative differential forms and complex polynomials_, Bulletin des Sciences Mathématiques **124** Issue 7 (2000) pp 557-571, doi:[10.1016/s0007-4497(00)01055-1](https://doi.org/10.1016/s0007-4497%2800%2901055-1), ([author pdf](https://web.archive.org/web/20150326174153/https://jones.math.unibas.ch/~bonnet/Publications/Bullscience.pdf))
[[!redirects vertical differential forms]]
|
vertical tangent Lie algebroid | https://ncatlab.org/nlab/source/vertical+tangent+Lie+algebroid | For $\pi : Y \to X$ a morphism of smooth [[manifold]]s, the **vertical tangent Lie algebroid** with respect to $\pi$ is the sub-[[Lie algebroid]] $T_{vert} Y \hookrightarrow T X $ of the [[tangent Lie algebroid]] $T Y$ of $Y$ whose [[Chevalley-Eilenberg algebra]] is the [[dg-algebra]] of horizontal differential forms on $Y$ with respect to $\pi$.
For more details on the construction see the examples section at [[exterior differential system]]s.
The vertical tangent Lie algebroid is the infinitesimal version of the [[schreiber:Ehresmann ∞-connection|vertical path ∞-groupoid]]. It plays a central role in the context of [[Ehresmann connection]]s and [[schreiber:Cartan-Ehresmann ∞-connection|Cartan-Ehresmann ∞-connections]].
|
vertical transformation | https://ncatlab.org/nlab/source/vertical+transformation | # Contents
* tic
{: toc}
## Idea
A **vertical transformation** is an analogue of a [[natural transformation]] which goes between [[double functors]] of [[double categories]], and whose components are vertical arrows and squares. There is a dual notion of **horizontal transformation.
## Definition
If $C$ and $D$ are strict double categories regarded as [[internal categories]] in $Cat$ and $F,G\colon C\to D$ are double functors regarded as internal functors in $Cat$, then a transformation between them is simply an internal natural transformation in $Cat$. Whether this is a vertical or horizontal transformation depends on how we identify double categories with internal categories in $Cat$ (there being two ways).
More explicitly, a vertical transformation $\alpha\colon F\to G$ consists of
* For every object $c\in C$, a vertical arrow
$$ \array{ F c \\ \downarrow ^{\alpha_c} \\ G c}$$
in $D$, which are natural with respect to vertical composition of vertical arrows in $C$.
* For every horizontal arrow $p\colon c_1 \to c_2$ in $C$, a square
$$\array{F c_1 & \overset{F p}{\to} & F c_2\\
^{\alpha_{c_1}}\downarrow & \Downarrow^{\alpha_p}& \downarrow^{\alpha_{c_2}}\\
G c_1& \underset{G p}{\to} & G c_2}$$
in $D$, which are natural with respect to vertical composition of squares in $C$.
* For each $c\in C$, if $1_c\colon c\to c$ is its horizontal identity, then the square $\alpha_{1_c}$ is equal to $1_{\alpha_c}$, the identity square on the arrow $\alpha_c$.
* For $p\colon c_1\to c_2$ and $q\colon c_2\to c_3$, the horizontal composite of $\alpha_p$ and $\alpha_q$ is equal to $\alpha_{q p}$.
The notion of horizontal transformation is dual.
## The double category of double functors
Another characterization of transformations between double categories comes from observing that the 1-category $DblCat$ is [[cartesian closed category|cartesian closed]], and so any two double categories have an exponential $D^C$. The objects of $D^C$ are double functors, its vertical arrows are vertical transformations, and its horizontal arrows are horizontal transformations. Its squares are a sort of "square modification" relating a pair of vertical and a pair of horizontal transformations.
## Generalizations
It is easy to modify the explicit definition to handle the cases when $C$ and $D$ are weak in one direction or the other, and/or when $F$ and $G$ are pseudo functors in one direction or the other, by composing with appropriate coherence constraints. In this way, we obtain many [[2-categories]] of double categories.
It is also easy to define *vertical* transformations between double functors which are *horizontally* lax or colax, and dually. In fact, given double categories $C,D,C',D'$, lax functors $F\colon C\to D$ and $F'\colon C'\to D'$, and colax functors $G\colon C\to C'$ and $G'\colon D\to D'$, we can define a vertical transformation having the shape
$$\array{C & \overset{F}{\to} & D \\
^G\downarrow & \Downarrow& \downarrow^{G'}\\
C'& \underset{F'}{\to} & D'}$$
despite the fact that the composites $G' \circ F$ and $F'\circ G$ do not exist as double functors of any sort. Such transformations are the squares of a large double category $Dbl$ whose objects are double categories, whose horizontal arrows are lax functors, and whose vertical arrows are colax functors. This, in turn, is a special case of a construction which works for algebras over any [[2-monad]].
Finally, we can also define vertical transformations between functors of (horizontally) [[virtual double categories]].
## References
* [[Robert Paré]], *Yoneda theory for double categories*, TAC, 2011 ([pdf](http://emis.library.cornell.edu/journals/TAC/volumes/25/17/25-17.pdf))
[[!redirects horizontal transformation]]
[[!redirects vertical transformations]]
[[!redirects horizontal transformations]]
[[!redirects vertical natural transformation]]
[[!redirects horizontal natural transformation]]
[[!redirects vertical natural transformations]]
[[!redirects horizontal natural transformations]]
|
vertical vector field | https://ncatlab.org/nlab/source/vertical+vector+field |
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#Contents#
* table of contents
{:toc}
## Definition
### Vertical vector fields
Let $\pi : P \to X$ be a [[bundle]] in the [[category]] of [[SmoothManifolds]]. A [[vector field]] $v \in \Gamma(T P)$ is _vertical_ with respect to this bundle if it is in the [[kernel]] of the [[derivative]] $d \pi \colon T P \to T X$.
### Vertical tangent bundle
{#VerticalTangentBundle}
The collection of vertical vectors forms the _vertical tangent bundle_ inside the full [[tangent bundle]], typically denoted $T_\pi P$.
For $\pi \colon P \to B$ a [[smooth function]] between [[smooth manifolds]], its _vertical tangent bundle_ is the [[fiber]]-wise [[kernel]] of the [[differential]] $d \pi$, as shown in the following diagram (e.g [Tu 17 (27.4)](#Tu17), [Berglund 20, p. 16](#Berglund20)):
\begin{tikzcd}[column sep=normal, row sep=small]
T_{\pi} P
\ar[
r,
"
\mathrm{ker}(d \pi)
"
]
&
T P
\ar[
r,
dashed,
"d \pi"
]
\ar[ddr]
&
\pi^\ast
\big(
T B
\big)
\ar[
r
]
\ar[
ddr,
phantom,
"
\mbox{\tiny\textrm(pb)}
"{
description
}
]
\ar[dd]
&
T B
\ar[
dd
]
\\
\\
&
&
P
\ar[
r,
"\pi"{
below
}
]
&
B
\end{tikzcd}
### Horizontal differential forms
A [[differential form]] on $P$ is a [[horizontal differential form]] with respect to $P \to X$ it it _vanishes_ on vertical vector fields.
## Properties
\begin{proposition}\label{SplittingOfTotalSpaceTangentBundle}
If $P \overset{\pi}{\longrightarrow} X$ is a [[surjective submersion]] (for instance a [[smooth manifold|smooth]] [[fiber bundle]]) then the full [[tangent bundle]] of its total space $P$ is [[isomorphism|isomorphic]] to the [[direct sum of vector bundles|direct sum]] of the vertical tangent bundle ([above](#VerticalTangentBundle)) with the [[pullback bundle|pullback]] of the [[tangent bundle]] of the base space:
$$
T P
\;\simeq\;
\big(
\pi^\ast T B
\big)
\oplus_P
\big(
T_\pi P
\big)
$$
\end{proposition}
\begin{proof}
The assumption that $\pi$ is a [[surjective submersion]] implies that $d \pi \colon T P \longrightarrow \pi^\ast T B$ is a [[surjection]] and hence that
$$
0
\to
T_\pi P
\longrightarrow
T P
\overset{d \pi}{\longrightarrow}
\pi^\ast T B
\to
0
$$
is a [[short exact sequence]] of smooth [[real vector bundles]].
Now all [[short exact sequences]] of [[real vector bundles]] over [[paracompact topological spaces]] (such as [[smooth manifolds]]) [[split exact sequence|split]] (by a choice of fiberwise metric, see at _[[short exact sequence of vector bundles]]_), which is the statement to be shown.
\end{proof}
## Examples
\begin{proposition}\label{VerticalTangentBundleOfRealVectorBundle}
**([[vertical tangent bundle]] of a [[real vector bundle]])**
Let $\pi \colon \mathcal{V} \longrightarrow M$ be a [[real vector bundle]]. Then its [[vertical tangent bundle]] is [[isomorphism|isomorphic]] to the its [[pullback bundle|pullback]] along itself:
$$
T_\pi \mathcal{V}
\;\simeq_{{}_{\mathcal{V}}}\;
\pi^\ast \mathcal{V}
\,.
$$
\end{proposition}
(e.g. [tomDieck 00 (6.9)](#tomDieck00), [tomDieck 08 (15.6.7)](#tomDieck2008), [Gollinger 16, inside the proof of Prop. 1.1.9](#Gollinger16))
\begin{prop}\label{StableTangentBundleOfUnitSphereBundle}
**([[stable tangent bundle]] of [[unit sphere bundle]])** \linebreak
The once-[[stable tangent bundle|stabilized]] [[tangent bundle]] of a [[unit sphere bundle]] $S(\mathcal{V})$ in a [[real vector bundle]] $\mathcal{V} \overset{p}{\longrightarrow} M$ (Example \ref{UnitSphereBundles}) over a [[smooth manifold]] $M$ is [[isomorphism|isomorphic]] to the [[pullback bundle|pullback]] of the [[direct sum of vector bundles|direct sum]] of the [[stable tangent bundle]] of the base manifold with that vector bundle:
$$
T S(\mathcal{V}) \times \mathbb{R}
\;
\simeq_{{}_M}
\;
S(p)^\ast
\big(
T M
\oplus_M
\mathcal{V}
\big)
\,.
$$
\end{prop}
This is stated without proof as [Crowley-Escher 03, Fact. 3.1](#CrowleyEscher03), apparently reading between the lines in [Milnor 56, p. 403](#Milnor1956). The key sub-statement that
$$
T_{S(p)} S(\mathcal{V}) \times \mathbb{R}
\;\simeq_{{}_M}\;
\mathcal{V}
$$
is made explicit in [Gollinger 16, Prop. 1.1.9](#Gollinger16)
\begin{proof}\label{ProofOfStableTangentBundleOfUnitSphereBundle}
Consider first the actual [[tangent bundle]] but to the [[open ball]]/[[disk]]-[[fiber bundle]] $D(\mathcal{V})$ that fills the given sphere-fiber bundle: By the standard splitting ([this Prop.](vertical+vector+field#SplittingOfTotalSpaceTangentBundle)) this is the [[direct sum of vector bundles|direct sum]]
$$
T
\big(
D(\mathcal{V})
\big)
\;\simeq\;
\big(
D(p)^\ast T M
\big)
\oplus_M
T_p D(\mathcal{V})
\,,
$$
where $T_p D(\mathcal{V})$ is the [[vertical tangent bundle]] of the disk bundle. But, by definition of disk bundles, that is the restriction of the vertical tangent bundle of the vector bundle $\mathcal{V}$ itself, and that is just the pullback of that vector bundle along itself (by [this Example](vertical+vector+field#SplittingOfTotalSpaceTangentBundle)):
$$
\begin{aligned}
\cdots
&
\simeq\;
\big(
D(p)^\ast T M
\big)
\oplus_M
\big(
D(p)^\ast \mathcal{V}
\big)
\\
& \simeq\;
D(p)^\ast
\big(
T M \oplus_M \mathcal{V}
\big)
\,.
\end{aligned}
$$
To conclude, it just remains to observe that the [[normal bundle]] of the [[n-sphere]]-boundary inside the $(n+1)$-ball is manifestly [[trivial bundle|trivial]], so that the restriction of the tangent bundle of $D(\mathcal{V})$ to $S(\mathcal{V})$ is the [[stable tangent bundle]] of $S(\mathcal{V})$.
\end{proof}
## Related concepts
* [[evolutionary vector field]]
## References
Textbook accounts:
* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], Section VII.1 in Volume 1 _De Rham Cohomology of Manifolds and Vector Bundles_, in: _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973) ([ISBN:978-0-12-302701-6](https://www.elsevier.com/books/connections-curvature-and-cohomology-v1/greub/978-0-12-302701-6))
* {#Tu17} [[Loring Tu]], Section 27.5 in: _Differential Geometry -- Connections, Curvature, and Characteristic Classes_, Springer 2017 ([ISBN:978-3-319-55082-4](https://www.springer.com/gp/book/9783319550824), [pdf](http://www.math.nagoya-u.ac.jp/~richard/teaching/f2018/Tu_geometry.pdf))
See also:
* [[Manifold Atlas]], _<a href="http://www.map.mpim-bonn.mpg.de/Tangent_bundles_of_bundles_(Ex)">Tangent bundles of bundles</a>_
* {#tomDieck00} [[Tammo tom Dieck]], around Satz 6.9 in: _Topologie_, De Gruyter (2000)([doi:10.1515/9783110802542](https://doi.org/10.1515/9783110802542))
* {#tomDieck2008} [[Tammo tom Dieck]], around (15.6.7) _Algebraic topology_, European Mathematical Society, Zürich (2008) ([doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf))
* {#Gollinger16} William Gollinger, Section 1.1.4 in: _Madsen-Tillmann-Weiss Spectra and a Signature Problem for Manifolds_, Münster 2016 ([pdf](https://repositorium.uni-muenster.de/document/miami/7369e8b5-6ae4-4e42-b6f3-4602ec24427a/diss_gollinger.pdf), [[GollingerMTWSpectra.pdf:file]])
* {#Berglund20} Alexander Berglund, _Characteristic classes for families of bundles_ ([arXiv:2012.12170](https://arxiv.org/abs/2012.12170))
* [MO discussion](https://math.stackexchange.com/a/251668/58526)
[[!redirects vertical vector fields]]
[[!redirects vertical vector]]
[[!redirects vertical vectors]]
[[!redirects vertical tangent bundle]]
[[!redirects vertical tangent bundles]]
[[!redirects vertical vector bundle]]
[[!redirects vertical vector bundles]]
[[!redirects vertical cotangent bundle]]
[[!redirects vertical cotangent bundles]]
|
very commutative object | https://ncatlab.org/nlab/source/very+commutative+object |
> under constrution
Given a [[symmetric monoidal (∞,1)-category]] $\mathcal{C}$ then one may consider [[E-∞ space]]-objects inside it.
If the monoidal structure happens to be Cartesian then one may ask for "more" commutativity than that. [[Jacob Lurie]] called that "very commutative" ([MO comment](http://mathoverflow.net/a/118528/381))
(should come down to whether one interprets commutative monoids as a 1-[[algebraic theory]] or an [[(∞,1)-algebraic theory]], i.e. [[(2,1)-algebraic theory of E-infinity algebras]]) |
very large (infinity,1)-sheaf (infinity,1)-topos | https://ncatlab.org/nlab/source/very+large+%28infinity%2C1%29-sheaf+%28infinity%2C1%29-topos | [[!redirects huge (infinity,1)-sheaf (infinity,1)-topos]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Fix a [[Grothendieck universe]] $\mathcal{U}$ and a smaller universe $\mathcal{V} \in \mathcal{U}$. Let $\mathcal{V}$ be the reference-universe, so that sets in $\mathcal{V}$ are called *[[small set]]s*, sets in $\mathcal{U}$ are called *large*, and sets not necessarily in $\mathcal{U}$ are called *very large*.
Write [[∞Grpd]] for the (large) [[(∞,1)-category]] of small [[∞-groupoid]]s and $\infty GRPD$ for the very-large $(\infty,1)$-category of large $\infty$-groupoids.
Then the general procedures of [[universe enlargement]] can be applied to any large $(\infty,1)$-category to produce a very-large one. Specifically, we have the *locally presentable enlargement*: for $C$ a large [[(∞,1)-category]] with small [[(∞,1)-colimit]]s, write
$$
\Uparrow C
\subset
Func(C^{op}, \infty GRPD)
$$
for the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-functors]] that preserves small [[(∞,1)-limit]]s. As described at [[universe enlargement]], if $C$ is locally presentable, then $\Uparrow C$ can be identified with the *naive enlargement*, which consists of the large models of the "theory" of which $C$ consists of the small models.
In particular, when $C$ is an [[(∞,1)-sheaf (∞,1)-topos]] $Sh(S)$ of small (∞,1)-sheaves, then $\Uparrow C$ can be identified with a (∞,1)-category of *large* (∞,1)-sheaves on the same site. (That is, with a suitable accessible left-exact-reflective subcategory of $PSH(S)$ rather than $Psh(S)$---it is not yet known how to specify such a reflective subcategory purely in terms of data on $S$.) Thus, we refer to $\Uparrow C$ as the **very large $(\infty,1)$-sheaf $(\infty,1)$-topos** on $C$.
Note that since every topos can be identified with the category of sheaves on itself for the [[canonical topology]], it is also reasonable to denote $\Uparrow \mathbf{H}$ by $SH(\mathbf{H})$. $\Uparrow C$ can also also be identified with the category of [[ind-object in an (∞,1)-category|ind-objects]] of $\mathbf{H}$, for a suitable [[regular cardinal]] $\kappa$ (namely, the cardinal of $\mathcal{V}$).
## Properties
+-- {: .num_lemma #LeftCosetsDisjoint}
###### Lemma
For every $(\infty,1)$-topos $\mathbf{H}$ there is an [[(∞,1)-functor]]
$$
\Uparrow ((\infty,1)Topos)
\to
\Uparrow \mathbf{H}
$$
that preserves large [[(∞,1)-colimit]]s and finite [[(∞,1)-limit]]s. It is defined by sending $F : (\infty,1)Topos^{op} \to \infty GRPD$ to the composite
$$
\mathbf{H}^{op}
\simeq
((\infty,1)Topos/\mathbf{H})_{et}^{op}
\to
(\infty,1)Topos^{op}
\xrightarrow{F}
\infty GRPD
$$
=--
This is [[Higher Topos Theory|HTT, lemma 6.3.5.21]].
## References
This is discussed in section 6.3 of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .
The definition of $\Uparrow\mathbf{H}$ is in Notation 6.3.5.16 and Remark 6.3.5.17. The relation to ind-objects appears as remark 6.3.6.18.
[[!redirects huge (infinity,1)-sheaf (infinity,1)-topos]]
[[!redirects huge (∞,1)-sheaf (∞,1)-topos]]
[[!redirects very large (∞,1)-sheaf (∞,1)-topos]]
|
Vesna Stojanoska | https://ncatlab.org/nlab/source/Vesna+Stojanoska |
* [webpage](http://math.mit.edu/~vstojanoska/)
category: people
|
Viatcheslav Mukhanov | https://ncatlab.org/nlab/source/Viatcheslav+Mukhanov |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Viatcheslav_Mukhanov)
## selected writings
* {#Mukhanov16} [[Viatcheslav Mukhanov]], _Quantum Universe_, Phys.Usp. 59 (2016) no.10, 1021-1027 ([spire:1507528](http://inspirehep.net/record/1507528), [doi:10.3367/UFNe.2016.07.037857](https://iopscience.iop.org/article/10.3367/UFNe.2016.07.037857/meta), [video recording](https://youtu.be/xUIrj_97i0A))
> In March 2013, following an accurate processing of available measurement data, the Planck Scientific Collaboration published the highest-resolution photograph ever of the early Universe when it was only a few hundred thousand years old. The photograph showed galactic seeds in sufficient detail to test some nontrivial theoretical predictions made more than thirty years ago. Most amazing was that all predictions were confirmed to be remarkably accurate. With no exaggeration, we may consider it established experimentally that quantum physics, which is normally assumed to be relevant on the atomic and subatomic scale, also works on the scale of the entire Universe, determining its structure with all its galaxies, stars, and planets.
## related $n$Lab entries
* [[standard model of cosmology]]
* [[Starobinsky model of cosmic inflation]]
category: people |
Vicente Cortés | https://ncatlab.org/nlab/source/Vicente+Cort%C3%A9s |
* [webpage](http://www.math.uni-hamburg.de/home/cortes/)
## selected writings
* {#Cotes05} [[Vicente Cortés]], _Quaternionic K ̈ahler manifolds_, 2005 ([pdf](https://www2.math.hu-berlin.de/gradkoll/Cortes_vorlesung1_handout.pdf))
(on [[quaternion-Kähler manifolds]])
category: people
|
Victor Bouniakowsky | https://ncatlab.org/nlab/source/Victor+Bouniakowsky | [[!redirects Bouniakowsky]]
Victor Bouniakowsky (Russian: Ви́ктор Я́ковлевич Буняко́вский) was a Russian mathematician.
He defended his PhD thesis in 1825 at Sorbonne supervised by [[Augustin Cauchy]].
He is known for publishing a proof of the [[Cauchy inequality]] in 1859,
also known as the Cauchy–Bouniakowsky–Schwarz inequality.
* [Wikipedia entry](https://en.wikipedia.org/wiki/Viktor_Bunyakovsky)
[[!redirects Bouniakowsky]]
category: people
|
Victor Buchstaber | https://ncatlab.org/nlab/source/Victor+Buchstaber |
* [webpage](http://www.mi.ras.ru/~buchstab/)
* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Victor_Buchstaber)
## Selected writings
Introducing the [[Chern-Dold character]] (specifically for [[complex cobordism cohomology]]):
* {#Buchstaber70} [[Victor Buchstaber]], _The Chern–Dold character in cobordisms. I_,
Russian original: Mat. Sb. (N.S.), 1970 Volume 83(125), Number 4(12), Pages 575–595 ([mathnet:3530](http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=3530&option_lang=eng))
English translation: Mathematics of the USSR-Sbornik, Volume 12, Number 4, AMS 1970 ([doi:10.1070/SM1970v012n04ABEH000939](https://iopscience.iop.org/article/10.1070/SM1970v012n04ABEH000939))
On [[formal groups]] and [[Adams operations]]:
* [[Victor M. Buchstaber]], [[Sergei P. Novikov]], _Formal groups, power systems and Adams operators_, Math. USSR-Sb.13 (1971) 80-116 [[doi:10.1070/SM1971v013n01ABEH001030](https://doi.org/10.1070/SM1971v013n01ABEH001030)]
* [[Victor Buchstaber]], A. P. Veselov, _Chern-Dold character in complex cobordisms and abelian varieties_ ([arXiv:2007.05782](https://arxiv.org/abs/2007.05782))
On [[cobordism cohomology theory]]:
* [[Victor Buchstaber]], _Cobordisms in problems of algebraic topology_, J Math Sci 7, 629–653 (1977) ([doi:10.1007/BF01084983](https://doi.org/10.1007/BF01084983))
On [[Hopf algebras]] and [[homology of H-spaces]] (such as [[homology of loop spaces]]):
* [[Victor Buchstaber]], [[Jelena Grbić]], *Hopf algebras and homology of loop suspension spaces* ([pdf](http://www.personal.soton.ac.uk/jg1u11/Papers/Buchstaber.Grbic.pdf), [[BuchstaberGrbicHopfAlgebrasAndHomology.pdf:file]]) in: V. Buchstaber et al. (eds.), *Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov’s Seminar 2012–2014*, American Mathematical Society Translations - Series 2
Advances in the Mathematical Sciences, 2014 ([ISBN:978-1-4704-1871-7](https://bookstore.ams.org/trans2-234))
## Related entries
* [[cobordism theory]]
* [[bordism ring]]
* [[Chern-Dold character]]
* [[multivalued group]]
[[!redirects Victor M. Buchstaber]]
[[!redirects Victor Buhštaber]]
[[!redirects Victor M. Buhštaber]]
category: people
|
Victor Carmona | https://ncatlab.org/nlab/source/Victor+Carmona |
* [personal page](https://personal.us.es/vcarmona1/)
## Selected writings
On the [[time slice axiom]] in [[homotopical AQFT]]:
* [[Marco Benini]], [[Victor Carmona]], [[Alexander Schenkel]], *Strictification theorems for the homotopy time-slice axiom* [[arxiv:2208.04344](https://arxiv.org/abs/2208.04344)]
category: people |
Victor Galitski | https://ncatlab.org/nlab/source/Victor+Galitski |
* [Institute page](https://umdphysics.umd.edu/people/faculty/current/item/164-galitski.html)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Victor_Galitski)
* [GoogleScholar page](https://scholar.google.com/citations?user=BJDC0QkAAAAJ&hl=en)
## Selected writings
On the [[adiabatic quantum computation|adiabaticity]] of [[topological quantum computation]] by [[braid group|braiding]] of [defect anyons](braid+group+statistics#AsBraidingOfDefects):
* {#ChengGalitskiDasSarma11} [[Meng Cheng]], [[Victor Galitski]], [[Sankar Das Sarma]], *Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors*, Phys. Rev. B **84** (2011) 104529 $[$[arXiv:1106.2549](https://arxiv.org/abs/1106.2549), [doi:10.1103/PhysRevB.84.104529](https://doi.org/10.1103/PhysRevB.84.104529)$]$
category: people |
Victor Ginzburg | https://ncatlab.org/nlab/source/Victor+Ginzburg | **Victor Ginzburg** (in some 1980s articles spelled Ginsburg) is a professor of mathematics at the University of Chicago. His thesis in Moscow was under [[Alexandre Kirillov]]. His main interests are [[representation theory]], especially [[geometric representation theory]], including more recently [[noncommutative algebraic geometry]].
Warning: there is another mathematician (global analysis, [[symplectic geometry]]), *Viktor* Ginzburg (note the English spelling).
* [U. Chicago: faculty research interests](http://www.math.uchicago.edu/research); wikipedia: [Victor Ginzburg](http://en.wikipedia.org/wiki/Victor_Ginzburg); an [article](http://chronicle.uchicago.edu/060608/gta-ginzburg.shtml) in Chicago Chronicle
* N. Chriss, V. Ginzburg, _Representation theory and complex geometry_, Birkhäuser 1997. x+495 pp.
* V. Ginzburg, _Geometric methods in representation theory of Hecke algebras and quantum groups_ (this survey is closely related to the book above), [math.AG/9802004](http://front.math.ucdavis.edu/math.AG/9802004)
* _Lectures on noncommutative geometry_, [math.AG/0506603](http://arxiv.org/abs/math/0506603)
* (with [[Alexander Beilinson|A. Beilinson]]), _[[wall crossing|Wall-crossing]] functors and $D$-modules_, Representation Theory __3__ (electronic), 1--31 (1999)
* A. A. Beĭlinson, V. A. Ginsburg, V. V. Schechtman, _Koszul duality_, J. Geom. Phys. 5 (1988), no. 3, 317–350.
* A. Beilinson, V. Ginzburg, W. Soergel, _Koszul duality patterns in representation theory_, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527.
* V. Ginsburg, _Characteristic varieties and vanishing cycles_, Inv. Math. __84__, 327--402 (1986) [MR87j:32030](http://www.ams.org/mathscinet-getitem?mr=833194), [doi](http://dx.doi.org/10.1007/BF01388811)
* V.G. _Lectures on D-modules_, 1998 Chicago notes, writeup by V. Baranovsky, [pdf](https://www.math.harvard.edu/~gaitsgde/grad_2009/Ginzburg.pdf)
* P. Etingof, V. Ginzburg, _Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism_, Invent. math. __147__, 243--348 (2002) [doi](https://doi.org/10.1007/s002220100171) arXiv:[math.AG/0011114](https://arxiv.org/abs/math/0011114)
* [[Pavel Etingof]], Wee Liang Gan, Victor Ginzburg, Alexei Oblomkov, _Harish-Chandra homomorphisms and [[symplectic reflection algebra]]s for wreath-products_, arXiv:[math.RT/0511489](https://arxiv.org/abs/math/0511489)
* Victor Ginzburg, [[Dmitry Kaledin]], _Poisson deformations of symplectic quotient singularities_, Adv. Math. __186__(1) 1--57 (2003) [doi](https://doi.org/10.1016/j.aim.2003.07.006)
* [[William Crawley-Boevey]], [[Pavel Etingof]], [[Victor Ginzburg]], _Noncommutative geometry and quiver algebras_, Adv. Math. __209__:1 (2007) 274--336 [doi](https://doi.org/10.1016/j.aim.2006.05.004)
* V. Ginzburg, [[T. Schedler]], _Differential operators and BV structures in noncommutative geometry_, Sel. Math. New Ser. 16, 673--730 (2010) [doi](https://doi.org/10.1007/s00029-010-0029-8)
* V. Ginzburg, [[T. Schedler]], _A new construction of cyclic homology_, Proc. Lon. Math. Soc 112 (2016), no. 3, 549--587; [doi](https://doi.org/10.1112/plms/pdw001; arXiv:1201.6635)
He introduced the notion of $d$-Calabi-Yau algebra (and also a construction of a class of examples, [[Ginzburg dg-algebra]]s) in
* _Calabi-Yau algebras_, [arXiv:math.AG/0612139](https://arxiv.org/abs/math/0612139)
category: people
[[!redirects Victor Ginzburg]]
[[!redirects Victor Ginsburg]]
[[!redirects V. Ginzburg]]
[[!redirects Viktor Ginzburg]]
|
Victor Gugenheim | https://ncatlab.org/nlab/source/Victor+Gugenheim |
* [obituatries](https://www.maths.ed.ac.uk/~v1ranick/surgery/uicc/vkamg.txt)
* [biographical information](http://www.maths.ed.ac.uk/~aar/surgery/uicc/vkamg.txt)
## Selected writings
On [[twisting functions]] and ([[semisimplicial set|semi]]-)[[simplicial fiber bundles]]:
* [[Michael G. Barratt]], [[Victor K.A.M. Gugenheim]], [[John C. Moore]], _On semisimplicial fibre-bundles_, Amer. J. Math. __81__ (1959) 639-657 [[doi:10.2307/2372920](https://doi.org/10.2307/2372920), [jstor:2372920](https://www.jstor.org/stable/2372920), MR0111028]
On [[rational homotopy theory]] and the [[fundamental theorem of dg-algebraic rational homotopy theory]]:
* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))
category: people
[[!redirects Victor K.A.M. Gugenheim]]
|
Victor Guillemin | https://ncatlab.org/nlab/source/Victor+Guillemin |
* [webpage](http://math.mit.edu/~vwg/)
* [Wikipedia entry](http://en.wikipedia.org/wiki/Victor_Guillemin)
## Selected writings
On [[differential geometry]] with focus of [[geometric optics]] and [[current (distribution theory)|currents]]:
* [[Victor Guillemin]], [[Shlomo Sternberg]], *Geometric asymptotics*, Mathematical Surveys and Monographs **14**, AMS (1977) [[ams:surv-14](https://bookstore.ams.org/surv-14)]
On [[equivariant de Rham cohomology]] and [[supersymmetry]]:
* {#GuilleminSternberg99} [[Victor Guillemin]], [[Shlomo Sternberg]], _Supersymmetry and equivariant de Rham theory_, Springer, (1999) ([doi:10.1007/978-3-662-03992-2](https://link.springer.com/book/10.1007/978-3-662-03992-2))
On [[differential geometry]] with emphasis on [[differential forms]] and their [[de Rham cohomology]]:
* {#GuilleminHaine19} [[Victor Guillemin]], [[Peter Haine]], _Differential Forms_, World Scientific (2019) [[doi:10.1142/11058](https://doi.org/10.1142/11058), draft:[pdf](https://math.mit.edu/classes/18.952/2018SP/files/18.952_book.pdf)]
## Related entries
* [[symplectic geometry]]
* [[geometric quantization]]
* [[deformation quantization]]
* [[Guillemin-Sternberg geometric quantization conjecture]]
* [[integrability of G-structures]]
* [[Weil algebra]]
* [[equivariant de Rham cohomology]]
category: people |
Victor Gurarie | https://ncatlab.org/nlab/source/Victor+Gurarie |
* [personal page](https://spot.colorado.edu/~gurarie/research.htm)
* [institute page](https://www.colorado.edu/physics/victor-gurarie)
## Selected writings
The triplet algebra of the $c = -2$ Virasoro model as a [[logarithmic CFT]]:
* [[Victor Gurarie]], *Logarithmic Operators in Conformal Field Theory*, Nucl.Phys. B **410** (1993) 535-549 $[$[arXiv:hep-th/9303160](https://arxiv.org/abs/hep-th/9303160), <a href="https://doi.org/10.1016/0550-3213(93)90528-W">doi:10.1016/0550-3213(93)90528-W</a>$]$
On [[2d CFT|2d]] [[logarithmic CFT]] for generalized [[Laughlin wavefunctions]]:
* {#GurarieFlohrNayak97} [[Victor Gurarie]], [[Michael Flohr]], [[Chetan Nayak]], *The Haldane-Rezayi Quantum Hall State and Conformal Field Theory*, Nucl. Phys. B **498** (1997) 513-538 $[$[arXiv:cond-mat/9701212](https://arxiv.org/abs/cond-mat/9701212), [doi:10.1016/S0550-3213%2897%2900351-9](https://doi.org/10.1016/S0550-3213%2897%2900351-9)$]$
category: people
|
Victor Kac | https://ncatlab.org/nlab/source/Victor+Kac |
* [wikipedia page](http://en.wikipedia.org/wiki/Victor_Kac)
## Selected writings
On [[super Lie algebras]], [[Jordan superalgebras]]:
* {#Kac77a} [[Victor Kac]], _Lie superalgebras_, Advances in Math. 26 (1977), no. 1, 8--96 (<a href="https://doi.org/10.1016/0001-8708(77)90017-2">doi:10.1016/0001-8708(77)90017-2</a>)
* {#Kac77b} [[Victor Kac]], _A sketch of Lie superalgebra theory_, Comm. Math. Phys. Volume 53, Number 1 (1977), 31-64. ([euclid:cmp/1103900590](https://projecteuclid.org/euclid.cmp/1103900590))
* {#Kac77} [[Victor Kac]], _Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras_, Comm. Algebra **5** (1977), 1375–1400, ([doi: 10.1080/00927877708822224]( https://doi.org/10.1080/00927877708822224))
See also:
* _The idea of locality_ [[q-alg/9709008](http://arxiv.org/abs/q-alg/9709008)]
On [[affine Lie algebras]]:
* [[Victor G. Kac]], *Infinite Dimensional Lie Algebras*, Progress in Mathematics **44** Springer 1983 [[doi:10.1007/978-1-4757-1382-4](https://link.springer.com/book/10.1007/978-1-4757-1382-4)), Cambridge University Press (1990) [[doi:10.1017/CBO9780511626234](https://doi.org/10.1017/CBO9780511626234)]
their relation to [[modular forms]]:
* [[Victor G. Kac]], [[Dale H. Peterson]], *Affine Lie algebras and Hecke modular forms*, Bull. Amer. Math. Soc. (N.S.) **3** 3 (1980) 1057-1061 [[bams:1183547694](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-3/issue-3/Affine-Lie-algebras-and-Hecke-modular-forms/bams/1183547694.full)]
* [[Victor G. Kac]], [[Dale H. Peterson]], *Infinite-dimensional Lie algebras, theta functions and modular forms*, Advances in Mathematics **53** 2 (1984) 125-264 [<a href="https://doi.org/10.1016/0001-8708(84)90032-X">doi:10.1016/0001-8708(84)90032-X</a>]
* [[Victor G. Kac]], [[Minoru Wakimoto]], *Modular and conformal invariance constraints in representation theory of affine algebras*, Advances in Mathematics **70** 2 (1988) 156-236 [<a href="https://doi.org/10.1016/0001-8708(88)90055-2">doi:10.1016/0001-8708(88)90055-2</a>, [spire:275458](https://inspirehep.net/literature/275458)]
On non-integrable but "admissible" [[irreps]] of [[affine Lie algebras]] (cf. *[fractional level WZW models](Wess-Zumino-Witten+model#FractionalLevelWZWModelReferences)*):
* [[Victor G. Kac]], [[Minoru Wakimoto]], *Modular invariant representations of infinite-dimensional Lie algebras and superalgebras*, PNAS **85** 14 (1988) 4956-4960 [[doi:10.1073/pnas.85.14.4956](https://doi.org/10.1073/pnas.85.14.4956)]
* [[Victor G. Kac]], [[Minoru Wakimoto]], *Classification of modular invariant representations of affine algebras*, p. 138-177 in V. G. Kač (ed.): *Infinite dimensional Lie algebras and groups*, Advanced series in Mathematical physics **7**, World Scientific (1989) [[pdf](https://math.mit.edu/~kac/not-easily-available/admissible.pdf), [cds:268092](https://cds.cern.ch/record/268092)]
On [[vertex operator algebras]]:
* [[Victor Kac]], *Vertex algebras for beginners*, Amer. Math. Soc., Univ. Lecture Series **10** (1998) [[ams:ulect-10-r](https://bookstore.ams.org/ulect-10-r), [ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0924.17023&format=complete)]
On [[vertex operator algebras]] over [[non-archimedean fields]]:
* [[Victor Kac]], _Non-Archimedean vertex algebras_ [[arXiv:2304.09651](https://arxiv.org/abs/2304.09651)]
category: people
[[!redirects V. G. Kac]]
[[!redirects Victor G. Kac]]
[[!redirects Victor Kac]]
[[!redirects V. G. Kac]]
[[!redirects Victor G. Kac]]
|
Victor Maslov | https://ncatlab.org/nlab/source/Victor+Maslov |
A Soviet and Russian mathematical physicist. Discovered
[[Maslov index]], [[Fourier integral operator]]s (in terms of "canonical operator"), [[Maslov-Feynman calculus]], systematic approach to [[WKB method]] to all orders, corresponding "Maslov" quantization conditions and the complex WKB method. He also works on idempotent analysis.
* [Wikipedia](https://en.wikipedia.org/wiki/Victor_Pavlovich_Maslov)
* _[Fiftieth anniversary of research and teaching by Victor Pavlovich Maslov]_, Theoretical and Mathematical Physics, 155(2): 674--677 (2008), transl. from Teoreticheskaya i Matematicheskaya Fizika __155__:2, pp. 197--201 (2008) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/maslovbio.pdf))
## Literature
### Related entries
* [[Lagrangian submanifold]]
* [[Maslov index]]
* [[Maslov dequantization]]
* [[semiclassical approximation]]
### Books
* V. P. Maslov, _Theory of perturbations and asymptotic methods_ (in Russian), Mos. Gos. Univ., Moscow (1965)
### Papers
* [[V. P. Maslov]], O. Yu. Shvedov, _Large-N expansion as a semiclassical approximation to the third-quantized theory_, Physical Review D60(10) 105012 [doi](https://doi.org/10.1103/PhysRevD.60.105012)
* G. L. Litvinov, V.P. Maslov, _The correspondence principle for Idempotent Calculus and some computer applications_, In book Idempotency J. Gunawardena (Editor), Cambridge University Press, Cambridge, 1998, p.420-443
category: people
[[!redirects Victor Pavlovich Maslov]]
[[!redirects V. P. Maslov]]
[[!redirects Maslov]] |
Victor Nistor | https://ncatlab.org/nlab/source/Victor+Nistor |
* [institute page](https://iecl.univ-lorraine.fr/membre-iecl/nistor-victor-2/)
* [Google Scholar page](https://scholar.google.com/citations?user=F9gokPUAAAAJ&hl=en)
## Selected writings
On the [[cyclic homology]] of the [[groupoid convolution algebra]] of [[étale groupoid|étale]] [[Lie groupoids]], hence of [[orbifolds]]:
* [[Jean-Luc Brylinski]], [[Victor Nistor]], *Cyclic cohomology of étale groupoids*, K-theory 8.4 (1994): 341-365 ([pdf](https://www.researchgate.net/profile/Victor-Nistor/publication/226520448_Cyclic_cohomology_of_etale_groupoids/links/5509d9d30cf20ed529e227b0/Cyclic-cohomology-of-etale-groupoids.pdf), [[BrylinskiNistor_CyclicCohomology.pdf:file]])
category: people |
Victor Ostrik | https://ncatlab.org/nlab/source/Victor+Ostrik | __Victor Ostrik__ is a mathematicain at University of Oregon ([web](http://pages.uoregon.edu/vostrik/)). He has been one of the main figures in the development of the theory of [[fusion categories]].
## Selected writings
On [[tensor categories]]:
* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], _Topics in Lie theory and Tensor categories -- 9 Tensor categories_, Lecture notes (spring 2009) ([pdf](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec09.pdf) [web](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/))
* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) $[$[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205)$]$
On [[Deligne's theorem on tensor categories]]:
* {#Ostrik04} [[Victor Ostrik]], _Tensor categories (after P. Deligne)_ ([arXiv:math/0401347](http://arxiv.org/abs/math/0401347))
On [[braided monoidal category|braided]] [[fusion categories]]:
* {#EtingofNikshychOstrik05} [[Pavel Etingof]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On fusion categories*, Annals of Mathematics Second Series, Vol. 162, No. 2 (Sep., 2005), pp. 581-642 ([arXiv:math/0203060](http://arxiv.org/abs/math/0203060), [jstor:20159926](https://www.jstor.org/stable/20159926))
* {#DrinfeldGelakiNikshychOstrik10} [[Vladimir Drinfeld]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On braided fusion categories I*, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 ([doi:10.1007/s00029-010-0017-z](https://doi.org/10.1007/s00029-010-0017-z))
On [[Frobenius exact categories]]:
* {#EtingofOstrik21} [[Pavel Etingof]], [[Victor Ostrik]], *On the Frobenius functor for symmetric tensor categories in positive characteristic*, J. für die reine und angewandte Mathematik **773** (2021) 165–198 $[$[arXiv:1912.12947](https://arxiv.org/abs/1912.12947)$]$
and 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)$]$
## Related entries
* [[monoidal category]],
* [[fusion category]], [[weak Hopf algebra]]
category: people
[[!redirects Ostrik]]
[[!redirects V. Ostrik]]
[[!redirects Viktor Ostrik]]
|
Victor Petrov | https://ncatlab.org/nlab/source/Victor+Petrov |
* [webpage](http://thd.pnpi.spb.ru/~victorp)
## Selected writings
* [[Dmitri Diakonov]], [[Victor Petrov]], _Exotic baryon resonances in the Skyrme model_ ([arXiv:0812.1212](https://arxiv.org/abs/0812.1212), [doi:10.1142/9789814704410_0004](https://doi.org/10.1142/9789814704410_0004)), Chapter 3 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))
> It is astounding that [[Tony Skyrme|Skyrme]] had suggested [[Skyrmion|his model]] as early as in 1961 before it has been generally accepted that [[pions]] are (pseudo) [[Goldstone bosons]] associated with the [[spontaneous symmetry breaking|spontaneous breaking]] of [[chiral symmetry]], and of course long before [[QCD|Quantum Chromodynamics (QCD)]] has been put forward as the microscopic theory of [[strong nuclear force|strong interactions]].
> The revival of the Skyrme idea in 1983 is [due to Witten](Skyrmion#Witten83) who explained the _raison d'ˆetre_ of the [[Skyrmion|Skyrme model]] from the viewpoint of [[QCD]]. In the [[chiral perturbation theory|chiral limit]] when the light [[quark]] masses $m_u$, $m_d$, $m_s$ tend to zero, such that the octet of the pseudoscalar mesons [[pion|π]], [[kaon|K]] , η become nearly massless (pseudo) [[Goldstone bosons]], they are the lightest degrees of freedom of [[QCD]]. The [[effective field theory|effective]] chiral Lagrangian (EχL) for pseudoscalar mesons, understood as an infinite expansion in the derivatives of the pseudoscalar (or chiral) fields, encodes, in principle, full information about QCD. The famous two-term Skyrme Lagrangian can be understood as a low-energy truncation of this infinite series. Witten has added an important four-derivative [[WZW-term|Wess–Zumino term]] to the original Skyrme Lagrangian and pointed out that the overall coefficient in front of the EχL is proportional to the number of quark [[color charge|colours]] $N_c$.
> $[...]$
> Soon after Witten's work it has been realized that it is possible to bring the Skyrme model and the Skyrmion even closer to QCD and to the more customary language of constituent quarks. It has been first noticed $[$[6](#DiakonovEides83), [7a](#DharWadia84), [7b](#DarShankarWadia85), [8](#DiakonovPetrov86)$]$ that a simple chiral invariant Lagrangian for massive (constituent) quarks $Q$ interacting with the octet chiral field $\pi_A$ $(A = 1, ..., 8)$,
> $$\mathcal{L} = \overline{Q} \left( \mathrlap{\partial}/ - M e^{ \tfrac{i \pi^A \lambda^A \gamma_5}{F_\pi} } \right) Q$$
> induces, via a quark loop in the external pseudoscalar fields (see Fig. 3.1), the EχL whose lowest-derivative terms coincide with the Skyrme Lagrangian, including
automatically the Wess–Zumino term, with the correct coefficient!
<center>
<img src="https://ncatlab.org/nlab/files/DiakonovPetrovQuarkLoop.jpg" width="600">
</center>
> $[...]$
> The condition that the winding number of the trial field is unity needs to be imposed to get a deeply bound state, that is to guarantee that the baryon number is unity. $[$[9](#DiakonovPetrovPobylitsa88)$]$ The Skyrmion is, thus, nothing but the **mean chiral field binding quarks in a baryon**.
<center>
<img src="https://ncatlab.org/nlab/files/DiakonovPetrovMeanField.jpg" width="600">
</center>
\linebreak
* {#DiakonovEides83} [[Dmitri Diakonov]], Michael I. Eides, _Chiral Lagrangian from a functional integral over quarks_, JETP Letters 38.7 (1983): 433-436 ([pdf](http://jetpletters.ac.ru/ps/1483/article_22635.pdf), [[DiakonovEides83.pdf:file]])
* {#DharWadia84} A. Dhar, Spenta R. Wadia, _Nambu—Jona-Lasinio Model: An Effective Lagrangian for Quantum Chromodynamics at Intermediate Length Scales_, Phys. Rev. Lett. 52, 959 (1984) ([doi:10.1103/PhysRevLett.52.959](https://doi.org/10.1103/PhysRevLett.52.959))
* {#DarShankarWadia85} Avinash Dhar, R. Shankar, Spenta R. Wadia, _Nambu–Jona-Lasinio–type effective Lagrangian: Anomalies and nonlinear Lagrangian of low-energy, large-N QCD_, Phys. Rev. D 31, 3256 (1985) ([doi:10.1103/PhysRevD.31.3256](https://doi.org/10.1103/PhysRevD.31.3256))
* {#DiakonovPetrov86} [[Dmitri Diakonov]], [[Victor Petrov]], _A theory of light quarks in the instanton vacuum_, Nuclear Physics B Volume 272, Issue 2, 21 July 1986, Pages 457-489 (<a href="https://doi.org/10.1016/0550-3213(86)90011-8">doi:10.1016/0550-3213(86)90011-8</a>)
* {#DiakonovPetrovPobylitsa88} [[Dmitri Diakonov]], [[Victor Petrov]], P.V. Pobylitsa, _A Chiral Theory of Nucleons_, Nucl. Phys. B306 (1988) 809 ([spire:247700](http://inspirehep.net/record/247700), <a href="https://doi.org/10.1016/0550-3213(88)90443-9">doi:10.1016/0550-3213(88)90443-9</a>)
category: people |
Victor Porton | https://ncatlab.org/nlab/source/Victor+Porton |
Victor Porton is the owner of a [math site](http://www.mathematics21.org/) with some research (mainly in [[general topology]]) and a [math weblog](https://portonmath.wordpress.com/2009/05/). He wishes to be nominated for the Abel Prize ([link](http://www.mathematics21.org/abel-prize.html)).
category: people
|
Victor Snaith | https://ncatlab.org/nlab/source/Victor+Snaith | __Victor P. Snaith__ was an algebraist, algebraic topologist and algebraic geometer (born 1944, died 2021), prof. emeritus at Sheffield
* [personal webpage](http://victor-snaith.staff.shef.ac.uk),
* [institute page](http://maths.dept.shef.ac.uk/maths/staff_info.php?id=34).
## Selected writings
On [[infinite loop spaces]]:
* [[Ib Madsen]], [[Victor Snaith]], [[Jørgen Tornehave]], *Infinite loop maps in geometric topology*, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 81, Issue 3, (1977)([doi:10.1017/S0305004100053482](https://doi.org/10.1017/S0305004100053482))
On [[algebraic cobordism]] and introducing [[Snaith's theorem]]:
* Algebraic cobordism and $K$-theory. Mem. Amer. Math. Soc. __21__ (1979), no. 221, vii+152 pp.
* Algebraic $K$-theory and localised stable homotopy theory. Mem. Amer. Math. Soc. __43__ (1983), no. 280, xi+102 pp.
* Stable homotopy around the Arf-Kervaire invariant, Progress in Mathematics __273__, Birkhäuser Basel, 2009, ISBN10:3764399031
* _Infinite loop maps and the complex J-homomorphism_, Bull. Amer. Math. Soc. __82__, 3 (1976), 508-510 [MR0410741](http://www.ams.org/mathscinet-getitem?mr=0410741), [euclid](http://projecteuclid.org/euclid.bams/1183537922)
On the [[J-homomorphism]]:
* _The complex J-homomorphism. I_, Proc. Lond. Math. Soc., III. Ser. 34, 269-302 (1977), (Zbl 0344.55016)
* _Geometric dimension of complex vector bundles_, Serie
notas у simposia del Мех. Mat. Soc. 1 (1975) 199-227.
* _Towards algebraic cobordism_, Bull. A, M.. Soc. 83, 3
(1977).
* D. Gepner, V. Snaith, _On the motivic spectra representing algebraic cobordism and algebraic K-theory_, Doc. Math., 14:359–396 (electronic), 2009, [pdf](http://www.math.uni-bielefeld.de/documenta/vol-14/14.pdf)
## Related pages
* [[algebraic cobordism]]
* [[Snaith's theorem]]
[[!redirects V. P. Snaith]]
[[!redirects Snaith]]
|
Victor Tourtchine | https://ncatlab.org/nlab/source/Victor+Tourtchine |
* [webpage](https://www.math.ksu.edu/~turchin/)
## Selected writings
On [[graph complex]]-models for the [[rational homotopy type]] of [[configuration spaces of points]]:
* [[Pascal Lambrechts]], [[Victor Tourtchine]], _Homotopy graph-complex for configuration and knot spaces_, Transactions of the AMS, Volume 361, Number 1, January 2009, Pages 207–222 ([arxiv:math/0611766](https://arxiv.org/abs/math/0611766))
[[!redirects Victor Turchin]]
[[!redirects Виктор Турчин]] |
Victor V. Albert | https://ncatlab.org/nlab/source/Victor+V.+Albert |
* [personal page](https://sites.google.com/site/victorvalbert)
* [GoogleScholar page](https://scholar.google.com/citations?user=hyddAfkAAAAJ&hl=en)
## Selected writings
On ([[quantum error correction|quantum]]) [[error correcting codes]]:
* [errorcorrectionzoo.org](https://errorcorrectionzoo.org)
Specifically on [[Absorption-Emission codes]]:
* Shubham Jain, Eric Hudson, Wesley Campbell, [[Victor V. Albert]], *Æ codes* [[arxiv:2311.12324](https://arxiv.org/abs/2311.12324)]
category: people
[[!redirects Victor Albert]] |
Victoria Lebed | https://ncatlab.org/nlab/source/Victoria+Lebed | [website](http://www.maths.tcd.ie/~lebed/)
##Slides:
[Leeds 5-8 July 2016](http://www1.maths.leeds.ac.uk/~matzk/Modelling%20TPM%20Resources_files/Crossed.pdf)
category:people
|
Vidas Regelskis | https://ncatlab.org/nlab/source/Vidas+Regelskis |
Vidas Regelski is an EPRSC research fellow in [[mathematics]] at University of Surrey.
* [webpage](http://www.surrey.ac.uk/maths/people/vidas_regelskis/)
## related $n$Lab entries
* [[AdS3-CFT2 and CS-WZW correspondence]]
category: people
|
Vidhyanath K. Rao | https://ncatlab.org/nlab/source/Vidhyanath+K.+Rao |
* [institute page](https://math.osu.edu/people/rao.3)
## Selected writings
On [[category theory and foundations]]:
* {#Rao06} [[Vidhyanath K. Rao]], _On Doing Category Theory within Set Theoretic Foundations_ in: Giandomenico Sico (ed.), _What is category theory?_, Polimetrica (2006) [[[Rao-CategoryTheory.pdf:file]], [semanticscholar](https://www.semanticscholar.org/paper/What-is-category-theory-Sica/7737401202b220f89ae9c11cf9af65995a4dcf50)],
category: people
[[!redirects Vidhyanath Rao]]
|
vielbein | https://ncatlab.org/nlab/source/vielbein |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _vielbein_ or _solder form_ on a [[manifold]] $X$ is a linear identification of the [[tangent bundle]] with a [[vector bundle]] with explicit ([[pseudo-orthogonal structure|pseudo-]])[[orthogonal structure]].
Any such choice encodes a ([[pseudo-Riemannian metric|pseudo-]])[[Riemannian metric]] on $X$.
## Definition
There are different equivalent perspectives on the notion of vielbein that are closely related:
* [In terms of Cartan geometry](#InTermsOfCartanGeometry)
* [In terms of orthogonal structure](#InTermsOfOrthogonalStructure)
### In terms of Cartan geometry
{#InTermsOfCartanGeometry}
Let $X$ be a [[smooth manifold]] of [[dimension]] $d$. For definiteness we assume here that $X$ is [[orientation|oriented]], but this is not necessary.
A [[Lie algebra valued 1-form]]
$$
(E,\Omega) : T X \to \mathfrak{iso}(d)
$$
with values in the [[Poincaré Lie algebra]] encodes a [[pseudo-Riemannian metric]] on $X$ (if non-degenerate, at least). In this context the component
$$
E : T X \to \mathbb{R}^d
$$
of the [[connection on a bundle|connection]] 1-form is called the _vielbein_ . It encodes the metric by
$$
g = \langle E \otimes E\rangle \in Sym^2_{C^\infty(X)} \Gamma(T^* X)
\,,
$$
where $\langle -,-\rangle : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ is the canonical [[bilinear form]].
In other words, Given an $(SO(d) \hookrightarrow ISO(d))$-[[Cartan connection]] on $X$, the _vielbein_ is the isomorphism in the definition of Cartan connection.
For $d=4$ this is the _vierbein_ , for $d = 3$ the _dreibein_ , etc.
This terminology is used notably in the context of the _[[first-order formulation of gravity]]_.
### In terms of orthogonal structure
{#InTermsOfOrthogonalStructure}
We discuss here how a choice of vielbein on a [[manifold]] is equivalently the [[reduction of structure groups|reduction of the structure group]] of the [[tangent bundle]] from the [[general linear group]] $GL(n)$ to its [[maximal compact subgroup]], the [[orthogonal group]].
The following also introduces the description of this in terms of [[smooth infinity-groupoid|smooth]] [[twisted cohomology|twisted]] [[cohomology]]. While of course this is not necessary to understand vielbeins, it does give a very natural conceptual description with the advantage that it seamlessly generalizes to notions of _[[generalized vielbein]]_ fields and generally to [[twisted differential c-structures]].
#### The class of the tangent bundle
{#ClassOfTheTangentBundle}
For completeness, we first review how the [[tangent bundle]] of a [[smooth manifold]] is naturally incarnated as a [[cocycle]] in $GL(n)$-valued [[Cech cohomology]] and how this in turn is naturally formulated in terms of [[Lie groupoids]]/[[differentiable stack|smooth]] [[moduli stacks]]. The reader familiar with these basics should skip to the [next section](#ReductionOfTheStructureGroup).
Let $X$ be a [[smooth manifold]] of [[dimension]] $n$.
By definition this means that there is an [[atlas]] $\{\phi_i^{-1} : \mathbb{R}^n \simeq U_i \hookrightarrow X\}$ of [[coordinate charts]]. On each overlap $U_i \cap U_j$ of two [[coordinate charts]] the [[partial derivatives]] of the corresponding [[coordinate transformations]]
$$
\phi_j\circ \phi_i^{-1}
:
U_i \cap U_j \subset \mathbb{R}^n \to \mathbb{R}^n
$$
form the [[Jacobian matrix]] of [[smooth functions]]
$$
((\lambda_{i j})^{\mu}{}_{\nu})
\coloneqq
\left[\frac{d}{d x^\nu} \phi_j \circ \phi_i^{-1} (x^\mu)
\right]
:
U_i \cap U_j
\to
GL_n
$$
with values in invertible [[matrices]], hence in the
[[general linear group]] $GL(n)$. By construction (by the [[chain rule]]), these functions satisfy on triple overlaps of coordinate charts the [[matrix product]] equations
$$
(\lambda_{i j})^\mu{}_\lambda (\lambda_{j k})^\lambda{}_{\nu}
=
(\lambda_{i k})^\mu{}_{\nu}
\,,
$$
hence the equation
$$
\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}
$$
in the [[group]] $C^\infty(U_i \cap U_j, GL(n))$ of smooth $GL(n)$-valued functions on the chart overlap.
This is the [[cocycle]] condition for a smooth [[Cech cohomology|Cech cocycle]] in degree 1 with coefficients in $GL(n)$ (precisely: with coefficients in the [[sheaf]] of smooth functions with values in $GL(n)$ ):
$$
[(\lambda_{i j})] \in H^1_{smooth}(X, GL_n)
\,.
$$
It is useful to formulate this statement in the language of [[Lie groupoids]]/[[differentiable stacks]].
* $X$ itself is trivially a Lie groupoid $(X \stackrel{\to}{\to} X)$;
* from the atlas $\{U_i \to X\}$ we get the corresponding [[Cech groupoid]]
$$
C(\{U_i\}) =
(\coprod_{i, j} U_i \cap U_j \stackrel{\to}{\to} \coprod_i U_i)
$$
* any [[Lie group]] $G$ induces its [[delooping]] [[Lie groupoid]]
$$
\mathbf{B}G
=
\left(
G \stackrel{\to}{\to} *
\right)
\,.
$$
The above situation is neatly encoded in the existence of a [[diagram]] of Lie groupoids of the form
$$
\array{
C(\{U_i\}) &\stackrel{\lambda}{\to}& \mathbf{B} GL(n).
\\
{}^{\mathllap{\simeq}}\downarrow
\\
X
}
\,,
$$
where
* the left morphism is [[stalk]]-wise (around small enough [[neighbourhoods]] of each point) an [[equivalence of groupoids]];
* the horizontal functor has as components the functions $\lambda_{i j}$ and its functoriality is the cocycle condition $\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}$.
We want to think of such a diagram as being directly a morphism of [[smooth infinity-groupoid|smooth groupoids]]
$$
T X : X \to \mathbf{B} GL(n) \;\; \in \mathbf{H}
\,.
$$
This is true in the [[(2,1)-category]] $\mathbf{H}$ in which stalkwise equivalences $W \subset Mor(PSh(SmthMfd, Grpd))$ have been [[simplicial localization|formally inverted]] to become [[homotopy equivalences]].
Since all real [[vector bundles]] on $X$ are encoded by such morphisms, as are their [[gauge transformations]], we say that $\mathbf{B} GL(n)$ is the [[moduli stack]] for real vector bundles.
Of course there is a "smaller" Lie groupoid that also classifies real vector bundles. Passing to this "smaller" Lie groupoid is what the choice of vielbein accomplishes, to which we now turn.
#### Reduction of the structure group
{#ReductionOfTheStructureGroup}
Consider the defining inclusion of the [[orthogonal group]] into the [[general linear group]]
$$
O(n) \hookrightarrow GL(n)
\,.
$$
We may understand this inclusion geometrically in terms of the canonical [[metric]] on $\mathbb{R}^n$, but we may also understand it purely Lie theoretically as the
the inclusion of the [[maximal compact subgroup]] of $GL(n)$. This makes manifest that the inclusion is trivial at the level of [[homotopy theory]] (it is a [[homotopy equivalence]]) and hence _only_ encodes geometric information.
The inclusion induces a corresponding morphism of moduli stacks
$$
\mathbf{c} : \mathbf{B} O(n) \to \mathbf{B} GL(n)
\,.
$$
A choice of [[orthogonal structure]] on $T X$ a [[G-structure]] for $G = O(n)$, hence is a factorization of the above $GL(n)$-valued cocycle through $\mathbf{c}$, up to a smooth [[homotopy]].
$$
\array{
X &&\stackrel{h}{\to}&& \mathbf{B} O(n)
\\
& {}_{\mathllap{\lambda}}\searrow &\swArrow_{E^{-1}}& \swarrow_{\mathrlap{\mathbf{c}}}
\\
&& \mathbf{B}GL(n)
}
\,.
$$
This consists of two pieces of data
* the morphism $h$ is a $O(n)$-valued 1-cocycle -- a collection of _orthogonal transition functions_ -- hence on each overlap of coordinate patches a smooth function
$$
((h_{i j}){}^a{}_b) : U_i \cap U_j \to O(n)
$$
such that
$$
h_{i j} \cdot h_{j k} = h_{i k}
\,;
$$
* the homotopy $E$ is on each chart a function
$$
E_i = ((E_i)^a{}_\mu) : U_i \to GL(n)
$$
* such that on each double overlap it intertwines the transition functions $\lambda$ of the tangent bundle with the new orthogonal transition functions, meaning that the equation
$$
(E_i)^a{}_{\mu} (\lambda_{i j})^{\mu}{}_\nu
=
(h_{i j})^a{}_b (E_j]^b{}_\nu
$$
holds. This exhibits the [[natural transformation|naturality]] [[diagram]] of $E$:
$$
\array{
* &\stackrel{E_i}{\to}& *
\\
{}^{\mathllap{h_{i j}}}\downarrow && \downarrow^{\mathrlap{\lambda_{i j}}}
\\
* &\stackrel{E_j}{\to}& *
}
$$
Such a lift $(h,E)$ is an [[orthogonal structure]] on $T X$.
The component $E$ is called the corresponding **vielbein**. It exhibits an [[isomorphism]]
$$
E : T X \stackrel{\simeq}{\to} V
$$
between a [[vector bundle]] $V \to X$ with [[structure group]] explicitly being the [[orthogonal group]] and the [[tangent bundle]], hence it exhibits the [[reduction of structure groups|reduction of the structure group]] of $T X$ from $GL(n)$ to $O(n)$.
#### Moduli space of orthogonal structures: twisted cohomology
{#ModuliSpaceOfOrhtogonalStructures}
In order to understand the space of choices of vielbein fields on a given tangent bundle, hence the _[[moduli space]]_ or _[[moduli stack]]_ of [[orthogonal structures]]/[[Riemannian metrics]] on $X$, it is useful to first consider the [[homotopy fiber]] of the morphism $\mathbf{c} : \mathbf{B}O(n) \to \mathbf{B}GL(n)$. One finds that this is the [[coset]] $O(n) \backslash GL(n)$. We may think of the [[fiber sequence]]
$$
\array{
GL(n)/O(n) &\to& \mathbf{B} O(n)
\\
&& \downarrow
\\
&& \mathbf{B} GL(n)
}
$$
as being a bundle in $\mathbf{H}$ over the [[moduli stack]] $\mathbf{B}GL(n)$ with typical fiber $GL(n)/O(n)$. It is the smooth [[associated infinity-bundle|associated bundle]] to the smooth [[universal principal bundle|universal GL(n)-bundle]] induced by the canonical [[action]] of $GL(n)$ on $O(n)\backslash GL(n)$.
This means that if the tangent bundle $T X$ is trivializable, then the coset space $O(n)\backslash GL(n)$ is the moduli space for vielbein fields on $T X$, in that the space of these is
$$
\mathbf{H}(X, O(n)\backslash GL(n)) = C^\infty(X, O(n)\backslash GL(n))
\,.
$$
However, if $T X$ is not trivial, then this is true only locally: there is then an [[atlas]] $\{U_i \to X\}$ such that restricted to each $U_i$ the moduli space of vielbein fields is $C^\infty(U_i, GL(n)/ O(n))$, but globally these now glue together in a non-trivial way as encoded by the tangent bundle: we may say that
the tangent bundle _twists_ the functions $X \to GL(n)/O(n)$. If -- as we may -- we think of an ordinary such function as a cocycle in degree-0 cohomology, then this means that a vielbein is a cocycle in $T X$-_[[twisted cohomology]]_ relative to the _twisting coefficient bundle_ $\mathbf{c}$.
We can make this more manifest by writing equivalently
$$
\array{
O(n)\backslash GL(n) &\to& (O(n)\backslash GL(n)) // GL(n)
\\
&& \downarrow
\\
&& \mathbf{B}GL(n)
}
\,,
$$
where now on the right we have inserted the [[fibration]] resolution of the morphism $\mathbf{c}$ as provided by the [[factorization lemma]]: this is the morphism out of the [[action groupoid]] of the action of $GL(n)$ on $O(n)\backslash GL(n)$.
The pullback
$$
\array{
T X \times_{GL(n)} (O(n)\backslash GL(n))
&\to&
O(n)\backslash GL(n) // GL(n)
\\
\downarrow && \downarrow
\\
X &\stackrel{T X}{\to}& \mathbf{B}GL(n)
}
$$
give the non-linear $T X$-associated bundle whose space of sections is the "twisted $O(n)\backslash GL(n)$-0-cohomology", hence the space of inequivalent vielbein fields.
#### Moduli stack of orthogonal structures
The above says that the space of vielbein fields is the [[cohomology]] of $T X$ in the [[over-(infinity,1)-topos|slice (2,1)-topos]] $\mathbf{H}_{/\mathbf{B}GL(n)}$ with coefficients in $\mathbf{c} : \mathbf{B}O \to \mathbf{B}GL(n)$
$$
O Struc_{TX}(X) \simeq \mathbf{H}_{/\mathbf{B}GL(n)}(T X, \mathbf{c})
\,.
$$
But also this space of choices of vielbein fields has a smooth structure, it is itself a smooth [[moduli stack]]. This is obtained by forming the [[internal hom]] in the slice over $\mathbf{B}GL(n)$ of the [[locally cartesian closed (infinity,1)-category|locally cartesian closed (2,1)-category]] $\mathbf{H}$.
$$
O \mathbf{Struc}_{T X}(X)
=
[X, \mathbf{B}O]_{\mathbf{B}GL(n)}
$$
#### Differential refinement: Spin connection
We may further lift this discussion to [[differential cohomology]] to get genuine _differential_ $T X$-twisted $\mathbf{c}$-structures.
Write $\mathbf{B}G_{conn}$ for [[groupoid of Lie-algebra valued forms]]. As an object of $\mathbf{H} = $ [[smooth infinity-groupoid|SmoothGrpd]] this the [[moduli stack]] of $G$-[[connection on a bundle|connections]].
The morphism $\mathbf{c}$ has an evident differential refinement
$$
\mathbf{c}_{conn} : \mathbf{B}O(n)_{conn} \to \mathbf{B}GL(n)_{conn}
\,.
$$
The [[homotopy fiber]] of this differential refinement is still the same as before
$$
\array{
GL(n)/ O(n) &\to& \mathbf{B} O(n)_{conn}
\\
&& \downarrow
\\
&& \mathbf{B} GL(n)_{conn}
}
\,,
$$
so that the moduli space of "differential vielbein fields" is the same as that of plain vielbein fields.
Consider an [[affine connection]]
$$
\nabla_{T X} : X \to \mathbf{B}GL(n)
$$
hence a $GL(n)$-[[principal connection]] which locally on out atas is given by the [[Christoffel symbols]]
$$
\Gamma_i = ((\Gamma_i)_\mu{}{}^{\alpha}{}_\beta)
\in
\Omega^1(U_i, \mathfrak{gl}(n))
\,.
$$
A lift $(\nabla_V, E)$ in
$$
\array{
X &&\stackrel{\nabla_{V}}{\to}&& \mathbf{B}O_{conn}
\\
& {}_{\mathllap{\nabla_{T X}}}\searrow
&\swArrow_{E^{-1}}&
\swarrow_{\mathbf{c}_{conn}}
\\
&& \mathbf{B}GL(n)_{conn}
}
$$
is in components a "[[spin connection]]"
$$
\omega_\mu = E d E^{-1} + E \Gamma_\mu E^{-1}
$$
$$
\omega_\mu{}^a{}_b
=
E^a{}_\nu \partial_\mu E^\nu{}_b
+
E^a{}_\nu \Gamma_\mu{}^\nu{}_\lambda E^\lambda{}_b
\,.
$$
This is the standard formula for the relation between the [[Christoffel symbols]] and the [[spin connection]] in terms of the vielbein.
#### Generalized and exceptional vielbein fields
The above discussion seamlessly generalizes to many other related cases. For instance
1. For the coefficient bundle
$$
\array{
O(n)\backslash O(n,n) /O(n) &\to& \mathbf{B} (O(n) \times O(n))
\\
&& \downarrow
\\
&& \mathbf{B}O(n,n)
}
$$
one gets the [[generalized vielbein]] of [[type II geometry]];
1. for the coefficient bundle
$$
\array{
H_n\backslash E_n &\to& \mathbf{B} H_n
\\
&& \downarrow
\\
&& \mathbf{B} E_n
}
$$
coming from the inclusion of the [[maximal compact subgroup]] of an [[exceptional Lie group]] one gets generalized vielbein fields for [[exceptional generalized geometry]];
1. for the coefficient bundle
$$
\array{
&& \mathbf{B} E_8
\\
&& \downarrow^{\mathbf{a}}
\\
&& \mathbf{B}^3 U(1)
}
$$
coming from the second smooth universal [[Chern class]] of [[E8]] one gets part of the geometry of the [[supergravity C-field]]
and so on. More examples are discussed for instance at
_[[twisted smooth cohomology in string theory]]_.
### In terms of basic forms on the frame bundle
{#InTermsOfBasicFormsOnFrameBundle}
A [[G-structure]] on $X$ for $G = O(n)$ the [[orthogonal group]] is equivalently an $O(n)$-[[principal bundle|principal]] subbundle of the [[frame bundle]] $\pi \colon Fr(X)\to X$.
This frame bundle carries a universal "basic" $\mathbb{R}^n$-valued differential form
$$
\tau_{b} \in \Omega^1(Fr(X), \mathbb{R}^n)
$$
defined on a [[tangent vector]] $v\in \Gamma_{f \in Fr(X)}$ by
$$
\tau_b(v) \coloneqq f^{-1}(d \pi(v))
\,,
$$
where $d\pi \colon T Fr(X)\to T X$ is the [[differential]] of the bundle projection $\pi$ and $f$ is the given frame regarded as a linear [[isomorphism]] $f\colon \mathbb{R}^n \stackrel{\simeq}{\longrightarrow} T_x X$.
Then given an orthogonal structure in the form of an $O(n)$-subbundle $i \colon Fr_O(X) \hookrightarrow Fr(X)$ and given finally a local section $\sigma$ of $Fr_O(X)$, then the vielbein field with respect to that local trivialization is the [[pullback of differential forms|pullback]] form
$$
\tau = \sigma^\ast i^\ast \tau_b
\,.
$$
(exposition of this in the wider context of [[integrability of G-structures]] includes [Lott 90, p. 4](#Lott90)).
## Related concepts
* [[reduction of structure groups]], [[orthogonal structure]]
* [integrability of G-structures -- Examples -- Orthogonal structure](integrability+of+G-structures#ExamplesOrthogonalStructure)
* [[super-vielbein]]
* [[generalized vielbein]], [[exceptional generalized geometry]]
See also at [[field (physics)]] the section on _[Ordinary gravity](field%20%28physics%29#OrdinaryGravity)_.
## References
Discussion in the general context of [[G-structures]] includes
* {#Lott90} [[John Lott]], _The Geometry of Supergravity Torsion Constraints_, Comm. Math. Phys. 133 (1990), 563–615, (exposition in [arXiv:0108125](http://arxiv.org/abs/math/0108125))
[[!redirects vielbeine]]
[[!redirects vielbeins]]
[[!redirects vierbein]]
[[!redirects vierbein field]]
[[!redirects soldering form]]
[[!redirects soldering forms]]
[[!redirects vielbein field]]
[[!redirects vielbein fields]]
[[!redirects solder form]]
[[!redirects solder forms]]
[[!redirects soldering form]]
[[!redirects soldering forms]] |
Vienna Circle | https://ncatlab.org/nlab/source/Vienna+Circle |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--
The **Vienna Circle** (in German, *Der Wiener Kreis*) were a group of philosophers, mathematicians, logicians and scientists who were active in Vienna in the 1920s and 1930s where they promoted a positivist view of the world and the use of formal languages to resolve or dissolve philosophical disputes, hence the name of their position _Logical Positivism_. Prominent members included [[Rudolf Carnap]], Moritz Schlick, Hans Hahn, Otto Neurath, Olga Hahn-Neurath, Friedrich Waismann, Philipp Frank, and Herbert Feigl.
In 1929, Hahn, Neurath and Carnap published the Circle's Manifesto, [The Scientific Conception of the World: The Vienna Circle](https://www.manchesterism.com/the-scientific-conception-of-the-world-the-vienna-circle/):
>The scientific world conception is characterised not so much by theses of its own, but rather by its basic attitude, its points of view and direction of research. The goal ahead is _unified science_. The endeavour is to link and harmonise the achievements of individual investigators in their various fields of science. From this aim follows the emphasis on _collective efforts_, and also the emphasis on what can be grasped intersubjectively; from this springs the search for a neutral system of formulae, for a symbolism freed from the slag of historical languages; and also the search for a total system of concepts. Neatness and clarity are striven for, and dark distances and unfathomable depths rejected. In science there are no 'depths'; there is surface everywhere: all experience forms a complex network, which cannot always be surveyed and, can often be grasped only in parts. Everything is accessible to man; and man is the measure of all things. ...The scientific world-conception knows _no unsolvable riddle_. Clarification of the traditional philosophical problems leads us partly to unmask them as pseudo-problems, and partly to transform them into empirical problems and thereby subject them to the judgment of experimental science. The task of philosophical work lies in this clarification of problems and assertions, not in the propounding of special 'philosophical' pronouncements. The method of this clarification is that of _logical analysis_; of it, Russell says (*Our Knowledge of the External World*, p. 4) that it "has gradually crept into philosophy through the critical scrutiny of mathematics… It represents, I believe, the same kind of advance as was introduced into physics by Galileo: the substitution of piecemeal, detailed and verifiable results for large untested generalities recommended only by a certain appeal to imagination."
## References
* SEP, _[Vienna Circle](https://plato.stanford.edu/entries/vienna-circle/)_
* Wikipedia, _[Vienna Circle](http://en.wikipedia.org/wiki/Vienna_Circle)_
|
Vietoris complex | https://ncatlab.org/nlab/source/Vietoris+complex |
#Contents#
* table of contents
{:toc}
##Idea##
In the 1920s homology and cohomology were known for simplicial complexes and there were attempts to extend the definitions to first of all compact metric spaces and then more general spaces. [[Leopold Vietoris]] (1927) came up with a construction and then shortly after Alexandrov and Čech gave a different one involving [[Čech nerve|the nerve that now bears Čech's name]]. The input of the two methods is the same, we have a space $X$ and an open cover $\mathcal{U}$ of $X$.
It was noted that all the calculations of Vietoris homology gave the same answer as Čech homology. In 1952, C. H. Dowker showed why. His [[Dowker's theorem|result]] is a beautiful mix of abstraction and concrete explicit calculation (and is not that well known unfortunately).
First some abstraction:
* We take the set $X$ and the open cover $\mathcal{U}$, together with the relation $x\in U$, i.e. an element $x$ will be considered to be _related to_ the set $U$ in the cover $\mathcal{U}$, if it is an element of it!
* We replace this by an abstract setting of a set $X$, another set $Y$ and a relation $R\subseteq X\times Y$, from $X$ to $Y$.
+--{: .query}
Thinks: does what follows have a good generalisation to [[spans]]?
=--
##Definition:##
The **Vietoris complex** of a relation $R$, as above, is the [[simplicial complex]] specified by
* Vertex set: the set of vertices is the set $X$;
* Simplices : a $p$-simplex of $K$ is a set $\{x_0, \ldots, x_p\} \subseteq X$ such that there is some $y \in Y$ with $x_i R y$ for $i = 0, 1, \ldots, p$
The Vietoris complex of $R$ will be denoted $V(R)$.
The special, and original, case where $R$ comes from an open cover $\mathcal{U}$ of $X$ will be denoted $V(X,\mathcal{U})$ or simply $V(\mathcal{U})$ and will be called the _Vietoris complex of the open covering $\mathcal{U}$_.
##Remarks##
In fact each relation will give **two** simplicial complexes, one as above, the other obtained by reversing the roles of $X$ and $Y$ in the above. In other words, in the usual way define the _[[opposite relation]]_ by ' $R^{op}\subseteq Y\times X$ is the relation in which $yR^{op}x$ if and only if $xRy$'.
We define $C(R) = V(R^{op})$ and call it the **Čech nerve** or **Čech complex** of the relation $R$.
If we look at this in the case of the relation coming from a space $X$ and an open covers $\mathcal{U}$ you get exactly the nerve of $\mathcal{U}$ as discussed in [[Čech methods]].
There are natural questions that arise here. One is how to turn these [[simplicial complexes]] into [[simplicial sets]], to which _one_
reply is take a total order (or at least pick a partial order that any simplex is ordered in that partial order; another reply would be to take each $n$-simplex $(n+1)!$-times, one for each possible order.
Another obvious question is to compare $V(R)$ and $C(R)$: are they closely related? That is discussed more in [[Dowker's Theorem]]. It looks very much a situation for a combinatorial duality result.
A final question (and one I can answer!) is: are there useful applications of this construction other than in (co)homology. The answer is most decidedly 'yes'.
##Volodin spaces
One of the ways of constructing the [[algebraic K-theory]] of a ring $R$ is to construct a simplicial set / complex from the stable [[general linear group]] and the family of cosets of subgroups of upper triangular matrices. (This method is due to _[[I.A. Volodin]].)
The method can be applied to other groups with given families of subgroups as is described at [[Volodin spaces]]. It uses the cosets of the subgroups in the family to get a covering of the set of elements and then applies the Vietoris construction (but without mention of that source).
##Higher generation by subgroups
It is notable that the same situation but with corresponding Čech complex is used by Abels and Holz, (1993) for the calculation of syzygies and is applied to various settings with linear groups. They give induction methods for homological finiteness criteria for the groups. This is discussed in more detail in [[higher generation by subgroups]].
##'Hidden Stasheff polytopes ... '
Kapranov and Saito have studied the syzygies of the usual presentation of the Steinberg group. These are expected to give a combinatorial construction of the classifying space for the algebraic K-theory of rings. They use Volodin's construction but their combinatorial construction would seem to be related to that of Abels and Holz. They conjecture various results in this. but there is still some mystery about what results have been proved, beyond those published in their paper.
##Gavin Wraith's puzzle##
Gavin Wraith posted a puzzle on the $n$Café whose solution uses the fact that the simplicial complexes $V(R)$ and $C(R)$ are homotopy equivalent:
* [A Puzzle From Gavin Wraith](http://golem.ph.utexas.edu/category/2009/07/a_puzzle_from_gavin_wraith.html)
Perhaps this fact is [[Dowker's Theorem]]?
## Related concepts
* [[persistent homotopy]]
## References
Discussion in [[persistent homotopy theory]]:
* [[J. F. Jardine]], *Data and homotopy types* $[$[arXiv:1908.06323](https://arxiv.org/abs/1908.06323)$]$
See also:
* Wikipedia, *[Vietoris-Rips complex](https://en.wikipedia.org/wiki/Vietoris–Rips_complex)*
* [[I. Volodin]], _Algebraic K-theory as extraordinary homology theory on the category of associative rings with unity_, Izv. Akad. Nauk. SSSR, 35, (Translation: Math. USSR Izvestija Vol. 5 (1971)
No. 4, 859-887).
* H. Abels and S. Holz, _Higher generation by subgroups_, J. Alg, 160, (1993), 311– 341
* M. Kapranov and M. Saito, 1999, Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions , in Higher homotopy structure in topology and mathematical physics
(Poughkeepsie, N.Y. 1996) , volume 227 of Contemporary Mathematics , 191–225, AMS.
[[!redirects Vietoris complexes]]
[[!redirects Vietoris-Rips complex]]
[[!redirects Vietoris-Rips complexes]]
|
Vigleik Angeltveit | https://ncatlab.org/nlab/source/Vigleik+Angeltveit |
* [webpage](http://maths-people.anu.edu.au/~angeltveit/)
## Selected writings
On [[enriched model category|enriched]] [[Reedy model categories]]:
* {#Angeltveit} [[Vigleik Angeltveit]], *Enriched Reedy categories*, Proceedings of the American Mathematical Society **136** 7 (2008) [[arXiv:math/0612137](http://arxiv.org/abs/math/0612137), [AMS:S0002-9939-08-09185-5](https://www.ams.org/journals/proc/2008-136-07/S0002-9939-08-09185-5/home.html), [jstor:20535419](https://www.jstor.org/stable/20535419)]
category: people
|
Vijay Balasubramanian | https://ncatlab.org/nlab/source/Vijay+Balasubramanian |
* [webpage](https://www.physics.upenn.edu/people/standing-faculty/vijay-balasubramanian)
## related $n$Lab entries
* [[moduli stabilization]]
category: people |
Vijay Patodi | https://ncatlab.org/nlab/source/Vijay+Patodi |
* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Vijay_Kumar_Patodi)
## Related $n$Lab entries
* [[index theory]]
* [[e-invariant]]
category: people |