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Whitney extension theorem | https://ncatlab.org/nlab/source/Whitney+extension+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A statement about sufficient data for extensions of a [[smooth function]] from a [[compact subset]] to an [[open neighbourhood]].
## Related concepts
[[!include extension theorems -- table]]
* [[Hadamard lemma]],
* [[Borel's theorem]]
* [[Tietze extension theorem]]
* [[Taimanov theorem]]
* [[Steenrod-Wockel approximation theorem]]
* [[embedding of smooth manifolds into formal duals of R-algebras]]
* [[smooth Serre-Swan theorem]]
* [[derivations of smooth functions are vector fields]]
## References
The original article is
* [[Hassler Whitney]], _Analytic extensions of differentiable functions defined in closed sets_, 1933 ([pdf](http://www.ams.org/journals/tran/1934-036-01/S0002-9947-1934-1501735-3/S0002-9947-1934-1501735-3.pdf))
Textbook accounts include
* {#Hoermander90} [[Lars Hörmander]], theore 2.3.6 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990
Enhancement to a linear splitting of [[restriction]] maps on Fréchet [[spaces of sections]] with compact support of [[vector bundles]]:
* {#RobertsSchmeding18} [[David Roberts]], [[Alexander Schmeding]], _Extending Whitney's extension theorem: nonlinear function spaces_ ([arXiv:1801.04126](https://arxiv.org/abs/1801.04126))
This is then used to show the restriction map to (suitable) regular closed subsets is a submersion of mapping spaces (with maps valued in an arbitrary manifold).
See also
* Wikipedia, _[Whitney extension theorem](https://en.wikipedia.org/wiki/Whitney_extension_theorem)_
* [[eom]], _[Whitney extension theorem](https://www.encyclopediaofmath.org/index.php/Whitney_extension_theorem)_
* Armin Rainer, *Ultradifferentiable extension theorems: a survey*, Expositiones Mathematicae (2021) [[arXiv:2107.01061](https://arxiv.org/abs/2107.01061), [doi:10.1016/j.exmath.2021.12.001](https://doi.org/10.1016/j.exmath.2021.12.001)]
|
Whittaker function | https://ncatlab.org/nlab/source/Whittaker+function |
## Related concepts
* [[geometric Langlands duality]]
## References
Name after [[Edmund T. Whittaker]].
See also:
* Wikipedia, *[Whittake function](https://en.wikipedia.org/wiki/Whittaker_function)*
Relation of Whittaker functions to [[equivariance|equivariant]] [[Gromov-Witten theory]] on [[flag varieties]] is discussed in
* [[Alexander Givental]], _Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture_, Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser., 2 vol. 180, Amer.Math.Soc.,Providence, Rhode Island, 1997, pp. 103–115; ([arXiv:alg-geom/9612001](http://arxiv.org/abs/alg-geom/9612001))
* [[Anton Gerasimov]], Dimitri Lebedev, [[Sergey Oblezin]], _New Integral Representations of Whittaker Functions for Classical Lie Groups_ ([arXiv:0705.2886](http://arxiv.org/abs/0705.2886))
* [[Sergey Oblezin]], _On parabolic Whittaker functions_ ([arXiv:1011.4250](http://arxiv.org/abs/1011.4250))
For relation to [[geometric Langlands duality]] see [Gerasimov-Lebedev-Oblezin 09](geometric+Langlands+correspondence#GerasimovLebedevOblezin09)
[[!redirects Whittaker functions]]
|
why (infinity,1)-categories? | https://ncatlab.org/nlab/source/why+%28infinity%2C1%29-categories%3F |
see at _[[(infinity,1)-category]]_ |
WhyNotZoidgebra? | https://ncatlab.org/nlab/source/WhyNotZoidgebra%3F |  |
Wick algebra | https://ncatlab.org/nlab/source/Wick+algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
A _Wick algebra_ is an [[algebra of quantum observables]] of [[free fields|free]] [[quantum field theory|quantum fields]].
In the [[quantum mechanics]] of [[harmonic oscillators]] or in [[quantum field theory]] of [[free fields]] on [[Minkowski spacetime]] one encounters [[linear operators]] $\{a_k, a^\ast_k\}_{k \in K}$ that satisfy the [[canonical commutation relation]] $[a_i, a^\ast_j] = diag((c_k))_{i j}$.
Then by a _normal ordered polynomial_ or _Wick polynomial_ ([Wick 50](#Wick50)) one means a [[polynomial]], denoted $:P:$, which is obtained from a polynomial $P((a_k, a^\ast_k))$ in these operators by ordering all "creation operators" $a_k^\ast$ to the left of all "annihiliation operators". For example focusing on a single mode $k$ we have:
$$
\array{
:a^\ast: = a^\ast
\\
:a: = a
\\
:a^\ast a: = a^\ast a
\\
:a a^\ast: = a^\ast a
\\
:a a^\ast a: = a^\ast a a
\\
etc.
}
\,
$$
The intuitive idea is that these operators span a [[Hilbert space]] $\mathcal{H}$ of [[quantum states]] from a [[vacuum state]] $\vert vac \rangle \in \mathcal{H}$ characterized by the condition
$$
a_k \vert vac \rangle = 0
\phantom{AAA}
\text{for all} \,\, k
$$
hence (if we think of $a_k$ as acting by "removing a quantum in mode $k$") by the condition that it contains no quanta. So the normal ordered Wick polynomials represent the [[quantum observables]] with vanishing [[vacuum expectation value]]. In [[quantum field theory]] they model [[scattering]] processes where quanta enter a reaction process (the modes corresponding to the "annihilation" operators $a_k$) and other particles come out of the reaction (the modes corresppnding to the "creation" operators $a^\ast_k$).
The product of two Wick polynomials, computed in the ambient operator algebra and
then re-expressed as a Wick polynomial, is given by computing the relevant sequence of [[commutators]] by [[Wick's lemma]], for example
$$
{:a^\ast a:} \, {:a^\ast a:}
=
:a^\ast a^\ast a a: + \hbar \, :a^\ast a:
\,,
$$
where $\hbar = [a, a^\ast]$ is the value of the [[canonical commutation relations|canonical commutator]].
The [[associative algebra]] thus obtained is hence called the _algebra of normal ordered operators_ or _Wick polynomial algebra_ or just _Wick algebra_.
This plays a central role in [[perturbative quantum field theory]], where the [[quantization]] of [[quantum observables]] of [[free fields]] is traditionally _defined_ as the corresponding Wick algebra.
But the Wick algebra in [[quantum field theory]] may also be understood more systematically from first principles of [[quantization]]. It turns out that it is [[Moyal deformation quantization]] of the canonical [[Poisson bracket]] on the [[covariant phase space]] of the [[free field]], which is the [[Peierls bracket]] modified to an [[almost Kähler structure]] by the [[2-point function]] of a [[quasi-free Hadamard state]] ([Dito 90](#Dito90), [Dütsch-Fredenhagen 01](#DutschFredenhagen01)).
In [[free field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations]] of motion, the analog of the star product tensor ([this equation](star+product#eq:InStarProductTensorInvertingHermitianForm))
$$
\pi \;=\; \tfrac{i}{2}\omega^{-1} + \tfrac{1}{2}g^{-1}
$$
is the [[Wightman propagator]] according to [this equation](Hadamard+distribution#eq:DeompositionOfHadamardPropagatorOnMinkowkski)
$$
\Delta_H
\;=\;
\tfrac{i}{2}\Delta_S + H
\,.
$$
Understood in this form the construction directly generalizes to [[quantum field theory on curved spacetimes]] ([Brunetti-Fredenhagen 95](#BrunettiFredenhagen95), [Brunetti-Fredenhagen 00](#BrunettiFredenhagen00), [Hollands-Wald 01](#HollandsWald01)).
Finally, the shift by the [[quasi-free Hadamard state]], which is the very source of the "normal ordering", was understood as an example of the almost-Kähler version of the quantization recipe of [[Fedosov deformation quantization]] ([Collini 16](#Collini16)). For more on this see at _[[locally covariant perturbative quantum field theory]]_.
$\,$
## Definition
{#Definition}
Traditionally the Wick algebra is regarded as an [[operator algebra]] acting on a [[Fock space]].
However, it is useful to realize the Wick algebra directly as an [[associative algebra]] [[structure]]
on the space of [[microcausal polynomial observables]]. This "abstract" Wick algebra (meaning: not [[representation|represented]]
yet on a [[Hilbert space]]) we discuss in
* _[Abstract Wick algebra](#AbstractWickAlgebra)_.
That the abstract Wick algebra indeed has a [[faithful representation]] on [[Fock space]] is _[[Wick's lemma]]_.
Similarly there is the
* _[Abstract time-ordered product](#AbstractTimeOrderedProduct)_
The traditional notation for the [[operator products]] on the [[Fock space]] may be carried across the [[representation]] map to the abstract Wick algebra:
* _[Operator product notation](#OperatorProductAndNormalOrderedProduct)_
The abstract Wick algebra carries a canonical [[state on a star-algebra]], whose [[2-point function]] is just the
[[Wightman propagator]] that the abstract Wick algebra structure is constructed from. This we discuss in
* _[Hadamard vacuum states](#HadamardVacuumStatesOnWickAlgebras)_.
The only non-trivial part of the proof of the [[state on a star-algebra|state]]-property (prop. \ref{WickAlgebraCanonicalState}) below is
positivity. This however is immediate from the [[representation]] on the [[Fock space]], observing that
under this identification the state is represented by the [[inner product]] of the [[Hilbert space]] ([Dütsch 18, remark 2.20](#Duetsch18)).
From the point of view of [[algebraic quantum field theory]], the [[Hilbert space]]-structure mainly just serves
as a technical tool for establishing this positivity property.
### Abstract Wick algebra
{#AbstractWickAlgebra}
The abstract [[Wick algebra]] of a [[free field theory]] with [[Green hyperbolic differential equation]] is directly analogous to the [[star product]]-algebra induced by a [[finite dimensional vector space|finite dimensional]] [[Kähler vector space]] ([this def.](star+product#WickAlgebraOfAlmostKaehlerVectorSpace)) under the following identification of the [[Wightman propagator]] with the [[Kähler space]]-[[structure]]:
+-- {: .num_remark #WightmanPropagatorAsKaehlerVectorSpaceStructure}
###### Remark
**([[Wightman propagator]] as [[Kähler vector space]]-[[structure]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] whose [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] is a [[Green hyperbolic differential equation]]. Then the corresponding [[Wightman propagator]] is analogous to the rank-2 tensor on a [[Kähler vector space]] as follows:
| [[covariant phase space]] <br/> of [[free field theory|free]] [[Green hyperbolic differential equation|Green hyperbolic]] <br/> [[Lagrangian field theory]] | [[finite dimensional vector space|finite dimensional]] <br/> [[Kähler vector space]] |
|-------------------|-------------------------|
| [[space of field histories]] <br/> $\Gamma_\Sigma(E)$ | $\mathbb{R}^{2n}$ |
| [[symplectic form]] <br/> $\tau_{\Sigma_p} \Omega_{BFV}$ | [[Kähler form]] $\omega$ |
| [[causal propagator]] $\Delta$ | $\omega^{-1}$ |
| [[Peierls-Poisson bracket]] <br/> $\{A_1,A_2\} = \int \Delta^{a_1 a_2}(x_1,x_2) \frac{\delta A_1}{\delta \mathbf{\Phi}^{a_1}(x_1)} \frac{\delta A_2}{\delta \mathbf{\Phi}^{a_2}(x_2)} dvol_\Sigma(x)$ | [[Poisson bracket]] |
| [[Wightman propagator]] <br/> $\Delta_H = \tfrac{i}{2} \Delta + H$ | [[Hermitian form]] <br/> $\pi = \tfrac{i}{2}\omega^{-1} + \tfrac{1}{2}g^{-1}$ |
=--
([Fredenhagen-Rejzner 15, section 3.6](pAQFT#FredenhagenRejzner15), [Collini 16, table 2.1](pAQFT#Collini16))
+-- {: .num_defn #MicrocausalObservable}
###### Definition
**([[microcausal observable|microcausal]] [[polynomial observables]])**
Let $E \overset{fb}{\to}$ be [[field bundle]] which is a [[vector bundle]]. An [[off-shell]] _[[polynomial observable]]_ is a [[smooth function]]
$$
A
\;\colon\;
\Gamma_\Sigma(E)
\longrightarrow
\mathbb{C}
$$
on the [[on-shell]] [[space of sections]] of the [[field bundle]] $E \overset{fb}{\to} \Sigma$ (space of field histories) which may be expressed as
$$
\begin{aligned}
A(\Phi)
& =
\phantom{+} \alpha^{(0)}
\\
& \phantom{=}
+
\int_\Sigma
\alpha^{(1)}_a(x) \Phi^a(x)
\,
dvol_\Sigma(x)
\\
& \phantom{=}
+
\int_\Sigma \int_\Sigma
\alpha^{(2)}_{a_1 a_2}(x_1, x_2)
\Phi^{a_1}(x_1) \Phi^{a_2}(x_2)
\,dvol_\Sigma(x_1)
\, dvol_\Sigma(x_2)
\\
& \phantom{=}
+
\cdots
\,,
\end{aligned}
$$
where
$$
\alpha^{(k)} \in \Gamma'_{\Sigma^k}\left((E^\ast)^{\boxtimes^k_{sym}} \right)
$$
is a [[compactly supported distribution]] [[distribution of two variables|of k variables]] on the $k$-fold graded-symmetric [[external tensor product of vector bundles]] of the [[field bundle]] with itself.
Write
$$
PolyObs(E) \hookrightarrow Obs(E)
$$
for the [[subspace]] of off-shell polynomial observables onside all off-shell [[observables]].
Let moreover $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] whose [[equations of motion]] are [[Green hyperbolic differential equations]]. Then an _[[on-shell]] polynomial observable_ is the [[restriction]] of an off-shell polynomial observable along the inclusion of the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E)$. Write
$$
PolyObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})
$$
for the subspace of all on-shell polynomial observables inside all on-shell [[observables]].
By [this prop.](Green+hyperbolic+partial+differential+equation#DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions) restriction yields an [[isomorphism]] between polynomial on-shell observables and polynomial off-shell observables modulo the image of the [[differential operator]] $P$:
$$
PolyObs(E,\mathbf{L})
\underoverset{\simeq}{\text{restriction}}{\longleftarrow}
PolyObs(E)/im(P)
\,.
$$
Finally a polynomial observable is a _[[microcausal observable]]_ if each [[coefficient]] $\alpha^{(k)}$ as above has [[wave front set]] away from those points where the $k$ [[wave vectors]] are all in the [[future cone]] or all in the [[past cone]]. We write
$$
\array{
PolyObs(E)_{mc}
&\hookrightarrow&
PolyObs(E)
\\
PolyObs(E,\mathbf{L})_{mc}
\simeq
PolyObs(E)_{mc}/im(P)
&\hookrightarrow&
PolyObs(E,\mathbf{L})
}
$$
for the [[subspace]] of [[off-shell]]/[[on-shell]] [[microcausal observables]] inside all [[off-shell]]/[[on-shell]] [[polynomial observables]].
=--
+-- {: .num_prop #MoyalStarProductOnMicrocausal}
###### Proposition
**([[Hadamard distribution|Hadamard]]-[[Moyal star product]] on [[microcausal observables]] -- [[abstract Wick algebra]])**
Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] $P \Phi = 0$. Write $\Delta$ for the [[causal propagator]] and let
$$
\Delta_H
\;=\;
\tfrac{i}{2}\Delta + H
$$
be a corresponding [[Wightman propagator]] ([[Hadamard 2-point function]]).
Then the [[star product]] induced by $\Delta_H$
$$
A \star_H A
\;\coloneqq\;
prod
\circ
\exp\left(
\int_{X^2} \hbar \Delta_H^{a b}(x_1, x_2) \frac{\delta}{\delta \mathbf{\Phi}^a(x_1)}
\otimes \frac{\delta}{\delta \mathbf{\Phi}^b(x_2)} dvol_g
\right)
(P_1 \otimes P_2)
$$
on [[off-shell]] [[microcausal observables]] $A_1, A_2 \in \mathcal{F}_{mc}$ (def. \ref{MicrocausalObservable}) is well defined in that the [[wave front sets]] involved in the [[products of distributions]] that appear in expanding out the [[exponential]] satisfy [[Hörmander's criterion]].
Hence by the general properties of [[star products]] ([this prop.](star+product#AssociativeAndUnitalStarProduct)) this yields a [[unital algebra|unital]] [[associative algebra]] [[structure]] on the space of [[formal power series]] in $\hbar$ of [[off-shell]] [[microcausal observables]]
$$
\left(
PolyObs(E)_{mc}[ [\hbar] ] \,,\, \star_H
\right)
\,.
$$
This is the _[[off-shell]] [[Wick algebra]]_ corresponding to the choice of [[Wightman propagator]] $H$.
Moreover the image of $P$ is an ideal with respect to this algebra structure, so that it descends to the [[on-shell]] [[microcausal observables]] to yield the _[[on-shell]] [[Wick algebra]]_
$$
\left(
PolyObs(E,\mathbf{L})_{mc}[ [ \hbar ] ] \,,\, \star_H
\right)
\,.
$$
Finally, under [[complex conjugation]] $(-)^\ast$ these are [[star algebras]] in that
$$
\left(
A_1 \star_H A_2
\right)^\ast
=
A_2^\ast \star_H A_1^\ast
\,.
$$
=--
([Dito 90](#Dito90), [Dütsch-Fredenhagen 00](#DuetschFredenhagen00) [Dütsch-Fredenhagen 01](#DuetschFredenhagen01), [Hirshfeld-Henselder 02](#HirschfeldHenselder02), see [Collini 16, p. 25-26](#Collini16))
+-- {: .proof}
###### Proof
By definition of the [[Wightman propagator]] (or else by [this prop.](Hadamard+distribution#WaveFronSetsForKGPropagatorsOnMinkowski)), the [[wave front set]] of powers of $\Delta_H$ has all cotangent [[wave vectors]] on the first variables in the [[closed future cone]] at the given base point (which itself is on the [[light cone]])
<center>
<img src="https://ncatlab.org/nlab/files/HadamardPropagator.png" width="60">
</center>
and hence all those on the second variables in the [[closed past cone]].
The first variables are integrated against those of $A_1$ and the second against $A_2$. By definition of [[microcausal observables]] (def. \ref{MicrocausalObservable}), the wave front sets of $A_1$ and $A_2$ are disjoint from the subsets where all components are in the [[closed future cone]] or all components are in the [[closed past cone]]. Therefore the relevant sum of of the wave front covectors never vanishes and hence [[Hörmander's criterion]] for partial [[products of distributions|products of]] [[distributions of several variables]] ([this prop.](product+of+distributions#PartialProductOfDistributionsOfSeveralVariables)) is met and the star product is well defined.
It remains to see that the star product $A_1 \star_H A_2$ is itself again a [[microcausal observable]]. It is clear that it is again a [[polynomial observable]] and that it respects the ideal generated by the equations of motion. That it still satisfies the condition on the [[wave front set]] follows directly from the fact that the wave front set of a [[product of distributions]] is inside the fiberwise sum of elements of the factor wave front sets ([this prop.](product+of+distributions#WaveFrontSetOfProductOfDistributionsInsideFiberProductOfFactorWaveFrontSets), []()).
Finally the [[star algebra]]-structure follows via remark \ref{WightmanPropagatorAsKaehlerVectorSpaceStructure} as in [this prop.](star+product#StarProductAlgebraOfKaehlerVectorSpaceIsStarAlgebra).
=--
+-- {: .num_remark #WickAlgebraIsFormalDeformationQuantization}
###### Remark
**([[Wick algebra]] is [[formal deformation quantization]] of [[Poisson-Peierls bracket|Poisson-Peierls algebra of observables]])**
Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] $P \Phi = 0$ with [[causal propagator]] $\Delta$ and let $\Delta_H \;=\; \tfrac{i}{2}\Delta + H$ be a corresponding [[Wightman propagator]] ([[Hadamard 2-point function]]).
Then the [[Wick algebra]] $\left( PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ] \,,\, \star_H \right)$ from prop. \ref{MoyalStarProductOnMicrocausal} is a [[formal deformation quantization]] of the [[Poisson algebra]] on the [[covariant phase space]] given by the [[on-shell]] [[polynomial observables]] equipped with the [[Poisson-Peierls bracket]] $\{-,-\} \;\colon\; PolyObs(E,\mathbf{L})_{mc} \otimes PolyObs(E,\mathbf{L})_{mc} \to PolyObs(E,\mathbf{L})_{mc}$ in that for all $A_1, A_2 \in PolyObs(E,\mathbf{L})_{mc}$ we have
$$
A_1 \star_H A_2
\;=\;
A_1 \cdot A_2
\;mod\;
\hbar
$$
and
$$
A_1 \star_H A_2
-
A_2 \star_H A_1
\;=\;
i \hbar \{A_1, A_2\}
\;mod\;
\hbar^2
\,.
$$
=--
([Dito 90](#Dito90), [Dütsch-Fredenhagen 01](#DutschFredenhagen01))
+-- {: .proof}
###### Proof
By prop. \ref{MoyalStarProductOnMicrocausal} this is immediate from the general properties of the [[star product]] ([this example](A+first+idea+of+quantum+field+theory+--+Quantization#MoyalStarProductIsFormalDeformationQuantization)).
Explicitly, consider, without restriction of generality, $A_1 = \int (\alpha_1)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ and $A_2 = \int (\alpha_2)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ be two linear observables. Then
$$
\begin{aligned}
A_1 \star_H A_2
& =
A_1 A_2
+
\hbar
\int
\left(
\tfrac{i}{2} \Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1,x_2)
\right)
\frac{\partial A_1}{\partial \mathbf{\Phi}^{a_1}(x_1)}
\frac{\partial A_2}{\partial \mathbf{\Phi}^{a_2}(x_2)}
\;mod\;
\hbar^2
\\
& =
A_1 A_2
+
\hbar
\left(
\int
(\alpha_1)_{a_1}(x_1)
\left(
\tfrac{i}{2}\Delta^{a_1 a_2}(x_1, x_2)
+ H^{a_1 a_2}(x_1, x_2)
\right)
(\alpha_2)_{a_2}(x_2)
\right)
\;mod\;
\hbar^2
\end{aligned}
$$
Now since $\Delta$ is skew-symmetric while $H$ is symmetric is follows that
$$
\begin{aligned}
A_1 \star_H A_2
-
A_2 \star_H A_1
& =
i \hbar
\left(
\int
(\alpha_1)_{a_1}(x_1)
\Delta^{a_1 a_2}(x_1, x_2)
(\alpha_2)_{a_2}(x_2)
\right)
\;mod\;
\hbar^2
\\
& =
i \hbar \, \left\{ A_1, A_2\right\}
\end{aligned}
\,.
$$
The right hand side is the [[integral kernel]]-expression for the [[Poisson-Peierls bracket]], as shown in the second line.
=--
### Abstract time-ordered product
{#AbstractTimeOrderedProduct}
+-- {: .num_defn #OnRegularPolynomialObservablesTimeOrderedProduct}
###### Definition
**([[time-ordered product]] on [[regular polynomial observables]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]]
over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H $ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]].
Then the _[[time-ordered product]]_ on the space of [[off-shell]] [[regular polynomial observable]] $PolyObs(E)_{reg}$ is the [[star product]] induced by the [[Feynman propagator]] (via [this prop.](star+product#PropagatorStarProduct)):
$$
\array{
PolyObs(E)_{reg}[ [\hbar] ]
\otimes
PolyObs(E)_{reg}[ [\hbar] ]
&\overset{}{\longrightarrow}&
PolyObs(E)_{reg}[ [\hbar] ]
\\
(A_1, A_2)
&\mapsto&
\phantom{\coloneqq} A_1 \star_F A_2
}
$$
hence
$$
A_1 \star_F A_2
\;
\coloneqq
\;
((-)\cdot(-))
\circ
\exp\left(
\underset{\Sigma \times \Sigma}{\int}
\Delta_F^{a b}(x,y)
\frac{\delta}{\delta \mathbf{\Phi}^a(x)}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}^b(y)}
\,
dvol_\Sigma(x)
\,
dvol_\Sigma(y)
\right)
$$
(Notice that this does not descend to the [[on-shell]] observables, since the [[Feynman propagator]] is not a solution to the _homogeneous_ [[equations of motion]].)
=--
+-- {: .num_prop #CausalOrderingTimeOrderedProductOnRegular}
###### Proposition
**([[time-ordered product]] is indeed causally ordered [[Wick algebra]] product)**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]]
over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H $ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]].
Then the [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is indeed a time-ordering of the [[Wick algebra]] product $\star_H$ in that for all [[pairs]] of [[regular polynomial observables]]
$$
A_1, A_2 \in PolyObs(E)_{reg}[ [\hbar] ]
$$
with [[disjoint subset|disjoint]] [[spacetime]] [[support]] we have
$$
A_1 \star_F A_2
\;=\;
\left\{
\array{
A_1 \star_H A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
\\
A_2 \star_H A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
}
\right.
\,.
$$
Here $S_1 {\vee\!\!\!\wedge} S_2$ is the [[causal order]] relation ("$S_1$ does not intersect the [[past cone]] of $S_2$"). Beware that for general [[pairs]] $(S_1, S-2)$ of subsets neither $S_1 {\vee\!\!\!\wedge} S_2$ nor $S_2 {\vee\!\!\!\wedge} S_1$.
=--
+-- {: .proof}
###### Proof
Recall the following facts:
1. the [[advanced and retarded propagators]] $\Delta_{\pm}$ by definition are [[support|supported]] in the [[future cone]]/[[past cone]], respectively
$$
supp(\Delta_{\pm}) \subset \overline{V}^{\pm}
$$
1. they turn into each other under exchange of their arguments ([this cor.](causal+propagator#CausalPropagatorIsSkewSymmetric)):
$$
\Delta_\pm(y,x) = \Delta_{\mp}(x,y)
\,.
$$
1. the real part $H$ of the [[Feynman propagator]], which by definition is the real part of the [[Wightman propagator]] is symmetric (by definition or else by [this prop.](Hadamard+distribution#SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime)):
$$
H(x,y) = H(y,x)
$$
Using this we compute as follows:
$$
\label{CausallyOrderedWickProductViaFeynmanPropagator}
\begin{aligned}
A_1 \underset{\Delta_{F}}{\star} A_2
\;
& =
A_1 \underset{\tfrac{i}{2}(\Delta_+ + \Delta_-) + H}{\star} A_2
\\
& =
\left\{
\array{
A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
\\
A_1 \underset{\tfrac{i}{2}\Delta_- + H}{\star} A_2 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
}
\right.
\\
& =
\left\{
\array{
A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
\\
A_2 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
}
\right.
\\
& =
\left\{
\array{
A_1 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
\\
A_2 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
}
\right.
\\
& =
\left\{
\array{
A_1 \underset{\Delta_H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
\\
A_2 \underset{\Delta_H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
}
\right.
\end{aligned}
$$
=--
+-- {: .num_prop #IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}
###### Proposition
**([[time-ordered product]] on [[regular polynomial observables]] [[isomorphism|isomorphic]] to pointwise product)
The [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is [[isomorphism|isomorphic]] to the pointwise product of [[observables]] ([this def.](A+first+idea+of+quantum+field+theory#Observable)) via the [[linear isomorphism]]
$$
\mathcal{T}
\;\colon\;
PolyObs(E)_{reg}[ [\hbar] ]
\longrightarrow
PolyObs(E)_{reg}[ [\hbar] ]
$$
given by
$$
\label{OnRegularPolynomialObservablesPointwiseTimeOrderedIsomorphism}
\mathcal{T}A
\;\coloneqq\;
\exp\left(
\tfrac{1}{2}
\hbar
\underset{\Sigma}{\int}
\Delta_F(x,y)^{a b}
\frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)}
\right)
A
$$
in that
$$
\begin{aligned}
T(A_1 A_2)
& \coloneqq
A_1 \star_{F} A_2
\\
& =
\mathcal{T}( \mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2) )
\end{aligned}
$$
hence
$$
\array{
PolyObs(E)_{reg}[ [\hbar] ]
\otimes
PolyObs(E)_{reg}[ [\hbar] ]
&\overset{(-)\cdot (-)}{\longrightarrow}&
PolyObs(E)_{reg}[ [\hbar] ]
\\
{}^{\mathllap{\mathcal{T} \otimes \mathcal{T}}}_\simeq\Big\downarrow
&&
\downarrow^{\mathrlap{\mathcal{T}}}_\simeq
\\
PolyObs(E)_{reg}[ [\hbar] ]
\otimes
PolyObs(E)_{reg}[ [\hbar] ]
&\overset{(-) \star_F (-)}{\longrightarrow}&
PolyObs(E)_{reg}[ [\hbar] ]
}
$$
=--
([Brunetti-Dütsch-Fredenhagen 09, (12)-(13)](time-ordered+product#BrunettiDuetschFredenhagen09), [Fredenhagen-Rejzner 11b, (14)](time-ordered+product#FredenhagenRejzner11b))
+-- {: .proof}
###### Proof
Since the [[Feynman propagator]] is symmetric ([this prop.](A+first+idea+of+quantum+field+theory#SymmetricFeynmanPropagator)), the statement is a special case of [this prop.](star+product#SymmetricContribution)).
=--
+-- {: .num_remark }
###### Remark
**([[renormalization]] of [[time-ordered product]])**
The [[time-ordered product]] on [[regular polynomial observables]] from prop. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} extends to a product on [[polynomial observable|polynomial]] [[local observables]], then taking values in [[microcausal observables]]:
$$
T
\;\colon\;
PolyLocObs(E)^{\otimes_n}[ [\hbar] ]
\longrightarrow
PolyObs(E)_{mc}[ [\hbar] ]
\,.
$$
This extension is not unique. A choice of such an extension, satisfying some evident compatibility conditions, is a choice of _[[renormalization scheme]]_ for the given [[perturbative quantum field theory]]. Every such choice corresponds to a choice of [[perturbative S-matrix]] for the theory. This construction is called _[[causal perturbation theory]]_.
=--
### Operator product notation
{#OperatorProductAndNormalOrderedProduct}
+-- {: .num_defn #NormalOrderedProductNotation}
###### Definition
**(notation for [[operator product]] and [[normal-ordered product]])**
It is traditional to use the following alternative notation for the product structures on [[microcausal polynomial observables]]:
1. The [[Wick algebra]]-product, hence the [[star product]] $\star_H$ for the [[Wightman propagator]] (def. \ref{MoyalStarProductOnMicrocausal}), is rewritten as plain juxtaposition:
$$
\text{"operator product"}
\phantom{AAA}
A_1 A_2 \phantom{AA} \coloneqq \phantom{AA} A_1 \star_H A_1
\phantom{AAAA}
\array{
\text{star product of}
\\
\text{Wightman propagator}
}
\,.
$$
1. The pointwise product of observables ([this def.](A+first+idea+of+quantum+field+theory#Observable)) $A_1 \cdot A_2$
is equivalently written as plain juxtaposition enclosed by colons:
$$
\array{
\text{"normal-ordered}
\\
\text{product"}
}
\phantom{AAAA}
:A_1 A_2:
\phantom{AA}\coloneqq\phantom{AA}
A_1 \cdot A_2
\phantom{AAAA} \phantom{AAa}\text{pointwise product}\phantom{AAa}
$$
1. The [[time-ordered product]], hence the [[star product]] for the [[Feynman propagator]] $\star_F$ (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is equivalently written as plain juxtaposition prefixed by a "$T$"
$$
\array{
\text{"time-ordered}
\\
\text{product"}
}
\phantom{AAAA}
T(A_1 A_2)
\phantom{AA}\coloneqq\phantom{AA}
A_1 \star_F A_2
\phantom{AAAA}
\array{
\text{star product of}
\\
\text{Feynman propagator}
}
$$
Under [[representation]] of the [[Wick algebra]] on a [[Fock space|Fock]] [[Hilbert space]] by [[linear operators]]
the first product become the _[[operator product]]_, while the second becomes the operator poduct applied after
suitable re-ordering, called "[[normal-ordered product|normal odering]]" of the factors.
Disregarding the [[Fock space]]-representation, which is [[faithful representation|faithful]], we may still
refer to these "abstract" products as the "operator product" and the "normal-ordered product", respectively.
=--
[[!include Wick algebra -- table]]
### Hadamard vacuum states
{#HadamardVacuumStatesOnWickAlgebras}
+-- {: .num_prop #WickAlgebraCanonicalState}
###### Proposition
**(canonical [[Hadamard vacuum state|vacuum]] [[state on a star-algebra|states]] on abstract [[Wick algebra]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]].
For
$$
\Phi_0 \in \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
$$
any [[on-shell]] [[field history]] (i.e. solving the [[equations of motion]]), consider the function from the [[Wick algebra]] to [[formal power series]] in $\hbar$ with [[coefficients]] in the [[complex numbers]]
which evaluates any [[microcausal polynomial observable]] on $\Phi_0$
$$
\array{
PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ]
&\overset{\langle -\rangle_{\Phi_0}}{\longrightarrow}&
\mathbb{C}[ [\hbar] ]
\\
A &\mapsto& A(\Phi_0)
}
$$
Specifically for $\Phi_0 = 0$ (which is a solution of the [[equations of motion]] by the assumption that $(E,\mathbf{L})$ defines a [[free field theory]]) this is the function
$$
\array{
PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ]
&\overset{\langle -\rangle_0}{\longrightarrow}&
\mathbb{C}[ [\hbar] ]
\\
\left.
\begin{aligned}
A & = \alpha^{(0)}
\\ & \phantom{=} + \underset{\Sigma}{\int} \alpha^{(1)}_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)
\\
& \phantom{=} + \cdots
\end{aligned}
\right\}
&\mapsto&
A(0) = \alpha^{(0)}
}
$$
which sends each [[microcausal polynomial observable]] to its value $A(\Phi = 0)$ on the zero [[field history]], hence to the constant contribution $\alpha^{(0)}$ in its [[polynomial]] expansion.
The function $\langle -\rangle_0$ is
1. [[linear function|linear]] over $\mathbb{C}[ [\hbar] ]$;
1. real, in that for all $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$
$$
\langle A^\ast \rangle = \langle A \rangle^\ast
$$
1. positive, in that for every $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$ there exist a $c_A \in \mathbb{C}[ [\hbar] ]$ such that
$$
\langle A^\ast \star_H A\rangle_{\Phi_0} = c_A^\ast \cdot c_A
\,,
$$
1. normalized, in that
$$
\langle 1\rangle_H = 1
$$
where $(-)^\ast$ denotes componet-wise [[complex conjugation]].
This means that $\langle -\rangle_{0}$ is a [[state on a star-algebra|states]] on the [[Wick algebra|Wick]] [[star-algebra]] $\left( (PolyObs(E,\mathbf{L}))_{mc}[ [\hbar] ], \star_H\right)$ (prop. \ref{MoyalStarProductOnMicrocausal}). One says that
* $\langle - \rangle_0$ is a _[[Hadamard vacuum state]]_;
and generally
* $\langle - \rangle_{\Phi_0}$ is called a _[[coherent state]]_.
=--
([Dütsch 18, def. 2.12, remark 2.20, def. 5.28, exercise 5.30 and equations (5.178)](#Duetsch18))
+-- {: .proof}
###### Proof
The properties of linearity, reality and normalization are obvious, what requires proof is positivity. This is proven by exhibiting a [[representation]] of the Wick algebra on a [[Fock space|Fock]] [[Hilbert space]] (this algebra [[homomorphism]] is _[[Wick's lemma]]_), with formal powers in $\hbar$ suitably taken care of, and showing that under this representation the function $\langle -\rangle_0$ is represented, degreewise in $\hbar$, by the [[inner product]] of the [[Hilbert space]].
=--
+-- {: .num_example #HadamardMoyalStarProductOfTwoLinearObservables}
###### Example
**([[operator product]] of two [[linear observables]])**
Let
$$
A_i \in LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc}
$$
for $i \in \{1,2\}$ be two [[linear observable|linear]] [[microcausal observables]] represented by [[distributions]] which in [[generalized function]]-notation are given by
$$
A_i
\;=\;
\int (\alpha_i)_{a_i}(x_i) \mathbf{\Phi}^{a_i}(x_i) \, dvol_\Sigma(x_i)
\,.
$$
Then their Hadamard-Moyal [[star product]] (prop. \ref{MoyalStarProductOnMicrocausal}) is the [[sum]] of their pointwise product
with their value
$$
\label{EvaluatingLinearObservablesInWightmanPropagator}
\langle A_1 \star_H A_2 \rangle_0
\;\coloneqq\;
i \hbar \int \int
(\alpha_1)_{a_1}(x_1) \Delta_H^{a_1 a_2}(x_1,x_2) (\alpha_2)_{a_2}(x_2)
\,dvol_\Sigma(x_1)
\,dvol_\Sigma(x_2)
$$
in the [[Wightman propagator]], which is the value of the [[Hadamard vacuum state]] from prop. \ref{WickAlgebraCanonicalState}
$$
A_1 \star_H A_2
\;=\;
A_1 \cdot A_2
\;+\;
\langle A_1 \star_H A_2 \rangle_0
$$
In the [[operator product]]/[[normal-ordered product]]-notation of def. \ref{NormalOrderedProductNotation} this reads
$$
A_1 A_2 \;=\; :A_1 A_2: \;+\; \langle A_1 A_2\rangle
\,.
$$
=--
+-- {: .num_example #WeylRelations}
###### Example
**([[Weyl relations]])**
Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] and with [[Wightman propagator]] $\Delta_H$.
Then for
$$
A_1, A_2
\;\in\;
LinObs(E,\mathbf{L})_{mc}
\hookrightarrow
PolyObs(E,\mathbf{L})_{mc}
$$
two [[linear observables|linear]] [[microcausal observables]], the Hadamard-Moyal star product (def. \ref{MoyalStarProductOnMicrocausal}) of their [[exponentials]] exhibits the [[Weyl relations]]:
$$
e^{A_1}
\star_H
e^{A_2}
\;=\;
e^{A_1 + A_2} \; e^{\langle A_1 \star_H A_2\rangle_0}
$$
where on the right we have the [[exponential]] of the value of the [[Hadamard vacuum state]] (prop. \ref{WickAlgebraCanonicalState}) as in example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.
=--
(e.g. [Dütsch 18, exercise 2.3](#Duetsch18))
+-- {: .num_example }
###### Example
**([[Wightman propagator]] is [[2-point function]] in the [[Hadamard vacuum state]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]].
With respect to the induced [[Hadamard vacuum state]] $\langle - \rangle_0$ from prop. \ref{WickAlgebraCanonicalState}, the [[Wightman propagator]] $\Delta_H(x,y)$ itself is the _[[2-point function]]_, namely the [[distribution|distributional]] [[vacuum expectation value]] of the operator product of two [[field observables]]:
$$
\left\langle
\mathbf{\Phi}^a(x) \star_H \mathbf{\Phi}^b(y)
\right\rangle_0
\;=\;
\underset{ = 0 }{
\underbrace{
\left\langle
\mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y)
\right\rangle
}}
+
\underset{ = \hbar \Delta^{a b}_H(x,y) }{
\underbrace{
\left \langle
\hbar
\underset{\Sigma \times \Sigma}{\int}
\delta(x-x') \Delta^{a b}_H(x,y) \delta(y-y')
\right\rangle
}}
$$
by example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.
Equivalently in the [[operator product]]-notation of def. \ref{NormalOrderedProductNotation} this reads:
$$
\left\langle
\mathbf{\Phi}^a(x)
\mathbf{\Phi}^b(y)
\right\rangle_0
\;=\;
\hbar \Delta_H(x,y)
\,.
$$
=--
Similarly:
+-- {: .num_example }
###### Example
**([[Feynman propagator]] is time-ordered [[2-point function]] in the [[Hadamard vacuum state]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]] with induced [[Feynman propagator]] $\Delta_F$.
With respect to the induced [[Hadamard vacuum state]] $\langle - \rangle_0$ from prop. \ref{WickAlgebraCanonicalState}, the [[Feynman propagator]] $\Delta_F(x,y)$ itself is the _time-ordered [[2-point function]]_, namely the [[distribution|distributional]] [[vacuum expectation value]] of the [[time-ordered product]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) of two [[field observables]]:
$$
\left\langle
T\left(
\mathbf{\Phi}^a(x) \star_F \mathbf{\Phi}^b(y)
\right)
\right\rangle_0
\;=\;
\underset{ = 0 }{
\underbrace{
\left\langle
\mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y)
\right\rangle
}}
+
\underset{ = \hbar \Delta^{a b}_H(x,y) }{
\underbrace{
\left \langle
\hbar
\underset{\Sigma \times \Sigma}{\int}
\delta(x-x') \Delta^{a b}_F(x,y) \delta(y-y')
\right\rangle
}}
$$
analogous to example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.
Equivalently in the [[operator product]]-notation of def. \ref{NormalOrderedProductNotation} this reads:
$$
\left\langle
T\left(
\mathbf{\Phi}^a(x)
\mathbf{\Phi}^b(y)
\right)
\right\rangle_0
\;=\;
\hbar \Delta_F(x,y)
\,.
$$
=--
[[!include propagators - table]]
## Related concepts
[[!include products in pQFT -- table]]
[[!include states and observables -- content]]
## References
The construction goes back to
* {#Wick50} [[Gian-Carlo Wick]], _The evaluation of the collision matrix_, Phys. Rev. 80, 268-272 (1950)
Its realization as the [[Moyal deformation quantization]] of the [[Peierls bracket]] shifted by a [[quasi-free Hadamard state]] is due to
* {#Dito90} Joseph Dito, _Star-product approach to quantum field theory: The free scalar field_. Letters in Mathematical Physics, 20(2):125–134, 1990 ([spire](https://inspirehep.net/record/303898/))
further amplified in
* {#DuetschFredenhagen00} [[Michael Dütsch]], [[Klaus Fredenhagen]], section 5.1 of _Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion_, Commun.Math.Phys. 219 (2001) 5-30 ([arXiv:hep-th/0001129](https://arxiv.org/abs/hep-th/0001129))
* {#DuetschFredenhagen01} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic field theory, and deformation quantization_, in [[Roberto Longo]] (ed.), _Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects_, volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001 ([arXiv:hep-th/0101079](https://arxiv.org/abs/hep-th/0101079))
* {#HirschfeldHenselder02} A. C. Hirshfeld, P. Henselder, _Star Products and Perturbative Quantum Field Theory_, Annals Phys. 298 (2002) 382-393 ([arXiv:hep-th/0208194](https://arxiv.org/abs/hep-th/0208194))
* {#Collini16} [[Giovanni Collini]], _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))
and the generalization to [[quantum field theory on curved spacetime]] is discussed in
* {#BrunettiFredenhagen95} [[Romeo Brunetti]], [[Klaus Fredenhagen]], M. Köhler, _The microlocal spectrum condition and Wick polynomials on curved spacetimes_, Commun. Math. Phys. 180, 633-652, 1996 ([arXiv:gr-qc/9510056](https://arxiv.org/abs/gr-qc/9510056))
* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661, 2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))
* {#HollandsWald01} [[Stefan Hollands]], [[Robert Wald]], _Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime_, Commun. Math. Phys. 223:289-326, 2001 ([arXiv:gr-qc/0103074](https://arxiv.org/abs/gr-qc/0103074))
Review is in
* {#Duetsch18} [[Michael Dütsch]], section 2.1 of _[[From classical field theory to perturbative quantum field theory]]_, 2018
[[!redirects Wick algebras]]
[[!redirects Wick polynomial]]
[[!redirects Wick polynomials]]
[[!redirects normal ordering]]
[[!redirects normal-ordered product]]
[[!redirects normal-ordered products]]
[[!redirects normal ordered product]]
[[!redirects normal ordered products]]
[[!redirects abstract Wick algebra]]
[[!redirects abstract Wick algebras]]
|
Wick algebra -- table | https://ncatlab.org/nlab/source/Wick+algebra+--+table | | | [[free field theory|free field]] [[algebra of quantum observables]] | [[physics]] terminology | [[mathematics|maths]] terminology |
|-|-|----------------------|--------------------------------|
| 1) | [[supercommutative algebra|supercommutative product]] | $\phantom{AA} :A_1 A_2:$ <br/> [[normal ordered product]] | $\phantom{AA} A_1 \cdot A_2$ <br/> pointwise product of functionals |
| 2) | [[non-commutative algebra|non-commutative product]] <br/> ([[deformation quantization|deformation]] induced by [[Poisson bracket]]) | $\phantom{AA} A_1 A_2$ <br/> [[operator product]] | $\phantom{AA} A_1 \star_H A_2$ <br/> [[star product]] for [[Wightman propagator]] |
| 3) | | $\phantom{AA} T(A_1 A_2)$ <br/> [[time-ordered product]] | $\phantom{AA} A_1 \star_F A_2$ <br/> [[star product]] for [[Feynman propagator]] |
| | [[perturbative quantum field theory|perturbative expansion]] <br/> of 2) via 1) | [[Wick's lemma]] <br/> <img src="https://ncatlab.org/nlab/files/WickTheorem.png" width="350"/> | [[Moyal star product|Moyal product]] for [[Wightman propagator]] $\Delta_H$<br/> $ \begin{aligned} & A_1 \star_H A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_H)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}$ |
| | [[perturbative quantum field theory|perturbative expansion]] <br/> of 3) via 1) | [[Feynman diagrams]] <br/> <img src="https://ncatlab.org/nlab/files/FeynmanDiagramGlobal.jpg" width="350"> | [[Moyal star product|Moyal product]] for [[Feynman propagator]] $\Delta_F$ <br/> $ \begin{aligned} & A_1 \star_F A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_F)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}$ |
|
Wick rotation | https://ncatlab.org/nlab/source/Wick+rotation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What is called _Wick rotation_ (after [[Gian-Carlo Wick]]) is a method in [[physics]] for finding a construction in [[relativistic field theory]] on [[Minkowski spacetime]] or, more generally, on [[Lorentzian manifolds]], from a related construction in [[Euclidean field theory]] on [[Riemannian manifolds]] in a way that involves or generalizes the idea of [[analytic continuation]] for the _[[time]]_ [[coordinates]].
<center>
<img src="https://ncatlab.org/nlab/files/WickRotation.jpg" width="400">
</center>
> The [[complex plane]] for a complexified [[time]] [[coordinate]] $t + i s$. After Wick rotation it is the [[imaginary part]] $s$ that replaces the "[[real part|real]]" time $t$.
> graphics grabbed from [Fulling-Ruijsenaars 87](#FullingRuijsenaars87)
This is motivated by the observation that the [[Minkowski metric]] (with the $(-1,1,1,1)$ convention for [[spacetime signature]]) and the four-Euclidean metric are equivalent if the [[time]] components of either are allowed to have [[imaginary part|imaginary]] values. Hence Wick rotation, when it applies, involves [[analytic continuation]] of [[n-point functions]] to [[complex number|complex valued]] [[time]] [[coordinates]].
For [[relativistic field theory|relativistic]]/[[Euclidean field theory|Euclidean]] [[quantum field theory]] on [[Minkowski spacetime]] $\mathbb{R}^{d,1}$/[[Euclidean space]] $\mathbb{R}^{d+1}$ Wick rotation is rigorously understood and takes the form of a [[theorem]] relating the [[Wightman axioms]] for [[n-point functions]] in [[relativistic field theory]] to the _[[Osterwalder-Schrader axioms]]_ for [[correlators]] in [[Euclidean field theory]].
See for instance [Fulling-Ruijsenaars 87, section 2](#FullingRuijsenaars87) for a clear account.
More generally, in this context Wick rotation applies to relativistic field theory in [[thermal equilibrium]] [[states]] at [[positive number|positive]] [[temperature]] $T$ ("[[KMS states]]"), and then relates this to [[Euclidean field theory]] with compact/periodic Euclidean time of length $\beta = 1/T$ (see [Fulling-Ruijsenaars 87, section 3](#FullingRuijsenaars87)):
$$
\array{
\left.
\array{
\text{relativistic field theory}
\\
\text{on Minkowski spacetime}
\\
\mathbb{R}^{d,1}
\\
\text{in a thermal equilibrium state}
\\
\text{at temperature}\; T
}
\right\}
&
\;\;\;\;
\overset{ \text{Wick rotation} }{\leftrightarrow}
\;\;\;\;
&
\left\{
\array{
\text{Euclidean field theory}
\\
\text{on Euclidean space}
\\
\mathbb{R}^d \times S^1_{\beta}
\\
\text{with compact/periodic Euclidean time}
\\
\text{of length} \; \beta = 1/T
}
\right.
\\
\phantom{A}
\\
\underset{
{
\text{equal-time n-point function}
\atop
\text{of relativistic fields}
}
\atop
\text{ in thermal equilibrium state } \; \vert T\rangle
}{
\underbrace{
\left\langle T\vert :\mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t) : \vert T \right\rangle_{\mathbb{R}^{d,1}}
}}
&\;=\;&
\underset{
\text{correlator of Euclidean fields}
\atop
\text{ for "Euclidean time" of periodicity}\; \beta = 1/T
}{
\underbrace{
\left\langle 0 \vert \mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t) \vert 0 \right\rangle_{\mathbb{R}^{d} \times S^1_{\beta}}
}}
}
$$
In this form, Wick rotation is also known as _[[thermal quantum field theory]]_. See there for more.
<center>
<img src="https://ncatlab.org/nlab/files/EuclideanTime.jpg" width="580">
</center>
> graphics grabbed form [Frolov-Zelnikov 11](#FrolovZelnikov11)
Wick rotation also applies on suitable [[black hole]]-[[spacetimes]] spring
thermodynamics, such as the [[Bekenstein-Hawking entropy]], find elegant explanations, at least at the level of the manipulation of formulas (see e.g. [Fulling-Ruijsenaars 87, section 4](#FullingRuijsenaars87)).
### Example
Consider the Minkowski metric with the $-1,1,1,1$ convention for the tensor:
$d s^{2}= -(d t)^{2} + (d x)^{2} + (d y)^{2} + (d z)^{2}$
and the four-dimensional Euclidean metric:
$d s^{2}= d \tau^{2} + (d x)^{2} + (d y)^{2} + (d z)^{2}$.
Notice that if $d t = i\cdot d \tau$, the two are equivalent.
### Method
A typical method for employing Wick rotation would be to make the substitution $t=i\tau$ in a problem in Minkowski space. The resulting problem is in Euclidean space and is sometimes easier to solve, after which a reverse substitution can (sometimes) be performed, yielding a solution to the original problem.
Technically, this works for any four-vector comparison between Minkowski space and Euclidean space, not just for space-time intervals.
## Related concepts
* [[reflection positivity]]
## References
Wick rotation between [[relativistic field theory]] in terms of [[Wightman axioms]] for [[n-point functions]] on [[Minkowski spacetime]] and [[Euclidean field theory]] in terms of [[Osterwalder-Schrader axioms]] for [[correlators]] on [[Euclidean space]] is due to
* {#OsterwalderSchrader73} [[Konrad Osterwalder]], [[Robert Schrader]], _Axioms for Euclidean Green's functions_, Comm. Math. Phys. Volume 31, Number 2 (1973), 83-112 ([Euclid:1103858969](https://projecteuclid.org/euclid.cmp/1103858969))
The generalization of this for [[positive number|positive]] [[thermal equilibrium]] [[vacuum states]], relating to [[thermal quantum field theory]] with compact/periodic Euclidean time is discussed in
* {#KleinLandau81} Abel Klein, Lawrence Landau, _Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle_, Pacific Journal of Mathematics, Vol. 94, No. 2, 1981 ([
DOI: 10.2140/pjm.1981.94.341](https://msp.org/pjm/1981/94-2/p12.xhtml), [pdf](https://msp.org/pjm/1981/94-2/pjm-v94-n2-p12-s.pdf))
The idea here apparently goes back to
* {#Bloch58} Claude Bloch, _Sur la détermination de l'état fondamental d'un système de particules_, Nucl. Phys. 7 (1958) 451
This has maybe first been made precise, for the case of 1+1 dimensions, in
* {#HoeghKrohn74} [[Raphael Høegh-Krohn]], _Relativistic Quantum Statistical Mechanics in two-dimensional Space-Time_, Communications in Mathematical Physics 38.3 (1974): 195-224 ([pdf](https://www.duo.uio.no/bitstream/handle/10852/44072/1973-22.pdf))
Good review on the relation to [[thermal quantum field theory]] and [[black hole thermodynamics]] is in
* {#FullingRuijsenaars87} S.A. Fulling, S.N.M. Ruijsenaars, _Temperature, periodicity and horizons_, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 ([pdf](https://www1.maths.leeds.ac.uk/~siru/papers/p26.pdf), <a href="https://doi.org/10.1016/0370-1573(87)90136-0">doi:10.1016/0370-1573(87)90136-0</a>)
* {#GibbonsPerry78} [[Gary Gibbons]], Malcolm J. Perry, _Black Holes and Thermal Green Functions_, Vol. 358, No. 1695 (1978) ([jstor:79482](https://www.jstor.org/stable/79482))
New formalization in terms of complex metrics:
* [[Maxim Kontsevich]], [[Graeme Segal]], *Wick rotation and the positivity of energy in quantum field theory*, The Quarterly Journal of Mathematics **72** 1-2 (2021) 673–699 [[arXiv:2105.10161](https://arxiv.org/abs/2105.10161), [doi:10.1093/qmath/haab027](https://doi.org/10.1093/qmath/haab027)]
review talks:
* [[Graeme Segal]], _Wick rotation and the positivity of energy in quantum field theory_, talk at Institut des Hautes Études Scientifiques (IHÉS), June 2014 ([video recording](https://www.youtube.com/watch?feature=player_embedded&v=vTvXHL6ZJik))
* [[Graeme Segal]], *Wick Rotation and the Positivity of Energy in Quantum Field Theory*, talk at IAS Physics Group Meeting, December 2021 ([video recording](https://www.ias.edu/video/wick-rotation-and-positivity-energy-quantum-field-theory))
Discussion of thermal Wick rotation on global [[anti-de Sitter spacetime]] (which is already periodic in _real_ time) is in
* {#AllenFolacciGibbons87} B. Allen, A. Folacci, [[Gary Gibbons]], _Anti-de Sitter space at finite temperature_, Physics Letters B Volume 189, Issue 3, 7 May 1987, Pages 304-310 (<a href="https://doi.org/10.1016/0370-2693(87)91437-7">doi:10.1016/0370-2693(87)91437-7</a>)
See also
* Dirk Schlingemann, _From euclidean field theory to quantum field theory_ ([arXiv:hep-th/9802035](http://arxiv.org/abs/hep-th/9802035))
* {#Witten13} [[Edward Witten]], _The Feynman $i \epsilon$ in String Theory_ ([arXiv:1307.5124](http://arxiv.org/abs/1307.5124))
On the (non-)existence of Wick rotation for [[quantum field theory on curved spacetimes]]:
MathOverflow comments:
* [[Igor Khavkine]] ([MO:a/165317](https://mathoverflow.net/a/165317/381))
* [[Robert Bryant]] ([MO:a/165345](https://mathoverflow.net/a/165345/381))
See also:
* {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], p. 7-8 in: _Quantum fields in curved spacetime_, Physics Reports Volume 574, 16 April 2015, Pages 1-35 ([arXiv:1401.2026](https://arxiv.org/abs/1401.2026), [doi:10.1016/j.physrep.2015.02.001](https://doi.org/10.1016/j.physrep.2015.02.001))
[[!redirects Wick rotations]]
|
Wick's lemma | https://ncatlab.org/nlab/source/Wick%27s+lemma |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_Wick's lemma_ is a [[combinatorics|combinatorial]] formula for the product in [[associative algebras]] that appear in the context of [[free field theory]].
From the point of view of [[path integral quantization]], Wick's lemma is about the [[moments]] of [[Gaussian probability distributions]]. See at _[[Feynman diagram]]_ for more on this.
From the point of view of [[causal perturbation theory]] Wick's lemma expresses the [[Moyal deformation quantization]] of a [[free field theory]] ([[Wick algebras]]) in terms of [[operator products]] on.
[[!include Wick algebra -- table]]
From the point of view of [[BV-quantization]] Wick's lemma arises as a consequence of the [[homological perturbation lemma]] ([Gwilliam, section 2.3](#Gwilliam)).
## Statement
Let $\mathcal{W}$ be the [[Wick algebra]] of the [[free field theory|free]] [[scalar field]], hence the space of [[microcausal observables]] with product the [[star product]] induced by the [[Wightman propagator]]:
$$
\mathcal{W}
\;\coloneqq\;
\left(
PolyObs(E,\mathbf{L})_{mc}, \star_{\Delta_H}
\right)
$$
Then the evident map from $\mathcal{W}$ to [[linear operators]] on the [[Fock space]] equipped with their [[operator product]] $\circ$ is an [[associative algebra]] [[isomorphism]] onto its image:
$$
\left(
PolyObs(E,\mathbf{L})_{mc}, \star_{\Delta_H}
\right)
\underoverset{\simeq}{\text{Wick's lemma}}{\hookrightarrow}
\left(
End(FockSpace), \circ
\right)
$$
([Dütsch 18, theorem 2.17](#Duetsch18), following [Dütsch-Fredenhagen 00,, pages 10-11](#DuetschFredenhagen00), [Dütsch-Fredenhagen 01](#DuetschFredenhagen01))
## Related concepts
* [[perturbation theory]]
* [[Feynman diagram]]
## References
Original articles include
* {#Wick50} [[Gian-Carlo Wick]], _The evaluation of the collision matrix_, Phys. Rev. 80, 268-272 (1950)
* {#Hepp69} [[Klaus Hepp]], _Théorie de la Renormalisation_ Lect. Notes in Physics, Springer 1969
* {#BrunettiFredenhagenVerch03} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Rainer Verch]], theorem 2.4 in _The generally covariant locality principle -- A new paradigm for local quantum physics_, Commun.Math.Phys.237:31-68, 2003 ([arXiv:math-ph/0112041](https://arxiv.org/abs/math-ph/0112041))
The interpretation as an algebra isomorphism to the [[star product]] with respect to the [[Wightman propagator]] is made explicit in
* {#DuetschFredenhagen00} [[Michael Dütsch]], [[Klaus Fredenhagen]], pages 10-11 of _Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion_, Commun.Math.Phys. 219 (2001) 5-30 ([arXiv:hep-th/0001129](https://arxiv.org/abs/hep-th/0001129))
* {#DuetschFredenhagen01} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic quantum field theory and deformation quantization_, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) ([arXiv:hep-th/0101079](http://xxx.uni-augsburg.de/abs/hep-th/0101079))
and [Dütsch 18, theorem 2.17](#Duetsch18)
Textbook accounts include
* {#Scharf95} [[Günter Scharf]], theorem 3.1 in _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition
* {#Duetsch18} [[Michael Dütsch]], E.26 - E.30 in _[[From classical field theory to perturbative quantum field theory]]_, 2018
See also
* Wikipedia, _[Wick's theorem](https://en.wikipedia.org/wiki/Wick%27s_theorem)_
Discussion of Wick's lemma as a consequence of the [[homological perturbation lemma]] for [[BV-complexes]] is in
* {#Gwilliam} [[Owen Gwilliam]], section 2.3 _Factorization algebras and free field theories_ PhD thesis ([pdf](http://math.berkeley.edu/~gwilliam/thesis.pdf))
[[!redirects Wick lemma]]
[[!redirects Wick's theorem]]
[[!redirects Wick theorem]]
|
wide pullback | https://ncatlab.org/nlab/source/wide+pullback |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
\tableofcontents
## Definition
A **wide pullback** or **wide fiber product** or **wide fibre product** in a [[category]] $\mathcal{C}$ is a [[product]] (of arbitrary [[cardinality]]) in a [[over category|slice category]] $\mathcal{C} \downarrow C$. In terms of $\mathcal{C}$, this can be expressed as a [[limit]] over a category obtained from a [[discrete category]] by adjoining a [[terminal object]].
Yet more explicitly, the wide pullback of a family of morphisms $f_i\colon A_i \to C$ (a **[[wide cospan]]**) is an object $P$ equipped with projection $p_i\colon P\to A_i$ such that $f_i p_i$ is independent of $i$, and which is universal with this property.
Binary wide pullbacks are the same as ordinary [[pullbacks]], a.k.a. fiber products.
Of course, a **wide pushout** is a wide pullback in the opposite category.
## Properties
* A category has wide pullbacks (of all [[small category|small]] cardinalities) if and only if it has (binary) [[pullbacks]] and [[cofiltered limits]].
* The [[saturated class of limits|saturation]] of the class of wide pullbacks is the class of limits over categories $C$ whose [[fundamental groupoid]] $\Pi_1(C)$ is trivial.
On the other hand, together with a terminal object, wide pullbacks generate all limits:
+-- {: .num_prop #WidePbToComplete}
###### Proposition
A [[category]] $C$ with all [[wide pullbacks]] and a [[terminal object]] $1$ is [[complete category|complete]]. If $C$ is complete and $F\colon C \to D$ preserves wide pullbacks and the terminal object, then it preserves all limits.
=--
+-- {: .proof}
######Proof
To build up arbitrary products $\prod_{i \in I} c_i$ in $C$, take the wide pullback of the family $c_i \to 1$. Then to build equalizers of diagrams $f, g\colon c \rightrightarrows d$, construct the pullback of the diagram
$$\array{
& & d \\
& & \downarrow \delta \\
c & \underset{\langle f, g \rangle}{\to} & d \times d
}$$
From products and equalizers, we can get arbitrary limits.
=--
## Related concepts
* [[connected limit]]
[[!include notions of pullback -- contents]]
**Analogues in [[dependent type theory]]**:
* [[wide pullback type]]
* [[wide pushout type]]
## References
The terminology *wide pullback* appears in:
* [[Paul Taylor]], _Quantitative domains, groupoids and linear logic_, Category Theory and Computer Science: Manchester, UK, September 5–8, 1989 Proceedings. Springer Berlin Heidelberg, 1989.
Wide pullbacks are considered under the term *fibred product* in:
* [[Robert Paré]], *Simply connected limits*. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, [CMS](http://math.ca/10.4153/CJM-1990-038-6)
[[!redirects wide pullback]]
[[!redirects wide pullbacks]]
[[!redirects wide fiber product]]
[[!redirects wide fiber products]]
[[!redirects wide fibre product]]
[[!redirects wide fibre products]]
[[!redirects wide pullback type]]
[[!redirects wide pullback types]]
[[!redirects wide pushout]]
[[!redirects wide pushouts]]
|
wide subcategory | https://ncatlab.org/nlab/source/wide+subcategory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
### Of a general category
Traditionally, a **wide subcategory** of a [[category]] $C$ is a [[subcategory]] containing all the [[objects]] of $C$.
Equivalently, it is a subcategory through which the canonical [[functor]] $disc(Obj(C)) \to C$ (from the [[discrete category]] on the collection of [[objects]]) factors, or whose inclusion functor is [[bo functor|bijective on objects]].
Notice that the condition to contain all the objects is not invariant under [[equivalence of categories]] and so the definition of wide subcategory above violates the [[principle of equivalence]]. A variant of the definition which fixes this is:
an **essentially wide subcategory** contains at least one object from each [[isomorphism class]] of objects; that is, its inclusion functor is [[essentially surjective functor|essentially surjective on objects]].
A wide subcategory is also called a **lluf subcategory** ("lluf" being "[[full subcategory|full]]" spelled backwards).
### Of an abelian category
An unrelated definition of "wide subcategory" is commonly used in the study of [[derived categories]] and [[stability conditions]].
In this context, a [[full subcategory]] $\mathcal{W} \hookrightarrow \mathcal{A}$ of an [[abelian category]] $\mathcal{A}$ is called **wide** if it is closed under [[kernels]], [[cokernels]] and [[extensions]].
See, for example, [Hovey 01](#Hovey01), [Ingalls-Thomas 09](#IngallsThomas09), [Marks-Stovicek 15](#MarksStovicek15).
Given a wide subcategory $\mathcal{W}$ in this sense, one can consider the minimal [[torsion theory|torsion class]] $T(\mathcal{W})$ containing it. Conversely, if $\mathcal{T}$ is a torsion class, define $W(\mathcal{T})$ to be the [[full subcategory]] on those [[objects]] $X \in \mathrm{Ob}(\mathcal{T})$ such that, for any $Y \in \mathrm{Ob}(\mathcal{T})$ and any $g: Y \to X$, the [[kernel]] of $g$ is in $\mathcal{T}$. The [[composition]] $W \circ T$ is the [[identity functor|identity]], thus exhibiting the [[poset]] of wide subcategories as a [[retract]] of the poset of [[torsion theory|torsion classes]].
The [[surjection]] $W$ becomes an [[injection]] when restricted to [[functorially finite]] torsion classes, and is often a [[bijection]] between functorially finite torsion classes and functorially finite wide subcategories; see [Marks-Stovicek 15](#MarksStovicek15).
## Examples
### General
Given any category $C$, the maximal sub-groupoid of $C$ is the subcategory consisting of all objects of $C$ but with morphisms only the isomorphisms of $C$. This is the [[core]] of a category, and it is a wide subcategory.
## References
### General
...
### For abelian categories
* {#Hovey01} [[Mark Hovey]], _Classifying subcategories of modules_, Trans. Amer. Math. Soc. 353 (2001), 3181-3191 ([doi:10.1090/S0002-9947-01-02747-7](https://doi.org/10.1090/S0002-9947-01-02747-7))
* {#IngallsThomas09} Colin Ingalls, Hugh Thomas, _Noncrossing partitions and representations of quivers_, Volume 145, Issue 6 November 2009 , pp. 1533-1562 ([doi:10.1112/S0010437X09004023](https://doi.org/10.1112/S0010437X09004023))
* {#MarksStovicek15} Frederik Marks, Jan Stovicek, _Torsion classes, wide subcategories and localisations_ [arXiv:1503.04639](https://arxiv.org/abs/1503.04639)
[[!redirects wide subcategory]]
[[!redirects wide subcategories]]
[[!redirects lluf subcategory]]
[[!redirects lluf subcategories]]
[[!redirects essentially wide subcategory]]
[[!redirects essentially wide subcategories]]
[[!redirects essentially lluf subcategory]]
[[!redirects essentially lluf subcategories]]
|
Wiebe van der Hoek | https://ncatlab.org/nlab/source/Wiebe+van+der+Hoek |
* [personal page](https://www.csc.liv.ac.uk/~wiebe/)
* [GoogleScholar page](https://scholar.google.com/citations?user=WNmaKaIAAAAJ&hl=en)
## Selected writings
On [[modal logic]] in [[game theory]] and [[information science]]:
* [[Wiebe van der Hoek]], [[Marc Pauly]], *Modal logic for games and information*, Ch. 20 in: *The Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 85-183 $[$<a href="https://doi.org/10.1016/S1570-2464(07)80023-1">doi:10.1016/S1570-2464(07)80023-1</a>, [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/)$]$
On [[epistemic logic]] and application to [[game theory]]:
* [[Hans Ditmarsch]], [[Wiebe Hoek]], Barteld Kooi, *Dynamic Epistemic Logic*, Studies In Epistemology, Logic, Methodology, And Philosophy Of Science (SYLI) **337**, Springer (2008) $[$[doi:10.1007/978-1-4020-5839-4_2](https://doi.org/10.1007/978-1-4020-5839-4_2), [pdf](http://www.ivanociardelli.altervista.org/wp-content/uploads/2018/04/Epistemic-logic.pdf)$]$
## Related entries
* [[epistemic logic]]
category: people
[[!redirects Wiebe Hoek]] |
Wiener measure | https://ncatlab.org/nlab/source/Wiener+measure |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Wiener measure_ is a [[measure]] on the [[topological space|space]] of [[continuous function|continuous paths]] in a given [[manifold]]. The [[Lebesgue integral]] with respect to Wiener's measure is called the __Wiener integral__.
The Wiener measure serves to make precise the [[path integral quantization]] for the (charged) non-relativistic [[particle]] (that of the [[relativistic particle]] may be amenable to Wiener measure methods via [[Wick rotation]], i.e. analytic continuation to [[imaginary number|imaginary]] time. ).
## Related concepts
* [[Brownian motion]]
## References
Named after [[Norbert Wiener]]'s discussion of [[Brownian motion]]:
* [[Norbert Wiener]], *The Average of an Analytic Functional and the Brownian Movement*, Proceedings of the National Academy of Sciences of the United States of America, **7** 10 (1921) 294-298 [[jstor:84434](https://www.jstor.org/stable/84434)]
Introduction:
* Tamas Szabados, *An elementary introduction to the Wiener process and stochastic integrals*, Studia Scientiarum Mathematicarum Hungarica **31** (1996) 249-297 [[arXiv:1008.1510](https://arxiv.org/abs/1008.1510)]
History:
* Arthur Genthon, *The concept of velocity in the history of Brownian motion -- From physics to mathematics and back*, Eur. Phys. J. H **45** (2020) 49-105 [[arXiv:2006.05399](https://arxiv.org/abs/2006.05399), [doi:10.1140/epjh/e2020-10009-8]( https://doi.org/10.1140/epjh/e2020-10009-8)]
See also:
* [[eom|Enc. Math
]], _[Wiener integral](https://encyclopediaofmath.org/wiki/Wiener_integral)_
* PlanetMath _[Wiener measure](http://planetmath.org/wienermeasure)_
A textbook account in the context of [[path integral quantization]]:
* [[Barry Simon]], around p. 49 in: _Functional integration and quantum physics_, AMS Chelsea Publ., Providence, 2005
[[!redirects Wiener measures]]
[[!redirects Wiener integral]]
[[!redirects Wiener integrals]]
[[!redirects Wiener process]]
[[!redirects Wiener processes]]
|
Wiener-Hopf decomposition | https://ncatlab.org/nlab/source/Wiener-Hopf+decomposition | Decomposition of an analytic function into a product of a function analytic in upper half-plane and analytic function in lower half-plane. There is a version for stochastic processes. It can be related tp Riemann-Hilbert-[[Birkhoff decomposition]].
* wikipedia: [Wiener-Hopf method](http://en.wikipedia.org/wiki/Wiener%E2%80%93Hopf_method)
* [[eom]]: [Wiener-Hopf method](http://www.encyclopediaofmath.org/index.php/Wiener%E2%80%93Hopf_method) |
Wieslaw Pawlowski | https://ncatlab.org/nlab/source/Wieslaw+Pawlowski |
* [personal page](https://inf.ug.edu.pl/~wpawlowski/)
* [GoogleScholar page](https://scholar.google.pl/citations?user=mr7fct0AAAAJ&hl=en)
## Selected writings
On generalized [[congruences]] and [[coequalizers]] in the [[1-category]] [[Cat]] of [[small categories]]:
* {#BednarczykBorzyszkowskiPawlowski99} [[Marek A. Bednarczyk]], [[Andrzej M. Borzyszkowski]], [[Wieslaw Pawlowski]], Section 4 of: *Generalized congruences -- epimorphisms in $\mathcal{C}at$*, Theory and Applications of Categories **5** 11 (1999) 266-280 [[tac:5-11](http://www.tac.mta.ca/tac/volumes/1999/n11/5-11abs.html), [dml:120226](https://eudml.org/doc/120226?lang=fr&limit=5)]
category: people |
Wightman axioms | https://ncatlab.org/nlab/source/Wightman+axioms |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **Wightman axioms** are an attempt to [[axiom|axiomatize]] and thus formalize the notion of a [[quantum field theory]] on [[Minkowski spacetime]] ([[relativistic quantum field theory]]) in the sense of [[AQFT]], i.e. in terms of the assignment of field [[quantum observables]] to points or subsets of [[spacetime]] ([[operator-valued distributions]]).
They serve as the basis of what is known as **constructive quantum field theory** which seeks to provide a mathematically sound framework for quantum theory over the Minkowski space background of [[special relativity]]. [[Arthur Wightman]] first formulated them in the 1950s but they were not published until 1964 after advances in scattering theory confirmed their applicability.
The Wightman axioms served to establish rigorously several basic structural properties of quantum field theories on Minkowski spacetime, such as the [[spin-statistics theorem]] or the [[Osterwalder–Schrader theorem]] relating Lorenzian and Euclidean quantum field theories ("Wick rotation").
They were later further abstracted to the [[Haag-Kastler axioms]] that characterize [[local net]]s of operator algebras and serve as the basis for [[AQFT|algebraic quantum field theory]]. See there for further details.
This page is a draft, see:
* Wikipedia on the [Wightman axioms] (http://en.wikipedia.org/wiki/Wightman_axioms)
## Wightman Axioms
Note: the numbering - and indeed the actual number - of axioms varies depending on the source. In many sources, several of the axioms below are combined.
+-- {.num_defn}
###### Axiom
There is a physical Hilbert space $\mathcal{H}$ in which a unitary representation $U(a,\Lambda)$ of the [[Poincaré spinor group]], $P_{0}$ acts.
=--
+-- {.num_defn}
###### Axiom
The spectrum of the energy-momentum operator P is concentrated in the closed upper (forward) light cone $V^{+}$.
=--
+-- {.num_defn}
###### Axiom
There exists in $\mathcal{H}$ a unique unit vector $|0\rangle$ (the _[[vacuum state]]_), which is invariant with respect to the space-time translations $U(a,1)$.
=--
+-- {.num_defn}
###### Axiom
The components $\phi_{i}$ of the quantum field $\phi$ are [[operator-valued distributions]] $\phi_{i}(x)$ over the [[Schwartz space]] $S(M)$ ([[tempered distributions]]) with domain of definition $D$ which is common to all the operators and is dense in $\mathcal{H}$. $|0\rangle$ is contained in $D$ and $D$ is taken into itself under the action of $\phi (f)$ and $U(a,\Lambda)$.
=--
Note: As in distribution theory it is custom to abuse the notation and write $\phi (x)$ for a point $x$ of the Minkowski spacetime and talk about the function $\phi$, rather than the value of the distribution $\phi(f)$ of a test function $f$.
+-- {.num_defn}
###### Axiom
$U(a,\Lambda)\phi_{i}(x)U(a,\Lambda)^{-1}=\sum_{j}V_{ij}(\Lambda^{-1})\phi_{j}(\Lambda x + a)$ where $V_{ij}(\Lambda)$ is a complex or real finite-dimensional matrix representation of $SL(2,C)$.
=--
+-- {.num_defn}
###### Axiom
Any two field components $\phi_{i}(x)$ and $\phi_{j}(y)$ either commute or anticommute under a space-like separation of the arguments $x$ and $y$.
=--
+-- {.num_defn}
###### Axiom
The set $D_{0}$ of finite linear combinations of vectors of the form $\phi_{i_1}(f_{1})\ldots\phi_{i_n}(f_{n})|0\rangle$ is dense in $\mathcal{H}$.
=--
## Properties
### The Wightman Reconstruction Theorem
The **[[vacuum expectation values]]** ([[n-point functions]]) of the theory are all (tempered, by the axioms) distributions of the form
$$
\langle0| \phi_{i_1}(f_{1})\ldots\phi_{i_n}(f_{n})|0\rangle
$$
The term "distribution" is used interchangably with "function" and "value" in the physics literature. The Wightman reconstruction theorem states properties that a set of tempered distributions $\{\mathcal{W}^n | n \in \N \}$ need to have to be the set of vacuum expectation values of a Wightman theory, and all such Wightman theories are then unitarily equivalent.
...to be continued...
### Equivalence to Euclidean Field Theory
See [[Osterwalder–Schrader theorem]]
### Equivalence to the Haag--Kastler Axioms
See [[Haag–Kastler axioms]]. Since both the Wightman and the Haag-Kastler approach try to formulate an axiomatic approach to quantum field theory on Minkowski spacetime, the natural question to ask is what is their relationship? Three possible answers come to mind:
* They are equivalent.
* One contains the other, i.e. one set of axioms implies the other.
* At least one is wrong (from the physical viewpoint).
Unfortunatly the situation does not seem to be as clear as this list suggests. The current state of the affair seems to be that
* "the Haag-Kastler approach is more convenient, because it deals with algebras of bounded operators, while Wightman fields are allowed to generate unbounded operators"
and
* "a Wightman field theory is equivalent to a Haag-Kastler theory, if some mild additional assumptions are made".
Reference:
* [[Hans-Jürgen Borchers]], [[Jakob Yngvason]], *From quantum fields to local von Neumann algebras*, Rev. Math. Phys. **4** spec01 (1992) 15-47 [[doi:10.1142/S0129055X92000145](https://doi.org/10.1142/S0129055X92000145)]
+--{: .query}
[[Tim van Beek]]: The two statemants above are a condensate of diverse papers I read, any input about the true state of the art would be most welcome. My educated guess is that the "mildness" of the assumptions indicates the fact that all physically realistic models are supposed to satisfy these.
=--
Should it be bounded or unbounded operators/observables?
* Pro: Some observables do not have an upper bound, e.g. you can always increase the impulse of an electron, no matter how big that already is (in theory).
* Contra: A detector has an upper bound for every observable it can detect, and that is what is actually measured in an experiment. The situation of the pro-bullet won't arise because we measure observables in bounded regions of spacetime, and the energy contained in a bounded region is always finite (we are excluding artifacts like black holes from our consideration, of course).
* Contra to even asking the question: Theories of bounded and unbounded operators/observables are physically equivalent modulo mathematical subtleties, meaning they produce the same numbers that can be compared to experiments.
One simple situation where the construction of a Haag-Kastler net out of Wightman fields is straight forward is this: Suppose that all smeared field operators of the Wightman theory are essentially self adjoint for real test functions and commute strongly (their spectral projections commute) if the test functions have space-like separated support.
Then we can define local algebras by
$\mathcal{M}(\mathcal{O}) := \{F(\Psi(f)) | f$ is a real test function with support contained in $\mathcal{O}$ and F is a bounded Borel measurable function on $\R \}$.
(Our assumptions allow us to use the [Borel functional calculus] (http://en.wikipedia.org/wiki/Borel_functional_calculus)).
## Examples
### Neutral Real (uncharged) Scalar Field
A neutral real quantum scalar field on Minkowski spacetime $M$ with mass parameter $m \gt 0$ can be defined as follows:
+-- {: .un_def}
###### Definition
The **positive mass shell** is the subset of Minkowski spacetime defined by
$$
X^+_m := \{p | p^2 = m^2, \; p_0 \gt 0 \}
$$
The **normalized Lorentz-invariant measure** on the positive mass shell is defined with respect to the Lesbegue measure by
$$
d\lambda(p) = d\lambda(\omega_p, \vec p) = \frac{d^3 \vec p}{(2\pi)^3 \omega_p} \; \text{with} \; \omega_p = p^0 = \sqrt{|\vec p|^2 + m^2}
$$
=--
Let $H = L^2(X^+_m, \lambda)$ be the Hilbert space with $X^+_m$ the positive mass shell and $\lambda$ the normalized Lorentz-invariant measure on it as defined above. Construct the Boson [[Fock space]] $F_s(H)$.
Define the operator $R$ to be the Lorentz invariant Fourier transform restricted to $X^+_m$:
$$
R: \mathcal{S}(\mathbb{R}^4) \to H
$$
$$
R(f) := \hat f |_{X^+_m} \; \text{with} \; \hat f := \mathcal{F}(f) = \int_M e^{i p_{\mu} x^{\mu}} f(x) dx
$$
The quantum field $\Psi$ is now a real tempered distribution on $M$ with values in the space of operators of $F_s(H)$.
$$
\Psi: \mathcal{S}(M) \to \mathcal{B}(F_s(H))
$$
$$
f \mapsto \Psi(f) = \frac{1}{\sqrt(2)} (a(Rf) + a^*(Rf))
$$
$a$ and $a^*$ are the annihilation and creation operators on $F_s(H)$, that is $a(v)$ anihilates a single particle state $v$ and $a^*(v)$ creates a single particle state $v$.
$\Psi$ is a distribution solution of the [[Klein-Gordon equation]] by construction, that is for every test function $f$ we get
$$
\Psi((\Box + m^2) f) = 0
$$
The reason for this is that
$$
\mathcal{F}{[(\Box + m^2) f]} = (-p^2 + m^2) f = 0 \; \text{on} \; X^+_m
$$
### Further examples
The Wightman axioms have been established for the following theories.
* Massive 2d scalar theories with polynomial interactions, see [this article](http://www.jstor.org/stable/1970959) by Glimm, Jaffe and Spencer.
* Massive $\phi^4$ in 3d, see [this article](http://www.sciencedirect.com/science/article/pii/0003491676902232) by Feldman and Osterwalder as well as this one by Magnen and Seneor.
* Massive Gross-Neveu in 2d see [this article](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104114300) by Gawedzki and Kupiainen and [this one](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104114625) by Feldman, Magnen, Rivasseau and Seneor.
* Massive Thirring model, see [this article](http://dx.doi.org/10.5169/seals-114796) by Frohlich and Seiler and this more recent one by Benfatto, Falco and Mastropietro.
## Related concepts
* [[AQFT]], [[local net]]
* [[conformal net]], [[conformal bootstrap]]
* analog for [[Euclidean field theory]]: [[Osterwalder-Schrader axioms]]
## References
The original texts are
* A. S. Wightman, L. Gårding, _Fields as Operator-valued Distributions in Relativistic Quantum Theory_, Arkiv f. Fysik, Kungl. Svenska Vetenskapsak. 28, 129–189 (1964).
* Raymond F. Streater, [[Arthur Wightman]], _PCT, spin and statistics, and all that_, Princeton University Press 1989 ([jstor](https://www.jstor.org/stable/j.ctt1cx3vcq), [ZMATH] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1026.81027&format=complete) of the newest edition).
Raymond Streater relates some historical background about the book and the approach [on his webpage] (http://www.mth.kcl.ac.uk/~streater/wightman.html).
A review of [[QFT]] via Wightman axioms and [[AQFT]] is in
* [[Franco Strocchi]], _Relativistic Quantum Mechanics and Field Theory_, Found.Phys. 34 (2004) 501-527 ([arXiv:hep-th/0401143](http://arxiv.org/abs/hep-th/0401143))
The list of examples above draws from
* Abdesselam, _[TP.SE comment](http://theoreticalphysics.stackexchange.com/questions/308/which-qfts-were-rigorously-constructed/350#350)_
Discussion of the axioms for $\phi^4$-theory in 3d is in
* Joel Feldman, [[Konrad Osterwalder]], _The Wightman axioms and the mass
gap for weakly coupled $phi_3^4$ quantum field theories_, Ann. Physics 97 (1976), 80–135
See also
* Wikipedia, _[Wightman axioms](https://en.wikipedia.org/wiki/Wightman_axioms)_
[[!redirects Wightman theory]]
[[!redirects Wightman approach]] |
Wigner 9j symbol | https://ncatlab.org/nlab/source/Wigner+9j+symbol |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
## Related concepts
* [[Wigner 3j symbols]] ([[Clebsch-Gordan coefficient]])
* [[Wigner 6j symbol]]
## References
* Wikipedia, _[9-j symbol](https://en.wikipedia.org/wiki/9-j_symbol)_
[[!redirects Wigner 9j symbols]]
[[!redirects Wigner 9-j symbol]]
[[!redirects Wigner 9-j symbols]]
[[!redirects 9-j symbol]]
[[!redirects 9-j symbols]]
[[!redirects 9j symbol]]
[[!redirects 9j symbols]]
|
Wigner classification | https://ncatlab.org/nlab/source/Wigner+classification | > not to be confused with [[Wigner's theorem]]
***
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The identification of [[particle]] species with [[irreducible representation|irreducible]] [[unitary representations]] of the [[symmetry group]] acting on [[matter]] [[field (physics)|fields]] over [[spacetime]]. Named after [[Eugene Wigner]].
For details see at
* _[[unitary representation of the Poincaré group]]_.
* _[[unitary representation of the super Poincaré group]]_.
It uses a method that goes back to Frobenius and came to be known as "Wigner's little group method" or the "Mackey machine" ([Mackey 68](#Mackey68)).
## Related concepts
* [[Doplicher-Roberts reconstruction theorem]]
* [[Deligne's theorem on tensor categories]] -- [Relevance for Supersymmetry](Deligne's+theorem+on+tensor+categories#RelevanceForSupersymmetry)
* [[supermultiplet]]
## References
The original article is
* {#Wigner39} [[Eugene Wigner]], _On unitary representations of the inhomogeneous Lorentz group_, Annals of Mathematics, 40 (1939) 149–204
The observation that Wigner's representations $U_{(-)}$ are those for which all the maps $g \mapsto \langle \Phi \vert U_g \Psi \rangle$ are [[measurable function|measurable]] and their correspondence of irreducible unitary representations to [[induced representations]] of irreps of the [[stabilizer group]] of a given momentum is (e.g. [Dragon 16](#Dragon16)) due to
* {#Mackey68} [[George Mackey]], _Induced Representations of Groups and Quantum Mechanics_, W. A. Benjamin, New York, 1968
Review includes
* {#Schroer96} [[Bert Schroer]], _Wigner Representation Theory of the Poincaré
Group, Localization, Statistics and the S-Matrix_, 1996 ([pdf](http://cds.cern.ch/record/308785/files/9608092.pdf))
* {#Dragon16} [[Norbert Dragon]], _Currents for Arbitrary Helicity_ ([arXiv:1601.07825](http://arxiv.org/abs/1601.07825))
* Wikipedia, _[Wigner's classification](http://en.wikipedia.org/wiki/Wigner%27s_classification)_
[[!redirects Wigner's classification]]
[[!redirects Wigner-Weyl theory]] |
Wigner function | https://ncatlab.org/nlab/source/Wigner+function | ## Idea
The usual quantum mechanics based on Schroedinger's wave-function can be reformulated using a different distribution which behaves in some aspects like a probability, however it is negative for some points in space.
## Literature
* related $n$Lab entries include [[quantum mechanics]], [[Weyl calculus]]
* wikipedia/[Wigner quasiprobability distribution](https://en.wikipedia.org/wiki/Wigner_quasiprobability_distribution)
* E. P. Wigner, _On the quantum correction for thermodynamic equilibrium_, Phys. Rev. 40 (5): 749–759 (1932)
* C. Zachos, D. Fairlie, T. Curtright, _Quantum mechanics in phase space: an overview with selected papers_, World Sci. 2005.
* Giuseppe Dito, Francisco J. Turrubiates, _The damped harmonic oscillator in deformation quantization_, Physics Letters A352:4–5 (2006) 309–316 [doi](http://dx.doi.org/10.1016/j.physleta.2005.12.013)
* B. Belchev, M.A. Walton, _On Wigner functions and a damped star product
in dissipative phase-space quantum mechanics_, Annals of Physics 324 (2009) 670–681
* B Molnár, M G Benedict, J Bertrand, _Coherent states and the role of the affine group in the quantum mechanics of the Morse potential_, J. Phys. A34, N.14 [doi](https://doi.org/10.1088/0305-4470/34/14/318)
category: physics
[[!redirects Wigner distribution]]
|
Wigner theorem | https://ncatlab.org/nlab/source/Wigner+theorem |
> not to be confused with _[[Wigner classification]]_
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### AQFT
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
{#Idea}
What is known as *Wigner's theorem* (in honor of the appendix of §20 in [Wigner 1931](#Wigner31)/[59](#Wigner1959)) is one of the basic results on the foundations of [[quantum mechanics]] (generally of [[quantum physics]]). The theorem characterizes the intrinsic notion of *[[symmetry|symmetries]]* in [[quantum physics]] ("quantum symmetries") assuming only observable properties of ([[pure state|pure]]) [[quantum states]] (namely their [[projective space|projectivity]] subject to the [[Born rule]]) and *deriving* from this that quantum symmetries are necessarily [[linear representation|represented]] by [[unitary operators]] *or* [[anti-unitary operators]] on the [[Hilbert space]] [[space of quantum states|of states]].
Notice that a prominent class of examples of [[anti-unitary operators]] appearing as [[quantum symmetries]] are [[time-reversal symmetry|time-reversal symmetries]] ([Wigner 1959, §26](#Wigner1959)) such as appearing in the [[CPT-theorem]].
A key applications of Wigner's theorem is within the [[K-theory classification of topological phases of matter]], where the (anti-)unitary quantum symmetries of [[effective field theory|effectively]] [[free field|free]] [[electrons]] in a [[crystal]] are identified with possible [[twisted equivariant K-theory|twistings]] of the [[equivariant K-theory]] of the crystal's [[Brillouin torus]] ([Freed & Moore 2013, §1](#FreedMoore13), see also [[schreiber:Anyonic topological order in TED K-theory|SS22, §2.2]]).
## Preliminaries
{#Preliminaries}
Let $\mathcal{H}$ be a [[complex vector space|complex]] [[separable Hilbert space|separable]] [[Hilbert space]] and write
\[
\label{ProjectiveSpaceOfHilbertSpace}
P\mathcal{H}
\;\coloneqq\;
\big(
\mathcal{H} \setminus \{0\}
\big)/ \mathbb{C}^\times
\]
for its [[complex projective space]] (here $\mathbb{C}^\times = \mathbb{C} \setminus \{0\}$ denotes the [[group of units]] of the [[ring]] of [[complex numbers]] and $\mathrm{U}(1) \subset \mathbb{C}^\times$ will denote the [[unitary group]] [[U(1)]]).
In [[quantum physics]] one often says that a [[pure state|pure]] [[quantum state]] is an element (hence a [[vector]], then often called a "[[wavefunction]]") $\psi \,\in\, \mathcal{H}$, but by the [[Born rule]] it is really only the corresponding [[complex line]]
$$
[\psi]
\;\coloneqq\;
\big\{
c \cdot \psi
\,\big\vert\,
c \in \mathrm{U}(1)
\big\}
\;\;\;
\in
\;
P\mathcal{H}
$$
which is physically observable. In fact, what is concretely observable (still according to the [[Born rule]]) are the transition [[probabilities]], given by the following [[function]] to the [[closed interval]] of possible [[probability]] values:
\[
\label{ProbabilityFunctionOnRays}
\array{
P\mathcal{H}
\times
P\mathcal{H}
&\xrightarrow{\;\;\;p\;\;\;}&
[0,1]
\\
\big(
[\psi_1]
,\,
[\psi_2]
\big)
&
\overset{\;\;\;\;}{\mapsto}
&
\frac{
\langle \psi_1 \vert \psi_2 \rangle
\langle \psi_2 \vert \psi_1 \rangle
}{
\langle \psi_1 \vert \psi_1 \rangle
\langle \psi_2 \vert \psi_2 \rangle
}
\mathrlap{\,,}
}
\]
where $\langle - \vert - \rangle$ denotes the given [[inner product]] on $\mathcal{H}$.
Hence a "quantum symmetry" in the sense of a re-shuffling of the [[pure states]] of a [[quantum system]] which leaves the observable physics [[invariant]] must be a [[bijection]] on the projective space $P\mathcal{H}$ (eq:ProjectiveSpaceOfHilbertSpace) which preserves the observable probabilities (eq:ProbabilityFunctionOnRays):
\[
\label{QuantumSymmetry}
S
\;\colon\;
P\mathcal{H}
\xrightarrow{\;\sim\;}
P\mathcal{H}
\,,
\;\;\;\;\;\;
p
\big(
S(-)
,\,
S(-)
\big)
\;=\;
p(-,\,-)
\,.
\]
An *example* of a quantum symmetry (eq:QuantumSymmetry) is provided by any [[complex numbers|complex]]-[[linear map|linear]] and [[unitary operator]] on the underlying [[Hilbert space]],
$$
U \,\colon\,
\mathcal{H}
\xrightarrow{\;\;}
\mathcal{U}
\,,
\;\;\;\;\;\;\;\;
\big\langle
U(-)
\big\vert
U(-)
\big\rangle
\;=\;
\big\langle
-
\big\vert
-
\big\rangle
\,,
$$
in that the induced projectivization
\[
\label{ProjectivisedUnitaryOperator}
\array{
P\mathcal{H}
&\xrightarrow{\;\;[U]\;\;}&
P\mathcal{H}
\\
[\psi]
&\mapsto&
\big[
U(\psi)
\big]
}
\]
evidently satisfies (eq:QuantumSymmetry).
On the one hand, there are more quantum symmetries than arise from unitary operators this way. Namely, if $U$ is an [[anti-linear map|anti-linear]] [[anti-unitary operator]], then (eq:ProjectivisedUnitaryOperator) still holds.
On the other hand, this does now already exhaust the most general situation: Wigner's theorem (Prop. \ref{WignerTheorem} below) says that for every quantum symmetry $S \,\colon\, P\mathcal{H} \xrightarrow{\;} P\mathcal{H}$ (eq:QuantumSymmetry) there exists a map $U \,\colon\, \mathcal{H} \xrightarrow{\;} \mathcal{H}$ which is either a [[unitary operator]] or an [[anti-unitary operator]], such that $S \,=\, [U]$ (eq:ProjectivisedUnitaryOperator).
## Statement
{#Statement}
\begin{proposition}\label{WignerTheorem}
**(Wigner's theorem)**
\linebreak
Every quantum symmetry $S \,\colon\, P\mathcal{H} \xrightarrow{\;} P\mathcal{H}$ (eq:QuantumSymmetry) is the projectivization (eq:ProjectivisedUnitaryOperator) of a map
$U \,\colon\, \mathcal{H} \xrightarrow{\;} \mathcal{H}$ which is either a [[unitary operator]] or an [[anti-unitary operator]].
\end{proposition}
The first full proof of this statement seems to be that due to [Bargmann 1964, §1.3](#Bargmann64), following indications in the appendix of §20 in [Wigner 1931](#Wigner31)/[59](#Wigner1959). A geometric proof via the [[Fubini-Study metric]] is given in [Freed 2012, Thm. 8](#Freed12). Statement and proof in the greater generality of possibly non-[[pure state|pure]] [[quantum states]] is given in [Moretti 2017, Thm. 12.11](#Moretti17).
## Related theorems
Other theorems about the foundations and [[interpretation of quantum mechanics]] include:
* [[order-theoretic structure in quantum mechanics]]
* [[Kochen-Specker theorem]]
* [[Alfsen-Shultz theorem]]
* [[Harding-Döring-Hamhalter theorem]]
* [[Fell's theorem]]
* [[Gleason's theorem]]
* [[Bell's theorem]]
* [[Bub-Clifton theorem]]
* [[Kadison-Singer problem]]
## References
{#References}
The original:
* {#Wigner31} [[Eugene P. Wigner]]: §20(app.) *Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren*, Springer (1931) [[doi:10.1007/978-3-663-02555-9](https://doi.org/10.1007/978-3-663-02555-9), [pdf](https://digbib.ubka.uni-karlsruhe.de/volltexte/wasbleibt/33355459/33355459.pdf)]
and its English translation:
* {#Wigner1959} [[Eugene P. Wigner]], §20(app.) and §26 in: *Group theory: And its application to the quantum mechanics of atomic spectra*, Academic Press (1959) [[doi:978-0-12-750550-3](https://www.elsevier.com/books/group-theory/wigner/978-0-12-750550-3)]
The first full proof:
* {#Bargmann64} [[Valentine Bargmann]], _Note on Wigner's theorem on symmetry transformations_, Journal of Mathematical Physics **5** 7 (1964) 862-868 [[doi:10.1063/1.1704188](https://doi.org/10.1063/1.1704188), [pdf](http://materias.df.uba.ar/t2a2019c2/files/2012/07/Bergman-1964.pdf), [[Bargmann-WignerTheorem.pdf:file]]]
New proof using the [[Fubini-Study metric]]:
* {#Freed12} [[Daniel Freed]], _On Wigner's theorem_, Geometry & Topology Monographs **18** (2012) 83–89 [[arXiv:1112.2133](http://arxiv.org/abs/1112.2133), [doi:10.2140/gtm.2012.18.83](http://dx.doi.org/10.2140/gtm.2012.18.83)]
A proof in the greater generality of possibly non-[[pure state|pure]] [[quantum states]]:
* {#Moretti17} [[Valter Moretti]], Thm. 12.11 of: *Spectral Theory and Quantum Mechanics*, Springer (2017) [[doi:10.1007/978-3-319-70706-8](https://doi.org/10.1007/978-3-319-70706-8)]
See also:
* Wikipedia, _[Wigner's theorem](https://en.wikipedia.org/wiki/Wigner%27s_theorem#Statement)_
Discussion for [[quaternions|quaternionic]] Hilbert spaces:
* C. S. Sharma and D. F. Almeida, _Additive isometries on a quaternionic Hilbert space_, Journal of Mathematical Physics 31, 1035 (1990) ([doi:10.1063/1.528779](https://doi.org/10.1063/1.528779))
In the context of the [[K-theory classification of topological phases of matter]]:
* {#FreedMoore13} [[Daniel Freed]], [[Gregory Moore]], §1 of: *Twisted equivariant matter*, Ann. Henri Poincaré **14** (2013) 1927-2023 [[arXiv:1208.5055](https://arxiv.org/abs/1208.5055), [doi:10.1007/s00023-013-0236-x](https://doi.org/10.1007/s00023-013-0236-x)]
[[!redirects Wigner theorem]]
[[!redirects Wigner's theorem]]
[[!redirects Wigner's theorem]]
[[!redirects quantum symmetry]]
[[!redirects quantum symmetries]]
|
wild category | https://ncatlab.org/nlab/source/wild+category |
## Definition
A **wild category** is a 1-dimensional approximation of an [[nlab:infinity-category]] that is definable in [[Book HoTT]]. It consists of
* A type $ob A$ of objects
* For objects $x,y$ a type $A(x,y)$ of morphisms
* For each object $x$ an identity $id_x : A(x,x)$
* For objects $x,y,z$ a composition function $\circ : A(y,z) \times A(x,y) \to A(x,z)$
* For objects $x,y$ and a morphism $f:A(x,y)$, equalities $f \circ id_x = f = id_y \circ f$
* For objects $x,y,z,w$ and morphisms $f:A(x,y)$ and $g:A(y,z)$ and $h:A(z,w)$, an equality $h\circ (g\circ f) = (h\circ g)\circ f$.
If each $A(x,y)$ is a [[set]], then a wild category reduces to a [[precategory]], but in general this condition is not imposed. This means that, for instance, there is a nontrivial pentagon identity for the associativities that does not necessarily commute, and so on. However, even lacking these coherence data, a wild category is sufficient for some purposes.
For example, we can define an *initial object* in a wild category to be an object $0$ such that $A(0,x)$ is contractible for all $x$. In cases when the wild category "is" actually a coherent higher category, this still gives the right answer, and it is sufficient for applications such as producing induction principles for [[higher inductive types]]. See the references for more specific examples.
## References
* Paolo Capriotti, Nicolai Kraus, _Univalent Higher Categories via Complete Semi-Segal Types_, [arxiv](https://arxiv.org/abs/1707.03693), 2017
* Nicolai Kraus, Jakob von Raumer, _Path Spaces of Higher Inductive Types in Homotopy Type Theory_, [arxiv](https://arxiv.org/abs/1901.06022), 2019
* Mike Shulman, _Impredicative Encodings, Part 3_, [blog post](https://homotopytypetheory.org/2018/11/26/impredicative-encodings-part-3/), 2018
|
Wiles' proof of Fermat's last theorem | https://ncatlab.org/nlab/source/Wiles%27+proof+of+Fermat%27s+last+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [Frey82](#Frey82) [[Gerhard Frey]] showed that, if the equation $a^{p}+b^{p}=c^{p}$ had any nontrivial integer solutions $(a,b,c)$ for primes $p\gt 2$, the elliptic curve (now known as the _Frey-Hellegouarch curve_ after Frey and [[Yves Hellegouarch]])
$$y^{2}=x(x-a^{p})(x-b^{p})$$
would exhibit certain unusual properties. In particular in [Serer87](#Serre87) [[Jean-Pierre Serre]] showed, assuming the "epsilon conjecture", that if such a solution existed then the Frey-Hellegouarch curve would not be modular.
[[Andrew Wiles]] with [[Richard Taylor]] proved the [[modularity theorem]] stating that all elliptic curves over $\mathbb{Q}$ are modular. By [[Ken Ribet]]'s proof of the epsilon conjecture (now known as [[Ribet's theorem]]) in [Ribet90](#Ribet90) this implied a proof of [[Fermat's last theorem]].
## References
* {#Frey82} [[Gerhard Frey]], 1982, _Rationale Punkte auf Fermatkurven und getwisteten Modulkurven_, J. reine angew. Math., 331: 185–191
* {#Serre87} [[Jean-Pierre Serre]], 1987, _Sur les représentations modulaires de degré 2 de Gal(Q/Q)_, Duke Mathematical Journal, 54 (1): 179–230
* {#Ribet90} [[Ken Ribet]], 1990, _On modular representations of Gal(Q/Q) arising from modular forms_, Inventiones Mathematicae. 100 (2): 431–476.
See also the references at *[[modularity theorem]]* ([here](modularity+theorem#References))
See also:
* Wikipdia, _<a href="https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem">Wiles's proof of Fermat's last theorem</a>_
[[!redirects Wiles's proof of Fremat's last theorem]] |
Wilfrid Hodges | https://ncatlab.org/nlab/source/Wilfrid+Hodges |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Wilfrid_Hodges)
## related $n$Lab entries
* [[model theory]]
category: people |
Wilhelm Ackermann | https://ncatlab.org/nlab/source/Wilhelm+Ackermann |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Wilhelm_Ackermann)
## related $n$Lab entries
* [[Ackermann function]]
* [[mathematical logic]]
category: people |
Wilhelm Killing | https://ncatlab.org/nlab/source/Wilhelm+Killing |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Wilhelm_Killing)
## Selected writings
On [[spherical space forms]]:
* {#Killing1891} [[Wilhelm Killing]], _Ueber die Clifford-Klein’schen Raumformen_, Math. Ann. 39 (1891), 257–278 ([doi:10.1007/BF01206655](https://doi.org/10.1007/BF01206655))
## Related $n$Lab entries
* [[Killing tensor]]
* [[Clifford-Klein space form]]
category: people |
Wilhelm Magnus | https://ncatlab.org/nlab/source/Wilhelm+Magnus | [[!redirects W. Magnus]]
Wilhelm Magnus was a German mathematician who worked in [[combinatorial group theory]].
* [Wikipedia page](http://en.wikipedia.org/wiki/Wilhelm_Magnus)
* [St. Andrews history page](http://www-history.mcs.st-andrews.ac.uk/Biographies/Magnus.html)
## Selected writings
On understanding the [[braid group]] as a [[mapping class group]]:
* [[Wilhelm Magnus]], *Über Automorphismen von Fundamentalgruppen berandeter Flächen*, Mathematische Annalen **109** (1934) 617–646 [[doi:10.1007/BF01449158](https://doi.org/10.1007/BF01449158)]
On [[combinatorial group theory]] ([[presentations of groups]]):
* [[Wilhelm Magnus]], A. Karras, D. Solitar, *Combinatorial group theory: Presentation of groups in terms of generators and relations*, Courier Corporation (1966, 2004) [[ISBN:0-486-43830-9](https://www.maa.org/press/maa-reviews/combinatorial-group-theory-presentations-of-groups-in-terms-of-generators-and-relations), [GoogleBooks](https://books.google.com/books?hl=en&lr=&id=1LW4s1RDRHQC&oi=fnd&pg=PP1&ots=WszNa-0hEz&sig=Hn_MTbyLcEreBBNZ8jJtQUVKlZk)]
[[!redirects W. Magnus]]
category:people |
Wilhemus A. J. Luxemburg | https://ncatlab.org/nlab/source/Wilhemus+A.+J.+Luxemburg |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Wilhelmus_Luxemburg)
## Selected writings
On [[monads in nonstandard analysis]]:
* [[Wilhemus A. J. Luxemburg]], *A General Theory of Monads*, in: *Applications of Model Theory to Algebra, Analysis and Probability*, Holt, Rinehart and Minston (1966) 18–86 [[ark](https://archive.org/stream/in.ernet.dli.2015.141486/2015.141486.Applications-Of-Model-Theory-To-Algebra-analysis-And-Probability_djvu.txt)]
category: people
[[!redirects Wilhemus Luxemburg]]
|
Willard Miller | https://ncatlab.org/nlab/source/Willard+Miller |
* [personal page](https://www-users.cse.umn.edu/~mille003/index.html)
* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=7741)
## Selected writings
On [[symmetry groups]] in [[physics]] (such as [[crystallographic groups]]) and their [[group theory]]:
* [[Willard Miller]], *Symmetry Groups and Their Applications*, Pure and Applied Mathematics **50** (1972) 16-60 [[ISBN:9780080873657](https://www.elsevier.com/books/symmetry-groups-and-their-applications/miller/978-0-12-497460-9)]
category: people
|
Willard Quine | https://ncatlab.org/nlab/source/Willard+Quine | Willard Van Orman Quine (1908–2000) was an American [[philosophy|philosopher]] who worked in theoretical philosophy and in [[logic]].
## Related nLab entries
* [[logicality and invariance]]
## Selected writings
* 1951, _Two Dogmas of Empiricism_, Philosophical Review, 60: 20–43; reprinted in From a Logical Point of View, pp. 20–46.
* 1953, _From a Logical Point of View_, Cambridge, Mass.: Harvard University Press, revised edition 1980. |
Willem van Est | https://ncatlab.org/nlab/source/Willem+van+Est |
* [German Wikipedia entry](http://de.wikipedia.org/wiki/Willem_Titus_van_Est)
## related $n$Lab entries
* [[van Est isomorphism]]
[[!redirects van Est]]
category: people
|
William Abram | https://ncatlab.org/nlab/source/William+Abram |
* [webpage](https://sites.google.com/site/abrammath/home)
## related entries
* [[equivariant cohomology]]
* [[equivariant complex cobordism cohomology theory]]
category: people
|
William Arveson | https://ncatlab.org/nlab/source/William+Arveson |
* [Wikipedia](https://en.wikipedia.org/wiki/William_Arveson)
## Selected writings
On [[completely positive maps]] on [[C-star algebras|$C^\ast$-algebras]]:
* {#Arveson69} [[William B. Arveson]], *On subalgebras of $C^\ast$-algebras*, Bull. Amer. Math. Soc. **75** (1969) 790-794 [[doi:1969-75-04/S0002-9904-1969-12293-7](https://www.ams.org/journals/bull/1969-75-04/S0002-9904-1969-12293-7), [pdf](https://www.ams.org/journals/bull/1969-75-04/S0002-9904-1969-12293-7/S0002-9904-1969-12293-7.pdf)]
On [[operator algebra]] and [[spectral theory]]:
* [[William Arveson]], *A Short Course on Spectral Theory*, Graduate Texts in Mathematics **209**, Springer (2002) [[doi:10.1007/b97227](https://link.springer.com/book/10.1007/b97227)]
category: people
[[!redirects William B. Arveson]] |
William Barlow | https://ncatlab.org/nlab/source/William+Barlow |
* <a href="https://en.wikipedia.org/wiki/William_Barlow_(geologist)">William Barlow (geologist)</a>
## Selected writings
Early discussion of the internal structure of [[crystals]]:
* [[William Barlow]], *Probable Nature of the Internal Symmetry of Crystals*, Nature **29** (1883) 186–188 $[$[doi:10.1038/029186a0](https://doi.org/10.1038/029186a0)$]$
category: people |
William Browder | https://ncatlab.org/nlab/source/William+Browder |
* [Wikipedia entry](https://en.m.wikipedia.org/wiki/William_Browder_(mathematician))
## Selected writings
On [[homology of H-spaces]] and their [[Pontryagin product]]:
* [[William Browder]], *Homology and Homotopy of H-Spaces*, Proceedings of the National Academy of Sciences of the United States of America **46** 4 (1960) 543-545 [[jstor:70867](https://www.jstor.org/stable/70867)]
* [[William Browder]], p. 36 of: *Torsion in H-Spaces*, Annals of Mathematics, Second Series **74** 1 (1961) 24-51 [[jstor:1970305](https://www.jstor.org/stable/1970305)]
* [[William Browder]], *Homology Rings of Groups*, American Journal of Mathematics **90** 1 (1968) [[jstor:2373440](https://www.jstor.org/stable/2373440)]
category: people |
William Burnside | https://ncatlab.org/nlab/source/William+Burnside |
* [Wikipedia entry](https://en.wikipedia.org/wiki/William_Burnside)
* [St. Andrews History page](http://www-groups.dcs.st-and.ac.uk/history/Biographies/Burnside.html)
## Selected writings
On [[group theory]] of [[finite groups]]:
* {#Burnside1897} [[William Burnside]], _Theory of Groups of Finite Order_, 1897 ([pdf](http://www.gutenberg.org/files/40395/40395-pdf.pdf)), reprinted by Cambridge University Press 2012 ([doi:10.1017/CBO9781139237253]( https://doi.org/10.1017/CBO9781139237253))
See also:
* {#NMT04} Peter M. Neumann, A.J.S. Mann, and Julia Tompson (eds.) _The Collected Papers of William Burnside_, Oxford University Press 2004 ([ISBN:9780198505853](https://global.oup.com/academic/product/the-collected-papers-of-william-burnside-9780198505853))
## Related $n$Lab entries
* [[Burnside ring]]
* [[Burnside ring is equivariant stable cohomotopy of the point]]
category: people |
William Clifford | https://ncatlab.org/nlab/source/William+Clifford |
Discoverer of _[[Clifford algebra]]_, following [[Hermann Grassmann]]'s [[Grassmann algebra]].
* [Wikipedia entry](http://en.wikipedia.org/wiki/William_Kingdon_Clifford)
## Selected writings
Early speculation on what much later came to be called *[[geometrodynamics]]*:
* [[William Clifford]], *[[On the Space-Theory of Matter]]*, Proceedings of the Cambridge Philosophical Society **2** (1864-1876 - Printed 1876) 157-158 reprinted in: *The Concepts of Space and Time* Boston Studies in the Philosophy of Science **22** Springer (1976) [[doi:10.1007/978-94-010-1727-5_50](https://doi.org/10.1007/978-94-010-1727-5_50)]
English translation of [[Bernhard Riemann]]'s, *[[Über die Hypothesen, welche der Geometrie zu Grunde liegen]]*, Göttingen (1845) [[doi:10.1007/978-3-642-35121-1](https://doi.org/10.1007/978-3-642-35121-1)]:
* [[William Clifford]]: *[[On the hypotheses which underlie geometry]]*, Nature **VIII** (1873) 183-184 [[doi:10.1007/978-3-319-26042-6](https://doi.org/10.1007/978-3-319-26042-6)]
Introducing the notion of [[Clifford algebra]]:
* [[William Clifford]], *Applications of Grassmann's extensive algebra*, American Journal of Mathematics **1** 4 (1878) 350-358 [[doi:10.2307/2369379](https://doi.org/10.2307/2369379), [jstor:2369379](https://www.jstor.org/stable/2369379)]
> (referring to [[Hermann Grassmann]]'s *[[Ausdehnungslehre]]*)
## Related entries
* _[[On the Space-Theory of Matter]]_
category: people
[[!redirects William Kingdon Clifford]] |
William Crawley-Boevey | https://ncatlab.org/nlab/source/William+Crawley-Boevey | __William Crawley-Boevey__ is an English mathematician, mainly algebraist. Topics he contributed to include representations of quivers, quiver varieties, [[Deligne-Simpson problem]], Kleinian singularities, [[double derivation]]s.
* [Webpage](https://www.math.uni-bielefeld.de/~wcrawley/)
* wikipedia:[William_Crawley-Boevey](https://en.wikipedia.org/wiki/William_Crawley-Boevey); Google Scholar [page](https://scholar.google.com/citations?user=Ozh9p84AAAAJ)
He introduced _deformed_ [[preprojective algebra]]s of quivers in connection with study of Kleinian singularities in
* W. Crawley-Boevey, Martin P. Holland, _Noncommutative deformations of Kleinian singularities_, Duke Math. J. 92 (3): 605–635 (1998) [doi](https://doi.org/10.1215/S0012-7094-98-09218-3) [MR1620538](https://mathscinet.ams.org/mathscinet-getitem?mr=1620538)
A construction generalizing deformed preprojective algebra of quivers and assigning to a $K$-algebra $A$ with an element $\lambda\in K\otimes_{\mathbf{Z}}K_0(A)$ a new algebra $\pi^\lambda(A)$ and a canonical homomorphism $A\to \pi^\lambda(A)$ is described in
* _Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities_, Comment. Math. Helv. 74 (1999) 548-574 [doi](https://doi.org/10.1007/S000140050105)
Flavour of noncommutative differential geometry based on [[double derivation]]s:
* [[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)
Some papers
* _Poisson structures on moduli spaces of representations_, J. Algebra 325, 205–215 (2011) [MR2745537](https://mathscinet.ams.org/mathscinet-getitem?mr=2745537) [doi](https://doi.org/10.1016/j.jalgebra.2010.09.033)
* _Geometry of the moment map for representations of quivers_, Comp. Math. __126__ (3): 257–293 [doi](https:/doi.org/10.1023/A:1017558904030) [MR1834739](https://mathscinet.ams.org/mathscinet-getitem?mr=1834739)
category: people
[[!redirects W. Crawley-Boevey]]
[[!redirects Bill Crawley-Boevey]]
|
William Detmold | https://ncatlab.org/nlab/source/William+Detmold |
* [personal page](https://wdetmold.mit.edu/welcome)
* [institute page](https://physics.mit.edu/faculty/william-detmold/)
* [inspire page](https://inspirehep.net/authors/1019639)
## Selected writings
On the [[pressure]] of the proton, via [[lattice QCD]]:
* [[Phiala Shanahan]], [[William Detmold]], *Pressure Distribution and Shear Forces inside the Proton*, Physical Review Letters **122** 7 (2019) [[doi:10.1103/PhysRevLett.122.072003](https://doi.org/10.1103/PhysRevLett.122.072003), [arXiv:1810.07589](https://arxiv.org/abs/1810.07589)]
category: people
|
William Dunbar | https://ncatlab.org/nlab/source/William+Dunbar |
* [Institute page](https://simons-rock.edu/academics/faculty-bios/science-mathematics-and-computing-faculty/william-dunbar.php)
## Selected writings
On the classification of 3-dimensional compact [[Riemannian orbifolds]]:
* {#Dunbar88} [[William Dunbar]], _Geometric orbifolds_, Rev. Mat. Univ. Complutense Madr. 1, No.1-3, 67-99 (1988)
category: people |
William Dwyer | https://ncatlab.org/nlab/source/William+Dwyer |
William G. Dwyer is a mathematician at the University of Notre Dame.
* [website](http://www.nd.edu/~wgd/)
* [Wikipedia entry](https://en.wikipedia.org/wiki/William_Gerard_Dwyer)
## Selected articles
On the $G$-[[Borel model structure]] and its [[Quillen equivalence]] with the [[slice model structure]] over the [[simplicial classifying space]] $\overline W G$:
* {#DDK80} [[Emmanuel Dror]], [[William Dwyer]], [[Daniel Kan]], _Equivariant maps which are self homotopy equivalences_, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 ([jstor:2043448](http://www.jstor.org/stable/2043448))
On [[simplicial localization]]:
* {#DwyerKanLocalizations} [[William Dwyer]], [[Daniel Kan]], _Simplicial localizations of categories_, J. Pure Appl. Algebra **17** 3 (1980), 267-284 [<a href="https://doi.org/10.1016/0022-4049(80)90049-3">doi:10.1016/0022-4049(80)90049-3</a>, [pdf](http://www.nd.edu/~wgd/Dvi/SimplicialLocalizations.pdf)]
* {#DwyerKanCalculating} [[William Dwyer]], [[Daniel Kan]], _Calculating simplicial localizations_, J. Pure Appl. Algebra 18 (1980), 17-35 [<a href="https://doi.org/10.1016/0022-4049(80)90113-9">doi:10.1016/0022-4049(80)90113-9</a>, [pdf](http://www.nd.edu/~wgd/Dvi/CalculatingSimplicialLocalizations.pdf)]
* {#DwyerKan80FunctionComplexes} [[William Dwyer]], [[Daniel Kan]], _Function complexes in homotopical algebra_, Topology 19 (1980), 427-440 [<a href="https://doi.org/10.1016/0040-9383(80)90025-7">doi:10.1016/0040-9383(80)90025-7</a>, [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/dwyer-kan-3.pdf)]
* {#DwyerKanEquivalences} [[William Dwyer]], [[Daniel Kan]], *Equivalences between homotopy theories of diagrams*, in: *Algebraic topology and algebraic K-theory*, Ann. of Math. Stud. **113**, Princeton University Press (1988) [[doi:10.1515/9781400882113-009](https://doi.org/10.1515/9781400882113-009)]
On [[derived hom-spaces]] ([[function complexes]]) in [[projective model structures on simplicial presheaves]]:
* [[William Dwyer]], [[Daniel Kan]], *Function complexes for diagrams of simplicial sets*, Indagationes Mathematicae (Proceedings) **86** 2 (1983) 139-147 [<a href="https://doi.org/10.1016/1385-7258(83)90051-3">doi:10.1016/1385-7258(83)90051-3</a>, [pdf](https://core.ac.uk/download/pdf/82652265.pdf)]
On (enhancement and generalization of) [[Elmendorf's theorem]] in [[equivariant homotopy theory]]:
* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]], _Singular functors and realization functors_, Indagationes Mathematicae (Proceedings) Volume 87, Issue 2, 1984, Pages 147-153 (<a href="https://doi.org/10.1016/1385-7258(84)90016-7">doi:10.1016/1385-7258(84)90016-7</a>)
Introducing the [[model structure on simplicial groupoids]]:
* [[William Dwyer]], [[Daniel Kan]], *Homotopy theory and simplicial groupoids*, Indagationes Mathematicae (Proceedings) **87** 4 (1984) 379-385 [<a href="https://doi.org/10.1016/1385-7258(84)90038-6">doi:10.1016/1385-7258(84)90038-6</a>]
On [[homotopy commutative diagrams]]:
* [[William Dwyer]], [[Dan Kan]], [[Jeff Smith]], _Homotopy commutative diagrams and their realizations_, Journal of Pure and Applied Algebra 57 (1989) 5-24 (<a href="https://doi.org/10.1016/0022-4049(89)90023-6">doi:10.1016/0022-4049(89)90023-6</a>)
On [[p-compact groups]]:
* {#DwyerWilkerson94} [[William Dwyer]], [[Clarence Wilkerson]], _Homotopy Fixed-point Methods for Lie Groups and Finite Loop Spaces_, Annals of Mathematics Second Series, Vol. 139, No. 2 (Mar., 1994), pp. 395-442 ([jstor:2946585](https://www.jstor.org/stable/2946585))
Introducing the [[Dwyer-Wilkerson H-space]]:
* {#DwyerWilkerson93} [[William Dwyer]], [[Clarence Wilkerson]], _A new finite loop space at the prime two_, J. Amer. Math. Soc. 6 (1993), 37-64 ([doi:10.1090/S0894-0347-1993-1161306-9](https://doi.org/10.1090/S0894-0347-1993-1161306-9))
On [[homotopy theory]] and [[model categories]]:
* {#DwyerSpalinski95} [[William Dwyer]], [[Jan Spalinski]], *[[Homotopy theories and model categories]]* ([[DwyerSpalinski_HomotopyTheories.pdf:file]])
in: [[Ioan Mackenzie James|I. M. James]], *[[Handbook of Algebraic Topology]]*, North Holland 1995 ([ISBN:9780080532981](https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2), [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))
On [[homotopy theory|homotopy theoretic]] methods in [[group cohomology]]:
* [[William G. Dwyer]], [[Hans-Werner Henn]], *Homotopy Theoretic Methods in Group Cohomology*, Advanced Courses in Mathematics - CRM Barcelona, Springer (2001) [[doi:10.1007/978-3-0348-8356-6](https://doi.org/10.1007/978-3-0348-8356-6)]
On [[derived functors]] such as [[homotopy limit]]-functors on [[model categories]] and more general [[homotopical categories]]:
* [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], **[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]**, Mathematical Surveys and Monographs **113**, AMS 2004 ([ISBN: 978-1-4704-1340-8](https://bookstore.ams.org/surv-113-s), [pdf](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf))
On [[localization]] in [[homotopy theory]]:
* [[William Dwyer]], _Localizations_, in: _Axiomatic, enriched and motivic homotopy theory_, NATO Sci. Ser. II **131**, Kluwer Acad. Publ. (2004) 3-28 [[doi:10.1007/978-94-007-0948-5](https://doi.org/10.1007/978-94-007-0948-5)]
On [[homotopy theory]] and [[classifying spaces]]:
* {#Dwyer2008} [[William Dwyer]], _Homotopy theory of classifying spaces_, Lecture notes, Copenhagen 2008, ([pdf](http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf), [[Dwyer_HomotopyTheoryOfClassifyingSpaces.pdf:file]])
## Related entries
* [[Dwyer-Kan localization]]
* [[Dwyer-Kan loop groupoid]]
[[!redirects William Gerard Dwyer]]
[[!redirects W. G. Dwyer]]
[[!redirects William G. Dwyer]]
[[!redirects Bill Dwyer]]
[[!redirects Dwyer]]
category: people
|
William F. Sawin | https://ncatlab.org/nlab/source/William+F.+Sawin |
* [personal page](https://williamsawin.com/)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Will_Sawin)
## Selected writings
On [[commutativity of limits with colimits]]:
* [[Marie Bjerrum]], [[Peter Johnstone]], [[Tom Leinster]], [[William F. Sawin]], *Notes on commutation of limits and colimits*, Theory and Applications of Categories **30** (2015), 527-532 [[arXiv:1409.7860](http://arxiv.org/abs/1409.7860), [tac:3015](http://www.tac.mta.ca/tac/volumes/30/15/30-15abs.html)]
category: people
[[!redirects William Sawin]]
[[!redirects Will Sawin]]
[[!redirects Will F. Sawin]]
|
William Fulton | https://ncatlab.org/nlab/source/William+Fulton |
* [webpage](http://www.math.lsa.umich.edu/~wfulton/)
* [Wikipedia entry](http://en.wikipedia.org/wiki/William_Fulton_%28mathematician%29)
## Selected writings
On [[representation theory]]:
* {#FultonHarris91} [[William Fulton]], [[Joe Harris]], _Representation Theory: a First Course_, Springer, Berlin, 1991 ([doi:10.1007/978-1-4612-0979-9](https://link.springer.com/book/10.1007/978-1-4612-0979-9))
The [[Fulton-MacPherson compactification]] of [[configuration spaces of points]]:
* {#FultonMacPherson94} [[William Fulton]], [[Robert MacPherson]], _A compactification of configuration spaces_, Ann. of Math. (2), 139(1):183–225, 1994 ([jstor:2946631](https://www.jstor.org/stable/2946631))
On [[algebraic topology]]:
* [[William Fulton]], *Algebraic Topology -- A First Course*, Graduate Texts in Mathematics **153**, Springer (1995) \[<a href="https://doi.org/10.1007/978-1-4612-4180-5">doi:10.1007/978-1-4612-4180-5</a>\]
On [[Young tableaux]] in [[representation theory]] and geometry:
* [[William Fulton]], _Young Tableaux, with Applications to Representation Theory and Geometry_, Cambridge U. Press, 1997 ([doi:10.1017/CBO9780511626241](https://doi.org/10.1017/CBO9780511626241))
## Related $n$Lab entries
* [[Fulton-MacPherson compactification]]
* [[configuration spaces of points]]
* [[Hilbert scheme]]
category: people
|
William G. Dwyer > history | https://ncatlab.org/nlab/source/William+G.+Dwyer+%3E+history |
> See at *[[William Gerard Dwyer]]* |
William Hodge | https://ncatlab.org/nlab/source/William+Hodge |
* [Wikipedia entry](http://en.wikipedia.org/wiki/W._V._D._Hodge)
## Related entries
* [[Hodge theory]]
[[Hodge theorem]]
category: people
[[!redirects W. V. D. Hodge]]
[[!redirects W.V.D. Hodge]]
[[!redirects Hodge]] |
William Howard | https://ncatlab.org/nlab/source/William+Howard |
* [Wikipedia entry](http://en.wikipedia.org/wiki/William_Alvin_Howard)
## Related entries
* [[propositions as types]]
category: people
[[!redirects Bill Howard]]
[[!redirects William Alvin Howard]] |
William J. Munro | https://ncatlab.org/nlab/source/William+J.+Munro |
* [personal page](https://www.brl.ntt.co.jp/people/bilmun/)
* [institute page](https://www.oist.jp/research/william-john-munro)
## Selected writings
On "continuous variable" or *analog* [[quantum computation]]:
* [[Vivien M. Kendon]], [[Kae Nemoto]], [[William J. Munro]], *Quantum Analogue Computing*, Phil. Trans. R. Soc. A **368** 1924 (2010) 3621-3632 [[arXiv:1001.2215](https://arxiv.org/abs/1001.2215), [doi:10.1098/rsta.2010.0017](https://doi.org/10.1098/rsta.2010.0017)]
On [[quantum error correction]]:
* {#DevittNemotoMunro09} [[Simon J. Devitt]], [[Kae Nemoto]], [[William J. Munro]], *Quantum Error Correction for Beginners*, Rep. Prog. Phys. **76** (2013) 076001 [[arXiv:0905.2794](https://arxiv.org/abs/0905.2794), [doi:10.1088/0034-4885/76/7/076001](https://iopscience.iop.org/article/10.1088/0034-4885/76/7/076001)]
category: people |
William K. Wootters | https://ncatlab.org/nlab/source/William+K.+Wootters |
* [Wikipedia entry](https://en.m.wikipedia.org/wiki/William_Wootters)
## Selected writings
Introducing the [[no-cloning theorem]] in [[quantum information theory]]:
* [[William Wooters]], [[Wojciech Zurek]], *A single quantum cannot be cloned*, Nature **299** (1982) 802-803 [[doi:10.1038/299802a0](https://doi.org/10.1038/299802a0)]
Introducing the notion of [[quantum teleportation]]:
* {#BBCJPW93} [[Charles H. Bennett]], [[Gilles Brassard]], [[Claude Crépeau]], [[Richard Jozsa]], [[Asher Peres]], [[William K. Wootters]]: *Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels*, Phys. Rev. Lett. **70** 1895 (1993) [[doi:10.1103/PhysRevLett.70.1895](https://doi.org/10.1103/PhysRevLett.70.1895)]
category: people |
William Keigher | https://ncatlab.org/nlab/source/William+Keigher |
* [Mathematics Genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=4264)
category: people
|
William Kinney | https://ncatlab.org/nlab/source/William+Kinney |
* [webpage](http://arts-sciences.buffalo.edu/physics/faculty/william-h-kinney.html)
## selected writings
* {#BarenboimKinneyPark16} Gabriela Barenboim, [[William Kinney]], Wan-Il Park, _Eternal Hilltop Inflation_, Journal of Cosmology and Astroparticle Physics, Volume 2016, May 2016 ([arXiv:1601.08140](https://arxiv.org/abs/1601.08140))
([[eternal inflation]] in [[Starobinsky model of cosmic inflation|Starobinsky-type models]] of [[cosmic inflation]])
## related $n$Lab entries
* [[Starobinsky model of cosmic inflation]]
* [[eternal inflation]]
category: people |
William Kirwin | https://ncatlab.org/nlab/source/William+Kirwin |
* [webpage](http://www.mi.uni-koeln.de/~wkirwin/index.php?nav=3)
## related $n$Lab entries
* [[coherent state in geometric quantization]]
category: people
|
William Kronholm | https://ncatlab.org/nlab/source/William+Kronholm |
* [webpage](http://www.whittier.edu/academics/math/Kronholm)
## Selected writings
On the [[Serre spectral sequence]] in [[equivariant cohomology]] ([[RO(G)-grading|RO(G)-graded]] [cohomology with coefficients in a Mackey functor](Mackey+functor#Cohomology)):
* {#Kronholm10} [[William C. Kronholm]], *The $RO(G)$-graded Serre spectral sequence*, Homology Homotopy Appl. **12** 1 (2010) 75-92. [[arXiv:0908.3827](https://arxiv.org/abs/0908.3827), [doi:10.4310/HHA.2010.v12.n1.a7](https://doi.org/10.4310/HHA.2010.v12.n1.a7), [euclid:hha/1296223823](https://projecteuclid.org/euclid.hha/1296223823)]
Introducing [[Kronholm's freeness theorem]] for [[cyclic group of order 2|$\mathbb{Z}/2$]]-[[equivariant cohomology|equivariant]] [[Bredon cohomology]]:
* [[William C. Kronholm]], *A Freeness Theorem for $RO(\mathbb{Z}/2)$-graded Cohomology*, Topology and its Applications **157** 5 (2010) 902-915 [[arXiv:0908.3825](https://arxiv.org/abs/0908.3825), [doi:10.1016/j.topol.2009.12.006](https://doi.org/10.1016/j.topol.2009.12.006)]
## Related entries
* [[Serre spectral sequence]]
[[!redirects William C. Kronholm]]
category: people |
William Lawvere | https://ncatlab.org/nlab/source/William+Lawvere |
[[!redirects Bill Lawvere]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabiJndex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Synthetic differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Overview
> Talking with Bill, I often feel like a fly buzzing around a cow. (It seems to me I can liken Bill to a cow, if I'm just a fly myself.) On any easy question, I'll probably see the answer first. But his thoughts seem to move on a level where I don't function, I can barely see down there. (From an [interview](http://at.yorku.ca/t/o/p/c/06.htm) with [[John Isbell]].)
F. William Lawvere (February 9, 1937 -- January 23, 2023)
* ([web site](http://www.acsu.buffalo.edu/~wlawvere/), [Wikipedia entry](http://en.wikipedia.org/wiki/William_Lawvere))
was an influential [[category theory|category theorist]].
See
* {#Interview07} _An interview with F. William Lawvere_, [Bulletin of the International Center for Mathematics](http://www.cim.pt/?q=publications) ([part 1, December 2007](http://www.cim.pt/files/publications/b23www.pdf), [part 2, June 2008](http://www.cim.pt/files/publications/b24www.pdf), [full pdf](http://www.mat.uc.pt/~picado/lawvere/interview.pdf), [[LawvereInterview.pdf:file]])
for a survey of his academic path and work. See also
* Interview by Felice Cardone, March 2013 ([link](https://dl.dropboxusercontent.com/u/92056191/Archive/temporary_new_material/lawvere-cardone-2000.html))
for an interview on his contributions to [[categorical logic]].
A recording of the commemorative special session in honour of Bill Lawvere from CT2023 can be found here: ([video](https://www.youtube.com/watch?v=x23KS3VOpRg&list=PLu4STGsfbix9l6rPxGsjG6Gl822k1hj2X&index=1&pp=iAQB))
For a (somewhat random) list of further links see also at "[conceptual mathematics](http://conceptualmathematics.wordpress.com/2013/02/09/happy-birthday-professor-f-william-lawvere/)".
## Topics
### Main contributions
Lawvere invented [[categorical logic]] and introduced the eponymous [[Lawvere theories]] as a [[category theory|category-theoretic]] way to describe finitary [[algebraic theories]]. He generalised [[Grothendieck toposes]] to [[elementary toposes]], revolutionising the [[foundations]] of [[mathematics]]; in this vein, he developed the [[foundation]] in [[structural set theory]] called [[ETCS]]. He also introduced and worked on [[synthetic differential geometry]] as a foundation for [[differential geometry]] and [[equations of motion]] in [[continuum physics]]. Later he introduced the notion of _[[cohesive topos]]_ as a more general foundation of [[geometry]].
### Mathematics relating to Physics
{#MotivationFromFoundationsOfPhysics}
A central motivation for Lawvere's work is the search for a good mathematical [[foundations]] of [[physics]], specifically of ([[classical physics|classical]]) [[continuum mechanics]] (or at least some [[kinematics|kinematical]] aspects thereof, Lawvere does not seem to mention [[Hamiltonians]], [[Lagrangians]] or [[action functionals]]).
In ([interview, p. 8](http://www.mat.uc.pt/~picado/lawvere/interview.pdf)) he recalls:
> I had been a student at Indiana University from 1955 to January 1960. I liked [[experiment|experimental]] [[physics]] but did not appreciate the imprecise reasoning in some theoretical courses. So I decided to study [[mathematics]] first. [[Clifford Truesdell|Truesdell]] was at the Mathematics Department but he had a great knowledge in Engineering Physics. He took charge of my education there. $[...]$ in 1955 (and subsequently) had advised me on pursuing the study of [[continuum mechanics]] and [[kinetic theory]].
> In Summer 1958 I studied [[topological dynamics|Topological Dynamics]] with George Whaples, with the agenda of understanding as much as possible in categorical terms. $[...]$ Categories would clearly be important for simplifying the foundations of [[continuum physics]]. I concluded that I would make category theory a central line of my study.
Then in ([interview, p. 11](http://www.mat.uc.pt/~picado/lawvere/interview.pdf)) about the early 1960s:
> I felt a strong need to learn more [[set theory]] and [[logic]] from experts in that field, still of course with the aim of clarifying the foundations of [[category theory]] and of [[physics]].
The title of the early text _[[Toposes of laws of motion]]_, which is often cited as the text introducing [[synthetic differential geometry]], clearly witnesses the origin and motivation of these ideas in [[classical mechanics]].
On this, in ([interview, p. 15](http://www.mat.uc.pt/~picado/lawvere/interview.pdf)):
> Q: As an assistant professor in Chicago, in 1967, you taught with Mac Lane a course on [[mechanics|Mechanics]], where you "started to think about the justification of older intuitive methods in geometry" You called it "[[synthetic differential geometry]]". How did you arrive at the program of [[Categorical dynamics|Categorical Dynamics]] and [[synthetic differential geometry|Synthetic Differential Geometry]]?
> A: From January 1967 to August 1967 I was Assistant Professor at the University of Chicago. Mac Lane and I soon organized to teach a joint course based on Mackey's book "Mathematical Foundations of [[quantum mechanics|Quantum Mechanics]]".
> Q: So, Mackey, a functional analyst from Harvard mainly concerned with the relationship between quantum mechanics and representation theory, had some relation to category theory.
Then ([interview, p. 16](http://www.mat.uc.pt/~picado/lawvere/interview.pdf)):
> Q: Back to the origins of Synthetic Differential Geometry, where did the idea of organizing such a joint course on Mechanics originate ? Apparently, Chandra had suggested that Saunders give some courses relevant to physics, and our joint course was the first of a sequence. Eventually Mac Lane gave a talk about the [[Hamilton-Jacobi equation]] at the Naval Academy in summer 1970 that was published in the American Mathematical Monthly
> A: $[...]$ I began to apply the Grothendieck topos theory that I had learned from Gabriel to the problem of simplied foundations of [[continuum mechanics]] as it had been inspired by [[Clifford Truesdell|Truesdell]]'s teachings, [[Walter Noll|Noll]]'s [[axiom|axiomatizations]], and by my 1958 efforts to render categorical the subject of [[topological dynamics]].
A review of this with more comments on more relations to physics is in the introduction to the book collection _[[Categories in Continuum Physics]]_, which is the proceedings of a meeting organized by Lawvere in 1982.
In the talk _[[Toposes of laws of motion]]_ in 1997, Lawvere starts with the following remark
> I read somewhere recently that the basic program of infinitesimal [[calculus]], [[continuum mechanics]], and [[differential geometry]] is that all the world can be reconstructed from the [[infinitesimal object|infinitely small]]. One may think this is not possible, but nonetheless it's certainly a program that has been very fruitful over the last 300 years. I think we are now finally in a position to actually make more explicit what that program amounts to.
> $[...]$ I think that on the basis of these developments we can focus on this question of making very explicit how [[continuum physics]] etc. can be built up mathematically from very simple ingredients.
In the same talk, a few lines later after discussion of [[infinitesimally thickened points]] $T$, it says:
> The basic [[spaces]] which are needed for [[functional analysis]] and theories of [[physical fields]] are thus in some sense available in any [[topos]] with a suitable [[object]] $T$.
In 2000 in the article _[[Comments on the development of topos theory]]_ Lawvere writes in the closing section 7 titled "From and to continuum physics":
> What was the impetus which demanded the simplification and generalization of [[Grothendieck]]'s concept of [[topos]], if indeed the application to [[logic]] and [[set theory]] were not decisive. $[...]$ My own motivation came from my earlier study of [[physics]]. The [[foundation]] of the [[continuum physics]] of general materials, in the spirit of [[Clifford Truesdell|Truesdell]], [[Walter Noll|Noll]] and others, involves powerful and clear physical ideas, which unfortunately have been submerged under a mathematical apparatus $[...]$. But, as Fichera $[25]$ lamented, all this apparatus gives often a very uncertain fit to the phenomena. The apparatus may well be helpful in the solution of certain problems, but can the problems themselves and the needed axioms be stated in a direct and clear manner? And might this not lead to a simpler, equally rigorous account? These were the questions to which I began to apply the [[topos]] method in my 1967 Chicago lectures $[$ _[[Categorical dynamics]]_ $]$. It was clear that work on the notion of topos itself would be needed to achieve the goal. I had spent 1961-62 with the Berkeley logicians, believing that listening to experts on [[foundations]] might be the road to clarifying foundational questions.
> $[...]$ Several books treating the simplified [[topos theory]] ([[Sheaves in Geometry and Logic|MacLane-Moerdijk]] being the most recent and readable text), together with the three excellent books on [[synthetic differential geometry]] $[...]$ provide a solid basis on which further treatment of [[functional analysis]] and the needed development of [[continuum physics]] can be based.
The [Wikipedia entry](http://en.wikipedia.org/wiki/William_Lawvere) concludes:
> Lawvere continues to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications.
See also at _[[higher category theory and physics]]_ for more on this.
### Mathematics relating to Philosophy
{#RelationToPhilosophy}
Lawvere has proposed formalizations in [[category theory]], [[categorical logic]] and [[topos theory]] of concepts which are motivated from [[philosophy]], notably in [[Georg Hegel]]'s _[[Science of Logic]]_ (see there for more). This includes for instance definitions of concepts found there such as:
* _[[objective and subjective logic]]_, _[[abstract general]]_, _[[concrete general]]_, _[[concrete particular]]_, _[[unity of opposites]]_, _[[Aufhebung]]_, _[[being]]_, _[[becoming]]_, _[[space and quantity]]_, _[[cohesion]]_, _[[intensive and extensive quantity]]_
(see the references in these entries for pointers).
In ([Lawvere 92](#Lawvere92)) it says:
> It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy.
A precursor to this undertaking is [[Hermann Grassmann]] with his _[[Ausdehnungslehre]]_ ([Lawvere 95](#Lawvere95)), see there for more.
## Talks, Lecture notes and Publications
{#Talks}
The following is a list of texts by Lawvere, equipped with brief comments and hyperlinks to further material on the $n$Lab. See also the
* [list of publications on Lawvere's website](http://www.acsu.buffalo.edu/~wlawvere/list.html)
and also this
* [repository of Lawvere's publications](https://github.com/mattearnshaw/lawvere), made available by [[Matt Earnshaw]].
(Some of Lawvere's writings don't exist as published articles, but circulate in some form or other. Notably the "[[Archive for Mathematical Sciences & Philosophy]]" run by [[Michael Wright]] has a lot of recordings or lectures by Lawvere, though presently few or none of the files in the archive are available online.)
* _[[Functorial Semantics of Algebraic Theories]]_ Originally published as: Ph.D. thesis, Columbia University, 1963 and
in Reports of the Midwest Category Seminar II, 1968, 41-61, Republished in:
Reprints in Theory and Applications of Categories, No. 5 (2004) pp 1-121 ([tac](http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html))
(on [[algebraic theories]], now also called [[Lawvere theories]])
* _An elementary theory of the category of sets_, 1964, Proceedings of the National Academy of Science of the U.S.A 52, 1506-1511. Expanded version with commentary by [[Colin McLarty]] and the author in:
Reprints in Theory and Applications of Categories, No. 11 (2005) pp. 1-35 ([tac](http://www.tac.mta.ca/tac/reprints/articles/11/tr11abs.html))
> (axiomatizing the category [[Set]] of sets; see at *[[ETCS]]*)
* _The Category of Categories as a Foundation for Mathematics_, pp. 1-20 in: [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]
> (axiomatizing the [[2-category]] [[Cat]] of categories; see at *[[ETCC]]*)
* _[[Categorical dynamics]]_, 1967 Chicago lectures ([pdf](http://www.mat.uc.pt/~ct2011/abstracts/lawvere_w.pdf))
_Categorical Dynamics Revisited_, talk at _Sets Within Geometry_, Nancy, France 26-29 July 2011 ([recording](https://www.youtube.com/watch?v=sXATg9_6iWE))
(on [[synthetic differential geometry]] as a context for axiomatizing [[ordinary differential equations]] as they appear in [[classical physics]])
* {#AdjointnessInFoundations} _[[Adjointness in Foundations]]_, ([TAC](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html)), Dialectica 23 (1969), 281-296
(on the interpretation of [[quantifiers]] in [[categorical logic]] as [[base change]] [[adjoints]])
* [[H. Applegate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]: _[[Seminar on Triples and Categorical Homology Theory]]_, ETH 1966/67, edited by [[Beno Eckmann]] and [[Myles Tierney]], **[[LNM 80]]**, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]
(on [[monads]] in [[universal algebra]] and ([[cohomology|co-]])[[homology]]-theory:
)
* (with [[Max Kelly]]) *On the complete lattice of essential localizations*, Bull. Soc. Math. Belgique **41** (1989) 289–319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]
> (on [[essential localizations]])
* _Equality in hyperdoctrines and comprehension schema as an adjoint functor_. In A. Heller, ed., _Proc. New York Symp. on Applications of Categorical Algebra_, pp. 1--14. AMS, 1970. ([[LawvereComprehension.pdf:file]])
(on [[comprehension]] in [[hyperdoctrines]])
* *Quantifiers and sheaves*, Actes Congrès intern. Math. **1** (1970) 329-334 [[pdf](https://raw.githubusercontent.com/mattearnshaw/lawvere/master/pdfs/1970-quantifiers-and-sheaves.pdf), [[Lawvere-QuantifiersAndSheaves.pdf:file]]]
on [[geometric modalities]]/[[Lawvere-Tierney topology]] and [[universal quantifier|universal]]/[[existential quantifier|existential]] [[quantifiers]] related to [[dependent product]]/[[dependent sum]] further developed in:
* _Equality in hyperdoctrines and
comprehension schema as an adjoint functor_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.
* _Teoria delle categorie sopra un topos di base_, lecture notes from Perugia (1972--73) ([pdf](https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1972-perugia-lecture-notes.pdf), [[LawverePerugia1972.pdf:file]])
(on basic category theory, the [[category of sets]] and [[elementary toposes]], and an approach to [[indexed categories]])
* _Metric spaces, generalized logic and closed categories_ Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 ([tac](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html))
(A classic text from 1973 on [[Cauchy complete categories]] and [[enriched category theory]])
* _Variable sets etendu and variable structure in topoi_ ,
Lecture notes University of Chicago 1975.
(a review of topos theory: on structural sets, role of [[forcing]] and non-standard analysis and [[étendue]] toposes)
* *Variable quantities and variable structures in topoi*, in [[Alex Heller]], [[Myles Tierney]] (eds.), *Algebra, Topology and Category Theory -- A Collection of Papers in Honor of [[Samuel Eilenberg]]*, Academic Press New York (1976) 101-131 [[doi:10.1016/C2013-0-10841-0](https://doi.org/10.1016/C2013-0-10841-0)]
(reviews toposes in [[algebraic geometry]], first attention to [[gros topos]])
* _Topos-theoretic methods in geometry_, Aarhus 1997
* _Extensive and intensive quantities_ Workshop on Categorical Methods in Geometry, Aarhus 1983
(a proposal for a general axiomatization of [[homotopy]]/[[homology]]-like "[[extensive]] quantities" and [[cohomology]]-like "[[intensive]] quantities")
* _State Categories, Closed Categories, and the Existence Semi-Continuous Entropy Functions_ , IMA reprint 86, 1984 ([pdf](https://conservancy.umn.edu/bitstream/handle/11299/4672/86s.pdf?sequence=1&isAllowed=y))
* _Functional Remarks on the General Concept of Chaos_ , IMA reprint 87, 1984 ([pdf](http://www.ima.umn.edu/sites/default/files/87s.pdf))
(on a formalization of the concept of [[chaos]])
* _Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body_, Cah.top.géo.diff.cat, 21 nr.4, 1980, pp.377-392 ([Numdam](http://www.numdam.org/item?id=CTGDC_1980__21_4_377_0))
(contains a section with some remarks on Lagrangian mechanics)
* with [[Stephen Schanuel]], _[[Categories in Continuum Physics]]_, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 1986
(on [[continuum mechanics]], [[variational calculus]] and [[laws of motion]] in [[synthetic differential geometry]] and in [[diffeological spaces]] and [[Frölicher spaces]], and on [[intensive and extensive]] properties in terms of [[ring objects]] and [[modules]] in a [[topos]].)
* {#TakingCategoriesSeriously} _Taking categories seriously_, Revista Colombiana de Matemáticas **XX** (1986) 147-178, Reprints in Theory and Applications of Categories, **8** (2005) 1-24. [[tac:tr8](http://www.tac.mta.ca/tac/reprints/articles/8/tr8abs.html), [pdf](http://www.emis.de/journals/TAC/reprints/articles/8/tr8.pdf)]
(on the relevance of fundamental concepts in [[category theory]], such as [[Isbell duality]])
* *Categories of spaces may not be generalized spaces, as exemplified by directed graphs*, Revista Colombiana de Matemáticas **XX** (1986) 179-186, reprinted as: Reprints in Theory and Applications of Categories, **9** (2005) 1-7 [[tac:tr9](http://www.tac.mta.ca/tac/reprints/articles/9/tr9abs.html)]
(on the notion of [[gros topos|gros toposes]], details of his claims are worked out at [[sufficiently cohesive topos]])
* *Qualitative Distinctions between some Toposes of Generalized Graphs*, in *Categories in Computer Science and Logic*, Cont. Math. **92** (1989) 261-299 [[doi:10.1090/conm/092](http://dx.doi.org/10.1090/conm/092), [pdf](https://github.com/mattearnshaw/lawvere/raw/master/pdfs/1989-qualitative-distinctions-between-some-toposes-of-generalized-graphs.pdf)]
(Uses toposes of graphs to introduce his ideas on petit/gros toposes)
* _Display of graphics and their applications, as exemplified by 2-categories and the Hegelian "taco"_, Proceedings of the first international Conference on algebraic methodology and software technology University of Iowa, May 22-24 1989, Iowa City pp. 51-74
(on [[Aufhebung]], [[graphic category|graphic categories]], the [[Hegelian taco]]; the most explicit text with Lawvere's ideas on [[Hegel]])
* _More on graphic toposes_, Cah. Top. Géom. Diff. Cat. **XXXII** no. 1 (1991) pp.5-10. ([Numdam](http://www.numdam.org/item?id=CTGDC_1991__32_1_5_0))
(on [[graphic category|graphic categories]] and their presheaf categories)
* _[[Some Thoughts on the Future of Category Theory]]_ in A. Carboni, M. Pedicchio, G. Rosolini (eds.), _Category Theory_ , [[Como|Proceedings of the International Conference held in Como]], LNM **1488** Springer Heidelberg 1991.
(on -- implicitly -- [[cohesive topos|cohesive toposes]]);
* with [[Stephen Schanuel]], _Conceptual Mathematics - A first introduction to categories_ , Cambridge UP 1997.
(first steps into category theory; the Spanish translation is available [here](http://www.acsu.buffalo.edu/~wlawvere/concep-3.pdf))
* {#Lawvere92} _Categories of space and quantity_ in: J. Echeverria et al (eds.), _The Space of mathematics_ , de Gruyter, Berlin, New York (1992)
(on [[space and quantity]] and on formalizing concepts from [[philosophy]] as in the _[[Science of Logic]]_ in terms of [[category theory]] and [[categorical logic]])
* {#IntroductionToLinearCategories} [[William Lawvere]], *Introduction to Linear Categories and Applications*, course lecture notes (1992) [[pdf](https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf), [[Lawvere-LinearCategories.pdf:file]]]
(on [[additive categories]])
* _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_ Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. ([[LawvereCohesiveToposes.pdf:file]])
(an early version of the notion of [[cohesive topos|cohesive toposes]])
* {#Lawvere94b} _[[Tools for the advancement of objective logic]]: closed categories and toposes_, in J. Macnamara and [[Gonzalo Reyes]] (Eds.), _The Logical Foundations of Cognition_, Oxford University Press 1993 (Proceedings of the Febr. 1991 Vancouver Conference "Logic and Cognition"),
pages 43-56, 1994.
(on formalizing [[philosophy]] as in the _[[Science of Logic]]_ in terms of [[category theory]] and [[categorical logic]].)
* {#Lawvere95} _A new branch of mathematics, "The Ausdehnungslehre of 1844," and other works. Open Court (1995), Translated by Lloyd C. Kannenberg, with foreword by Albert C. Lewis_, _Historia Mathematica Volume 32, Issue 1, February 2005, Pages 99–106_ ([publisher](http://www.sciencedirect.com/science/article/pii/S031508600400059X))
(on [[Hermann Grassmann]]'s _[[Ausdehnungslehre]]_ and hence also on _[[extensive and intensive quantity]]_)
* _Grassmann's Dialectics and Category Theory_, in _Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar_, Boston Studies in the Philosophy of Science Volume 187, 1996, pp 255-264 ([publisher](http://link.springer.com/chapter/10.1007%2F978-94-015-8753-2_21))
* _Volterra's functionals and covariant cohesion of space_ Perugia Studies in Mathematics (Proceedings of the May 1997 Meeting in Perugia) ([pdf](http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf))
(with first remarks on [[cohesive topos|cohesion]])
* _[[Toposes of laws of motion]]_ , transcript of a talk in Montreal, Sept. 1997 ([pdf](http://www.acsu.buffalo.edu/~wlawvere/ToposMotion.pdf))
(on the description of [[differential equations]] in terms of [[synthetic differential geometry]] and the notion of _[[toposes of laws of motion]]_)
* _Outline of synthetic differential geometry_ , seminar notes (1998) ([pdf](http://www.acsu.buffalo.edu/~wlawvere/SDG_Outline.pdf))
(the origin of the concept of [[synthetic differential geometry]])
* _Kinship and Mathematical Categories_ , pp.411-425 in Jackendoff, Bloom, Wynn (eds.), _Language, Logic, and Concepts_ , MIT Press Cambridge 1999.
(Lawvere's venture into anthropology; a summary is at [[kinship]])
* _[[Comments on the development of topos theory]]_ , pp.715-734 in Jean-Paul Pier (ed.), _Development of mathematics 1950-2000_, Birkhäuser Basel 2000.
(review of the history of topos theory)
* _Linearization of graphic toposes via Coxeter groups_, JPAA **168** (2002) pp. 425-436. ([pdf](http://www.sciencedirect.com/science/article/pii/S0022404901001074/pdfft?md5=a4ca9bc67df6ae63ddf53c559bd71315&pid=1-s2.0-S0022404901001074-main.pdf))
* _Categorical algebra for continuum micro physics_, JPAA **175** (2002) pp. 267-287. ([pdf](http://www.sciencedirect.com/science/article/pii/S002240490200138X/pdfft?md5=ebd435105778d597f25141bf065978e3&pid=1-s2.0-S002240490200138X-main.pdf))
* _Foundations and Applications: Axiomatization and Education_, Bulletin of Symbolic Logic **9** no.2 (2003) pp.213-224. ([ps-preprint](https://www.math.ucla.edu/~asl/bsl/0902/0902-006.ps))
(a review of (not only set-theoretic) foundations in mathematics)
* {#LawvereRosebrugh} with [[Robert Rosebrugh]], _[[Sets for Mathematics]]_ , Cambridge UP 2003. ([web](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics))
(on [[set theory]] from a practical [[category theory|theoretic]] point of view (informal [[ETCS]]))
* _Axiomatic cohesion_, Theory and Applications of Categories **19** 3 (2007) 41-49 [[tac:19-03](http://www.tac.mta.ca/tac/volumes/19/3/19-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf)]
(a more formal definition of [[cohesive topos|cohesive toposes]], [[sufficiently cohesive topos]] works out details of his claims concerning 'sufficient cohesion')
* _[[Cohesive Toposes -- Combinatorial and Infinitesimal Cases]]_ , lecture in Como (2008)
(on [[cohesive topos|cohesive toposes]])
* _Open problems in topos theory_, April 2009 ([pdf](https://web.archive.org/web/20170421103453/http://cheng.staff.shef.ac.uk/pssl88/lawvere.pdf))
(problem 7 presented there "The algebra of time", concerns the characterization of _[[Toposes of laws of motion]]_)
* _The Dialectic of the Continuous and the Discrete in the history of the struggle for a usable guide to mathematical thought_, talk at _Sets Within Geometry_, Nancy, France 26-29 July 2011 ([recording](https://www.youtube.com/watch?v=OE-R3565jrs))
* _What are Foundations of Geometry and Algebra_, Keynote lecture at the _[Fifty Years of Functorial Semantics](http://www.math.union.edu/~niefiels/13conference/Web/)_ conference, Union College, October 2013 ([recording](https://www.youtube.com/watch?v=ZYGyEPXu8as))
* _Alexander Grothendieck and the Concept of Space_, address CT15 Aveiro 2016. ([link](http://www.acsu.buffalo.edu/~wlawvere/GrothendiekAveiro15.htm))
(a talk given at CT15 reviewing 50 years of developments since the 1965 La Jolla conference)
* {#Lawvere21} _Toposes generated by codiscrete objects in combinatorial topology and functional analysis_, Reprints in Theory and Applications of Categories, No. 27 (2021) pp. 1-11, [pdf](http://tac.mta.ca/tac/reprints/articles/27/tr27abs.html).
category: people
[[!redirects William Lawvere]]
[[!redirects Lawvere]]
[[!redirects F. W. Lawvere]]
[[!redirects F.W. Lawvere]]
[[!redirects W. Lawvere]]
[[!redirects F. William Lawvere]] |
William Linch III | https://ncatlab.org/nlab/source/William+Linch+III |
* [webpage](http://people.physics.tamu.edu/wdlinch3)
## Selected writings
On [[D=5 supergravity]]:
* [[William Linch III]], Markus A. Luty, J. Phillips, _Five dimensional supergravity in $N = 1$ superspace_, Phys.Rev.D68:025008,2003 ([arXiv:hep-th/0209060](https://arxiv.org/abs/hep-th/0209060))
* [[Katrin Becker]], [[Melanie Becker]], Daniel Butter, [[William Linch III]], Stephen Randall, _Five-dimensional Supergravity in N = 1/2 Superspace_ ([arXiv:1909.09208](https://arxiv.org/abs/1909.09208))
category: people
[[!redirects William Linch]]
|
William Massey | https://ncatlab.org/nlab/source/William+Massey |
* [Wikipedia entry](https://en.wikipedia.org/wiki/William_S._Massey)
## Selected writings
On [[algebraic topology]]:
* [[William S. Massey]], *Algebraic Topology: An Introduction*, Harcourt Brace & World 1967, reprinted in: Graduate Texts in Mathematics, Springer 1977 ([ISBN:978-0-387-90271-5](https://link.springer.com/book/9780387902715))
The [[Arnold-Kuiper-Massey theorem]]:
* [[William Massey]], _The quotient space of the complex projective space under conjugation is a 4-sphere_, Geometriae Didactica 1973 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/massey5.pdf))
## Related $n$Lab entries
* [[Massey product]]
* [[Blakers-Massey theorem]]
* [[covering space]], [[universal cover]]
category: people
[[!redirects William S. Massey]]
|
William Messing | https://ncatlab.org/nlab/source/William+Messing |
+ [website](http://www.math.umn.edu/~messing/)
#related $n$Lab entries#
* [[gerbe]]
* [[infinitesimal singular simplicial complex]]
category: people |
William Mycroft | https://ncatlab.org/nlab/source/William+Mycroft |
## Selected writings
On [[cohomology operations]]:
* [[William Mycroft]], _Unstable Cohomology Operations: Computational Aspects of Plethories_, 2017 ([pdf](https://etheses.whiterose.ac.uk/20524/1/wm-thesis-final.pdf), [[MycroftUnstableCohomologyOperations.pdf:file]])
specifically on [[topological K-theory]]:
* [[William Mycroft]], [[Sarah Whitehouse]], _The plethory of operations in complex topological K-theory_ ([arXiv:2001.01608](https://arxiv.org/abs/2001.01608))
category: people |
William Rowan Hamilton | https://ncatlab.org/nlab/source/William+Rowan+Hamilton |
* [Wikipedia entry](http://en.wikipedia.org/wiki/William_Rowan_Hamilton)
## related $n$Lab entries
* [[Hamiltonian mechanics]]
* [[Hamiltonian]], [[Hamiltonian vector field]]
* [[quaternions]]
* [[nabla]]
category: people
[[!redirects William Hamilton]] |
William S. Massey | https://ncatlab.org/nlab/source/William+S.+Massey | [[!redirects W. S. Massey]]
William S. Massey was an American algebraic topologist (born 1920). He is best known for the development of [[Massey products]] and for contributions to the theory of spectral sequences, via his work on [[exact couples]]. He wrote several influential textbooks.
* [Wikipedia entry](http://en.wikipedia.org/wiki/William_S._Massey)
category:people |
William Stein | https://ncatlab.org/nlab/source/William+Stein |
William Stein is lead developer of *[[SageMath]]*
* [personal page](https://wstein.org/)
* [Wikipedia entry](https://en.wikipedia.org/wiki/William_A._Stein)
* [MathematicsGenealogy page](https://mathgenealogy.org/id.php?id=36697)
category: people
|
William T. Tutte | https://ncatlab.org/nlab/source/William+T.+Tutte | [[!redirects W. T. Tutte]]
Bill Tutte (b. May 14, 1917; d. May 2, 2002) was a British and Canadian mathematician.
Tutte made seminal contributions to [[graph theory]] and the theory of [[matroids]]. He [published](#Tutte1958I) the [[Tutte homotopy theorem]] which can be regarded as a generalization of the notion of [[paths]] to matroids, and about which [[Gian-Carlo Rota]] wrote
> The (...) motivation, (...) in Tutte's vertiginously profound papers, was graph-theoretic. Matroids are objects that play the role of the dual graph of a graph when the graph is not [[planar]]. Almost every fact about graphs that can be formulated without using the term "vertex" has a matroidal analogue. The deepest insight obtained from this yoga is Tutte's homotopy theorem, a unique combinatorial achievement whose far reaching implications are a long way from being fully understood. -- Gian-Carlo Rota: Indiscrete Thoughts. Modern Birkhäuser Classics. Chapter "Book Reviews"
His writing style is sometimes described to be extraordinarily compressed, in cautionary remarks like
> If you read Tutte, remember: it's all there, but take a semester off -- and plan the first ten pages.
(Personal recollections from a seminar)
## References
* William T. Tutte: _A homotopy theorem for matroids. I_, Transactions of the American Mathematical Society, 88: 144–160 (1958)
{#Tutte1958I}
* William T. Tutte: _A homotopy theorem for matroids. II_, Transactions of the American Mathematical Society, 88: 161–174, (1958)
{#Tutte1958II}
* W. T. Tutte (1984), _Graph Theory_, Addison-Wesley.
* W. T. Tutte (1998), _Graph Theory As I Have Known It_, Oxford.
* [Wikipedia page](https://en.wikipedia.org/wiki/W._T._Tutte)
* [St. Andrews History page](http://www-history.mcs.st-and.ac.uk/Biographies/Tutte.html)
category:people |
William Tait | https://ncatlab.org/nlab/source/William+Tait |
* [webpage](http://home.uchicago.edu/~wwtx/)
## Related entries
* [[propositions as types]]
category: people
[[!redirects Bill Tait]]
|
William Thurston | https://ncatlab.org/nlab/source/William+Thurston |
Bill Thurston was an American mathematician (Fields Medalist, 1982) who made important contributions to the theory of [[orbifolds]], the theory of [[foliation|foliations]], [[low-dimensional topology]] (especially in three dimensions), and to [[geometric group theory]]. He is perhaps best known for his [[geometrization conjecture]] (which was proved by Thurston in special cases, and by [[Richard Hamilton]] and [[Grigori Perelman]] in complete generality, and which in turn implies the [[Poincaré conjecture]]).
* [wikipedia entry](http://en.wikipedia.org/wiki/William_Thurston)
* [AMS obituary](http://www.ams.org/news?news_id=1602)
* [NYT obituary](http://www.nytimes.com/2012/08/23/us/william-p-thurston-theoretical-mathematician-dies-at-65.html?_r=1)
* [St Andrew's History entry](http://www-history.mcs.st-and.ac.uk/Biographies/Thurston.html)
* [[C. T. C. Wall]], _On the work of W Thurston_, Proceedings of the International Congress of Mathematicians, Warsaw 1983 1 (Warsaw, 1984), 11-14. ([pdf](www.mathunion.org/ICM/ICM1983.1/ICM1983.1.ocr.pdf))
## Selected publications
* The theory of foliations of codimension greater than one, Comm. Math. Helv. 49 (1974), 214-231.
* Existence of codimension-one foliations, Annals of Math. (2), 104 (1976), 249-268.
* Three-Dimensional Geometry and Topology, Vol. I (ed. Silvio Levy), Princeton University Press, 1997.
* On proof and progress in mathematics, Bull. AMS 30 (1994), 161-177. ([web](http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/home.html))
On [[3-manifolds]]:
* Geometry and topology of three-manifolds (1980), electronic version 1.1 (2002) available from MSRI ([web](http://library.msri.org/books/gt3m/))
On [[hyperbolic 3-manifolds]]:
* [[William Thurston]], _Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds_, Annals of Math, 124 (1986), 203--246 ([jstor:1971277](https://www.jstor.org/stable/1971277), [arXiv:math/9801019](https://arxiv.org/abs/math/9801019))
* [[William Thurston]], _Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle_ ([arXiv:math/9801045](https://arxiv.org/abs/math/9801045))
* [[William Thurston]], _Three dimensional manifolds, Kleinian groups and hyperbolic geometry_, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 3 (1982), 357-381 ([euclid.bams/1183548782](https://projecteuclid.org/euclid.bams/1183548782))
category: people
[[!redirects William P. Thurston]]
[[!redirects William P Thurston]]
[[!redirects William Thurston]]
[[!redirects Bill Thurston]]
[[!redirects Thurston]]
|
William Wooters | https://ncatlab.org/nlab/source/William+Wooters |
* [Wikipedia entry](http://en.wikipedia.org/wiki/William_Wootters)
## related $n$Lab entries
* [[no-cloning theorem]]
category: people
|
Willis Lamb | https://ncatlab.org/nlab/source/Willis+Lamb |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Willis_Lamb)
## related $n$Lab entries
* [[Lamb shift]]
category: people |
Willy Fischler | https://ncatlab.org/nlab/source/Willy+Fischler |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Willy_Fischler)
## Selected writings
Introducing the [[BFSS matrix model]] ([[M-theory]] as [[D0-brane]] [[quantum mechanics]] in [[non-perturbative effect|non-perturbative]] [[type IIA string theory]]):
* {#BanksFischlerShenkerSusskind96} [[Tom Banks]], [[Willy Fischler]], [[Stephen Shenker]], [[Leonard Susskind]], _M Theory As A Matrix Model: A Conjecture_ Phys. Rev. D55 (1997). ([arXiv:hep-th/9610043](http://arxiv.org/abs/hep-th/9610043))
On [[black holes in string theory]] via the [[BFSS matrix model]]:
* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzschild Black Holes from Matrix Theory_, Phys.Rev.Lett.80:226-229,1998 ([arXiv:hep-th/9709091](https://arxiv.org/abs/hep-th/9709091))
* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzchild Black Holes in Matrix Theory II_, JHEP 9801:008,1998 ([arXiv:hep-th/9711005](https://arxiv.org/abs/hep-th/9711005))
## Related $n$Lab entries
* [[axion]]
* [[BFSS matrix model]]
category: people |
Wilson line > history | https://ncatlab.org/nlab/source/Wilson+line+%3E+history |
see _[[Wilson loop]]_. |
Wilson loop | https://ncatlab.org/nlab/source/Wilson+loop |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
A _Wilson loop_ or _Wilson line_ is an [[observable]] in (both classical and quantum) [[gauge theory]] obtained from the [[holonomy]] of the [[gauge field|gauge]] [[connection on a bundle|connection]].
Hence if the gauge connection is given by a globally defined 1-form $A$, then the __Wilson loop__ along a closed [[loop]] $C$ is the trace of the [[Dyson formula|path-ordered exponential]]
$$
W_C = Tr(\mathcal{P}exp(i\oint_C A_\mu d x^\mu))
$$
where $\mathcal{P}$ is the "path-ordering operator" and $A_\mu$ are the components of the connection.
## Properties
### Relation to 1d Chern-Simons theory
For $G$ a suitable [[Lie group]] (compact, semi-simple and simply connected) the Wilson loops of $G$-[[principal connections]] are equivalently the [[partition functions]] of a [[1-dimensional Chern-Simons theory]].
This appears famously in the formulation of [[Chern-Simons theory]] [with Wilson lines](Chern-Simons+theory#WithWilsonLineObservables). More detailes are at _[[orbit method]]_.
### As the universal Vassiliev invariant
The un-traced [[Wilson loop observable]] of [[perturbative quantization of 3d Chern-Simons theory|perturbative Chern-Simons theory]] is the [[universal Vassiliev invariant]] (see there for more):
<center>
<img src="https://ncatlab.org/nlab/files/UniversalWilsonLoopObservable.jpg" width="700">
</center>
<center>
<img src="https://ncatlab.org/nlab/files/TheGrandStoryOfVassilievKnotInvariants.jpg" width="800">
</center>
> from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
### Relation to defects
Wilson loop insertions may be thought of or at least related to _defects_ in the sense of _[[QFT with defects]]_.
### Duality with 't Hooft operators under S-duality and geometric Langlands
[[S-duality]] of 4d [[super Yang-Mills theory]] may exchange Wilson loop operators with [['t Hooft operators]], in an incarnation of the [[geometric Langlands correspondence]] ([Kapustin-Witten 06](#KapustinWitten06))
[[!include geometric Langlands QFT -- table]]
## Examples
### In Chern-Simons theory
In $SU(2)$-[[Chern-Simons theory]] the Wilson line observables compute the [[Jones polynomial]] of the given curve. See there for more details.
### In Rozansky-Witten theory
+-- {: .num_prop #RozanskyWittenWilsonLoopOfUnknotIsSquareRootOfAHat}
###### Proposition
**([[Rozansky-Witten Wilson loop of unknot is A-hat genus|Rozansky-Witten Wilson loop of unknot is square root of A-hat genus]])**
For $\mathcal{M}^{4n}$ a [[hyperkähler manifold]] (or just a [[holomorphic symplectic manifold]]) the [[Rozansky-Witten invariant]] [[Wilson loop observable]] associated with the [[unknot]] in the [[3-sphere]] is the [[square root]] $\sqrt{{\widehat A}(\mathcal{M}^{4n})}$ of the [[A-hat genus]] of $\mathcal{M}^{4n}$.
=--
This is [Roberts-Willerton 10, Lemma 8.6](Rozansky-Witten+Wilson+loop+of+unknot+is+A-hat+genus#RobertsWillerton10), using the [[Wheels theorem]] and the [[Hitchin-Sawon theorem]].
## Related entries
* [[flux tube]]
* [[Polyakov gauge-string duality]]
* [['t Hooft operator]]
* [[Wilson surface]]
## References
### General
* [[Kenneth Wilson]], _Confinement of quarks_, Physical Review __D 10__ (8): 2445. [doi](http://dx.doi.org/10.1103/PhysRevD.10.2445) (1974)
* [[Yuri Makeenko]], *Methods of contemporary gauge theory*, Cambridge Monographs on Math. Physics, Cambridge University Press (2002) [[doi:10.1017/CBO9780511535147]( https://doi.org/10.1017/CBO9780511535147), [gBooks](http://books.google.com/books?id=9W-W2w75ulAC)]
* Wikipedia [Wilson loop](http://en.wikipedia.org/wiki/Wilson_loop)
* R. Giles, _Reconstruction of gauge potentials from Wilson loops_, Physical Review __D 24__ (8): 2160, [doi](http://dx.doi.org/10.1103/PhysRevD.24.2160)
* A. Andrasi, J. C. Taylor, _Renormalization of Wilson operators in Minkowski space_, Nucl. Phys. B516 (1998) 417, [hep-th/9601122](http://arxiv.org/abs/hep-th/9601122)
* Amit Sever, Pedro Vieira, Luis F. Alday, Juan Maldacena, [[Davide Gaiotto]], _An Operator product expansion for polygonal null Wilson loops_, [arxiv.org/abs/1006.2788](http://arxiv.org/abs/1006.2788)
On super-[[Wilson lines]] via [[integration over supermanifolds]]:
* C.A. Cremonini, [[Pietro Grassi]], S. Penati, _Supersymmetric Wilson Loops via Integral Forms_ ([arXiv:2003.01729](https://arxiv.org/abs/2003.01729))
### In Chern-Simons theory
See at [[perturbative quantization of 3d Chern-Simons theory]]/[[Vassiliev knot invariants]].
Relation between [[Dehn surgery]] and [[Wilson loop observables]] in [[Chern-Simons theory]]:
* E. Guadagnini, _Surgery rules in quantum Chern-Simons field theory_, Nuclear Physics B
Volume 375, Issue 2, 18 May 1992, Pages 381-398 (<a href="https://doi.org/10.1016/0550-3213(92)90037-C">doi:10.1016/0550-3213(92)90037-C</a>)
* Boguslaw Broda, _Chern-Simons theory on an arbitrary manifold via surgery_ ([arXiv:hep-th/9305051](https://arxiv.org/abs/hep-th/9305051))
The [[Poisson bracket]] of Wilson line observables in [[3d Chern-Simons theory]] was obtained in
* W. Goldman, _Invariant functions on Lie groups and Hamiltonian flow of surface
group representations_, Inventiones Math., 85 (1986), 263–302.
For more see
* {#GelcaUribe10b} [[Razvan Gelca]], [[Alejandro Uribe]], section 4 of _Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$_ ([arXiv:1007.2010](http://arxiv.org/abs/1007.2010))
In [[Chern-Simons theory as a topological string theory]]:
* [[Hirosi Ooguri]], [[Cumrun Vafa]], _Knot Invariants and Topological Strings_, Nucl. Phys. B 577:419-438, 2000 ([arXiv:hep-th/9912123](https://arxiv.org/abs/hep-th/9912123))
[[!include Jones polynomial as Wilson loop observable -- references]]
[[!include Chern-Simons Wilson lines in AdS3-CFT2 -- references]]
### In QFT with defects
Relation to [[QFT with defects]] is discussed in
slide 17 of
* [[Constantin Bachas]], _Conformal defects in/and String theory_ ([pdf](http://www.ggi.fi.infn.it/talks/talk106.pdf))
slide 5 of
* [[Constantin Bachas]], _Conformal defects in gauged WZW models_ ([pdf](http://hep.physics.uoc.gr/conf09/TALKS/Bachas.pdf))
### As partition functions of 1d Chern-Simons theory
Expression of Wilson loops as [[partition functions]] of [[1-dimensional Chern-Simons theories]] by the [[orbit method]] (as used notably in [[Chern-Simons theory]]) is in section 4 of
* [[Chris Beasley]], _Localization for Wilson Loops in Chern-Simons Theory_, in J. Andersen, H. Boden, A. Hahn, and B. Himpel (eds.) _Chern-Simons Gauge Theory: 20 Years After_, AMS/IP Studies in Adv. Math., Vol. 50, AMS, Providence, RI, 2011. ([arXiv:0911.2687](http://arxiv.org/abs/0911.2687))
referring to
* S. Elitzur, [[Greg Moore]], A. Schwimmer, [[Nathan Seiberg]], _Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory_, Nucl. Phys. B 326 (1989) 108–134.
in the context of [[Chern-Simons theory]] and in more general gauge theory to
* A. P. Balachandran, S. Borchardt, and A. Stern, _Lagrangian And Hamiltonian Descriptions of Yang-Mills Particles_, Phys. Rev. D 17 (1978) 3247–3256
### In S-duality and relation to t Hoofts operators
* {#KapustinWitten06} [[Anton Kapustin]], [[Edward Witten]], _Electric-Magnetic Duality And The Geometric Langlands Program_, Communications in Number Theory and Physics Volume 1 (2007) Number 1 ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))
[[!redirects Wilson loop operator]]
[[!redirects Wilson loop operators]]
[[!redirects Wilson loop]]
[[!redirects Wilson loops]]
[[!redirects Wilson-loop]]
[[!redirects Wilson-loops]]
[[!redirects Wilson line]]
[[!redirects Wilson-line]]
[[!redirects Wilson lines]]
[[!redirects Wilson-lines]]
[[!redirects Wilson operator]]
[[!redirects Wilson operators]]
[[!redirects Wilson loop observable]]
[[!redirects Wilson loop observables]]
[[!redirects Wilson line observable]]
[[!redirects Wilson line observables]]
|
Wilson surface | https://ncatlab.org/nlab/source/Wilson+surface |
#Contents#
* table of contents
{:toc}
## Idea
In [[physics]] the term _[[Wilson loop]]_ refers to an [[observable]] in [[gauge field theory]] given by the [[holonomy]] of the given [[gauge field]] [[connection on a principal bundle|connection]] along a fixed curve.
In [[higher gauge theory]] one correspondingly has [[observables]] given by [[higher holonomy]] around higher dimensional manifolds, such as [[surfaces]] etc. These are hence sometimes called "Wilson surfaces" etc.
## References
e.g.
* [[John Baez]], _[TWF 235](http://math.ucr.edu/home/baez/week235.html)_
* [[Anton Alekseev]], Olga Chekeres, [[Pavel Mnev]], _Wilson surface observables from equivariant cohomology_, JHEP11(2015)093 ([arXiv:1507.06343](https://arxiv.org/abs/1507.06343))
[[!redirects Wilson surfaces]] |
Wim Hordijk | https://ncatlab.org/nlab/source/Wim+Hordijk |
* [personal page](https://worldwidewanderings.net/)
## Selected writings
On [[Cayley graph spectra]]:
* {#RockmoreKostelecHordijkStadler02} [[Dan Rockmore]], [[Peter Kostelec]], [[Wim Hordijk]], [[Peter F. Stadler]], *Fast Fourier Transform for Fitness Landscapes*, Applied and Computational Harmonic Analysis Volume 12, Issue 1, January 2002, Pages 57-76 ([doi:10.1006/acha.2001.0346](https://doi.org/10.1006/acha.2001.0346))
category: people |
Wim Ruitenberg | https://ncatlab.org/nlab/source/Wim+Ruitenberg |
* [personal page](https://www.mscsnet.mu.edu/~wim/)
## Selected writings
On [[roots]] of [[polynomials]] in [[constructive mathematics]]:
* {#Ruitenberg91} [[Wim Ruitenberg]], *Constructing Roots of Polynomials over the Complex Numbers*, Computational Aspects of Lie Group Representations and Related Topics, CWI Tract, **84** Centre for Mathematics and Computer Science, Amsterdam (1991) 107–128 [[pdf](https://www.mscsnet.mu.edu/~wim/publica/roots_new.pdf), [[Ruitenberg-Roots.pdf:file]]]
category: people |
Wim van Dam | https://ncatlab.org/nlab/source/Wim+van+Dam |
* [institute page](https://engineering.ucsb.edu/people/wim-van-dam)
## Selected writings
On [[adiabatic quantum computation]]:
* [[Dorit Aharonov]], [[Wim van Dam]], [[Julia Kempe]], [[Zeph Landau]], [[Seth Lloyd]], [[Oded Regev]], *Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation*, SIAM Journal of Computing **37** 1 (2007) 166-194 [[arXiv:quant-ph/0405098](https://arxiv.org/abs/quant-ph/0405098), [jstor:20454175](https://www.jstor.org/stable/20454175), [doi:10.1109/FOCS.2004.8](https://doi.org/10.1109/FOCS.2004.8), [doi:10.1137/080734479](https://doi.org/10.1137/080734479)]
category: people |
winding number | https://ncatlab.org/nlab/source/winding+number |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[continuous function]] between two [[topological spaces]] that are each [[weak homotopy equivalence|weakly homotopy equivalent]] to the [[circle]] is, up to homotopy, defined by how much it "winds" one circle around the other. This is its _winding number_, an [[integer]]. Equivalently this is the element in the [[fundamental group]] of the codomain which is represented by the map.
A special case of the _[[degree of a continuous function]]_.
## Related construction
* [[clutching construction]]
* [[fundamental group of the circle is the integers]]
[[!redirects winding numbers]]
|
Winter 2016-2017 seminar on higher structures | https://ncatlab.org/nlab/source/Winter+2016-2017+seminar+on+higher+structures | ## Seminar on Higher Structures at Goettingen: 2016 Fall - 2017 Winter ##
***
**Place: Hörsaal 6.
|
Winter 2017-2018 seminar on higher structures (Higher Lie Groupoids) | https://ncatlab.org/nlab/source/Winter+2017-2018+seminar+on+higher+structures+%28Higher+Lie+Groupoids%29 | First all, we have our [studip link](
https://www.studip.uni-goettingen.de/dispatch.php/course/overview?cid=cd3c8d093de4d8fd9c8245f25e75f851). You need to register in Flexnow and present at least once, if you want to obtain your credit for this seminar.
Our seminar follows mostly Goerss-Jardine (GJ)
[Simplicial Homotopy Theory](http://dodo.pdmi.ras.ru/~topology/books/goerss-jardine.pdf)
1. Simplicial set (GJ I.1) by Daniel (02.11.2017)
2. Simplicial set (GJ I.1) by Daniel and realization by Laura (09.11.2017, 16.11.2017)
2.5 Kan Complex I by Laura(GJ I.3, first three pages) (23.11.2017)
3. Kan Complex II towards n-groupoids (GJ I.3, the rest of it) by Richard (23.11.2017)
4. [n-groupoid objects in a category with topology] (https://arxiv.org/pdf/0801.2057.pdf) section 1, 2.1, 2.2
5. simplicial homotopy group (GJ I.7) Laura 2 (but maybe we need more people, it's a long chapter)
6. Category of fibrant objects (GJ I.9 + factorization Lemma) Dennis
7. [Category of fibrant objects for n-groupoids with topology](https://arxiv.org/abs/1609.01394) Chenchang
8. Dold-Kan (GJ III.2) Madleine
9. Lie n-algebroids Malte |
Wirthmüller context | https://ncatlab.org/nlab/source/Wirthm%C3%BCller+context |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _Wirthmüller context_ is a pair of two [[symmetric monoidal category|symmetric]] [[closed monoidal categories]] $(\mathcal{X}, \otimes_X, 1_{X})$, $(\mathcal{Y}, \otimes_Y, 1_Y)$ which are connected by an [[adjoint triple]] of [[functors]] such that the middle one is a [[closed monoidal functor]].
This is the variant/special case of the [[yoga of six operations]] consisting of two [[adjoint pairs]] $(f_! \dashv f^!)$ and $(f^\ast \dashv f_\ast)$ and the [[tensor product]]/[[internal hom]] [[adjunctions]] $((-)\otimes B \dashv [B,-])$, specialized to the case that $f^! \simeq f^\ast$:
\begin{tikzcd}
\mathcal{X}
\ar[
rr,
shift left=20pt,
"f_!"{above}
]
\ar[
rr,
phantom,
shift left=10.5pt,
"\scalebox{.62}{$\bot$}"{description}
]
\ar[
rr,
phantom,
shift right=10.5pt,
"\scalebox{.62}{$\bot$}"{description}
]
\ar[
rr,
shift right=20pt,
"f_\ast"{below}
]
&&
\mathcal{Y}
\ar[
ll,
"f^! = f^\ast"{description}
]
\end{tikzcd}
(The other specialization of the [[six operations]] to $f_\ast \simeq f_!$ is called the _[[Grothendieck context]]_).
Often one is interested in the case that there is an [[object]] $C \in \mathcal{X}$ and an [[equivalence]]
$$
f_\ast 1_{X} \simeq f_! C
\,.
$$
If this induces a [[natural equivalence]]
$$
f_\ast A \simeq f_!(A \otimes_X C)
$$
for $A \in \mathcal{X}$, then one says this is a _Wirthmüller isomorphism_, following ([Wirthmueller 74](#Wirthmueller74)).
In particular there is a canonical [[natural transformation]]
$$
f_\ast A \longrightarrow f_!(A \otimes_X C)
$$
and one can ask this to be an equivalence, hence a Wirthmüller isomorphism ([May 05](#May05)).
## Definition
+-- {: .num_defn #WirthmullerContext}
###### Defininition
Let $(\mathcal{X}, \otimes_X, 1_{X})$, $(\mathcal{Y}, \otimes_Y, 1_Y)$ be two [[symmetric monoidal category|symmetric]] [[closed monoidal categories]] and let
\begin{tikzcd}
\mathcal{X}
\ar[
rr,
shift left=20pt,
"f_!"{above}
]
\ar[
rr,
phantom,
shift left=10.5pt,
"\scalebox{.62}{$\bot$}"{description}
]
\ar[
rr,
phantom,
shift right=10.5pt,
"\scalebox{.62}{$\bot$}"{description}
]
\ar[
rr,
shift right=20pt,
"f_\ast"{below}
]
&&
\mathcal{Y}
\ar[
ll,
"f^! = f^\ast"{description}
]
\end{tikzcd}
be an [[adjoint triple]] of [[functors]] between them. We call this setup
* a _pre-Wirthmüller context_ if $f^\ast$ is a [[strong monoidal functor]]:
* a _Wirthmüller context_ if $f^\ast$ is in addition a [[strong closed functor]], hence a strongly [[closed monoidal functor]].
=--
([May 05, def. 2.12](#May05))
+-- {: .num_defn }
###### Definition (Notation)
We write $[-,-]$ for the [[internal hom]] functors. For $A \in \mathcal{X}$ we write
$$
\mathbb{D}A \coloneqq [A,1_X]
$$
for the [[internal hom]] from $A$ into the [[unit object]], hence for dual of $A$ with respect to the [[closed category]] structure, its [[dual object in a closed category]].
We say "$A$ is dualizable" to mean that it is a [[dualizable object]] with respect, instead, to the (symmetric) [[monoidal category]] structure $\otimes_X$. If $A$ is dualizable we write $A^\vee$ for its monoidal [[dual object]].
Similarly for $B \in \mathcal{Y}$.
=--
+-- {: .num_remark }
###### Remark
If all objects in $\mathcal{X}$ and $\mathcal{Y}$ are [[dualizable object|dualizable]], hence if they are [[compact closed categories]], then they are in particular also [[star-autonomous categories]] with [[dualizing object]] the [[tensor unit]]. As such their [[internal logic]] is _[[linear logic]]_ and their [[type theory]] is _[[linear type theory]]_. In terms of this a Wirthmüller morphism, def. \ref{WirthmullerContext}, is the linear analog of a [[context extension]] morphism in a [[hyperdoctrine]]: it defines a _[[dependent linear type theory]]_. See there for more.
=--
## Properties
{#Properties}
### The comparison maps
{#TheComparisonMaps}
+-- {: .num_remark #QuasiMonoidalnessOfLeftAdjoint}
###### Remark
In a pre-Wirthmüller context, def. \ref{WirthmullerContext},
there is a canonical [[natural transformation]]
$$
f_!(A \otimes_X B)
\longrightarrow
(f_! A) \otimes_Y (f_! B)
\,,
$$
(not necessarily an [[equivalence]]) being the [[adjunct]] of the composite
$$
A \otimes_X B
\overset
{ \eta_A \otimes \eta_B }
{\longrightarrow}
(f^\ast f_! A) \otimes_X (f^\ast f_! B)
\overset
{\simeq}
{\longrightarrow}
f^\ast ( (f_! A) \otimes_Y (f_! B) )
\,,
$$
where the first morphism is the [[tensor product]] of two copies of the
[[adjunction unit]] and where the second
is the [[equivalence]] that exhibits $f^\ast$ as a [[strong monoidal functor]].
=--
+-- {: .num_defn #ComparisonMaps}
###### Definition
In a pre-Wirthmüller context, def. \ref{WirthmullerContext},
write $\overline {\pi}$ for the [[natural transformation]]
$$
\overline{\pi}
\;\colon\;
f_! ((f^\ast B) \otimes A)
\longrightarrow
B \otimes f_! A
$$
given as the composite
$$
\overline{\pi}
\;\colon\;
f_!
\big(
(f^\ast B) \otimes A
\big)
\overset
{ \widetilde{ \eta_{f^\ast B} \otimes \eta_A } }
{\longrightarrow}
(f_! f^\ast B) \otimes_Y (f_! A)
\overset
{ \epsilon_B \otimes_Y id }
{\longrightarrow}
B \otimes f_! A
\,,
$$
where
* the first morphism is that of remark \ref{QuasiMonoidalnessOfLeftAdjoint}
* the second is the $(f_! \dashv f^\ast)$ [[counit of an adjunction|counit]] (tensored with an [[identity morphism]]).
Also write
$$
\overline{\gamma}
\;\colon\;
[f_! A ,\, B]
\longrightarrow
f_\ast [A ,\, f^\ast B]
$$
for the $(f^\ast \dashv f_\ast)$ [[adjunct]] of the [[natural transformation]] given as the composite
\[
\label{MayProp211AdjunctMorphism}
f^\ast [f_! A ,\, B]
\stackrel{\simeq}{\longrightarrow}
[f^\ast f_! A ,\, f^\ast B]
\longrightarrow
[A ,\, f^\ast B]
\,,
\]
where the first map exhibits $f^\ast$ as a [[closed monoidal functor]] and where the second is the $(f_! \dashv f^\ast)$-[[unit of an adjunction|unit]] (under the [[internal hom]]).
=--
see ([May 05, prop. 2.11](#May05))
+-- {: .num_prop #ComparisonIsEquivalenceOnDualizables}
###### Proposition
In a pre-Wirthmüller context, def. \ref{WirthmullerContext}, the comparison maps of def. \ref{ComparisonMaps} are [[equivalences]]
when the argument $B \in (\mathcal{Y}, \otimes_Y, 1_Y)$ is a [[dualizable object]].
If either of the two happens to be a [[natural equivalence]] (hence an equivalence for all arguments), then so is the other.
=--
([May 05, prop. 2.8 and prop. 2.11](#May05))
+-- {: .num_prop #InWirthmuellerContextProjectionIsEquivalence}
###### Proposition
Precisely when the pre-Wirthmüller context is a Wirthmüller context, def. \ref{WirthmullerContext}, both comparison maps of def. \ref{ComparisonMaps} are natural equivalences.
=--
+-- {: .proof}
###### Proof
For all $A \in \mathcal{X}$ and $B,C \in \mathcal{Y}$ we have
by the $(f_! \dashv f^\ast)$-[[adjunction]] and the
tensor$\dashv$hom-adjunction a [[commuting diagram]] of the form
$$
\array{
\mathcal{Y}(B \otimes f_! A, \, C )
&
\stackrel{
\mathcal{Y}(\overline{\pi}(A,B), C)
}{
\longrightarrow
}
&
\mathcal{Y}(f_! ((f^\ast B) \otimes A),\, C)
\\
\downarrow^{\mathrlap{\simeq}}
&&
\downarrow^{\mathrlap{\simeq}}
\\
\mathcal{X}(A, f^\ast [B,C])
&\stackrel{}{\longrightarrow}&
\mathcal{X}(A, [(f^\ast B), (f^\ast C)])
}
\,.
$$
By naturality in $A$ and by the [[Yoneda lemma]] this shows
that $\overline{\pi}$ is an equivalence precisey if
$f^\ast$ is strong closed.
For $\overline{\gamma}$ the same statement follows from this with
prop. \ref{ComparisonIsEquivalenceOnDualizables}.
=--
+-- {: .num_remark }
###### Remark
The first [[natural equivalence]] of prop. \ref{InWirthmuellerContextProjectionIsEquivalence}
$$
f_!(f^\ast A \otimes B) \simeq A \otimes f_!(B)
$$
is often called the *[[projection formula]]*. In [[representation theory]] this is also sometimes called *[[Frobenius reciprocity]]*, though mostly that term is used for (just) the existence of the $(f_! \dashv f^\ast)$-[[adjunction]], where in representation theory the [[left adjoint]] $f_!$ forms [[induced representations]].
=--
### Consequences
{#Consequences}
Assuming a Wirthmüller context, the [[projection formula]] has the following implications.
\begin{proposition}\label{ProjectionComposedWithCounitInRemainingVariable}
In the special case that both of its variables are in the image of $f^\ast$, then
composing the [[projection formula]]-map from Def. \ref{ComparisonMaps} with the [[adjunction counit]] in the first variable equals the adjunction unit of the full term, in that the following [[commuting diagram|diagram commutes]] (where we are leaving the structure isomorphism of the [[strong monoidal functor]] $f^\ast$ notationally implicit):
$$
\array{
f_!
\big(
(f^\ast A) \otimes (f^\ast B)
\big)
&\overset{ \overline{\pi} }{\longrightarrow}&
(f_! f^\ast A) \otimes B
\\
\mathllap{{}^{\epsilon_{A \otimes B}}}\Big\downarrow
&&
\Big\downarrow
\mathrlap{ {}^{ \epsilon \otimes id } }
\\
A \otimes B
&=&
A \otimes B
}
$$
\end{proposition}
\begin{proof}
The [[adjunct]] of the composite of the top and the right morphism is the composite of
* the [[adjunct]] of the top morphism, which by Def. \ref{ComparisonMaps}, is
$f^\ast A \otimes f^\ast B \overset{\eta_{f^\ast A} \otimes \eta_{f^\ast B}}{\longrightarrow} f^\ast f_! f^\ast A \otimes f^\ast f_! f^\ast B \overset{id \otimes f^\ast \epsilon_B}{\longrightarrow} f^\ast f_! f^\ast A \otimes f^\ast B$
* the image under $f^\ast$ of the right morphism, which is $f^\ast f_! f^ \ast A \otimes f^\ast B \overset{f^\ast \epsilon_A \otimes id}{\longrightarrow} f^\ast A \otimes f^\ast B$.
That composite is the identity by the [[triangle identity]] in each of the two variables.
In conclusion, the left vertical morphism is the adjunct of the identity, and hence the counit as claimed.
\end{proof}
\begin{proposition}
\label{AdjunctionCounitByTensoringWithCounitOnTensorUnit}
The $(f_! \dashv f^\ast)$-[[adjunction counit]] on any object is, up to [[isomorphism]] given by the [[tensor product]] with the adjunction counit on the [[tensor unit]]:
$$
\epsilon_{\mathscr{V}}
\,\colon\,
f_! f^\ast \mathscr{V}
\,\simeq\,
\mathscr{V} \otimes f_! f^\ast(\mathscr{V})
\overset{
id_{\mathscr{V}}
\,\otimes\,
\epsilon_{\mathbb{1}}
}{\longrightarrow}
\mathscr{V} \,\otimes\, \mathbb{1}
\,\simeq\,
\mathscr{V}
$$
\end{proposition}
\begin{proof}
The identification is given by the following [[commuting diagram]]:
\begin{tikzcd}
f_! f^\ast \mathcal{V}
\ar[
rr,
"{ \epsilon_{\mathcal{V}} }"
]
\ar[
d,
"{ f_! f^\ast (r_{\mathcal{V}}) }",
"{\sim}"{sloped, swap}
]
&&
\mathcal{V}
\ar[
d,
"{ r_{\mathcal{V}} }",
"{ \sim }"{sloped, swap}
]
\\
f_! f^\ast
\big(
\mathcal{V}_{\mathcal{Y}}
\otimes
1
\big)
\ar[
rr,
"{ \epsilon_{\mathcal{V} \otimes 1} }"
]
\ar[
d,
"{ \sim }"{sloped, swap}
]
&&
\mathcal{V} \otimes 1
\ar[d, equals]
\\
f_!
\big(
f^\ast \mathcal{V}
\otimes
f^\ast 1
\big)
\ar[
d,
"{ \overline{\pi} }",
"{ \sim }"{sloped}
]
\ar[rr]
&&
\mathcal{V} \otimes 1
\ar[d, equals]
\\
\mathcal{V} \otimes f_! f^\ast 1
\ar[
rr,
"{
\mathcal{V}
\,\otimes\,
\epsilon_{1}
}"
]
&&
\mathcal{V} \otimes 1
\end{tikzcd}
Here
1. the top square is the [[naturality square]] of the [[adjunction counit]] on the right [[unitor]],
1. the bottom square is that from Prop. \ref{ProjectionComposedWithCounitInRemainingVariable}.
\end{proof}
### Comparison of push-forwards and Wirthmüller isomorphism
{#ComparisonOfPushForwardsAndWirthmuelleriso}
+-- {: .num_cor #PushforwardsIntertwinedByDuality}
###### Corollary
In a pre-Wirthmüller context, def. \ref{WirthmullerContext}, the functors $f_!$ and $f_\ast$ are intertwined by dualization, in that there is a [[natural equivalence]]
$$
\mathbb{D}(f_! A) \simeq f_\ast(\mathbb{D} A)
\,.
$$
=--
+-- {: .proof}
###### Proof
This is the special case of the property of $\overline{\gamma}$ in prop. \ref{ComparisonIsEquivalenceOnDualizables} for $B = 1_Y$:
$$
\mathbb{D}(f_! A)
=
[f_! A, 1_Y]
\underoverset{\simeq}{\overline{\gamma}}{\longrightarrow}
f_\ast [A, f^\ast 1_Y]
\simeq
f_\ast [A, 1_Y]
=
f_\ast \mathbb{D} A
\,.
$$
=--
+-- {: .num_remark #LinearDeMorganDualityForPushForwards}
###### Remark
With a Wirthmüller context regarded as [[categorical semantics]] for [[dependent linear type theory]] (see there), then the statement of cor. \ref{PushforwardsIntertwinedByDuality} is an instance of [[de Morgan duality]] where linear [[dual object|dualization]] intertwines linear [[dependent sum]] $\sum$ and [[dependent product]] $\prod$
$$
\prod_f \mathbb{D} \simeq \mathbb{D} \sum_f
\,.
$$
For more on this see also ([[schreiber:Quantization via Linear homotopy types|Schreiber 14, section 3.3]]).
=--
+-- {: .num_example}
###### Example
In a pre-Wirthmüller context
$$
f_\ast 1_X \simeq \mathbb{D}(f_! 1_X)
\,.
$$
=--
+-- {: .proof}
###### Proof
By cor. \ref{PushforwardsIntertwinedByDuality},
since $\mathbb{D} 1_X \simeq 1_X$.
=--
+-- {: .num_prop #AbstractWirthmuellerIso}
###### Proposition
**(Wirthmüller isomorphism)**
In a Wirthmüller context, def. \ref{WirthmullerContext} if $f_! 1_X$ is a [[dualizable object]] with dual $f_! C$, then there is a [[natural equivalence]]
$$
\omega
\;\colon\;
f_\ast f^\ast A \stackrel{\simeq}{\longrightarrow} f_!((f^\ast A) \otimes C)
\,.
$$
=--
([May 05, prop. 4.13](#May05)).
+-- {: .num_remark #MonadCommutesWithDualityUpTotwist}
###### Remark
In particular if there is $D \in \mathcal{Y}$ with
$$
\mathbb{D}(f_! f^\ast 1_Y) \simeq f_! f^\ast D
$$
(hence if $C \simeq f^\ast D$ in the notation of prop. \ref{AbstractWirthmuellerIso}) and using that by cor. \ref{PushforwardsIntertwinedByDuality},
$
f_\ast f^\ast \mathbb{D}B
\simeq
f_\ast \mathbb{D} f^\ast B
\simeq
\mathbb{D}(f_! f^\ast B)$
then prop. \ref{AbstractWirthmuellerIso} gives a natural equivalence of the form
$$
\mathbb{D}(f_! f^\ast B)
\stackrel{\simeq}{\longrightarrow}
f_! f^\ast ((\mathbb{D}B) \otimes D)
\,,
$$
saying that the [[comonad]] $f_! f^\ast$ commutes with dualization up to a "twist" given by tensoring with $D$.
=--
## Examples
### Cartesian Wirthmüller contexts in toposes
{#CartesianWirthmuellerContexts}
For $\mathbf{H}$ a [[topos]] and $f \colon X \longrightarrow Y$
any [[morphism]], then in the induced [[base change]] [[etale geometric morphism]]
$$
(\sum_f \dashv f^\ast \dashv \prod_f)
\;\colon\;
\mathbf{H}_{/X}
\longrightarrow
\mathbf{H}_{/Y}
$$
the [[inverse image]]/[[context extension]] is a [[cartesian closed functor]] (see there for the proof). Therefore any base change of [[toposes]] constitutes a cartesian Wirthmüller context.
Conversely, this means that systems of Wirthmüller contexts are generalizations of [[categorical logic]] ([[hyperdoctrines]]) to non-cartesian contexts (see at _[[dependent linear type theory]]_).
Notice that in a cartesian Wirthmüller context duality is trivial, in that $\mathbb{D}X \simeq 1$ for all objects $X$. Therefore to the extent that the [[six operations]] yoga involves duality, it is interesting only the more non-cartesian (non-classical) the ambient Wirthmüller context is.
For instance the [[projection formula]] $\overline{\gamma}$ in def. \ref{ComparisonMaps} for base change along a pointed connected type $\mathbf{B}G \to \ast$ equivalently says that genuine ([[Bredon cohomology|Bredon]]) [[equivariant cohomology]] reduces to [[Borel equivariant cohomology]] when the action on the coefficients is trivial. See at _[[equivariant cohomology]]_ for more on this.
### Pointed objects with smash product
{#PointedObjects}
A first step away from the Cartesian example [above](#CartesianWirthmuellerContexts) is the following.
Let $\mathbf{H}$ be a [[topos]]. For $X \in \mathbf{H}$ any [[object]], write
$$
\mathcal{C}_X \coloneqq \mathbf{H}_{/X}^{X/}
$$
for the [[category of pointed objects]] in the [[slice topos]] $\mathbf{H}_{/X}$. Equipped with the [[smash product]] $\wedge_X$ this is a [[closed monoidal category|closed]] [[symmetric monoidal category]] $(\mathcal{C}_X, \wedge_X, X \coprod X)$.
+-- {: .num_prop}
###### Proposition
For $f \colon X \longrightarrow Y$ any [[morphism]] in $\mathbf{H}$, the [[base change]] [[inverse image]] $f^\ast$ restricts to a functor $f^\ast \colon \mathcal{C}_Y \longrightarrow \mathcal{C}_X$ which is a Wirthmüller context.
=--
This appears as ([Shulman 07, examples 12.13 and 13.7](#Shulman07)) and ([Shulman 12, example 2.33](#Shulman12)).
+-- {: .proof}
###### Proof
For $f \colon X \longrightarrow Y$ any [[morphism]] in $\mathbf{H}$ then the [[base change]] [[inverse image]] $f^\ast \colon \mathbf{H}_{/Y} \longrightarrow \mathbf{H}_{/X}$ preserves pointedness, and the [[pushout]] functor $f_! \colon \mathbf{H}^{X/} \longrightarrow \mathbf{H}^{/Y}$ preserves co-pointedness. These two functors hence form an [[adjoint pair]]
$(f_1 \dashv f^\ast) \colon \mathcal{C}_X \longrightarrow \mathcal{C}_Y$.
Moreover, since [[colimits]] in the under-over category $\mathbf{H}_{/X}^{X/}$ are computed as colimits in $\mathbf{H}$ of [[diagrams]] with an [[initial object]] adjoined, and since by the [[Giraud axioms]] in the [[topos]] $\mathbf{H}$ [[pullback]] preserves these colimits, it follows that $f^\ast \colon \mathcal{C}_Y \to \mathcal{C}_X$ preserves colimits.
Finally by the discussion at _[[category of pointed objects]]_ we have that $\mathcal{C}_X$ and $\mathcal{C}_Y$ are [[locally presentable categories]], so that by the [[adjoint functor theorem]] it follows that $f^\ast$ has also a [[right adjoint]] $f_\ast \colon \mathcal{C}_X \to \mathcal{C}_Y$.
To see that $f^\ast$ is a [[strong monoidal functor]] observe that the [[smash product]] is, by the discussion there, given by a [[pushout]] over [[coproducts]] and [[products]] in the [[slice topos]]. As above these are all preserved by [[pullback]]. Finally to see that $f^\ast$ is also a [[strong closed functor]] observe that the [[internal hom]] on [[pointed objects]] is, by the discussion there, a [[fiber product]] of cartesian internal homs. These are preserved by [the above case](#CartesianWirthmuellerContexts), and the fiber product is preserved since $f^\ast$ preserves all limits. Hence $f^\ast$ preserves also the internal homs of pointed objects.
=--
### Bundles of modules
For $R$ a [[ring]], $R Mod$ its [[category of modules]], there is a [[functor]]
$$
[-, R Mod] \;\colon\; Set^{op} \longrightarrow ClMonCat
$$
which sends a [[set]] $X$ to the [[closed monoidal category]] of $R$-[[modules]] parameterized over $X$.
This takes values in Wirthmüller morphisms. ([Shulman 12, example 2.2, 2.17](#Shulman12)).
### Bundles of $\infty$-modules
for [[(infinity,1)-module bundles]]: ([[schreiber:master thesis Nuiten|Nuiten 13]], [Hopkins-Lurie](#HopkinsLurie), [[schreiber:Homotopy-type semantics for quantization|Schreiber 14]])
### Becker-Gottlieb transfer
A _[[transfer context]]_ is a Wirthmüller context in which also $f_\ast$ satisfies its [[projection formula]]. In this context there is an abstract concept of _[[Becker-Gottlieb transfer]]_. See there for more.
### In equivariant stable homotopy theory
In [[equivariant stable homotopy theory]], see ([May 05b](#May05b)).
### For quasicoherent sheaves (in $E_\infty$-geometry)
{#ForQuasicoherentSheaves}
Pull-push of [[quasicoherent sheaves]] is usually discussed as a [[Grothendieck context]] of [[six operations]], but under some conditions it also becomes a Wirthmüller context.
Using results of Lurie this follows in the full generality of [[E-∞ geometry]] ([[spectral geometry]]).
Consider quasi-compact and quasi-separated [[E-∞ algebraic spaces]] ([[spectral algebraic spaces]]). (This includes precisely those [[spectral Deligne-Mumford stacks]] which have a [[scallop decomposition]], see [here](derived+Deligne-Mumford+stack#RelationToDerivedAlgebraicSpaces).)
If $f \;\colon\; X \longrightarrow Y$ is a map between these which is
1. locally almost of finite presentation;
1. strongly proper;
1. has finite [[Tor-amplitude]]
then the left adjoint to pullback of [[quasicoherent sheaves]] exists
$$
(f_! \dashv f^\ast)
\;\colon\;
QCoh(X)
\stackrel{\overset{f_!}{\longrightarrow}}{\underset{f^\ast}{\longleftarrow}}
QCoh(Y)
\,.
$$
([[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, proposition 3.3.23]])
If $f$ is
* quasi-affine
then the right adjoint exists
$$
(f^\ast \dashv f_\ast)
\;\colon\;
QCoh(X)
\stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}}
QCoh(Y)
\,.
$$
([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|LurieQC, prop. 2.5.12]], [[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, proposition 2.5.12]])
The [[projection formula]] in the dual form
$$
f_\ast A \otimes B \longrightarrow f_\ast (A\otimes f^\ast B)
$$
for $f$ quasi-compact and quasi-separated appears as ([[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, remark 1.3.14]]).
Now if all the conditions on $f$ hold, so that $(f_! \dashv f^\ast \dashv f_\ast) \;\colon\; QCoh(X) \longrightarrow QCoh(Y)$, then passing to opposite categories $QCoh(X)^{op} \longrightarrow QCoh(Y)^{op}$ exchanges the roles of $f_!$ and $f_\ast$, makes the projection formula be as in the above discussion and hence yields a Wirthmüller context.
The existence of [[dualizing modules]] $K$
$$
D X = [X,K]
$$
is discussed in ([[Representability theorems|Lurie, Representability theorems, section 4.2]].)
## Related concepts
* [[Verdier-Grothendieck context]]
* [[Grothendieck context]]
* [[Beck-Chevalley condition]]
* [[ambidextrous adjunction]]
## References
The Wirthmüller isomorphism in [[equivariant stable homotopy theory]] goes back to and is named after
* {#Wirthmueller74} [[Klaus Wirthmüller]], _Equivariant homology and duality_. Manuscripta Math. 11(1974), 373-390
see Theorem II.6.2 in
* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], and Mark Steinberger (with contributions by J.E. McClure), _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics Vol.1213. 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf))
see also
* {#Nardin12} [[Denis Nardin]], section 2.6 and A.3 of _Stability and distributivity over orbital ∞-categories_, 2012 ([hdl.handle.net/1721.1/112895]( http://hdl.handle.net/1721.1/112895), [pdf](https://www.math.univ-paris13.fr/~nardin/thesis.pdf))
A clear discussion of axioms of [[six operations]] and their consequences, with emphasis on the Wirthmüller isomorphisms, 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)]
* [[Paul Balmer]], [[Ivo Dell'Ambrogio]], [[Beren Sanders]], _Grothendieck-Neeman duality and the Wirthmüller isomorphism_, Compositio Mathematica 152.8 (2016): 1740-1776 ([arXiv:1501.01999](http://arxiv.org/abs/1501.01999))
Discussion of the original Wirthmüller isomorphism in [[equivariant stable homotopy theory]], based on this, is in
* {#May05b} [[Peter May]], _The Wirthmüller isomorphism revisited_, Theory and Applications of Categories 11.5 (2003): 132-142 ([TAC:11/5/11-05](http://www.tac.mta.ca/tac/volumes/11/5/11-05abs.html), [K-theory:0574](http://www.math.uiuc.edu/K-theory/0574/WirthRev.pdf))
More elaboration of the Wirthmüller context is in
* Baptiste Calmès, Jens Hornbostel, section 4 of _Tensor-triangulated categories and dualities_, Theory and Applications of Categories, Vol. 22, 2009, No. 6, pp 136-198 ([TAC](http://www.tac.mta.ca/tac/volumes/22/6/22-06abs.html), [arXiv:0806.0569](http://arxiv.org/abs/0806.0569))
Discussion in the context of [[pure motives]] includes
* {#Deglise} [[Frédéric Déglise]], around prop. 1.34 of _Finite correspondences and transfers over a regular base_, [pdf](http://www.math.uiuc.edu/K-theory/0765/regular_base.pdf).
Wirthmüller morphisms between pointed objects and between bundles of modules are discussed in
* {#Shulman07} [[Mike Shulman]], _Framed bicategories and monoidal fibrations_, in Theory and Applications of Categories, Vol. 20, 2008, No. 18, pp 650-738. ([arXiv:0706.1286](http://arxiv.org/abs/0706.1286), [TAC](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html))
* {#Shulman12} [[Mike Shulman]], _Enriched indexed categories_ ([arXiv:1212.3914](http://arxiv.org/abs/1212.3914))
A twisted version of the Wirthmüller isomorphism is discussed in
* Bastiaan Cnossen, _Twisted ambidexterity in equivariant homotopy theory_ ([arXiv:2303.00736](https://arxiv.org/abs/2303.00736))
Discussion in [[E-∞ geometry]] is in
* {#Lurie} [[Jacob Lurie]], section 3.3. of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_
* [[Jacob Lurie]], section 4.2 of _[[Representability theorems]]_
* {#HopkinsLurie} [[Michael Hopkins]], [[Jacob Lurie]], _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_
[[!redirects Wirthmüller contexts]]
[[!redirects Wirthmueller context]]
[[!redirects Wirthmueller contexts]]
[[!redirects Wirthmuller context]]
[[!redirects Wirthmuller contexts]]
[[!redirects Wirthmüller isomorphism]]
[[!redirects Wirthmüller isomorphisms]]
[[!redirects Wirthmueller isomorphism]]
[[!redirects Wirthmueller isomorphisms]]
[[!redirects Wirthmuller isomorphism]]
[[!redirects Wirthmuller isomorphisms]] |
WISC | https://ncatlab.org/nlab/source/WISC |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
The assumption that every [[set]] has a **Weakly Initial Set of Covers**, or $WISC$, is a weak form of the [[axiom of choice]]. Like the [[axiom of multiple choice]] and the axiom of [[small violations of choice]] (which both imply it), it says intuitively that "$AC$ fails to hold only in a small way" (i.e. not in a [[proper class|proper-class]] way).
## Statement (for Sets)
Precisely, $WISC$ is the statement that for any [[set]] $X$, the [[full subcategory]] $(Set/X)_{surj}$ of the [[slice category]] $Set/X$ consisting of the [[surjections]] has a [[weakly initial set]]. In other words, there is a family of surjections $\{f_i\colon P_i \twoheadrightarrow X\}_{i\in I}$ such that for any surjection $Q\twoheadrightarrow X$, there exists some $f_i$ which factors through $Q$.
## Relationships to other axioms
* WISC is implied by [[COSHEP]], since any surjection $P\twoheadrightarrow X$ such that $P$ is projective is necessarily a weakly initial (singleton) set in $(Set/X)_{surj}$.
* WISC is also implied by the [[axiom of multiple choice]] (which is in turn implied by COSHEP). For if $X$ is in some collection family $\{D_c\}_{c\in C}$, then the family of all surjections of the form $D_c \twoheadrightarrow X$ is weakly initial in $(Set/X)_{surj}$.
* A [[ΠW-pretopos]] satisfying WISC is a _[[predicative topos]]_.
* Since [[Michael Rathjen]] proves that [[SVC]] implies [[AMC]] (at least in [[ZF]]), SVC therefore also implies WISC.
* WISC also follows from the assertion that the [[free exact completion]] of $Set$ is [[well-powered category|well-powered]], which in turn follows from assertion that $Set$ has a [[generic proof]] (so that $Set_{ex/lex}$ is a [[topos]]). Both of these can also be regarded as saying that choice is only violated "in a small way."
* WISC implies that the category of [[anafunctors]] between any two [[small categories]] is [[essentially small category|essentially small]]; see [here](/nlab/show/anafunctor#SizeQuestions), or below.
* WISC implies (in [[ZF]]) that there exist arbitrarily large [[regular cardinals]]. Therefore, WISC is not provable in ZF, as Moti Gitik constructed a model of ZF with only one regular cardinal, using large cardinal assumptions. A proof without large cardinals was given in ([Karagila](#Karagila14)).
### Applications
#### Local smallness of anafunctor categories
+-- {: .num_prop}
######Proposition
WISC implies the local essential smallness of $Cat_ana$, the bicategory of categories and [[anafunctors]].
=--
+-- {: .proof}
######Proof
Let $X,Y$ be small categories and consider the category $Cat_{ana}(X,Y)$, with objects which are spans
$$
(j,f) : X \stackrel{j}{\leftarrow} X[U] \stackrel{f}{\to} Y
$$
where $X[U] \to X$ is a surjective-on-objects, [[fully faithful]] functor. The underlying map on object sets is $U \to X_0$. By WISC there is a surjection $V \to X_0$ and a map $V\to U$ over $X_0$. We can thus define a commuting triangle of functors
$$
\array{
X[V] & \to & X[U] \\
& k \searrow & \downarrow j\\
&& X
}
$$
where $X[V] \to X$ is the canonical fully faithful functor arising from $V\to X_0$ (the arrows of $X[V]$ are given by $V^2 \times_{X_0^2} X_1$). This gives rise to a transformation from $(j,f)$ to a span with left leg $k$. Thus $Cat_{ana}(X,Y)$ is equivalent to the [[full subcategory]] of anafunctors where the left leg has as object component an element of the weakly initial set of surjections. Since there is only a set of functors $X[V] \to Y$ for each $V\to X_0$, this subcategory is small.
=--
#### Existence of higher inductive types
[Swan](#Swan18) showed that WISC implies the existence of [[W-types with reductions]], a kind of simple [[higher inductive type]].
## In other sites - external version
Let $(C,J)$ be a [[site]] with a singleton [[Grothendieck pretopology]] $J$. It makes sense to consider a version of WISC for $(C,J)$, along the lines of the following: Let $(C/a)_{cov}$ be the full subcategory of the slice category $C/a$ consisting of the covers. WISC then states that
* For all objects $a$ of $C$, $(C/a)_{cov}$ has a weakly initial set.
This definition is called _external_ because it refers to an external category of sets. This is to be contrasted with the _internal_ version of WISC, discussed below.
+-- {: .num_example}
###### Example
Assuming AC for $Set$, the category $Top$ with any of its usual pretopologies satisfies \'internal WISC\'. Consider, for instance, the pretopology in which the covers are the maps admitting local sections, i.e. those $p\colon Y\to X$ such that for any $x\in X$ there exist an open set $U\ni x$ such that $p^{-1}(U)\to U$ is split epic. If $Set$ satisfies AC, then a weakly initial set in $Top/_{cov}X$ is given by the set of all maps $\coprod_{U\in \mathcal{U}} U \to X$ where $\mathcal{U}\subset \mathcal{P}(X)$ is an open cover of $X$. For if $p\colon Y\to X$ admits local sections, then for each $x\in X$ we can choose an $U_x \ni x$ over which $p$ has a section, resulting in an open cover $\mathcal{U} = \{U_x \mid x\in X\}$ of $X$ for which $\coprod_{U\in \mathcal{U}} U \to X$ factors through $p$. (If $Set$ merely satisfies WISC itself, then a more involved argument is required.)
=--
And now a non-example
+-- {: .num_example}
###### Example
The category of [[affine schemes]] can be equipped with the [[fpqc topology]] (so this is the [[fpqc site]] over $Spec(\mathbb{Z})$). This does not satisfy WISC. Namely, given any set of fpqc covers of $Spec(R)$, there is a surjective fpqc map which is refined by none of the given covering families ([Stacks Project Tag 0BBK](#Stacks0BBK)).
=--
More generally, for a non-singleton pretopology on $C$, we can reformulate WISC along the lines of \'there is a set of covering families weakly initial in the category of all covering families of any object\'.
Given a site $(C,J)$ with $J$ subcanonical, and $C$ finitely complete, we can define a (weak) 2-category $Ana(C,J)$ of internal categories, anafunctors and transformations. If WISC holds for $(C,J)$, then $Ana(C,J)$ is locally essentially small.
## In other categories - internal version
To consider an _internal_ version of WISC, which doesn't refer to an external notion of set, one needs to assume that the ambient category $C$ has a strong enough [[internal logic]], such as a pretopos (this is the context in which van den Berg and Moerdijk work). Then the ordinary statement of WISC in set can be written in the internal logic, using the [[stack semantics]], as a statement about the objects and arrows of $C$. It is in this form that WISC is useful as a replacement choice principle in intuitionistic, constructive or predicative set theory, as these are modelled on various topos-like categories (or in the case of [van den Berg and Moerdijk](#vdBM12), a [[algebraic set theory|category of classes]], although this is not necessary for the approach).
Importantly, the internal form of WISC is stable under many more categorical constructions than other forms of choice. For instance, any [[Grothendieck topos]] or [[realizability topos]] inherits WISC from its base topos of sets (even if the latter is constructive or even predicative); see [van den Berg](#vdBerg12). This is in contrast to nearly all other choice principles, including weaker forms such as [[COSHEP]] and [[SVC]], which fail in at least some Grothendieck toposes even when the base is a model of ZFC.
## References
{#References}
* {#vdBM12} [[Benno van den Berg]], [[Ieke Moerdijk]], *The Axiom of Multiple Choice and Models for Constructive Set Theory*, [arXiv](http://arxiv.org/abs/1204.4045). In this paper WISC is called the "axiom of multiple choice".
* [[Thomas Streicher]], *Realizability Models for $CZF + \neg Pow$*, [unpublished note](http://www.mathematik.tu-darmstadt.de/~streicher/CIZF/rmczfnp.pdf). In this note WISC is called $TTCA_f$ ($TTCA$ stands for "type-theoretic collection axiom).
* [[Benno van den Berg]], *WISC is independent from ZF*, [PDF](http://www.staff.science.uu.nl/~berg0002/papers/WISC.pdf)
* {#vdBerg12} [[Benno van den Berg]], _Predicative toposes_ ([arXiv:1207.0959](http://arxiv.org/abs/1207.0959)). In this paper WISC is called the "axiom of multiple choice".
* {#Swan18} [[Andrew Swan]], *W-Types with Reductions and the Small Object Argument*, 2018, [arxiv:1802.07588](https://arxiv.org/abs/1802.07588)
The following two papers give models of set theory (without large cardinals) in which WISC fails.
* {#Karagila14} Asaf Karagila, *Embedding Orders Into Cardinals With $DC_\kappa$*, Fund. Math. 226 (2014), 143-156, doi:[10.4064/fm226-2-4](http://dx.doi.org/10.4064/fm226-2-4), [arXiv:1212.4396](http://arxiv.org/abs/1212.4396).
{#Karagila}
* {#Roberts15} [[David Roberts]], _The weak choice principle WISC may fail in the category of sets_, [Studia Logica](http://link.springer.com/journal/11225) Volume 103 (2015) Issue 5, pp 1005-1017, doi:[10.1007/s11225-015-9603-6](http://dx.doi.org/10.1007/s11225-015-9603-6) [arXiv:1311.3074](http://arxiv.org/abs/1311.3074)
The Stacks Project shows how to construct a counterexample to WISC from any set of fpqc covers of an affine scheme.
* {#Stacks0BBK} [[The Stacks Project]], [Tag 0BBK](http://stacks.math.columbia.edu/tag/0BBK)
category: foundational axiom
[[!redirects WISC]]
[[!redirects WISCs]]
[[!redirects weakly initial set of covers]]
[[!redirects weakly initial sets of covers]]
[[!redirects axiom of weakly initial sets of covers]]
[[!redirects SOSHWIS]] |
Witold Hurewicz | https://ncatlab.org/nlab/source/Witold+Hurewicz |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Witold_Hurewicz)
## Selected writings
Introducing the [[Hurewicz theorem]]:
* {#Hurewicz35} [[Witold Hurewicz]], *Beiträge zur Topologie der Deformationen (I. Höherdimensionale Homotopiegruppen)*, [[Koninklijke Akademie van Wetenschappen]] **38** (1935) 112–119. $[$[pdf](https://dwc.knaw.nl/DL/publications/PU00016667.pdf)$]$
Introducing [[compactly generated topological spaces]]:
* unrecorded lecture introducing [[compactly generated topological spaces]], Princeton 1948-49 ([communicated](compactly+generated+topological+space#fnref:2) by [[David Gale]])
Origin of the notion of [[Hurewicz fibrations]]:
* [[Witold Hurewicz]], _On the concept of fiber space_, Proc. Nat. Acad. Sci. USA __41__ (1955) 956--961; ([doi:10.1073%2Fpnas.41.11.956](https://dx.doi.org/10.1073%2Fpnas.41.11.956), [jstor:89187](https://www.jstor.org/stable/89187), [pdf](http://www.pnas.org/content/41/11/956.full.pdf). MR0073987)
## Related $n$Lab entries
* [[Hurewicz theorem]]
* [[Hurewicz cofibration]]
* higher [[homotopy groups]]
* [[long exact sequence of homotopy groups]]
category: people
[[!redirects Hurewicz]] |
Witt cohomology | https://ncatlab.org/nlab/source/Witt+cohomology |
# Contents
* table of contents
{: toc}
## Idea
Serre decided to try taking coefficients in the Witt vectors as an early attempt at a [[Weil cohomology theory]]. Ultimately, it wasn't successful for this purpose, but has been generalized in several ways for other purposes with great success.
## Sheaf of Witt vectors
Let $X$ be a [[scheme]] over a [[perfect field]] $k$ of positive [[characteristic]] $p$. Let $W$ and $W_n$ be the functors of [[Witt vectors]] and truncated Witt vectors respectively. The functorial nature allows us to define a [[sheaf]] of Witt vectors $\mathcal{W}$ and $\mathcal{W}_n$ just by taking Witt vectors of the rings of sections of $\mathcal{O}_X$.
Note that as a sheaf of sets $\mathcal{W}_n$ is just $\mathcal{O}_X^n$. The ring structure is just the addition and multiplication of the Witt vectors. The operations on the Witt vectors sheafify as well. When $n\geq m$ we have the exact sequence $0\to \mathcal{W}_m\stackrel{V}{\to} \mathcal{W}_n\stackrel{R}{\to}\mathcal{W}_{n-m}\to 0$. If we take $m=1$, then we get the sequence $0\to \mathcal{O}_X\to \mathcal{W}_n\to \mathcal{W}_{n-1}\to 0$
## Definition
The sheaf of Witt vectors is an abelian sheaf, so we just define cohomology $H^q(X, \mathcal{W}_n)$ as the standard sheaf cohomology (on the [[Zariski site]] of $X$). Let $\Lambda=W(k)$, then since $\mathcal{W}_n$ are $\Lambda$-modules annihilated by $p^n\Lambda$, we get that $H^q(X, \mathcal{W}_n)$ are also $\Lambda$-modules annihilated by $p^n\Lambda$.
In fact, all of our old operators $F$, $V$, and $R$ still act on $H^q(X, \mathcal{W}_n)$. They are easily seen to satisfy the formulas $F(\lambda w)=F(\lambda)F(w)$, $V(\lambda w)=F^{-1}(\lambda)V(w)$, and $R(\lambda w)=\lambda R(w)$ for $\lambda\in \Lambda$. If $X$ is projective then $H^q(X, \mathcal{W}_n)$ is a finite $\Lambda$-module.
What is usually referred to with Witt cohomology is $H^q(X, \mathcal{W})$ which is defined to be $lim H^q(X, \mathcal{W}_n)$. Note that even if $X$ is projective, this limit does not have to be a finite type $\Lambda$-module.
## References
* J.P. [[Serre]], Sur la topologie des variétés algébriques en caractéristique p, Symposium de Topologie Algébrique, Mexico (August, 1956)
[[!redirects Witt cohomology]]
[[!redirects Witt Cohomology]]
|
Witt ring | https://ncatlab.org/nlab/source/Witt+ring | There are two different notions called *Witt ring*:
* [[ring of Witt vectors]]
* [[Witt ring of equivalence classes of bilinear forms]] |
Witt ring of equivalence classes of bilinear forms | https://ncatlab.org/nlab/source/Witt+ring+of+equivalence+classes+of+bilinear+forms | ## References
* T.Y. Lam, Introduction To Quadratic Forms Over Fields, Graduate Studies in Mathematics, 2004
* M. Marshall, Abstract Witt Rings, Queens Papers in Pure and Applied Mathematics, 1980
* Baptiste Calmès, Jens Hornbostel, push-forwards for Witt groups of schemes, [
arXiv:0806.0571v2
] (http://arxiv.org/abs/0806.0571v2) |
Witt vectors > history | https://ncatlab.org/nlab/source/Witt+vectors+%3E+history | < [[Witt vectors]]
|
Witt-Burnside functor | https://ncatlab.org/nlab/source/Witt-Burnside+functor | [[!redirects Witt-Burnside functors]]
##Idea
Dress and Siebeneicher ([DS88](#DS88)) generalized the [[Witt vector]] construction to the
case of pro-$p$ groups for prime integer $p$. These functors, $W_G$, were originally defined for all profinite groups, $G$, and are now called **Witt-Burnside functors** due to the fact that they generalize both the $p$-typical (recovered when $G = \mathbb{Z}_p$ as an additive group) and 'big' Witt vector construction (recovered when $G = \hat{\mathbb{Z}}$), as well as Burnside functors (recovered as $W_G(\mathbb{Z})$).
Witt-Burnside functors have been used to study equivariant ring spectra as they arise as a left adjoint for [[Tambara functors]].
##References
* {#DS88} A. Dress, C. Siebeneicher, _The Burnside ring of profinite groups and
the Witt vector construction_, Adv. Math., 70, (1988), 87–132.
* Lance Edward Miller, _On the structure of Witt-Burnside rings attached to pro-p groups_, ([ arXiv:1103.4644](http://arxiv.org/abs/1103.4644))
[[!redirects Witt-Burnside functors]] |
Witten conjecture | https://ncatlab.org/nlab/source/Witten+conjecture |
#Contents#
* table of contents
{:toc}
## Idea
There are many [[conjectures]] due to [[Edward Witten]]; but by _the Witten conjecture_ one usually refers to the statement proven in ([Kontsevich 92](#Kontsevich92)) with new proofs due to [Mirzakhani 07a](#Mirzakhani07a), [Mirzakhani 07b](#Mirzakhani07b) and others.
This is about a [[generating function]] for [[intersection pairing]] numbers of Mumford-Morita-Miller stable classes on the compactification of a [[moduli space of punctured curves]]. The conjecture relates them to the [[Korteweg-de Vries integrable hierarchy]]. The conjecture was formulated by [[Edward Witten]] based on the conjectured equivalence of the [[partition function]] of two models of 2-dimensional [[quantum gravity]] (i.e. [[string]] [[worldsheet]] [[field theories]]), which are not manifestly equivalent.
## References
Reviews include
* Claude Itzykson in [MR93e:32027](http://www.ams.org/mathscinet-getitem?mr=93e:32027)
* Ali Ulas Ozgur Kisisel, _Integrable systems and Gromov-Witten theory_, p. 135--161 in: Topics in cohomological studies of algebraic varieties,
Impanga lecture notes. Edited by Piotr Pragacz. Trends Math., Birkhäuser, Basel, 2005, [MR2006d:14066](http://www.ams.org/mathscinet-getitem?mr=2006d:14066)
* {#DijkgraafWitten18} [[Robbert Dijkgraaf]], [[Edward Witten]], _Developments in Topological Gravity_ ([arXiv:1804.03275](https://arxiv.org/abs/1804.03275))
The original proof is due to
* {#Kontsevich92} [[Maxim Kontsevich]], _Intersection theory on the moduli space of curves and the matrix Airy function_, Comm. Math. Phys. __147__ (1992), no. 1, 1--23, [euclid](http://projecteuclid.org/euclid.cmp/1104250524), [author's pdf](http://193.51.104.7/~maxim/TEXTS/intersection_theory_6.pdf)
A celebrated new proof is due to
* {#Mirzakhani07a} [[Maryam Mirzakhani]], _Simple Geodesics and Weil-Petersson Volumes Of Moduli Spaces of Bordered Riemann Surfaces_, Invent. Math. 167 (2007) 179-222
* {#Mirzakhani07b} [[Maryam Mirzakhani]], _Weil-Petersson Volumes And Intersection Theory On The Moduli Space Of Curves_, Journal of the American Mathematical Society 20 (2007) 1-23
Some of the other newer approaches to the proof:
* M. E. Kazarian, S. K. Lando, _An algebro-geometric proof of Witten's conjecture_
and also versions for higher spin
* Tyler J.Jarvis, Takashi Kimura, Arkady Vaintrob, _Moduli spaces of higher spin curves and integrable hierarchies_, Compositio Math. __126__ (2001), no. 2, 157--212.
Gregory Naber's lectures on Witten conjecture (this is another Witten conjecture, about Seiberg-Witten invariants) can be found at his [homepage](http://www.pages.drexel.edu/~gln22).
See also
* Wikipedia, _[Witten conjecture](https://en.wikipedia.org/wiki/Witten_conjecture)_
[[!redirects Witten's conjecture]]
|
Witten genus | https://ncatlab.org/nlab/source/Witten+genus |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Witten genus_ is a [[genus]] with [[coefficients]] in [[power series]] in one variable, playing the role of a universal [[elliptic genus]].
This arises ([Witten 87](#Witten87)) as the [[large volume limit]] of the [[partition function]] of the [[superstring]] (hence in the string worldsheet [[perturbation theory]] about constant [[worldsheet]] configurations). Specifically, for the [[type II superstring]] this reproduces the universal [[elliptic genus]] as previously introduced by [[Serge Ochanine]], while for the [[heterotic string]] it yields what is now called the Witten genus proper.
Concretely, as Witten argued, this is a formal [[power series]] in string oscillation modes of the [[A-hat genus]] of the symmetric [[tensor powers]] of the [[tangent bundle]] that these modes take values in.
In ([Witten 86](#Witten86)) it is suggested, by regarding the [[superstring]] [[sigma-model]] as [[quantum mechanics]] on the [[smooth loop space]] of its [[target space]], that the Witten genus may be thought of as the [[large volume limit]] of an $S^1$-equivariant [[A-hat genus]] on [[smooth loop space]], hence the [[index]] of the [[Dirac-Ramond operator]] in that limit. (Ever since this suggestion people have tried to make precise the concept of Dirac operator on a [[smooth loop space]] (e.g. [Alvarez-Killingback-Mangano-Windey 87](#AlvarezKillingbackManganoWindey87)). But notice that, by the above, only the [[formal loop space]] and the [[Dirac-Ramond operator]] really appears in the definition of the Witten genus.)
A priori the [[coefficients]] of the Witten genus as a genus on [[orientation|oriented]] manifolds are [[formal power series]] over the [[rational numbers]]
$$
w \;\colon\; M SO_\bullet \longrightarrow \mathbb{Q}[ [ q ] ]
\,.
$$
In the construction from [[string theory|string physics]] this map is interpreted as sending a [[target space|target]] [[spacetime]] $X$ of the [[superstring]] to the function $w_X(q) = w_X(e^{2 \pi i \tau})$ which to each modulus $\tau \in \mathbb{C}$ characterizing a toroidal [[Riemann surface]] assigns the [[partition function]] of the [[superstring]] with [[worldsheet]] the [[torus]] $\mathbb{C}/(\mathbb{Z} + \mathbb{Z}\tau)$ and propagating on [[target space]] $X$.
On manifolds with [[spin structure]] the genus refines to integral power series (via the integrality of the [[A-hat genus]] ([Chudnovsky-Chudnovsky 88](#ChudnovskyChudnovsky88), [Kreck-Stolz 93](#KreckStolz93), [Hovey 91](#Hovey91)). Moreover on manifolds with rational string structure it takes values in [[modular forms]] ([Zagier 86](#Zagier86)) and crucially, on manifolds with [[string structure]] it takes values in [[topological modular forms]]
$$
\array{
M String_\bullet &\longrightarrow&
tmf_\bullet
\\
\downarrow && \downarrow
\\
\Omega_\bullet^{String, rat} &\longrightarrow& MF_\bullet
\\
\downarrow && \downarrow
\\
M Spin_\bullet &\longrightarrow& \mathbb{Z}[[ q ] ]
\\
\downarrow && \downarrow
\\
M SO_\bullet &\stackrel{w}{\longrightarrow}& \mathbb{Q}[ [ q ] ]
}
\,.
$$
(On the left is the image under forming [[Thom spectra]]/[[cobordism rings]] of the first stages in the [[Whitehead tower]] of $BO$, see also at _[[higher spin structure]]_.)
Observe here that [[topological modular forms]] are the [[coefficient]] [[ring]]
of the [[E-∞ ring]] [[spectrum]] known as _[[tmf]]_. By the general way in which [[genera]] (see there) tend to appear as [[decategorifications]] of [[homomorphisms]] of [[E-∞ rings]] out of a [[Thom spectrum]], this suggests that the Witten genus is the value on [[homotopy groups]] of a homomorphism of [[E-∞ rings]] of the form
$$
\sigma \colon M String \longrightarrow tmf
$$
from the [[Thom spectrum]] of [[String bordism]] to the [[tmf]]-spectrum. This lift of the Witten genus to a universal [[orientation in generalized cohomology|orientation]] in universal [[elliptic cohomology]] indeed exists and is called the _[[sigma-orientation]]_, or the _[[string orientation of tmf]]_.
This construction has been the central motivation behind the search for and construction of [[tmf]] ([Hopkins 94](#Hopkins94)). A construction of the [[string orientation of tmf]] is given in ([Ando-Hopkins-Rezk 10](#AndoHopkinsRezk)) and it is shown that indeed it refines the Witten genus ([Ando-Hopkins-Rezk 10, prop. 15.3](#AndoHopkinsRezk)).
It is maybe noteworthy that [[tmf]] (and hence its universal string orientation) also arises canonically from just studying [[chromatic homotopy theory]] (see [Mazel-Gee 13](#tmf#MazelGee13) for a nice survey of this) a fundamental topic in [[stable homotopy theory]], hence a fundamental topic in [[mathematics]]. Therefore in the Witten genus some very fundamental pure mathematics happens to equivalently incarnate as some conjecturally very fundamental [[physics]] ([[string theory]]).
## Properties
### Characteristic series
{#CharacteristicSeries}
The [characteristic series](genus#LogarithmAndCharacteristicSeries) of the Witten genus as a [[power series]] in $z$ with [[coefficients]] in formal power series in $q$ over $\mathbb{Q}$ is
$$
\begin{aligned}
K_w(z)(q)
& =
\frac{z}{\exp_w(z)(q)}
\\
& =
\frac{z}{\sigma_L(z)(q)}
\\
& =
\frac{z/2}{sinh(z/2)}
\prod_{n \geq 1}
\frac{(1-q^n)^2}{(1-q^n e^z)(1-q^n e^{-z})}
\\
& =
\exp\left(
\sum_{k \geq 2} 2G_k(q) \frac{z^k}{k!}
\right)
\end{aligned}
\,,
$$
where
* $\sigma_L$ is the [[Weierstrass sigma-function]] (see e.g. [Ando Basterra 00, section 5.1](#AndoBasterra00));
* $G_k$ are the [[Eisenstein series]] ([Zagier 86, equation (14)](#Zagier86), [Ando-Hopkins-Rezk 10, prop. 10.9](#AndoHopkinsRezk10)).
This is a [[modular form]] with respect to the variable $q$, see also the the discussion below at _[Integrality and modularity](#IntegralityAndModularity)_ . Such functions which are power series of two variables $z$ and $q$ with elliptic nature in $z$ and modular nature in $q$ are called _[[Jacobi forms]]_ ([Zagier 86, p. 8](#Zagier86), [Ando-French-Ganter 08](#AndoFrenchGanter08)).
There are various further ways to equivalently re-express the above in terms of other special [[modular forms]]. Here are some:
#### In terms of Kac-Weyl characters
{#CharacteristicSeriesInTermsOfKacWeylCharacters}
The Witten genus has a close relation to the [[Kac-Weyl character]] of [[loop group representations]].
Consider of four [[irreducible representations|irreducible]] [[level]]-1 positive energy [[Spin group|Spin]]$(2k)$-[[loop group representation]] the one denoted
$$
\tilde S_+ - \tilde S_- \in Rep(\tilde L Spin(2k))
$$
and write its [[Kac-Weyl character]] as
$$
\chi(\tilde S_+ - \tilde S_-) \in Rep(Spin(2k))[ [ q^{1/12} ] ]
\,.
$$
Under passing to [[group characters]] this is ([Brylinski 90, p. 7(467)](#Brylinski90), reviewed in [KL 96, section 1.2](#KL96)) equivalently
$$
\chi(\tilde S_+ - \tilde S_-) = \frac{\prod_{1}^k \theta}{\eta^k}
\,,
$$
where on the right we have the [[Jacobi theta-function]] $\theta$ divided by the [[Dedekind eta-function]] $\eta$.
Comparison shows that in terms of this the exponential series of the Witten genus is equivalently (by the [[splitting principle]] the $k$-fold products are left implicit):
$$
\exp_w = z/K_w = \eta^2 \, \chi(\tilde S_+ - \tilde S_-)
\,.
$$
Notice that, by the relation (see [here](equivariant+elliptic+cohomology#RelationToLoopGroupRepresentations)) between [[equivariant elliptic cohomology]] and [[loop group representations]], over the complex numbers $\chi(\tilde S_+ - \tilde S_-)$ may be regarded as an element of the $Spin(2k)$-[[equivariant elliptic cohomology]] of the point (at the [[Tate curve]], see at [[twisted ad-equivariant Tate K-theory]]).
### Integrality and modularity
{#IntegralityAndModularity}
A priori, the Witten genus has [[coefficients]] the [[power series]] ring $\mathbb{Q}[ [q] ]$ over the [[rational numbers]]. But under suitable conditions ([[quantum anomaly cancellation]]) it takes values in more interesting subrings.
#### For the type II superstring
{#ModularityForTypeIISuperstring}
The genus obtained from the [[type II superstring]] in the NS-R sector is a [[modular form]] for the [[congruence subgroup]] $\Gamma_2(2)$. ([Witten 87a, below (13)](Witten87a)) See at _[congruence subgroup -- Relation to spin structures](level+structure+on+an+elliptic+curve#RelationToSpinStructures)_ for more.
Hence, with suitable normalization, the universal Witten-Ochanine genus takes values in the subring $MF_\bullet^{\mathbb{Q}}(\Gamma_0(2)) \hookrightarrow \mathbb{Q}[ [q] ]$ of [[modular forms]] for $\Gamma_0(2)\subset SL_2(\mathbb{Z})$ with rational coefficients ([Zagier 86, item d) on page 2](#Zagier86) based on [Chudnovsky-Chudnovsky 88](#ChudnovskyChudnovsky88)).
#### For the heterotic superstring
{#ModularityForHeteroticString}
On manifolds with [[spin structure]] the heterotic string Witten genus has integral coeffcients, hence in the ring $\mathbb{Z}[ [ q ] ]$ ([Chudnovsky-Chudnovsky 88](#ChudnovskyChudnovsky88), [Landweber 88](#Landweber88)), see also ([Kreck-Stolz 93](#KreckStolz93), [Hovey 91](#Hovey91)).
On manifolds with rational [[string structure]] (meaning spin structure and the [[first fractional Pontryagin class]] is at most [[torsion subgroup|torsion]]), then the Witten genus takes values in actual [[modular forms]] $MF_\bullet$ ([Zagier 86, page 6](#Zagier86)).
On manifolds with actual [[string structure]], finally, the Witten genus factors through [[topological modular forms]] ([Hopkins 94](#Hopkins94), [Ando-Hopkins-Rezk 10](#AndoHopkinsRezk)).
### Relation to Dirac operators and supersymmetric QM on loop space
{#RelationToSuSyQMOnLoopSpace}
Originally in ([Witten 87a](#Witten87a)) the elliptic genus was derived as the [[large volume limit]] of the [[index]] of the [[supercharge]] of the [[superstring]] [[worldsheet]] [[2d SCFT]]. Here the "large volume limit" is what restricts the oscillations of the string to be "small". But then in ([Witten87b](#Witten87b)) it was observed that if this supercharge -- the [[Dirac-Ramond operator]] -- would really behave like a [[Dirac operator on smooth loop space]], then the elliptic genus would be the $S^1$-equivariant [[index of a Dirac operator]], where $S^1$ acts by rigid rotationl of the parameterization of the loops, and by analogy standard formulas for equivariant indices in K-theory would imply the localization to the tangent spaces to the space of constant loops.
Notice that the would-be [[Dirac operator on smooth loop space]] is what would realize the [[superstring]] quantum dynamics as [[supersymmetric quantum mechanics]] on [[smooth loop space]]. This observation was the original motivation for the study of [[supersymmetric quantum mechanics]] in ([Witten 82](#Witten82), [Witten 85](#Witten85)) in the presence of a given [[Killing vector field]] (correspinding to the $S^1$-action on loop space ).
### Heterotic (twisted) Witten genus, loop group representations and parameterized WZW models
{#TwistedWittenGenus}
If the [[superstring]] in question is the [[heterotic string]] then generally there is a "[[twisted differential string-structure|twist]]" of its [[background fields]] by a [[gauge field]], hence by a $G$-[[principal bundle]] for $G$ some simply connected [[compact Lie group]] (notably [[E8]]). The partition function in this case is a "twisted Witten genus" ([Witten 87, equations (30), (31)](#Witten87), [Brylinski 90](#Brylinski90), [KL 95](#KL95)). The modularity condition then is no longer just that the [[tangent bundle]] has [[string structure]], but that together with the gauge bundle it has [[twisted string structure]], hence [[string^c 2-group|String^c]]-structure for $c$ the $G$-[[second Chern class]] (explicitly identified as such in ([Chen-Han-Zhang 10](#ChenHanZhang10)).
An elegant formulation of twisted Witten genera (and proof of their rigidity) in terms of highest weight [[loop group representations]] is given in ([KL 95](#KL95)) along the lines of ([Brylinski 90](#Brylinski90)). In ([Distler-Sharpe 07](#DistlerSharpe07)), following suggestions around ([Ando 07](#Ando07)) this is interpreted geometrically in terms of fiberwise [[index|indices]] of [[parameterized WZW models]] [[associated bundle|associated]] to the given [[string 2-group|String]]-[[principal 2-bundle]].
What should be a concrete computation of the twisted Witten genus specifically for $G =$ [[E8]] is in ([Harris 12, section 4](#Harris12)).
### As the global character of sheaves of vertex operator algebras
For $U \subset \mathbb{C}$ an [[open subset]] of the [[complex plane]] then the space $\mathcal{D}^{ch}(U)$ of [[chiral differential operators]] on $U$ is naturally a [[super vertex operator algebra]]. For $X$ a [[complex manifold]] such that its [[first Chern class]] and [[second Chern class]] vanish over the [[rational numbers]], then this assignment gives a [[sheaf of vertex operator algebras]] $\mathcal{D}^{ch}_X(-)$ on $X$. Its [[cochain cohomology]] $H^\bullet(\mathcal{D}^{ch}_X)$ is itself a [[super vertex operator algebra]] and its super-[[Kac-Weyl character]] is proportional to the [[Witten genus]] $w(X)$ of $X$:
$$
char H^\bullet(\mathcal{D}^{ch}_X)\propto w(X)
\,.
$$
Physically this result is understood by observing that $\mathcal{D}^{ch}_X$is the sheaf of [[quantum observables]] of the [[topological twist|topologically twisted]] [[2d (2,0)-superconformal QFT]] (see there for more on this) of which the [[Witten genus]] is (the [[large volume limit]] of) the [[partition function]].
As highlighted in ([Cheung 10, p. 2](#Cheung10)), there is a [[resolution]] by the [[chiral Dolbeault complex]] which gives a precise sense in which over a [[complex manifold]] the Witten genus is a stringy analog of the [[Todd genus]]. See ([Cheung 10](#Cheung10)) for a brief review, where furthermore the problem of generalizing of this construction to [[sheaves of vertex operator algebras]] over more general [[string structure]] manifolds is addressed.
### Stolz conjecture
The _[[Stolz conjecture]]_ due to ([Stolz 96](#Stolz96)) asserts that if $X$ is a [[closed manifold]] with [[String structure]] which furthermore admits a [[Riemannian metric]] with [[positive number|positive]] [[Ricci curvature]], then its Witten genus vanishes.
### Relation to BPS state counting on target space
{#RelationToBPSStateCounting}
By [[supersymmetry]] and by the same argument that controls the expression of the [[index of a Dirac operator]] in terms of [[supersymmetric quantum mechanics]], the Witten genus may be thought of as counting those string states on which the left moving [[supercharge]] acts trivially. In terms of the [[target space]] theory these are the [[BPS states]]. (reviews include [Dijkgraaf 98](#Dijkgraaf98)).
Therefore the Witten genus may also be used as a generating function for BPS state counting. As such it has for instance been used in the microscopic explanation of [[Bekenstein-Hawking entropy]] of [[black holes]], see at _[[black holes in string theory]]_.
### Relation to Cayley plane bundles
The rational Witten genus vanishes on total spaces of [[Cayley plane]]-[[fiber bundles]], and is indeed characterized by this property ([McTague 10](#McTague10), [McTague 11](#McTague11)).
## Related concepts
* [[Witten index]]
* [[Tate K-theory]]
* [[elliptic Chern character]]
* [[M5-brane elliptic genus]]
[[!include genera and partition functions - table]]
## References
{#References}
[[!include elliptic cohomology -- references]]
[[!redirects Witten genera]]
[[!redirects twisted Witten genus]]
[[!redirects twisted Witten genera]] |
Witten index | https://ncatlab.org/nlab/source/Witten+index |
+-- {: .rightHandSide}
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###Context###
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Witten index_ or _supersymmetric index_ of a [[supersymmetry|supersymmetric]] [[quantum field theory]] is its [[partition function]] with the [[trace]]-operation replaced by the [[super-trace]].
## Examples
The Witten index of the [[superstring]] is the [[Witten genus]] of the corresponding [[target space]].
## References
### General
See also
* Wikipedia, _[Witten index](#https://en.m.wikipedia.org/wiki/Witten_index)_
### For $D=2$
For the [[superstring]] [[D=2 CFT]]:
See at _[[Witten genus]]_.
### For $D=3$ $\mathcal{N}=1$ SYM
For [[D=3 N=1 super Yang-Mills theory]]:
* [[Edward Witten]], _Supersymmetric Index Of Three-Dimensional Gauge Theory_, in: [[The many faces of the superworld]]_ ([arXiv:hep-th/9903005](https://arxiv.org/abs/hep-th/9903005))
### For $D=3$ $\mathcal{N}=2$ SYM
For [[D=3 N=2 super Yang-Mills theory]]:
* [[Mathew Bullimore]], Andrea E. V. Ferrari, Heeyeon Kim, _The 3d Twisted Index and Wall-Crossing_ ([arXiv:1912.09591](https://arxiv.org/abs/1912.09591))
### For $D=3$ $\mathcal{N}=4$ SYM
For [[D=3 N=4 super Yang-Mills theory]]:
* [[Mathew Bullimore]], Andrea E.V. Ferrari, Heeyeon Kim, _Twisted Indices of 3d N=4 Gauge Theories and Enumerative Geometry of Quasi-Maps_ ([arXiv:1812.05567](https://arxiv.org/abs/1812.05567))
* [[Davide Gaiotto]], Tadashi Okazaki, _Sphere correlation functions and Verma modules_ ([arXiv:1911.11126](https://arxiv.org/abs/1911.11126))
using discussion in
* [[Andrei Okounkov]], Section 3.4 of: _Lectures on K-theoretic computations in enumerative geometry_ ([arXiv:1512.07363](https://arxiv.org/abs/1512.07363))
### For $D=4$ $\mathcal{N}=1$ SYM
For [[D=4 N=1 super Yang-Mills theory]]:
* [[Leonardo Rastelli]], Shlomo S. Razamat, _The supersymmetric index in four dimensions_, Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 44 ([arXiv:1608.02965](https://arxiv.org/abs/1608.02965))
[[!redirects Witten indices]]
[[!redirects supersymmetric index]]
[[!redirects supersymmetric indices]]
|
Witten's Dark Fantasy | https://ncatlab.org/nlab/source/Witten%27s+Dark+Fantasy |
+-- {: .rightHandSide}
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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Cosmology
+-- {: .hide}
[[!include cosmology -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What has been called _Witten's Dark Fantasy_ (in [Heckmann-Lawrie-Lin-Zoccarato 19, Section 8](#HeckmannLawrieLinZoccarato19)) is an argument, going back to [Witten 95a](#Witten95a), [Witten 95b, Sec. 3](#Witten95b), [Witten 00, p. 7](#Witten00), for the existence of [[non-perturbative effect|non-perturbative]] _non-_[[supersymmetry|supersymmetric]] 4d [[string vacua]]/[[string phenomenology]] with fundamentally _vanishing_ [[cosmological constant]] (i.e. vanishing "[[dark energy]]").
The original idea was formulated in terms of 3d [[M-theory on 8-manifolds]] decompactified at strong coupling to 4d via [[duality between M-theory and type IIA string theory]] (recall the [[super 2-brane in 4d]]).
Based on an observation of [Vafa 96, Section 4.3](#Vafa96) that the argument should have a natural realization in 4d [[F-theory on Spin(7)-manifolds]] ([[T-duality|T-dual]] to the previous perspective), a detailed construction was finally laid out in [Bonetti-Grimm-Pugh 13](#BonettiGrimmPugh13), [Heckmann-Lawrie-Lin-Zoccarato1 18](#HeckmannLawrieLinZoccarato118), [Heckman-Lawrie-Lin-Sakstein-Zoccarato 19](#HeckmanLawrieLinSaksteinZoccarato19).
## Properties
### $1/2$-Supersymmetry
The key technical point is the claim that a careful analysis of [[D=4 N=1 supergravity]] obtained after [[KK-compactification]] of [[F-theory on Spin(7)-manifolds]] ([[T-duality|T-dual to]] [[M-theory on Spin(7)-manifolds]]) reveals, in contrast to the [[number of supersymmetries|N=1 supersymmetry]] of [[F-theory on CY4-folds]], an "$N= 1/2$ supersymmetry", where:
1. the [[vacuum state]] is [[supersymmetry|supersymmetric]] and hence has vanishing [[cosmological constant]];
1. but no [[finite number|finite]]-[[energy]]-excitation of the vacuum appears supersymmetrically,
hence [[fermions]] and [[bosons]] in the model do _not_ appear in supersymmetric spectra.
([Vafa 96, Sec. 4.3](#Vafa96) [BGP 13](#BonettiGrimmPugh13), [HLLZ 18, Sec. 4](#HeckmannLawrieLinZoccarato118))
### Closed spatial slices
{#ClosedSpatialSlices}
The concrete realization of [[Witten's Dark Fantasy]] in the [[F-theory]] [[model (in theoretical physics)|model]] of [Heckmann-Lawrie-Lin-Zoccarato1 18](#HeckmannLawrieLinZoccarato118) is a [[cosmology]] where spatial slices are [[closed manifold|closed]] and in fact of the [[topology]] of the [[3-sphere]]:
<center>
<img src="https://ncatlab.org/nlab/files/PixelatedSpatialSliceInWittenDarkFantasyModel.jpg" width="700">
</center>
> graphics from [Heckman-Lawrie-Lin-Sakstein-Zoccarato 19](#HeckmanLawrieLinSaksteinZoccarato19)
### Phenomenology
{#Phenomenology}
When the idea of "Witten's Dark Fantasy" was proposed in [Witten 95a](#Witten95a), [Witten 95b](#Witten95b), [Witten 00](#Witten00) it was right before observation of [[red shifts]] of [[supernovae]] convinced cosmologists, in 2001, of a relatively small but [[positive number|positive]] [[cosmological constant]]. When this result became enshrined in what is now the [[standard model of cosmology]], the idea of vanishing cosmological constant in [[string theory]] fell out of favor, and a vocal sub-community instead embarked on arguing that [[de Sitter spacetime]]-[[string vacua]] with [[positive number|positive]] [[cosmological constant]] had to be searched at random in a large [[landscape of string theory vacua]].
However, debate remains over whether the apparently observed [[cosmological constant]] is actually real:
1. The authors of [KLKCR 19](#KLKCR19) claim that temporal evolution of [[supernovae]] luminosity had been underappreciated, which makes the apparent evidence for a [[positive number|positive]] [[cosmological constant]] completely go away. Earlier, [NGS 16](#NielsenGuffantiSarkar16) had pointed out that even with the established interpretation of the data, a vanishing [[cosmological constant]] is not excluded by the data.
1. Since around 2000 authors have argued that the apparent [[cosmological constant]] may be an artifact of the usual [[FRW model]]-cosmologies not taking sizeable backreaction of [[inhomogeneous cosmology|cosmic inhomogeneities]] into account. The situation with this debate currently remains open (see at _[[inhomogeneous cosmology]]_).
While it is uncontroversial that [[inhomogeneous cosmology|cosmic inhomogeneity]] does have a measurable effect on cosmic expansion, the general current consensus seems to be that it is too small to explain all of the dark energy of the [[standard model of cosmology]]. But in view of the first item above, this would be a moot point.
<center>
<img src="https://ncatlab.org/nlab/files/SNLuminosityEvolutionCancelsApparentCC.jpg" width="740">
</center>
> from [KLKCR 19](#KLKCR19)
In the extreme case, if re-analysis of the data, combined with effects of [[inhomogeneous cosmology|cosmic inhomogeneity]], and possibly combined with [[higher curvature corrections]] to [[gravity]] (such as control the observationally preferred [[Starobinsky model of cosmic inflation]]), would explain all of the apparently observed [[cosmological constant]], then _Witten's dark fantasy_ would again appear to be viable [[string phenomenology]].
It is then interesting to notice that also the [[closed manifold|closed]] spatial slices found in the model [above](#ClosedSpatialSlices) have recently been argued to be preferred by observational data ([VMS 19](#VMS19)).
That available experimental data supports neither a cosmological constant nor open spatial slices is further argued in [Di Valentino-Melchiorri-Silk 20](#ValentinoMelchiorriSilk20).
## Related concepts
* [[cosmology]], [[inhomogeneous cosmology]]
* [[M-theory]], [[landscape of string theory vacua]]
## References
### Theory
#### General idea
The idea in rough form goes back to
* {#Witten95a} [[Edward Witten]], _Strong coupling and the cosmological constant_, Mod. Phys. Lett. A 10:2153-2156, 1995 ([arXiv:hep-th/9506101](https://arxiv.org/abs/hep-th/9506101))
* {#Witten95b} [[Edward Witten]], Section 3 of _Some Comments On String Dynamics_, talk at [Strings95](https://cds.cern.ch/record/305869) ([arXiv:hep-th/9507121](http://arxiv.org/abs/hep-th/9507121))
* {#Witten00} [[Edward Witten]], p. 7 of: _The Cosmological Constant From The Viewpoint Of String Theory_, lecture at [DM2000](http://inspirehep.net/record/972507), in: David Kline (ed.) _Sources and detection of dark matter and dark energy in the universe 2000_, Springer 2001. 27-36. ([arXiv:hep-ph/0002297](https://arxiv.org/abs/hep-ph/0002297), [doi:10.1007/978-3-662-04587-9](https://link.springer.com/book/10.1007/978-3-662-04587-9))
The observation that the idea should naturally embed in [[F-theory]], namely as [[F-theory on Spin(7)-manifolds]] is due to
* {#Vafa96} [[Cumrun Vafa]], Section 4.3 of: _Evidence for F-Theory_, Nucl. Phys. B469:403-418, 1996 ([arxiv:hep-th/9602022](https://arxiv.org/abs/hep-th/9602022))
#### Detailed implementation
{#DetailedImplementation}
A detailed implementation of the idea in [[F-theory on Spin(7)-manifolds]] is developed in:
* {#BonettiGrimmPugh13} [[Federico Bonetti]], [[Thomas Grimm]], [[Tom Pugh]], _Non-Supersymmetric F-Theory Compactifications on $Spin(7)$ Manifolds_, JHEP 01 (2014) 112 ([arXiv:1307.5858](https://arxiv.org/abs/1307.5858))
* {#BonettiGrimmPugh13b} [[Federico Bonetti]], [[Thomas Grimm]], [[Eran Palti]], [[Tom Pugh]], _F-Theory on $Spin(7)$ Manifolds: Weak-Coupling Limit_, JHEP 02 (2014) 076 ([arXiv:1309.2287](https://arxiv.org/abs/1309.2287))
* {#HeckmannLawrieLinZoccarato118} [[Jonathan Heckman]], Craig Lawrie, Ling Lin, [[Gianluca Zoccarato]], _F-theory and Dark Energy_, Fortschritte der Physik ([arXiv:1811.01959](https://arxiv.org/abs/1811.01959), [doi:10.1002/prop.201900057]( https://doi.org/10.1002/prop.201900057))
* {#HeckmanLawrieLinSaksteinZoccarato19} [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Jeremy Sakstein, [[Gianluca Zoccarato]], _Pixelated Dark Energy_, Fortsch. d. Physik ([arXiv:1901.10489](https://arxiv.org/abs/1901.10489), [doi:10.1002/prop.201900071]( https://doi.org/10.1002/prop.201900071))
### Phenomenology
{#ReferencesPhenomenology}
The [[standard model of cosmology]], as per 2020, with its [[positive number|positive]] [[dark energy]]-density and open spatial slices contradicts the vanishing [[cosmological constant]] and preferred [[closed manifold|closed]] ([[3-sphere|spherical]]) spatial slices of [[Witten's Dark Fantasy]].
It may very well be that [[Witten's Dark Fantasy]] is [[phenomenology|phenomenologicaly]] unviable. But it is interesting to notice that there is recent and very recent astrophysical analysis which claims problems with exactly these two aspects of the [[standard model of cosmology]]. If these contrarian authors are actually right, then [[Witten's Dark Fantasy]] is exactly the kind of model needed to match observation.
#### Dark energy or not?
{#ReferencesDarkEnergy}
Argument that the observed type Ia [[supernovae]] are actually consistent with a vanishing [[cosmological constant]]:
* {#NielsenGuffantiSarkar16} J. T. Nielsen, A. Guffanti, [[Subir Sarkar]], _Marginal evidence for cosmic acceleration from Type Ia supernovae_, Nature Scientific Reports volume 6, Article number: 35596 (2016) ([arXiv:1506.01354](https://arxiv.org/abs/1506.01354), [web discussion](https://4gravitons.wordpress.com/2016/11/11/a-response-from-nielsen-guffanti-and-sarkar/))
* {#ColinMoymhayeeRamezSarkar19} Jacques Colin, Roya Mohayaee, [[Mohamed Rameez]], [[Subir Sarkar]], _A response to Rubin & Heitlauf: "Is the expansion of the universe accelerating? All signs still point to yes"_ ([arXiv:1912.04257](https://arxiv.org/abs/1912.04257))
Stronger argument that the observed type Ia [[supernovae]] in fact _prefer_ a vanishing [[cosmological constant]] (due to time-dependency of SN brightness that had been missed):
* {#KLKCR19} Yijung Kang, Young-Wook Lee, Young-Lo Kim, Chul Chung, Chang Hee Ree, _Early-type Host Galaxies of Type Ia Supernovae. II. Evidence for Luminosity Evolution in Supernova Cosmology_, Astrophysical Journal ([arXiv:1912.04903](https://arxiv.org/abs/1912.04903))
exposition:
_[New evidence shows that the key assumption made in the discovery of dark energy is in error](https://phys.org/news/2020-01-evidence-key-assumption-discovery-dark.amp?__twitter_impression=true)_
#### Open spatial slices or not?
{#ReferencesOpenSlices}
Arguments that the [[Planck collaboration|PLANCK satellite]] data actually prefers a [[closed manifold|closed]] spatial slices (contrary to the assumption in the current [[standard model of cosmology]]):
* {#Handley19} Will Handley, _Curvature tension: evidence for a closed universe_ ([arXiv:1908.09139](https://arxiv.org/abs/1908.09139), [spire:1751120](http://inspirehep.net/record/1751120))
* {#VMS19} Eleonora Di Valentino, Alessandro Melchiorri, Joseph Silk, _Planck evidence for a closed Universe and a possible crisis for cosmology_, Nature Astronomy 2019 ([arXiv:1911.02087](https://arxiv.org/abs/1911.02087),
[doi:s41550-019-0906-9](https://www.nature.com/articles/s41550-019-0906-9))
A critique of these arguments is given in
* George Efstathiou, Steven Gratton, _The evidence for a spatially flat Universe_ ([arXiv:2002.06892](https://arxiv.org/abs/2002.06892))
but this critique again rests on just the combination [[Planck collaboration]] & baryon acoustic peak (BAO) & [[supernova]]-data which the above references argue cannot sensibly be combined.
From the abstract of [Handley 19](#Handley19):
> The curvature parameter tension between Planck 2018, cosmic microwave background lensing, and baryon acoustic oscillation data is measured using the suspiciousness statistic to be 2.5 to 3σ. Conclusions regarding the spatial curvature of the universe which stem from the combination of these data should therefore be viewed with suspicion. Without CMB lensing or BAO, Planck 2018 has a moderate preference for closed universes, with Bayesian betting odds of over 50:1 against a flat universe, and over 2000:1 against an open universe.
#### Both or neither?
Combined argument that experimental data disfavours both a [[cosmological constant]] (favoring instead effective [[phantom dark energy]]) as well as an open universe:
* {#ValentinoMelchiorriSilk20} Eleonora Di Valentino, Alessandro Melchiorri, Joseph Silk, _Cosmic Discordance: Planck and luminosity distance data exclude LCDM_ ([arXiv:2003.04935](https://arxiv.org/abs/2003.04935))
[[!redirects Witten's dark fantasy]]
[[!redirects Witten's Dark Phantasy]]
|
WKB method | https://ncatlab.org/nlab/source/WKB+method |
## Idea
A method for [[semiclassical approximation]]...
## References
A general survey is in
* Wikipedia, _[WKB approximation](http://en.wikipedia.org/wiki/WKB_approximation)_
An introduction to the formulation of the WKB method in [[symplectic geometry]]/[[geometric quantization]] is in the first section of
* Sean Bates, [[Alan Weinstein]], _Lectures on the geometry of quantization_, [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)
[[!redirects WKB approximation]] |
Wlodzimierz M. Tulczyjew | https://ncatlab.org/nlab/source/Wlodzimierz+M.+Tulczyjew |
* [Research Gate page](https://www.researchgate.net/scientific-contributions/Wlodzimierz-M-Tulczyjew-77896531)
## Selected writings
On [[symplectic manifold|symplectic]] [[phase spaces]] of [[classical field theories]]:
* [[Jerzy Kijowski]], [[Wlodzimierz M. Tulczyjew]], *A Symplectic Framework for Field Theories*, Lecture Notes in Physics **107** (1979, 2005) [[doi:10.1007/3-540-09538-1](https://doi.org/10.1007/3-540-09538-1)]
category: people
[[!redirects Wlodzimierz Tulczyjew]] |
Wojciech Chachólski | https://ncatlab.org/nlab/source/Wojciech+Chach%C3%B3lski | [[!redirects Wojciech Chacholski]]
* [webpage](http://www.math.kth.se/~wojtek/)
## Selected writings
On [[homotopy pullbacks]]:
* {#CPS05} [[Wojciech Chacholski]], Wolfgang Pitsch, and [[Jerome Scherer]], _Homotopy pullback squares up to localization_, in [[Dominique Arlettaz]], [[Kathryn Hess]] (eds.) _An Alpine Anthology of Homotopy Theory_ ([arXiv:math/0501250](http://arxiv.org/abs/math/0501250), [pdf](http://sma.epfl.ch/~jscherer/articles/hopullbacks2.pdf), [doi:10.1090/conm/399](http://dx.doi.org/10.1090/conm/399))
On [[posets]] in [[persistent homology theory]]:
* [[Wojciech Chacholski]], Alvin Jin, Francesca Tombari,, *Realisations of posets and tameness* $[$[arXiv:2112.12209](https://arxiv.org/abs/2112.12209), [talk](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#ChacholskiTalkAtHomotopicalPerspectivesOnTDA2022)$]$
On [[topological data analysis]]:
* Jens Agerberg, [[Wojciech Chacholski]], Ryan Ramanujam, *Data, geometry and homology* $[$[arXiv:2203.08306](https://arxiv.org/abs/2203.08306)$]$
On [[persistent homology]]:
* [[Wojciech Chachólski]], Alessandro De Gregorio, Nicola Quercioli, Francesca Tombari: *Symmetries of data sets and functoriality of persistent homology*, Theory and Applications of Categories **39** 23 (2023) 667-686 [[tac:39/23](http://www.tac.mta.ca/tac/volumes/39/23/39-23abs.html)]
## Related entries
* [[homotopy theory]]
* [[homotopy pullback]]
* [[topological data analysis]]
category:people
|
Wojciech De Roeck | https://ncatlab.org/nlab/source/Wojciech+De+Roeck |
* [GoogleScholar page](https://scholar.google.be/citations?user=72_fCE0AAAAJ&hl=en)
## Selected writings
On the [[quantum adiabatic theorem]] in [[condensed matter theory]] with an eye towards [[topological phases of matter]]:
* [[Sven Bachmann]], Spyridon Michalakis, [[Bruno Nachtergaele]], Robert Sims, *Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems*, Commun. Math. Phys. **309** (2012) 835-871 ([arXiv:1102.0842](https://arxiv.org/abs/1102.0842), [doi:10.1007/s00220-011-1380-0](https://doi.org/10.1007/s00220-011-1380-0))
* [[Sven Bachmann]], [[Wojciech De Roeck]], [[Martin Fraas]], *Adiabatic Theorem for Quantum Spin Systems*, Phys. Rev. Lett. **119** (2017) 060201 ([doi:10.1103/PhysRevLett.119.060201](https://doi.org/10.1103/PhysRevLett.119.060201))
* [[Sven Bachmann]], [[Wojciech De Roeck]], [[Martin Fraas]], *The adiabatic theorem in a quantum many-body setting*, in: *Analytic Trends in Mathematical Physics*, Contemporary Mathematics **741** (2020) 43-58 ([arXiv:1808.09985](https://arxiv.org/abs/1808.09985), [ISBN:978-1-4704-5388-6](https://bookstore.ams.org/conm-741))
category: people
|
Wojciech Zurek | https://ncatlab.org/nlab/source/Wojciech+Zurek |
* [Wikipedia entry](www.winedt.com/registration.html)
## Selected writings
Introducing the [[no-cloning theorem]] in [[quantum information theory]]:
* [[William Wooters]], [[Wojciech Zurek]], *A single quantum cannot be cloned*, Nature **299** (1982) 802-803 [[doi:10.1038/299802a0](https://doi.org/10.1038/299802a0)]
Mathematical argument that [[quantum computation]] with [[quantum error correction]] is practically feasible:
* [[Emanuel Knill]], [[Raymond Laflamme]], [[Wojciech H. Zurek]], *Resilient Quantum Computation: Error Models and Thresholds*, Proceedings of the Royal Society A **454** 1969 (1998) [[arXiv:quant-ph/9702058](https://arxiv.org/abs/quant-ph/9702058)]
Realization of [[quantum error correction]] in [[experiment]]:
* D. G. Cory, M. D. Price, W. Maas, [[Emanuel Knill]], [[Raymond Laflamme]], [[Wojchiek H. Zurek]], T. F. Havel, and S. S. Somaroo, *Experimental Quantum Error Correction*, Phys. Rev. Lett. 81, 2152 (1998) ([doi:10.1103/PhysRevLett.81.2152](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.2152))
On [[decoherence]] in [[quantum physics]] and the [[interpretation of quantum mechanics]]:
* [[Wojciech Zurek]], *Decoherence, einselection, and the quantum origins of the classical*, Rev. Mod. Phys. **75** (2003) 715-775 [[quant-ph/0105127](http://arxiv.org/abs/quant-ph/0105127), [doi:10.1103/RevModPhys.75.715](http://dx.doi.org/10.1103/RevModPhys.75.715)]
## Related $n$Lab entries
* [[no-cloning theorem]]
* [[einselection]]
category: people
[[!redirects Wojchiek H. Zurek]]
[[!redirects Wojciech H. Zurek]] |
Wolf space | https://ncatlab.org/nlab/source/Wolf+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[space]] which is both a [[quaternion-Kähler manifold]] as well as a [[symmetric space]]. Also known as a _Wolf space_.
## Examples
* [[quaternionic projective plane]]
## Properties
### As positive quaternion-Kähler manifolds
> under construction
+-- {: .num_prop}
###### Proposition
Every [[Wolf space]] is a [[positive quaternion-Kähler manifold]].
=--
In fact the [[Wolf spaces]] are the only known examples of [[positive quaternion-Kähler manifold]] (which is not hyper-Kähler ?!), as of today (e.g. [Salamon 82, Section 5](#Salamon82)).
This leads to the **conjecture** that un every dimension, the [[Wolf spaces]] are the only [[positive quaternion-Kähler manifolds]].
The conjecture has been proven for the following [[dimensions]]
* $d = 4$ (Hitchin)
* $d = 8 $ ([Poon-Salamon 91](#PoonSalamon91), [LeBrun-Salamon 94](#LeBrunSalamon94))
## References
* [[Joseph K. Wolf]], _Complex homogeneous contact manifolds and quaternionic symmetric spaces_, Journal of Math. and Mech., 14 (1965), p. 166 ([jstor:24901319](https://www.jstor.org/stable/24901319))
* {#Salamon82} [[Simon Salamon]], _Quaternionic Kähler manifolds_, Invent Math (1982) 67: 143. ([doi:10.1007/BF01393378](https://doi.org/10.1007/BF01393378))
* {#PoonSalamon91} Y. S. Poon, [[Simon Salamon]], _Quaternionic Kähler 8-manifolds with positive scalar curvature_, J. Differential Geom. Volume 33, Number 2 (1991), 363-378 ([euclid:1214446322](https://projecteuclid.org/euclid.jdg/1214446322))
* {#LeBrunSalamon94} Claude LeBrun, [[Simon Salamon]], _Strong rigidity of positive quaternion Kähler manifolds_, Inventiones Mathematicae 118, 1994, 109–132 ([dml:144231](https://eudml.org/doc/144231), [doi:10.1007/BF01231528](https://doi.org/10.1007/BF01231528))
* {#Amann09} Amann, _Positive Quaternion Kähler Manifolds_, 2009 ([pdf](https://d-nb.info/996176438/34))
See also
* Wikipedia, _[Quaternion-Kähler symmetric space](https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_symmetric_space)_
[[!redirects Wolf spaces]]
[[!redirects quaternion-Kähler symmetric space]]
[[!redirects quaternion-Kähler symmetric spaces]]
[[!redirects quaternion-Kaehler symmetric space]]
[[!redirects quaternion-Kaehler symmetric spaces]]
[[!redirects quaternion Kähler symmetric space]]
[[!redirects quaternion Kähler symmetric spaces]]
[[!redirects quaternion Kaehler symmetric space]]
[[!redirects quaternion Kaehler symmetric spaces]] |
Wolfgang Altmannshofer | https://ncatlab.org/nlab/source/Wolfgang+Altmannshofer |
* [webpage](https://www.perimeterinstitute.ca/people/wolfgang-altmannshofer)
## Selected writints
On the [[flavour problem]] and [[flavour anomalies]]:
* [[Wolfgang Altmannshofer]], _The Flavor Puzzle_, Aspen 2014 ([pdf](https://www.aspenphys.org/science/colloquia/2014_slides/altmannshofer.pdf), [[AltmannshoferFlavorProblem.pdf:file]])
On [[flavour anomalies]]:
* Jason Aebischer, [[Wolfgang Altmannshofer]], Diego Guadagnoli, Meril Reboud, [[Peter Stangl]], David M. Straub, _B-decay discrepancies after Moriond 2019_ ([arXiv:1903.10434](https://arxiv.org/abs/1903.10434))
* {#AltmannshoferStangl21} [[Wolfgang Altmannshofer]], [[Peter Stangl]], _New Physics in Rare B Decays after Moriond 2021_ ([arXiv:2103.13370](https://arxiv.org/abs/2103.13370))
On [[Z' bosons]] in view of the [[flavour anomalies]]:
* [[Wolfgang Altmannshofer]], Joe Davighi, Marco Nardecchia, _Gauging the accidental symmetries of the Standard Model, and implications for the flavour anomalies_ ([arXiv:1909.02021](https://arxiv.org/abs/1909.02021))
category: people |
Wolfgang Dür | https://ncatlab.org/nlab/source/Wolfgang+D%C3%BCr |
* [Institute page](https://www.uibk.ac.at/th-physik/staff/duer/)
* [GoogleScholar page](https://scholar.google.at/citations?user=YZZSOsoAAAAJ&hl=en)
## Selected writings
On [[measurement-based quantum computation]]:
* [[Hans J. Briegel]], [[Dan E. Browne]], [[Wolfgang Dür]], [[Robert Raussendorf]], [[Maarten Van den Nest]], *Measurement-based quantum computation*, Nature Physics **5** 1 (2009) 19-26 [[arXiv:0910.1116](https://arxiv.org/abs/0910.1116), [doi:10.1038/nphys1157](https://doi.org/10.1038/nphys1157)]
category: people |
Wolfgang L. Reiter | https://ncatlab.org/nlab/source/Wolfgang+L.+Reiter |
* [ResearchGate page](https://www.researchgate.net/profile/Wolfgang-Reiter-3)
## Selected writings
On [[experiment|experimental]] [[quantum physics]] and its [[epistemology|epistemological]] [[interpretation of quantum mechanics|interpretation]]:
* [[Daniel Greenberger]], [[Wolfgang L. Reiter]], [[Anton Zeilinger]], *Epistemological and Experimental Perspectives on Quantum Physics*, Vienna Circle Institute Yearbook (VCIY) **7** (1999) [[doi:10.1007/978-94-017-1454-9](https://doi.org/10.1007/978-94-017-1454-9)]
category: people
[[!redirects Wolfgang Reiter]] |
Wolfgang Lerche | https://ncatlab.org/nlab/source/Wolfgang+Lerche |
* [webpage](http://ph-dep-th.web.cern.ch/ph-dep-th/?site=php/viewprofile.php&id=73)
## Selected writings
Relation of [[Niemeier lattices]] to [[heterotic string theory]]:
* [[Wolfgang Lerche]], [[Dieter Lüst]], [[Adrian Norbert Schellekens]], *Ten-dimensional heterotic strings from Niemeier lattices*, Physics Letters B **181** 1–2 (1986) 71-75 [<a href="https://doi.org/10.1016/0370-2693(86)91257-8">doi:10.1016/0370-2693(86)91257-8</a>, [inspire:233203](https://inspirehep.net/literature/233203)]
On the [[partition function]] of the [[superstring]] ([[heterotic string theory|heterotic string]] and [[type II string theory|type II string]]) as a [[modular form]] with values in the [[Chern character]] of the [[background field|background]] [[field strengths]] ("character-valued partition function", then also called the *[[elliptic genus]]*/*[[Witten genus]]*) and relation to [[Green-Schwarz anomaly cancellation]]:
* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]], *Heterotic string-loop calculation of the anomaly cancelling term*, Nuclear Physics B Volume 289, 1987, Pages 609-627 (<a href="https://doi.org/10.1016/0550-3213(87)90397-X">doi:10.1016/0550-3213(87)90397-X</a>)
* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomaly cancelling terms from the elliptic genus*, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (<a href="https://doi.org/10.1016/0550-3213(88)90468-3">doi:10.1016/0550-3213(88)90468-3</a>)
On the [[duality between F-theory and heterotic string theory]]:
* {#Lerche99} [[Wolfgang Lerche]], _On the Heterotic/F-Theory Duality in Eight Dimensions_, In: Baulieu L., Green M., Picco M., Windey P. (eds.) _Progress in String Theory and M-Theory_, NATO Science Series (Series C: Mathematical and Physical Sciences), vol 564. Springer 2001 ([arXiv:hep-th/9910207](https://arxiv.org/abs/hep-th/9910207), [doi:10.1007/978-94-010-0852-5_2](https://doi.org/10.1007/978-94-010-0852-5_2))
On [[D-branes]] at an [[ADE-singularity]] and their [[open string]] [[worldsheet]] [[2d CFT]]s (see also [here](ADE+singularity#ReferencesTypeIIStringsOnADE)):
* [[Wolfgang Lerche]], [[Carsten Andrew Lütken]], [[Christoph Schweigert]], *D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories*, Nucl.Phys. B **622** (2002) 269-278 ([doi:10.1016/S0550-3213%2801%2900613-7](https://doi.org/10.1016/S0550-3213%2801%2900613-7), [arXiv:hep-th/0006247](https://arxiv.org/abs/hep-th/0006247))
On how this $\widehat{\mathfrak{su}(2)}^{\kappa-2}$-CFT encodes the [[BPS states]] of [[special unitary group|$SU(\kappa)$]]-[[SYM]] on [[D3-branes]] transverse to the singularity:
* [[Wolfgang Lerche]], *On a Boundary CFT Description of Nonperturbative $\mathcal{N} = 2$ Yang-Mills Theory* ([arXiv:hep-th/0006100](https://arxiv.org/abs/hep-th/0006100))
category: people
|
Wolfgang Lueck | https://ncatlab.org/nlab/source/Wolfgang+Lueck |
* [webpage](http://www.him.uni-bonn.de/lueck/)
* [webpage at Hausdorff Centre](https://www.hcm.uni-bonn.de/people/profile/wolfgang-lueck/)
## Selected writings
On [[equivariant homotopy theory]] and [[equivariant algebraic K-theory]]:
* {#Lueck89} [[Wolfgang Lück]], _Transformation Groups and Algebraic K-Theory_, Lecture Notes in Mathematics **1408** (Springer 1989) ([doi:10.1007/BFb0083681](https://doi.org/10.1007/BFb0083681))
On [[equivariant K-theory]] as a [[Spectra]]-valued [[presheaf]] on the [[orbit category]] (a [[naive G-spectrum]]):
* [[James Davis]], [[Wolfgang Lück]], _Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory_, K-Theory 15:201–252, 1998 ([pdf](https://jfdmath.sitehost.iu.edu/teaching/m721F19/assembly.pdf))
On [[equivariant principal bundles]] and their [[classifying spaces]]:
* [[Wolfgang Lück]], *Survey on Classifying Spaces for Families of Subgroups*, In: *Infinite Groups: Geometric, Combinatorial and Dynamical Aspects* Progress in Mathematics, **248** Birkhäuser (2005) ([arXiv:math/0312378](https://arxiv.org/abs/math/0312378), [doi:10.1007/3-7643-7447-0_7](https://doi.org/10.1007/3-7643-7447-0_7))
* [[Bernardo Uribe]], [[Wolfgang Lück]], _Equivariant principal bundles and their classifying spaces_, Algebr. Geom. Topol. 14 (2014) 1925-1995 ([arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925))
On [[Euler characteristics]] of [[categories]] and [[homotopy colimits]]:
* [[Thomas M. Fiore]], [[Wolfgang Lück]], [[Roman Sauer]], *Euler Characteristics of Categories and Homotopy Colimits* [[arXiv:1007.3868](http://arxiv.org/abs/1007.3868)]
On [[proper equivariant homotopy theory]]:
* {#DHLPS19} [[Dieter Degrijse]], [[Markus Hausmann]], [[Wolfgang Lück]], [[Irakli Patchkoria]], [[Stefan Schwede]], _Proper equivariant stable homotopy theory_ ([arXiv:1908.00779](https://arxiv.org/abs/1908.00779))
On the [[equivariant Chern character]] in [[equivariant K-theory]]:
* [[Wolfgang Lück]], [[Bob Oliver]], _Chern characters for the equivariant K-theory of proper G-CW-complexes_, In: Aguadé J., Broto C., Casacuberta C. (eds.) _Cohomological Methods in Homotopy Theory_ Progress in Mathematics, vol 196. Birkhäuser 2001 ([doi:10.1007/978-3-0348-8312-2_15](https://doi.org/10.1007/978-3-0348-8312-2_15))
On the [[equivariant Chern-Dold character]] in generalized [[equivariant cohomology]]:
* [[Wolfgang Lück]], _Chern characters for proper equivariant homology theories and applications to K- and L-theory_, Journal für die reine und angewandte Mathematik, Volume 2002: Issue 543 ([doi:10.1515/crll.2002.015](https://doi.org/10.1515/crll.2002.015), [pdf](https://www.him.uni-bonn.de/lueck/data/eh.pdf))
* [[Wolfgang Lück]], _Equivariant Cohomological Chern Characters_, International Journal of Algebra and Computation, Vol. 15, No. 05n06, pp. 1025-1052 (2005) ([arXiv:math/0401047](https://arxiv.org/abs/math/0401047), [doi:10.1142/S0218196705002773](https://doi.org/10.1142/S0218196705002773))
* [[Wolfgang Lück]], _Equivariant Chern characters_, 2006 ([pdf](https://www.him.uni-bonn.de/lueck/data/goettingen_equi_Chern061130trans.pdf), [[LuckEquivariantChernCharacters.pdf:file]])
On [[equivariant stable cohomotopy]] and the [[Burnside ring]]:
* {#Lueck05} [[Wolfgang Lück]], _The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups_, Pure Appl. Math. Q. 1 (2005), no. 3, Special Issue: In memory of Armand Borel. Part 2, 479--541 ([arXiv:math/0504051](https://arxiv.org/abs/math/0504051))
## Related entries
* [[global equivariant stable homotopy theory]]
* [[equivariant stable cohomotopy]]
* [[Atiyah-Hirzebruch spectral sequence]]
* [[smash product]]
category: people
[[!redirects Wolfgang Lück]] |
Wolfgang Pauli | https://ncatlab.org/nlab/source/Wolfgang+Pauli |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Wolfgang_Pauli)
## Selected writings
Introducing the [[Pauli exclusion principle]]
* {#Pauli25} [[Wolfgang Pauli]], *Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren*, Zeitschrift für Physik, **31** 1 (1925) 765-783 [[doi:10.1007/BF02980631](https://doi.org/10.1007/BF02980631)]
and the [[Born-Pauli rule]]:
* [[Wolfgang Pauli]], *Über Gasentartung und Paramagnetismus*, Zeitschrift für Physik: A Hadrons and nuclei **41** (1927) 81–102 [[doi:10.1007/BF01391920](https://doi.org/10.1007/BF01391920)]
## Concepts
* [[Pauli exclusion principle]]
category: people |
Wolfgang Polak | https://ncatlab.org/nlab/source/Wolfgang+Polak |
* [GoogleScholar page](https://scholar.google.com/citations?user=91RtF7oAAAAJ&hl=en)
## Selected writings
On [[quantum computing]]:
* [[Eleanor Rieffel]], [[Wolfgang Polak]], *An Introduction to Quantum Computing for Non-Physicists*, ACM Comput. Surveys **32** (2000) 300-335 [[arXiv:quant-ph/9809016](https://arxiv.org/abs/quant-ph/9809016), [doi:10.1145/367701.367709](https://dl.acm.org/doi/10.1145/367701.367709)]
* [[Eleanor Rieffel]], [[Wolfgang Polak]], *Quantum Computing -- A gentle introduction*, MIT Press (2011) [[ISBN:9780262526678](https://mitpress.mit.edu/9780262526678/quantum-computing/), [pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608150244118.pdf)]
category: people
|
Wolfgang Rindler | https://ncatlab.org/nlab/source/Wolfgang+Rindler |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Wolfgang_Rindler)
## related $n$Lab entries
* [[Lorentzian manifold]]
* [[spinor]]
category: people |
Wolfgang Rump | https://ncatlab.org/nlab/source/Wolfgang+Rump | Wolfgang Rump is a mathematician working at the University of Stuttgart.
He got his PhD degree in 1978 from the University of Stuttgart, advised by Klaus Wilhelm Roggenkamp.
## Selected writings
On [[quasi-abelian categories]]:
* [[Wolfgang Rump]], _Almost abelian categories_, Cahiers de topologie et géométrie différentielle catégoriques, tome 42, no 3 (2001) 163--225 [[numdam:CTGDC_2001__42_3_163_0](http://www.numdam.org/item?id=CTGDC_2001__42_3_163_0)]
On [[brace]]s, [[$L$-algebra]]s and related algebraic structures:
* W. Rump, _Braces, radical rings, and the quantum Yang-Baxter equation_, J. Algebra __307__ (2007), no. 1, 153--170
* Wolfgang Rump, _Generalized radical rings, unknotted biquandles, and quantum groups_, Colloquium Mathematicum __109__ (2007), 85-100 [doi](https://doi.org/10.4064/cm109-1-7)
* W. Rump, _The brace of a classical group_, Note di Matematica __34__ (2014) n.1, 115--144 [doi](https://doi.org/10.1285/i15900932v34n1p115)
* W. Rump, _Construction of finite braces_, Ann. Comb. 23, 391--416 (2019) [doi](https://doi.org/10.1007/s00026-019-00430-1)
* W. Rump, _The category of $L$-algebras_, Theory App. Cat. __39__ (2023), n. 21, 598--624 [pdf](http://www.tac.mta.ca/tac/volumes/39/21/39-21.pdf)
category: people |