Wess-Zumino-Witten model

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#### $\infty$-Wess-Zumino-Witten theory
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[[!include infinity-Wess-Zumino-Witten theory - contents]]
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#### Quantum field theory
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[[!include functorial quantum field theory - contents]]
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#### Physics
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#Contents#
* table of contents
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## Idea

The **Wess-Zumino-Witten model** (or **WZW model** for short, also called  **Wess-Zumino-Novikov-Witten** model, or short **WZNW** model) is a 2-dimensional [[sigma-model]] [[quantum field theory]] whose target space is a [[Lie group]].

This may be regarded as the boundary theory of [[Chern-Simons theory]] for Lie group $G$.

The [[vertex operator algebra]]s corresponding to the WZW model are [[current algebra]]s.

## Action functional
 {#ActionFunctional}

For $G$ a [[Lie group]], the [[configuration space]] of the WZW over a 2-[[dimension]]al [[manifold]] $\Sigma$ is the space of [[smooth function]]s $g : \Sigma \to G$.

The [[action functional]] of the WZW [[sigma-model]] is the sum of two terms, a kinetic term and a topological term

$$
  S_{WZW} = S_{kin} + S_{top}
  \,.
$$

### Kinetic term
 {#KineticTerm}

The Lie group canonically carries a [[Riemannian metric]] and the kinetic term is the standard one for [[sigma-model]]s on Riemannian [[target space]]s.

### Topological term -- WZW term
 {#TopologicalTerm}

#### For the 2d WZW model
 {#WZWTermFor2dModel}

In [[higher differential geometry]], then given any closed [[differential n-form|differential (p+2)-form]] $\omega \in \Omega^{p+2}_{cl}(X)$, it is natural to ask for a [[prequantization]] of it, namely for a [[circle n-bundle with connection|circle (p+1)-bundle with connection]] $\nabla$ (equivalently: [[cocycle]] in degree-$(p+2)$-[[Deligne cohomology]]) on $X$ whose [[curvature]] is $F_\nabla = \omega$. In terms of [[moduli stacks]] this means asking for lifts of the form

$$
  \array{
    && \mathbf{B}^{p+1}U(1)_{conn}
    \\
    &{}^{\mathllap{\nabla}}\nearrow& \downarrow^{\mathrlap{F_{(-)}}}
    \\
    X
    &\stackrel{\omega}{\longrightarrow}&
   \mathbf{\Omega}^{p+2}_{cl}
  }
$$

in the [[homotopy theory]] of [[smooth homotopy types]].

This immediately raises the question for natural classes of examples of such prequantizations.

One such class arises in [[infinity-Lie theory]], where $\omega$ is a [[left invariant form]] on a [[smooth infinity-group]] given by a [[cocycle]] in [[L-infinity algebra cohomology]]. The [[prequantum n-bundles]] arising this way are the higher [[WZW terms]] discussed here.

In low degree of traditional [[Lie theory]] this appears as follows: On [[Lie groups]] $G$, those closed $(p+2)$-forms $\omega$ which are [[left invariant forms]] may be identified, via the general theory of [[Chevalley-Eilenberg algebras]], with degree $(p+2)$-[[cocycles]] $\mu$ in the [[Lie algebra cohomology]] of the [[Lie algebra]] $\mathfrak{g}$ corresponding to $G$. These in turn may arise, via the [[van Est map]], as the [[Lie differentiation]] of a degree-$(p+2)$-[[cocycle]] $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^{p+2}U(1)$ in the [[Lie group cohomology]] of $G$ itself, with [[coefficients]] in the [[circle group]] $U(1)$.

This happens to be the case notably for $G$ a [[simply connected topological space|simply connected]] [[compact Lie group|compact]] [[semisimple Lie group]] such as [[special unitary group|SU]] or [[spin group|Spin]], where $\mu = \langle -,[-,-]\rangle$ is the [[Lie algebra cohomology|Lie algebra 3-cocycle]] in [[transgression]] with the [[Killing form]] [[invariant polynomial]] $\langle -,-\rangle$. This is, up to normalization, a representative of the de Rham image of a generator $\mathbf{c}$ of $H^3(\mathbf{B}G, U(1)) \simeq H^4(B G, \mathbb{Z}) \simeq \mathbb{Z}$.

Generally, by the discussion at _[[geometry of physics -- principal bundles]]_, the cocycle $\mathbf{c}$ [[modulating morphism|modulates]] an [[infinity-group extension]] which is a [[circle n-group|circle p-group]]-[[principal infinity-bundle]]

$$
  \array{
     \mathbf{B}^p U(1) &\longrightarrow& \hat G
     \\
     && \downarrow
     \\
     && G &\stackrel{\Omega\mathbf{c}}{\longrightarrow}& \mathbf{B}^{p+1}U(1)
  }
$$

whose higher [[Dixmier-Douady class]] class $ \int \Omega \mathbf{c} \in H^{p+2}(X,\mathbb{Z})$ is an integral lift of the real cohomology class encoded by $\omega$ under the [[de Rham isomorphism]]. In the example of [[spin group|Spin]] and $p = 1$ this extension is the [[string 2-group]].

Such a [[Lie theory|Lie theoretic]] situation is concisely expressed by a diagram of [[smooth homotopy types]] of the form

$$
  \array{
     && &\longrightarrow& \mathbf{B}^{p+1}U(1)
     \\
     &{}^{\mathllap{\Omega \mathbf{c}}}\nearrow& &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}}
     \\
     G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R}
  }
  \,,
$$

where $\flat_{dR}\mathbf{B}^{p+2}\mathbb{R} \simeq \flat_{dR}\mathbf{B}^{p+2}U(1)$ is the [de Rham coefficients](cohesive+infinity-topos+--+structures#deRhamCohomology) (see also at _[[geometry of physics -- de Rham coefficients]]_) and where the homotopy filling the diagram is what exhibits $\omega$ as a de Rham representative of $\Omega \mathbf{c}$.

Now, by the very [[homotopy pullback]]-characterization of the [[Deligne complex]] $\mathbf{B}^{p+1}U(1)_{conn}$ ([here](Deligne+cohomology#TheExactDifferentialCohomologyHexagon)), such a diagram is equivalently a [[prequantization]] of $\omega$:

$$
  \array{
     && \mathbf{B}^{p+1}U(1)_{conn} &\longrightarrow& \mathbf{B}^{p+1}U(1)
     \\
     &{}^\mathllap{\nabla}\nearrow& \downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}}
     \\
     G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R}
  }
  \,.
$$

For $\omega = \langle -,[-,-]\rangle$ as above, we have $p= 1$ and so $\nabla$ here is a [[circle n-bundle with connection|circle 2-bundle with connection]], often referred to as a [[bundle gerbe]] [[connection on a bundle gerbe|with connection]]. As such, this is also known as the _WZW gerbe_ or similar.

This terminology arises as follows. In ([Wess-Zumino 71](Wess-Zumino-Witten+model#WessZumino71)) the [[sigma-model]] for a [[string]] propagating on the [[Lie group]] $G$ was considered, with only the standard [[kinetic action]] term. Then in ([Witten 84](Wess-Zumino-Witten+model#Witten84)) it was observed that for this [[action functional]] to give a [[conformal field theory]] after [[quantization]], a certain [[higher gauge theory|higher gauge]] [[interaction term]] has to the added. The resulting [[sigma-model]] came to be known as the _[[Wess-Zumino-Witten model]]_ or _WZW model_ for short, and the term that Witten added became the _WZW term_. In terms of [[string theory]] it describes the propagation of the [[string]] on the group $G$ subject to a [[force]] of [[gravity]] given by the [[Killing form]] [[Riemannian metric]] and subject to a [[B-field]] [[higher gauge field|higher gauge force]] whose [[field strength]] is $\omega$.  In ([Gawedzki 87](Wess-Zumino-Witten+model#Gawedzki87)) it was observed that when formulated properly and generally, this WZW term is the [[surface holonomy]] functional of a [[connection on a bundle gerbe]] $\nabla$ on $G$. This is equivalently the $\nabla$ that we just motivated above.

Later WZW terms, or at least their curvature forms $\omega$, were recognized all over the place in [[quantum field theory]]. For instance the [[Green-Schwarz sigma-models for super p-branes]] each have an [[action functional]] that is the sum of the standard [[kinetic action]] plus a WZW term of degree $p+2$.

In general WZW terms are "[[gauged WZW model|gauged]]" which means, as we will see, that they are not defined on the give [[smooth infinity-group]] $G$ itself, but on a bundle $\tilde G$ of differential moduli stacks over that group, such that a map $\Sigma \to \tilde G$ is a pair consisting of a map $\Sigma \to G$ and of a [[higher gauge field]] on $\Sigma$ (a "tensor multiplet" of fields).

#### Generally
 {#FormalizationGenerally}

The following ([FSS 12](#FSS12), [dcct](#dcct)) is a general axiomatization of WZW terms in [[cohesive (infinity,1)-topos|cohesive homotopy theory]].

In an ambient [[cohesive (∞,1)-topos]] $\mathbf{H}$, let $\mathbb{G}$ be a [[sylleptic ∞-group]], equipped with a [[Hodge filtration]], hence in particular with a chosen morphism

$$
  \iota \colon
  \mathbf{\Omega}^2_{cl}(-,\mathbb{G})
  \longrightarrow
  \flat_{dR} \mathbf{B}^2 \mathbb{G}
$$

to its [de Rham coefficients]()

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

Given an [[∞-group]] object $G$ in $\mathbf{H}$ and given a [[group cohomology|group cocycle]]

$$
  \mathbf{c} \colon \mathbf{B}G \longrightarrow \mathbf{B}^2 \mathbb{G}
  \,,
$$

then a _refinement of the [[Hodge filtration]]_ of $\mathbb{G}$ along $\mathbf{c}$ is a completion of the [[cospan]] formed by $\flat_{dR}\mathbf{c}$ and by $\iota$ above to a [[diagram]] of the form

$$
  \array{
     \mathbf{\Omega}^1_{flat}(-,G)
     &\stackrel{\mu}{\longrightarrow}&
     \mathbf{\Omega}^2_{cl}(-,\mathbb{G})
     \\
     \downarrow && \downarrow^{\mathrlap{\iota}}
     \\
     \flat_{dR}\mathbf{B}G
     &\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}&
     \flat_{dR}\mathbf{B}^2 \mathbb{G}
  }
  \,.
$$

We write $\tilde G$ for the [[homotopy pullback]] of this refinement along the [[Maurer-Cartan form]] $\theta_G$ of $G$

$$
  \array{
    \tilde G &\stackrel{\theta_{\tilde G}}{\longrightarrow}& \mathbf{\Omega}^1_{flat}(-,G)
    \\
    \downarrow && \downarrow
    \\
    G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR}\mathbf{B}G
  }
  \,.
$$

=--

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

Let $\mathbf{H} = $ [[Smooth∞Grpd]] and $\mathbb{G} = \mathbf{B}^p U(1)$ the [[circle n-group|circle (p+1)-group]].

For $G$ an ordinary [[Lie group]], then $\mu$ may be taken to be the [[Lie algebra cohomology|Lie algebra cocycle]] corresponding to $\mathbf{c}$ and $\tilde G \simeq G$.

On the opposite extreme, for $G = \mathbf{B}^p U(1)$ itself with $\mathbf{c}$ the identity, then $\tilde G = \mathbf{B}^p U (1)_{conn}$ is the [[coefficients]] for [[ordinary differential cohomology]] (the [[Deligne complex]] under [[Dold-Kan correspondence]] and [[infinity-stackification]]).

Hence a more general case is a fibered product of these two, where $\tilde G$ is such that a map $\Sigma \longrightarrow \tilde G$ is equivalently a pair consisting of a map $\Sigma \to G$ and of differential $p$-form data on $\Sigma$. This is the case of relevance for WZW models of [[super p-branes]] with "tensor multiplet" fields on them, such as the [[D-branes]] and the [[M5-brane]].

=--

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

In the situation of def. \ref{RefinementOfHodgeFiltration} there is an essentially unique [[prequantum n-bundle|prequantization]]

$$
  \mathbf{L}_{WZW}
  \colon
  \tilde G
  \longrightarrow
  \mathbf{B}^2 \mathbb{G}_{conn}
$$

of the closed differential form

$$
  \mu(\theta_{\tilde G})
  \colon
  \tilde G
  \stackrel{\theta_{\tilde G}}{\longrightarrow}
  \mathbf{\Omega}^1_{flat}(-,G)
  \stackrel{\mu}{\longrightarrow}
  \mathbf{\Omega}^2_{cl}(-,\mathbb{G})
$$

whose underlying $\mathbb{G}$-[[principal ∞-bundle]] is [[modulating morphism|modulated]] by the [[looping and delooping|looping]] $\Omega \mathbf{c}$ of the original cocycle.

This we call the _WZW term_ of $\mathbf{c}$ with respect to the chosen refinement of the Hodge structure.

=--

## Properties
 {#Properties}

### Equations of motion
 {#EquationsOfMotion}

The [[variational calculus|variational derivative]] of the WZW [[action functional]] is

$$
  \delta S_{WZW}(g)
  =
  -\frac{k}{2 \pi i }
 \int_\Sigma \langle (g^{-1}\delta g), \partial (g^{-1}\bar \partial g)  \rangle
  \,.
$$

Therefore the classical [[equations of motion]] for function $g \colon \Sigma \to G$ are

$$
  \partial(g^{-1}\bar \partial g)
  = 0
  \;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;
  \bar \partial(g \partial g^{-1}) = 0
 \,.
$$

The space of solutions to these equations is small. However, discussion of the [[quantization]] of the theory ([below](#HolographyAndRigorousConstruction)) suggests to consider these equations with the real [[Lie group]] $G$ replaced by its [[complexification]] to the  [[complex Lie group]] $G({\mathbb{C}})$. Then the general solution to the equations of motion above has the form

$$
  g(z,\bar z)
  =
  g_{\ell}(z) g_r(\bar z)^{-1}
$$

where hence $g_{\ell} \colon \Sigma \to G(\mathbb{C})$ is any [[holomorphic function]] and $g_r$ similarly any anti-holomorphic function.

(e.g. [Gawedzki 99 (3.18), (3.19)](#Gawedzki99))

### Holography and Rigorous construction
 {#HolographyAndRigorousConstruction}

By  the [[AdS3-CFT2 and CS-WZW correspondence]] (see there for more details) the 2d WZW [[CFT]] on $G$ is related to $G$-[[Chern-Simons theory]] in $3d$.

In fact a rigorous constructions of the $G$-WZW model as a [[rational 2d CFT]] is via the [[FRS-theorem on rational 2d CFT]], which constructs the model as a [[boundary field theory]] of the $G$-[[Chern-Simons theory]] as a [[3d TQFT]] incarnated via a [[Reshetikhin-Turaev construction]].

[[!include hypergeometric KZ-solutions -- references]]

### D-branes for the WZW model
 {#DBranes}

The characterization of [[D-brane]] [[submanifolds]] for the [[open string]] WZW model on a [[Lie group]] $G$ comes from two consistency conditions:

1. geometrical condition:

   For the open string [[CFT]] to still have [[current algebra]] [[worldsheet]] symmetry, hence for half the current algebra symmetry of the closed WZW string to be preserved, the [[D-brane]] [[submanifolds]] need to be [[conjugacy classes]] of the group manifold (see e.g. [Alekseev-Schomerus](#AlekseevSchomerus) for a brief review and further pointers). These conjugacy classes are therefore also called the **symmetric D-branes**.

   Notice that these conjugacy classes are equivalently the [[leaves]] of the [[foliation]] induced by the canonical [[Cartan-Dirac structure]] on $G$, hence (by the discussion at [[Dirac structure]]), the leaves induced by the  [[Lagrangian dg-submanifold|Lagrangian sub-Lie 2-algebroids]] of the [[Courant Lie 2-algebroid]]  which is the [[higher gauge groupoid]] (see there) of the background [[B-field]] on $G$.(It has been suggested by [[Chris Rogers]] that such a foliation be thought of as a higher real [[polarization]].)

1. cohomological condition:

   In order for the Kapustin-part of the [[Freed-Witten-Kapustin anomaly]] of the [[worldsheet]] [[action functional]] of the open WZW string to vanish, the D-brane must be equipped with a [[Chan-Paton gauge field]], hence a [[twisted unitary bundle]] ([[bundle gerbe module]]) of some rank $n \in \mathbb{N}$ for the restriction of the ambient [[B-field]] to the brane.

   For [[simply connected topological space|simply connected]] Lie groups only the rank-1 Chan-Paton gauge fields and their direct sums play a role, and their existence corresponds to a trivialization of the underlying $\mathbf{B}U(1)$-[[principal 2-bundle]] ($U(1)$-[[bundle gerbe]]) of the restriction of the [[B-field]] to the brane. There is then a discrete finite collection of symmetric D-branes = [[conjugacy classes]] satisfying this condition, and these are called the **integral symmetric D-branes**. ([Alekseev-Schomerus](#AlekseevSchomerus), [Gawedzki-Reis](#GW)). As observed in [Alekseev-Schomerus](#AlekseevSchomerus), this may be thought of as identifying a D-brane as a variant kind of a [[Bohr-Sommerfeld leaf]].

   More generally, on non-simply connected group manifolds there are nontrivial higher rank [[twisted unitary bundles]]/[[Chan-Paton gauge fields]] over conjugacy classes and hence the cohomological "integrality" or "Bohr-Sommerfeld"-condition imposed on symmetric D-branes becomes more refined ([Gawedzki 04](#Gawedzki04)).

In summary, the [[D-brane]] [[submanifolds]] in a Lie group which induce an [[open string]] WZW model that a) has one [[current algebra]] symmetry and b) is [[Freed-Witten-Kapustin anomaly|Kapustin-anomaly]]-free are precisely the [[conjugacy class]]-submanifolds $G$ equipped with a [[twisted unitary bundle]] for the restriction of the background [[B-field]] to the conjugacy class.

### Quantization

on [[quantization]] of the WZW model, see at

* [[quantization of loop groups]],

* [[equivariant elliptic cohomology]]

* [[quantization of Chern-Simons theory]]

## Related concepts

* [[exponentiated pion field]]

* [[geometry of physics -- WZW terms]]

* [[basic bundle gerbe]]

* [[current algebra]], [[affine Lie algebra]]

* [[Knizhnik-Zamolodchikov equation]]

* [[coset WZW model]]

* [[gauged WZW model]]

* [[parameterized WZW model]]

* [[higher dimensional WZW model]]

* [[Green-Schwarz action functional]]

* [[analytically continued Wess-Zumino-Witten theory]]

* [[Gepner model]]

## References
 {#References}

### Introduction and survey
 {#IntroductionsAndSurveys}

* [[Peter Goddard]], [[David Olive]], *Kac-Moody and Virasoro algebras in relation to quantum physics*,  International Journal of Modern Physics A **01** 02 (1986) 303-414 [[doi:10.1142/S0217751X86000149](https://doi.org/10.1142/S0217751X86000149), [spire:18583](https://inspirehep.net/literature/18583)]

Textbook accounts:

* [[Philippe Di Francesco]], Pierre Mathieu, David Sénéchal, Part C of: *Conformal field theory*, Springer (1997) [[doi:10.1007/978-1-4612-2256-9](https://doi.org/10.1007/978-1-4612-2256-9)]

* [[Bojko Bakalov]], [[Alexander Kirillov]], *Wess-Zumino-Witten model*, chapter 7 of: *Lectures on tensor categories and modular functors*, University Lecture Series **21**, Amer. Math. Soc. (2001) [[[BakalovKirillov-WZWModel-Ch7OfTensorCat.pdf:file]], [web](http://www.math.stonybrook.edu/~kirillov/tensor/tensor.html), [ams:ulect/21](https://bookstore.ams.org/view?ProductCode=ULECT/21)]

* [[Ralph Blumenhagen]], [[Erik Plauschinn]], Chapter 3 of: *Introduction to Conformal Field Theory -- With Applications to String Theory*, Lecture Notes in Physics **779**, Springer (2009) [[doi:10.1007/978-3-642-00450-6](https://doi.org/10.1007/978-3-642-00450-6)]

Lecture notes:

* [[Patrick Meessen]], *Strings Moving on Group Manifolds: The WZW Model* [[pdf](http://www.unioviedo.es/hepth/people/Patrick/fysica/zooi/WZW_ClassMunoz.pdf), [[Meessen-WZWModel.pdf:file]]]

* [[Lorenz Eberhardt]], *Wess-Zumino-Witten models*, lecture notes at *[YRISW 2019: A modern primer for 2D CFT](https://conf.itp.phys.ethz.ch/esi-school/)*, Vienna (2019) [[pdf](https://conf.itp.phys.ethz.ch/esi-school/Lecture_notes/WZW%20models.pdf), [[Eberhardt-WZWModels.pdf:file]]]

A basic introduction also to the super-WZW model (and with an eye towards the corresponding [[2-spectral triple]]) is in

* {#FroehlichGawedzki93} [[Jürg Fröhlich]], [[Krzysztof Gawedzki]], _Conformal Field Theory and Geometry of Strings_,  extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 ([arXiv:hep-th/9310187](http://arxiv.org/abs/hep-th/9310187))

A useful account of the WZW model that encompasses both its [[action functional]] and [[path integral]] quantization as well as the [[current algebra]] aspects of the QFT is in

* {#Gawedzki99} [[Krzysztof Gawędzki]], *Conformal field theory: a case study*, in Y. Nutku, C. Saclioglu, T. Turgut (eds.) *Conformal Field Theory -- New Non-perturbative Methods In String And Field Theory*, CRC Press (2000) [[arXiv:hep-th/9904145](https://arxiv.org/abs/hep-th/9904145), [doi:10.1201/9780429502873](https://doi.org/10.1201/9780429502873)]

This starts in section 2 as a warmup with describing the 1d QFT of a particle propagating on a group manifold. The [[Hilbert space]] of [[states]] is expressed in terms of the [[Lie theory]] of $G$ and its [[Lie algebra]] $\mathfrak{g}$.

In section 4 the [[quantization]] of the 2d WZW model is discussed in analogy to that. In lack of a full formalization of the quantization procedure, the author uses the loop algebra $\mathcal{l} \mathfrak{g}$ --  the [[affine Lie algebra]] -- of $\mathfrak{g}$ as the evident analog that replaces $\mathfrak{g}$ and discusses the [[Hilbert space]] [[space of states|of states]] in terms of that. He also indicates how this may be understood as a space of [[sections]] of a ([[prequantum line bundle|prequantum]]) [[line bundle]] over the [[loop group]].

See also

* L. Fehér, L. O'Raifeartaigh, P. Ruelle, I. Tsutsui, A. Wipf, _On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories_, Phys. Rep. __222__ (1992), no. 1, 64 pp. [MR93i:81225](http://www.ams.org/mathscinet-getitem?mr=1192998), <a href="http://dx.doi.org/10.1016/0370-1573(92)90026-V">doi</a>

* [[Matthias Blau]], [[George Thompson]], *Equivariant Kähler Geometry and Localization in the $G/G$ Model*, Nucl. Phys. B **439** (1995) 367-394 [<a href="https://doi.org/10.1016/0550-3213(95)00058-Z">doi:10.1016/0550-3213(95)00058-Z</a>, [arXiv:hep-th/9407042](https://arxiv.org/abs/hep-th/9407042)]

  > ([[supersymmetry|supersymmetric]] and using [[group algebra]])

* [[Krzysztof Gawedzki]], Rafal Suszek, [[Konrad Waldorf]], _Global gauge anomalies in two-dimensional bosonic sigma models_ ([arXiv:1003.4154](http://arxiv.org/abs/1003.4154))

* Paul de Fromont, [[Krzysztof Gawędzki]], Clément Tauber, _Global gauge anomalies in coset models of conformal field theory_ ([arXiv:1301.2517](http://arxiv.org/abs/1301.2517))

[[!include WZW term of QCD chiral perturbation theory -- references]]

### Interpretation via CFT and gerbes

Interpretation of the 3d WZW term as defining a [[2d CFT]]

* {#Witten84} [[Edward Witten]], _Non-Abelian bosonization in two dimensions_ Commun. Math. Phys. 92, 455 (1984)

* {#KnizhnikZamolodchikov85} [[Vadim Knizhnik]], [[Alexander Zamolodchikov]], _Current algebra and Wess-Zumino model in two dimensions_, Nucl. Phys. B 247, 83-103 (1984)

and hence as part of a [[perturbative string theory vacuum]]/[[target space]]

* [[Doron Gepner]], [[Edward Witten]], _String theory on group manifolds_, Nucl. Phys. B 278, 493-549 (1986) ([spire:230076](http://inspirehep.net/record/230076))

The WZ term on $\Sigma_2$ was understood in terms of an integral of a 3-form over a cobounding manifold $\Sigma_3$ in

* [[Edward Witten]], _Global aspects of current algebra_, Nucl. Phys. B223, 422 (1983) ([spire:13234](http://inspirehep.net/record/13234), <a href="https://doi.org/10.1016/0550-3213(83)90063-9">doi:10.1016/0550-3213(83)90063-9</a>, [pdf](https://www.phys.sinica.edu.tw/~spring8/users/jychen/pub/reference/NPB_v223_422.pdf))

for the case that $\Sigma_2$ is [[closed manifold|closed]], and generally, in terms of [[surface holonomy]] of [[bundle gerbes]]/[[circle 2-bundles with connection]] in

* {#Gawedzki87} [[Krzysztof Gawedzki]], _Topological Actions in two-dimensional Quantum Field Theories_, in [[Gerard 't Hooft]] et. al (eds.) _Nonperturbative quantum field theory_ Cargese 1987 proceedings,  ([web](http://inspirehep.net/record/257658?ln=de))

* [[Giovanni Felder]] , [[Krzysztof Gawedzki]], A. Kupianen, _Spectra of Wess-Zumino-Witten models with arbitrary simple groups_. Commun. Math. Phys. 117, 127 (1988)

* [[Krzysztof Gawedzki]], _Topological actions in two-dimensional quantum field theories_. In: Nonperturbative quantum field theory. 'tHooft, G. et al. (eds.). London: Plenum Press 1988

as the [[surface holonomy]] of a [[circle 2-bundle with connection]]. See also the references at _[[B-field]]_ and at _[[Freed-Witten anomaly cancellation]]_.

See also

* {#DeligneFreed99} [[Pierre Deligne]], [[Daniel Freed]], chapter 6 of _Classical field theory_ (1999) ([pdf](https://publications.ias.edu/sites/default/files/79_ClassicalFieldTheory.pdf))

  this is a chapter in

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

For the fully general understanding as the [[surface holonomy]] of a [[circle 2-bundle with connection]] see the references [below](#ReferencesRelationToGerbesAndCS).

See also

* [[Edward Witten]], _On holomorphic factorization of WZW and coset models_, Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. ([Euclid](http://projecteuclid.org/euclid.cmp/1104249222))

### Relation to gerbes and Chern-Simons theory
 {#ReferencesRelationToGerbesAndCS}

Discussion of [[circle n-bundle with connection|circle 2-bundles with connection]] (expressed in terms of [[bundle gerbes]]) and discussion of the WZW-background [[B-field]] ([[WZW term]]) in this language (cf. *[[basic bundle gerbe]]*)

* {#GW} [[Krzysztof Gawędzki]], [[Nuno Reis]], *WZW branes and gerbes*, Rev. Math. Phys. **14** (2002) 1281-1334 [[arXiv:hep-th/0205233](https://arxiv.org/abs/hep-th/0205233), [doi:10.1142/S0129055X02001557](https://doi.org/10.1142/S0129055X02001557)]

* {#SchweigertWaldorf07} [[Christoph Schweigert]], [[Konrad Waldorf]], _Gerbes and Lie Groups_, in _Trends and Developments in Lie Theory_, Progress in Math., Birkhäuser ([arXiv:0710.5467](http://arxiv.org/abs/0710.5467))

Discussion of how this 2-bundle arises from the [[Chern-Simons circle 3-bundle]] is in

* [[Alan Carey]], Stuart Johnson, [[Michael Murray]], [[Danny Stevenson]], [[Bai-Ling Wang]], _Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories_ 	Commun.Math.Phys. 259 (2005) 577-613 ([arXiv:math/0410013](http://arxiv.org/abs/math/0410013))

and related discussion is in

* [[Konrad Waldorf]], _Multiplicative Bundle Gerbes with Connection_ , 	Differential Geom. Appl. 28(3), 313-340 (2010) ([arXiv:0804.4835](http://arxiv.org/abs/0804.4835))

See also Section 2.3.18 and section 4.7 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_  ([pdf slides](http://ncatlab.org/schreiber/files/Erlangen2011Schreiber.pdf)).

### Partition functions

* [[Terry Gannon]], _Partition Functions for Heterotic WZW Conformal Field Theories_, Nucl.Phys. B402 (1993) 729-753 ([arXiv:hep-th/9209042](http://arxiv.org/abs/hep-th/9209042))

### D-branes for the WZW model
 {#ReferencesDBranes}

A characterization of [[D-branes]] in the WZW model as those [[conjugacy classes]] that in addition satisfy an integrality ([[Bohr-Sommerfeld quantization|Bohr-Sommerfeld]]-type) condition missed in other parts of the literature is stated in

* {#AlekseevSchomerus} [[Anton Alekseev]], [[Volker Schomerus]], _D-branes in the WZW model_, Phys.Rev.D60:061901,1999 ([arXiv:hep-th/9812193v2](http://arxiv.org/abs/hep-th/9812193v2))

The refined interpretation of the integrality condition as a choice of trivialization of the underling [[principal 2-bundle]]/[[bundle gerbe]] of the [[B-field]] over the brane was then noticed in section 7 of

* [[Krzysztof Gawedzki]], Nuno Reis, _WZW branes and gerbes_, Rev.Math.Phys. 14 (2002) 1281-1334 ([arXiv:hep-th/0205233](http://arxiv.org/abs/hep-th/0205233))

The observation that this is just the special rank-1 case of the more general structure provided by a [[twisted unitary bundle]] of some rank $n$ on the D-brane ([[gerbe module]]) which is twisted by the restriction of the [[B-field]] to the D-brane -- the [[Chan-Paton gauge field]] --  is due to

* {#Gawedzki04} [[Krzysztof Gawedzki]], _Abelian and non-Abelian branes in WZW models and gerbes_, Commun.Math.Phys. 258 (2005) 23-73 ([arXiv:hep-th/0406072](http://arxiv.org/abs/hep-th/0406072)).

The observation that the "multiplicative" structure of the WZW-[[B-field]] (induced from it being the [[transgression]] of the [[Chern-Simons circle 3-bundle|Chern-Simons circle 3-connection]] over the [[moduli stack]] of $G$-[[principal connections]]) induces the [[Verlinde ring]] fusion product structure on symmetric D-branes equipped with [[Chan-Paton gauge fields]] is discussed in

* [[Alan Carey]], [[Bai-Ling Wang]], _Fusion of symmetric $D$-branes and Verlinde rings_, Commun. Math. Phys.277:577-625 (2008) ([arXiv:math-ph/0505040](http://arxiv.org/abs/math-ph/0505040))

The image in [[K-theory]] of these [[Chan-Paton gauge fields]] over conjugacy classes is shown to generate the [[Verlinde ring]]/the [[positive energy representations]] of the [[loop group]] in

* [[Eckhard Meinrenken]], _On the quantization of conjugacy classes_,  	Enseign. Math. (2) 55 (2009), no. 1-2, 33-75 ([arXiv:0707.3963](http://arxiv.org/abs/0707.3963))

Formalization of WZW terms in [[cohesive (infinity,1)-topos|cohesive homotopy theory]] is in

* {#dcct} _[[schreiber:differential cohomology in a cohesive topos]]_

### Relation to dimensional reduction of Chern-Simons

One can also obtain the WZW-model by [[KK-reduction]]
from [[Chern-Simons theory]].

E.g.

* [[Matthias Blau]], G. Thompson, _Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G model_, Nucl.Phys. B408 (1993) 345-390 ([arXiv:hep-th/9305010](http://arxiv.org/abs/hep-th/9305010))

A discussion in [[higher differential geometry]] via [[transgression]] in [[ordinary differential cohomology]] is in

* _[[schreiber:Extended higher cup-product Chern-Simons theories]]_

* _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_

### Relation to extended TQFT

Relation to [[extended TQFT]] is discussed in

* [[Dan Freed]], _[[4-3-2 8-7-6]]_

For a formulation of the WZW term in the presence of D-branes as an open-closed smooth [[functorial field theory]]:

* {#BunkWaldorf21a} [[Severin Bunk]], [[Konrad Waldorf]], *Transgression of D-branes*, Adv. Theor. Math. Phys. **25** 5 (2021) 1095-1198 [[arXiv:1808.04894] (https://arxiv.org/abs/arXiv:1808.04894), [doi:10.4310/ATMP.2021.v25.n5.a1](https://dx.doi.org/10.4310/ATMP.2021.v25.n5.a1)]

* {#BunkWaldorf21b} [[Severin Bunk]], [[Konrad Waldorf]], *Smooth functorial field theories from B-fields and D-branes*, J. Homot. Rel. Struc. **16** 1 (2021) 75-153 [[doi:10.1007/s40062-020-00272-2](https://doi.org/10.1007/s40062-020-00272-2), [arXiv:1911.09990] (https://arxiv.org/abs/arXiv:1911.09990)]

The formulation of the [[Green-Schwarz action functional]] for [[superstrings]] (and other [[branes]] of [[string theory]]/[[M-theory]]) as WZW-models (and [[schreiber:∞-Wess-Zumino-Witten theory|∞-WZW models]]) on ([[super L-∞ algebra]] [[L-∞ extensions]] of) the [[super translation group]] is in

* {#FSS12} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_, International Journal of Geometric Methods in Modern Physics, Volume 12, Issue 02 (2015) 1550018  ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

### In solid state physics

The low-energy physics of a Heisenberg antiferromagnetic spin chain is argued to be described by a WZW model in

* Zheng-Xin Liu, Guang-Ming Zhang, _Classification of quantum critical states of integrable antiferromagnetic spin chains and their correspondent two-dimensional topological phases_ ([arXiv:1211.5421](http://arxiv.org/abs/1211.5421))

See also section 7.10 of Fradkin's book.

Discussion of [[symmetry protected topological order]] phases of matter in [[solid state physics]] via [[higher dimensional WZW models]] is in

* {#CGLW11} Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, [[Xiao-Gang Wen]], _Symmetry protected topological orders and the group cohomology of their symmetry group_, Phys. Rev. B 87, 155114 (2013) [arXiv:1106.4772](http://arxiv.org/abs/1106.4772); A short version in Science __338__, 1604-1606 (2012) [pdf](http://dao.mit.edu/~wen/pub/dDSPTsht.pdf)

[[!include fractional-level WZW model -- references]]

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Weyl algebra

> To be distinguished from _[[Weil algebra]]_.

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## Idea
 {#Idea}

In general, the term *Weyl algebra* refers to [[noncommutative algebra|noncommutative]] [[associative algebras]] controlled by [[canonical commutation relations]] (CCR) which are the hallmark of [[quantum mechanics]].

More specifically, by the Weyl algebra with *[[Weyl relations]]* one refers to the *exponentiated* form of these CCR where the algebra generators are ([[linear representation|represented]] by) [[unitary operators]], introduced by [Weyl 1927, pp. 27](#Weyl27) and further highlighted in [von Neumann 1931](#vonNeumann31). It is this form of "Weyl algebra" that the [[Stone-von Neumann theorem]] directly applies to.

## Definition

Given a [[field]] $k$, the $n$-th __Weyl algebra__ $A_{n,k}$ is an [[associative unital algebra]] over $k$ generated by the symbols $x^1,\ldots,x^n,\partial_1,\ldots,\partial_n$ modulo the "[[canonical commutation relations]]" $x^i x^j = x^j x^i$, $\partial_i\partial_j = \partial_j\partial_i$ and $\partial_i x^j - x^j \partial_i = \delta_i^j$ (the [[Kronecker delta]]).

In [[characteristic zero]], this agrees with the algebra of [[regular differential operators]] on the $n$-dimensional [[affine space]].

Sometimes one considers the Weyl algebras over an arbitrary $k$-algebra $R$, including noncommutative $R$, when the definition is simply $A_{n,k}\otimes_k R$. Another generalization are the symplectic Weyl algebras.

In [[quantum physics]], one often studies Weyl algebras over the [[complex numbers]] (see [below](#RelationToHeisenbergLieAlgebra)); the usual notation there is $p_j$ for $- \mathrm{i} \partial_j$ (where $\mathrm{i}$ is the imaginary unit).

## Properties

### Relation to Heisenberg Lie algebra
 {#RelationToHeisenbergLieAlgebra}

Consider the standard [[symplectic form]] on the [[Cartesian space]] $\mathbb{R}^{2n}$, making a [[symplectic vector space]]. This gives rise to the corresponding [[Heisenberg Lie algebra]].

Depending on conventions, the [[universal enveloping algebra]] of the [[Heisenberg Lie algebra]] either already is the [[Weyl algebra]] on $2n$ generators or else it becomes so after after forming the [[quotient algebra]] in which the central generator is identified with the [[unit element]] of the [[ground field]] -- whereas in the former case (considered eg. in [Kravchenko 2000, Def. 2.1](#Kravchenko00); [Bekaert 2005, p. 9](#Bekaert05)) the central generator plays the role of the formal [[Planck constant]] $\hbar$ with the Weyl algebra regarded as a [[formal deformation quantization]] of the [[symplectic manifold]] $\mathbb{R}^{2m}$.

Accordingly, given a [[Heisenberg Lie n-algebra|Heisenberg Lie $n$-algebra]] it makes sense to call its [[universal enveloping E-n algebra|universal enveloping $E_n$-algebra]] a _[[Weyl n-algebra|Weyl $n$-algebra]]_.

## Related concepts

* [[Weyl relations]]

* [[canonical commutation relations]]

* [[Heisenberg algebra]]

## References

The term "Weyl algebra" for algebras freely generated subject to [[canonical commutation relations]] is due to

* [[Jacques Dixmier]], *Sur les algèbres de Weyl*, Bulletin de la Société Mathématique de France, **96** (1968) 209-242 [[numdam:BSMF_1968__96__209_0](http://www.numdam.org/item/?id=BSMF_1968__96__209_0)]

(there attributed to a suggestion by [[Irving Segal]]) and referring to the original discussion in

* {#Weyl27} [[Hermann Weyl]], (36) in: *Quantenmechanik und Gruppentheorie*, Zeitschrift für Physik **46** (1927) 1–46 [[doi:10.1007/BF02055756](https://doi.org/10.1007/BF02055756)]

However, beware that the invention of [Weyl 1927](#Weyl27) was not the [[canonical commutation relations]] but their exponential reformulation via the [[Weyl relations]], whose relevance was then picked up by

* {#vonNeumann31} [[John von Neumann]], *Die Eindeutigkeit der Schrödingerschen Operatoren*, Mathematische Annalen **104**  (1931) 570–578 [[doi:10.1007/BF01457956](https://doi.org/10.1007/BF01457956)]

  > (proving the [[Stone-von Neumann theorem]])

More on the history:

* [[Severino C. Coutinho]], Introduction to: *A primer of algebraic D-modules*, London Math. Soc. Stud. Texts **33**, Cambridge University Press (1995) [[doi:10.1017/CBO9780511623653](https://doi.org/10.1017/CBO9780511623653)]

* [[Severino C. Coutinho]], *The Many Avatars of a Simple Algebra*, The American Mathematical Monthly **104** 7 (1997) 593-604 [[doi:10.2307/2975052](https://doi.org/10.2307/2975052), [jstor:2975052](https://www.jstor.org/stable/2975052)]

Further discussion (of either notion):

* [[Alan Weinstein]], p. 392 of: *Deformation quantization*, Séminaire Bourbaki volume 1993/94, exposés 775-789, Astérisque, no. 227 (1995), Talk no. 789 [[numdam:SB_1993-1994__36__389_0](http://www.numdam.org/item/?id=SB_1993-1994__36__389_0)]

* {#Kravchenko00} Olga Kravchenko, *Deformation Quantization of Symplectic Fibrations*, Compositio Mathematica **123** (2000) 131–165 [[arXiv:math/9802070](https://arxiv.org/abs/math/9802070), [doi:10.1023/A:1002452002677](https://doi.org/10.1023/A:1002452002677)]

* {#Bekaert05} [[Xavier Bekaert]], *Universal enveloping algebras and some applications in physics* (2005) [[cds:904799](https://cds.cern.ch/record/904799), [pdf](https://cds.cern.ch/record/904799/files/cer-002575006.pdf)]

* [[Markus Pflaum]], *From Weyl quantization to modern algebraic index theory*, in *Groups and Analysis -- The Legacy of Hermann Weyl*, Cambridge University Press (2008) 84-99 [[doi:10.1017/CBO9780511721410.005](https://doi.org/10.1017/CBO9780511721410.005)]

* Jason Gaddis, *The Weyl algebra and its friends: a survey* [[arXiv:2305.01609](https://arxiv.org/abs/2305.01609)]

On [[continuous field of C-star algebras|continuous fields]] of [[Weyl algebras]] as [[continuous deformation quantizations]] of [[symplectic vector spaces|symplectic]] [[topological vector spaces]]:

* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization*, Annales Henri Poincaré **5** (2004) 327–346 [[doi:10.1007/s00023-004-0171-y](https://doi.org/10.1007/s00023-004-0171-y)]

On Weyl algebras as [[groupoid algebras]] being [[strict deformation quantizations]] of [[Lie-Poisson structures]] given by [[tangent Lie algebroids]]:

* [[Nicolaas P. Landsman]], B. Ramazan, Ex. 11.3 in: *Quantization of Poisson algebras associated to Lie algebroids*, in: *Groupoids in Analysis, Geometry, and Physics*, Contemporary Mathematics **282** (2001) [[arXiv:math-ph/0001005](https://arxiv.org/abs/math-ph/0001005), [ams:conm/282](http://www.ams.org/books/conm/282)]

On [[group algebras]] of ([[underlying]] [[discrete group|discrete]]) [[Heisenberg groups]] as [[strict deformation quantizations]] of [[presymplectic manifold|pre-]][[symplectic vector spaces|symplectic]] [[topological vector spaces]] by [[continuous field of C-star algebras|continuous fields of]] [[Weyl algebras]]:

* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization*, International Journal of Pure and Applied Mathematics **38** 1 (2007) [[ijpam:2007-38-1/6](https://ijpam.eu/contents/2007-38-1/6/index.html), [pdf](https://www.ijpam.eu/contents/2007-38-1/6/6.pdf)]

* [[Reinhard Honegger]], [[Alfred Rieckers]], *Heisenberg Group Algebra and Strict Weyl Quantization*, Chapter 23 in: *Photons in Fock Space and Beyond, Volume I: From Classical to Quantized Radiation Systems*, World Scientific (2015) [chapter:[doi;10.1142/9789814696586_0023](https://doi.org/10.1142/9789814696586_0023), book:[doi:10.1142/9251-vol1](https://doi.org/10.1142/9251-vol1)]

See also:

* [[eom]]: J.-E. Björk, [Weyl algebra](http://eom.springer.de/w/w097670.htm)

A [[categorification]] of the Weyl algebra is introduced in operadic language in

* Nikita Markarian, _Weyl $n$-algebras_, [arxiv/1504.01931](http://arxiv.org/abs/1504.01931)

[[!redirects Weyl algebras]]


Weyl bundle

If $(M,\omega)$ is a symplectic manifold then the completed symmetric power of the cotangent bundle $W = \hat{S}(T^* M)$, and sometimes also $W_h = \hat{S}(T^* M)[[h]]$ are called the __Weyl bundle__. (The same term is used for some other, quite different, notions!)
In addition to the commutative symmetric algebra structure, there is a noncommutative product due symplectic structure.
If $a,b\in \Gamma_U(W_h)$ are sections of
$W_h$ above open $U\subset M$ then their noncommutative Moyal-Weyl product is
$$
a \ast b = \left. exp \left(\frac{h}{2}\omega_{j l}(x) \frac{\partial}{\partial y}\frac{\partial}{\partial z}\right) a(y) b(z) \right|_{y=z}
$$

There is also a grading where $deg h = 2$ and $deg w = l$ for $w\in S^l(T^* M)$. So we get a bundle of noncommutative associative algebras.

[[Fedosov connection]] is a connection on $W_h$ (depending on a choice of a cocycle, the Weyl curvature $\Omega\in Z^2(M)[[h]]$). It has the property that the exponential map identifies the smooth functions on $M$ with horizontal sections of $W_h$ for the connection.

Related entries are [[deformation quantization]]

See section 2.2 of

* [[Nicolai Reshetikhin]], Milen Yakimov, _Deformation quantization of Lagrangian fiber bundles_, Conference Moshe Flato 1999, vol. 2, 269-288, Kluwer 2000, [math.QA/9907164](http://arxiv.org/abs/math/9907164)

and section 6 of

* Simone Gutt, John Rawnsley, _Natural star products on symplectic manifolds
and quantum moment maps_, [pdf](https://homepages.warwick.ac.uk/~marke/research/files/naturalstar.pdf)


Weyl character formula

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## Idea

A formula for the [[character]] of [[irreducible representations]] of a [[compact Lie group]] in terms of highest [[weights]].

## Related concepts

* [[Kac character formula]]

## References

* Wikipedia, _[Weyl character formula](http://en.wikipedia.org/wiki/Weyl_character_formula)_


Weyl functional calculus

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## Idea

What is called _Weyl quantization_ is a method of _[[quantization]]_ applicable to [[symplectic manifolds]] which are [[symplectic vector spaces]] or [[quotients]] of these by [[discrete groups]] ([[tori]]).

In Weyl quantization of the flat space $\mathbf{R}^n$, the classical observables of the form $f(x,p)$ are replaced by suitable operators which in the case when $f$ is a polynomial correspond to writing $f$ with $x$ and $p$ replaced by noncommutative variables $x$ and $i h\frac{\partial}{\partial x}$
in symmetric or Weyl ordering. This means that all possible orderings between $x$ and $i h\frac{\partial}{\partial x}$ are summed with an equal weight. More generally, one can extend this rule to more general functions via integral formulas due Weyl and Wigner. This is also useful in fundations of the theory of [[pseudodifferential operator]]s.

## Related concepts

* [[free field theory]]

## References

* [[Lars Hörmander]], _The Weyl calculus of pseudodifferential operators_, Comm. Pure Appl. Math. __32__ (1979), no. 3, 360–444. [MR80j:47060](http://www.ams.org/mathscinet-getitem?mr=517939), [doi](http://dx.doi.org/10.1002/cpa.3160320304)

* Robert F. V. Anderson, _The Weyl functional calculus_, J. Functional Analysis __4__:240-267, 1969, [MR635128](http://www.ams.org/mathscinet-getitem?mr=635128);

* _On the Weyl functional calculus_, J. Functional Analysis __6__:110–115, 1970, [MR262857](http://www.ams.org/mathscinet-getitem?mr=262857)

* E. M. Stein, _Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals_, Princeton University Press 1993

* M. W. Wong, _Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator_, Annals Global Anal. Geom. 28 (2005) 271–283

* [[Thomas L. Curtright]], [[David B. Fairlie]], [[Cosmas K. Zachos]], *A Concise Treatise on Quantum Mechanics in Phase Space*, World Scientific (2014) [[doi:10.1142/8870](https://doi.org/10.1142/8870)]

Discussion of [[quantization of Chern-Simons theory]] in terms of Weyl quantization is in

* [[Jørgen Andersen]], _Deformation quantization and geometric quantization of abelian moduli spaces_, Commun. Math. Phys., 255 (2005), 727–745

* {#GelcaUribe02} [[Razvan Gelca]], [[Alejandro Uribe]], _The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same_, 	Commun.Math.Phys. 233 (2003) 493-512 ([arXiv:math-ph/0201059](http://arxiv.org/abs/math-ph/0201059))

* {#GelcaUribe10a} [[Razvan Gelca]], [[Alejandro Uribe]], _From classical theta functions to topological quantum field theory_  ([arXiv:1006.3252](http://arxiv.org/abs/1006.3252), [slides pdf](http://www.math.ttu.edu/~rgelca/berk.pdf))

* {#GelcaUribe10b} [[Razvan Gelca]], [[Alejandro Uribe]], _Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$_ ([arXiv:1007.2010](http://arxiv.org/abs/1007.2010))

Discussion of the generalization to [[BV-quantization]] is in

* [[Owen Gwilliam]], [[Rune Haugseng]], _Linear Batalin-Vilkovisky quantization as a functor of ∞-categories_ ([arXiv:1608.01290](https://arxiv.org/abs/1608.01290))

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[[!redirects Weyl quantizations]]


Weyl group

> Not to be confused with [[Weil group]].

***

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## Idea

### In Lie theory

In [[Lie theory]], a _Weyl group_ is a [[group]] associated with a [[compact Lie group]] that can either be abstractly defined in terms of a [[root system]] or in terms of a [[maximal torus]]. More generally there are Weyl groups associated with [[symmetric spaces]].

The Weyl group of a [[compact Lie group]] $G$ is equivalently the [[quotient group]] of the [[normalizer]] of any [[maximal torus]] $T$ by that torus.

$$
  W \simeq N_G T / T
  \,.
$$

### In equivariant homotopy theory
  {#InEquivariantHomotopyTheory}

In [[equivariant homotopy theory]] one uses the term _Weyl group_ more generally for the [[quotient group]]

$$
  W_G H = (N_G H) / H
$$

of the [[normalizer]] of a given [[subgroup]] $H \hookrightarrow G$ by that subgroup (e.g. [May 96, p. 13](#May96)).

The relevance of the Weyl group in this sense is that it is the maximal group which canonically [[action|acts]] on $H$-[[fixed points]] of a [[topological G-space]]. (See at _[Change of equivariance group and fixed loci](topological+G-space#ChangeOfGroupsAndFixedLoci)_ for details and, at, e.g., _[[tom Dieck splitting]]_ for applications.)

This may be seen from the fact that the Weyl group of $H \subset G$ is the [[automorphism group]] of the [[coset space]] $G/H$ in the [[orbit category]] of $G$ (and in fact the [[endomorphism monoid]] of $G/H$, since the orbit category is an [[EI-category]], see [there](EI-category#OrbitCategory)):

\[
  \label{AsAutomorphismGroupInOrbitCategory}
  End_{G Orbits}
  \big(
    G/H
  \big)
  \;\;
  =
  Aut_{G Orbits}
  \big(
    G/H
  \big)
  \;\;
    \simeq
  \;\;
  W_G(H)
  \,.
\]

Notice that $W_G G = 1$ and $W_G 1 = G$.

{#WeylGroupOfNormalSubgroup} On the other hand, if $H  = N \subset G$ is a [[normal subgroup]], then its [[normalizer]] is $G$ itself, in which case the Weyl group is just the plain [[quotient group]]

$$
  W_G N \;\simeq\; G/N
  \,.
$$

## Definition

Given a [[compact Lie group]] $G$ with chosen
[[maximal torus]] $T$, its __Weyl group__ $W(G)=W(G,T)$ is the [[group of automorphisms]] of $T$ which are restrictions of [[inner automorphisms]] of $G$.

This is the [[quotient group]] of the [[normalizer subgroup]] of $T \subset G$ by $T$

$$
  W \simeq N_G(T)/T
  \,.
$$

## Properties

* The [[maximal torus]] is of [[finite index subgroup|finite index]] in its [[normalizer]]; the [[quotient]] $N(T)/T$ is [[isomorphism|isomorphic]] to $W(G)$.

* The [[cardinality]] of $W(G)$ for a compact connected $G$, equals the [[Euler characteristic]] of the [[homogeneous space]] $G/T$ ("[[flag variety]]").

* An important approach to the representations of the Weyl groups is the [[Springer theory]].

## Related concepts

* [[Schubert calculus]]

## References

* [[eom]]: [Weyl group](http://eom.springer.de/W/w097710.htm); wikipedia [Weyl group](http://en.wikipedia.org/wiki/Weyl_group)
* N. Chriss, V. Ginzburg, _Representation theory and complex geometry_, Birkhäuser 1997. x+495 pp.
* Walter Borho, Robert MacPherson, _Représentations des groupes de Weyl et homologie d'intersection pour les variétés nilpotentes_, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 15, 707–710 [MR82f:14002](http://www.ams.org/mathscinet-getitem?mr=618892)

* {#May96} [[Peter May]], _Equivariant homotopy and cohomology theory_ CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996.
With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/alaska1.pdf), [cbms-91](https://bookstore.ams.org/cbms-91))

[[!redirects Weyl groups]]


Weyl n-algebra

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## Idea

A concept of _Weyl n-algebra_ is to be a refinement to [[higher algebra]] of the concept of _[[Weyl algebra]]_.

## Definitions

Since ordinary [[Weyl algebras]] are [[universal enveloping algebras]] of [[Heisenberg Lie algebras]] of [[symplectic vector spaces]] (see [here](universal+enveloping+algebra#WeylAlgebraAndHeisenbergAlgebra)), and since there is a sensible notion of [[Heisenberg Lie n-algebra|Heisenberg Lie $n$-algebra]] for all $n \in \mathbb{N}$, it makes sense to defines _Weyl $n$-algebras_ to be the [[universal enveloping E-n algebra|universal enveloping $E_n$-algebras]] of [[Heisenberg Lie n-algebra|Heisenberg Lie $n$-algebras]].

Another definition is considered by [Markarian 2015](#Markarian15).

## References

* {#Markarian15} [[Nikita Markarian]], _Weyl $n$-algebras_ ([arXiv:1504.01931](http://arxiv.org/abs/1504.01931))

[[!redirects Weyl n-algebra]]


Weyl relation

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## Idea

What are called _Weyl relations_ is the incarnation of [[canonical commutation relations]] under passing to [[exponentials]], constituting the *[[Weyl algebra]]*.

For example if $a, a^\ast$ are two elements of an [[associative algebra]] with [[commutator]]

$$
  [a,a^\ast] = \hbar
$$

then the corresponding Weyl relation is, by the [[Baker-Campbell-Hausdorff formula]],

$$
  e^{z a}
  e^{z^\ast a^\ast}
  \;=\;
  e^{z^\ast a^\ast}
  e^{z a}
  e^{\hbar z z^\ast}
$$

for $z,z^\ast \in \mathbb{C}$.

## In the Wick algebra of free quantum fields

+-- {: .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 $\Delta_H$

$$
  A \star_H A
  \;\coloneqq\;
  prod
   \circ
  \exp\left(
    \int_{X^2} \hbar \Delta_H^{a b}(x_1, x_2) \frac{\delta}{\delta \Phi^a(x_1)}
    \otimes \frac{\delta}{\delta \Phi^b(x_2)} dvol_g
  \right)
  (P_1 \otimes P_2)
$$

on [[off-shell]] [[microcausal observables]] $A_1, A_2 \in \mathcal{F}_{mc}$ 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

$$
  \left(
    A_1 \star_H A_2
  \right)^\ast
  =
  A_2^\ast \star_H A_1^\ast
  \,.
$$

=--

For **proof** see at _[[Wick algebra]]_ [this prop.](Wick+algebra#MoyalStarProductOnMicrocausal).

+-- {: .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](Wick+algebra#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.

=--

+-- {: .num_example}
###### Example
**([[Hadamard vacuum state]] [[2-point function]])**

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 $\tfrac{1}{2} i \hbar$ times the evaluation

$$
  \begin{aligned}
    \langle A_1 A_2\rangle
    & \coloneqq
    \int \int
      (\alpha_1)_{a_1}(x_1)
      \,
      \left\langle \mathbf{\Phi}^{a_1}(x_1) \mathbf{\Phi}^{a_2}(x_2)\right\rangle
     \,
     (\alpha_2)_{a_2}(x_2)
     \, dvol_\Sigma(x_1)
     \, dvol_\Sigma(x_2)
    \\
    & \coloneqq
      \tfrac{1}{2}
     i \hbar \int \int
     (\alpha_1)_{a_1}(x_1) \Delta_H^{a_1 a_2}(x_1,x_2) (\alpha_2)_{a_2}(x_2)
      \,dvol_\Sigma(x_1)
      \,dvol_\Sigma(x_2)
  \end{aligned}
$$

of the [[Wightman propagator]] $\Delta_H$:

$$
  \label{StarProductOfTwoLinearObservables}
  A_1 \star_H A_2
  =
  A_1 \cdot A_2
  +
  \langle A_1 A_2\rangle
$$

Further [below](#HadamardVacuumStatesOnWickAlgebras) we see that this evaluation is the [[2-point function]] of a [[state on a star-algebra|state]] on the [[Wick algebra]].

=--

+-- {: .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]]:

$$
  e^{A_1}
  \star_H
  e^{A_2}
  \;=\;
  e^{A_1 + A_2} \; e^{\langle A_1 A_2\rangle}
$$

where on the right we have the [[exponential]] [[Wightman 2-point function]] (eq:StarProductOfTwoLinearObservables).

=--

(e.g. [Dütsch 18, exercise 2.3](#Duetsch18))

## Related concepts

* [[Weyl algebra]]

* [[canonical commutation relations]]

* [[Heisenberg algebra]]

## References

> For more references see at *[[Weyl algebra]]*.

The notion goes back to

* {#Weyl27} [[Hermann Weyl]], (46) in: *Quantenmechanik und Gruppentheorie*, Zeitschrift für Physik **46** (1927) 1–46 [[doi:10.1007/BF02055756](https://doi.org/10.1007/BF02055756)]

* {#vonNeumann31} [[John von Neumann]], p. 571 of: *Die Eindeutigkeit der Schrödingerschen Operatoren*, Mathematische Annalen **104**  (1931) 570–578 [[doi:10.1007/BF01457956](https://doi.org/10.1007/BF01457956)]

  > (proving the [[Stone-von Neumann theorem]])

See also:

* {#Duetsch18} [[Michael Dütsch]], exercise 2.3 in: _[[From classical field theory to perturbative quantum field theory]]_, 2018

[[!redirects Weyl relations]]


Weyl ring

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# Contents
* table of contents
{: toc}

## Idea

A $\mathbb{Z}$-[[Weyl algebra]].

## Definition

### Finitely generated Weyl rings

Given an [[abelian group]] $G$, the **$n$-th Weyl ring** is a [[ring]] $A_n(G)$ with

* an abelian group [[homomorphism]] $g:G \to A_n(G)$

* a function $x:[1, n] \to A_n(G)$

* a function $\partial:[1, n] \to A_n(G)$

such that

* for every number $i, j \in [1,n]$, $x(i) \cdot x(j) = x(j) \cdot x(i)$

* for every number $i, j \in [1,n]$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$

* for every number $i \in [1,n]$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$

* for every number $i, j \in [1,n]$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$

* for every other ring $R$ with abelian group homomorphism $h:G \to R$ with

  * a function $x:[1, n] \to R$

  * a function $\partial:[1, n] \to R$

where

  * for every number $i, j \in [1,n]$, $x(i) \cdot x(j) = x(j) \cdot x(i)$

  * for every number $i, j \in [1,n]$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$

  * for every number $i \in [1,n]$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$

  * for every number $i, j \in [1,n]$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$

there is a unique ring homomorphism $i:A_n(G) \to R$ such that $i \circ g = h$.

### General Weyl rings

Given an [[abelian group]] $G$ and a set $S$ with [[stable equality]], the **$S$-generated Weyl ring** is a [[ring]] $A(S,G)$ with

* an abelian group [[homomorphism]] $g:G \to A(S,G)$

* a function $x:S \to A(S,G)$

* a function $\partial:S \to A(S,G)$

such that

* for every number $i, j \in S$, $x(i) \cdot x(j) = x(j) \cdot x(i)$

* for every number $i, j \in S$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$

* for every number $i \in S$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$

* for every number $i, j \in S$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$

* for every other ring $R$ with abelian group homomorphism $h:G \to R$ with

  * a function $x:S \to R$

  * a function $\partial:S \to R$

where

  * for every number $i, j \in S$, $x(i) \cdot x(j) = x(j) \cdot x(i)$

  * for every number $i, j \in S$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$

  * for every number $i \in S$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$

  * for every number $i, j \in S$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$

there is a unique ring homomorphism $i:A(S,G) \to R$ such that $i \circ g = h$.

## See also

* [[ring]]

* [[tensor ring]]

* [[symmetric ring]]

* [[exterior ring]]

* [[Clifford ring]]

* [[Weyl algebra]]

[[!redirects Weyl rings]]


Weyl semimetal

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#### Topological physics
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{:toc}

## Idea

A type of [[topological phase of matter]] exhibiting [[Weyl spinor|Weyl]] [[fermion]] excitations.

## Related concepts

* [[strange metal]]

* [[topological insulator]]

* [[Chern insulator]]

## References

Review:

* Binghai Yan and Claudia Felser, *Topological Materials: Weyl Semimetals*, Annual Review of Condensed Matter Physics, Vol. 8:337-354 (2017) ([doi:10.1146/annurev-conmatphys-031016-025458](https://doi.org/10.1146/annurev-conmatphys-031016-025458))

* Satyaki Kar, Arun M Jayannavar, *A Primer on Weyl Semimetals: Down the Discovery of Topological Phases*, Asian Journal of Research and Reviews in Physics, 4(1), 34-45 (2021) ([arXiv:1902.01620](https://arxiv.org/abs/1902.01620))

See also:

* Wikipedia, *[Weyl semimetal](https://en.wikipedia.org/wiki/Weyl_semimetal)*

[[!redirects Weyl semimetals]]

[[!redirects Weyl semi-metal]]
[[!redirects Weyl semi-metals]]


Weyl spinor

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#### Representation theory
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[[!include representation theory - contents]]#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A type of [[spin representation]].

For the moment see _[here](Majorana+spinor#DiracAndWeylRepresentations)_.

## Related concepts

* [[Dirac spinor]], [[Majorana spinor]]

* [[Weyl semimetal]]

[[!redirects Weyl spinors]]
[[!redirects Weyl representation]]
[[!redirects Weyl representations]]

[[!redirects chiral spinor]]
[[!redirects chiral spinors]]
[[!redirects chiral spin representation]]
[[!redirects chiral spin representations]]


Weyl tensor

## Related concepts

* [[Riemann tensor]]

* [[conformal geometry]]

## References

* Wikipedia, _[Weyl tensor](http://en.wikipedia.org/wiki/Weyl_tensor)_

[[!redirects Weyl tensors]]


What is an elliptic object?

This entry records the article

* [[Stephan Stolz]], [[Peter Teichner]]:

  \linebreak

  **What is an elliptic object?**

  \linebreak

  in: _Topology, geometry and quantum field theory_

  London Math. Soc. LNS **308**

  Cambridge Univ. Press (2004) 247-343

  [[pdf](https://math.berkeley.edu/~teichner/Papers/Oxford.pdf), [[Stolz-Teichner_EllipticObject.pdf:file]]]

on [[geometric models for elliptic cohomology]] with speculations about the formalization of 2-dimensional [[CFT]] as an extended ([[2-functor]]ial) [[FQFT]], motivated by the [[Witten genus]].

Related discussion of the [[stringor bundle]]:

* {#StolzTeichner2005} [[Stephan Stolz]], [[Peter Teichner]], _The spinor bundle on loop space_ (2005) [[pdf](http://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/MPI.pdf), [[Stolz-Teichner-SpinorOnLoopSpace.pdf:file]]]

In more recent developments the authors exchanged, for technical convenience, the conformal structure in favor of flat [[Riemannian metric|Riemannian structure]] in

* [[Stefan Stolz]], [[Peter Teichner]], _Supersymmetric field theories and generalized cohomology_ , in: [[Hisham Sati]], [[Urs Schreiber]] (eds.), *[[Mathematical Foundations of Quantum Field and Perturbative String Theory]]*, Symposia in Pure Mathematics (2011) [[arXiv:1108.0189](http://arxiv.org/abs/1108.0189)]

See also:

* [[(1,1)-dimensional Euclidean field theories and K-theory]]

* [[(2,1)-dimensional Euclidean field theories and tmf]]

category: reference

[[!redirects What is an elliptic object?]]


what to contribute

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This page is supposed to give some idea about which kind of contributions to the [[HomePage|nLab]] are appropriate or desired.

* **small contributions**: While it is certainly possible that any entry could _eventually_ grow into a comprehensive discussion, everything needs to start small. If you feel like creating an entry but only have a single sentence to say or a single reference to record for the moment, please do so! One also speaks of _stub entries_ for such entries that are waiting to be expanded into something more substantial. Their existence can be very helpful to the general process of creating a good wiki. Just be sure to announce the creation of the entry on the [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/).

* **large contributions**: There is no lack of virtual room. If you feel you need to add a lot of material to do your topic justice, that's fine. We have entries that are the size of lengthy research articles. However, if a single entry grows too large, it can be useful to start breaking it up into a series of separate entries, all linked to each other. One can use "floating tables of contents" to collect groups of entries into "topic clusters". Eventually at [[HowTo]] there should be some information on how to do this.

* **standard material**: While the $n$Lab is meant to eventually push in a certain direction ("[[nPOV]]") the scope of this direction is vast and developing it requires background in all possible subjects. So all standard material in [[mathematics]] and [[physics]] is suitable and welcome for inclusion in to the $n$Lab. Your contribution need not have any visible relation to [[higher algebra]], [[homotopy theory]], [[type theory]], [[category theory]], or [[higher category theory]] at all! But after you announce it on the [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/) it might be that others join in and point out such a relation.

* **original research**: The $n$Lab is expressly meant to be a tool for researchers to make notes they find relevant for their work. This means that if you have an original insight and feel like recording it in context, you are welcome to do so on the $n$Lab. Just follow common good scientific practice: clearly indicate for every non-evident statement that you add where it comes from and what the available proof is (if one is available, that is; conjectures are welcome too). And be prepared to have your contributions discussed on the [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/).

* **expositional as well as specialized technical material**

  The $n$Lab differs from other wikis out there by containing a considerable amount of technical material and abstract mathematics. This does not mean, however, that expositions and friendly pedagogical lead-ins are discouraged. On the contrary! As far as they seem to be lacking, this is only because nobody has yet found the time to add them.

In general: if some topic that seems to be worthwhile is missing from the $n$Lab (and a lot is) this is with high probability not by intent, but by lack of manpower. Be the first to improve on the situation!

category: meta

[[!redirects What to Contribute]]


wheel

# Wheels
* table of contents
{: toc}

## Idea

A wheel is an algebraic concept similar to a [[field]] but in which division is *always* defined, even division by zero.  This naturally requires modifications to the usual axioms.  Every [[commutative ring]] gives rise to a wheel using a simple modification of the construction of the [[field of fractions]], although there are other examples.

The terminology 'wheel' comes from the example of the wheel of [[real numbers]], which geometrically consists of a circle (the circle of [[extended real numbers]]) together with an additional point (thought of as the hub of the wheel), thus looking like $\odot$.

## Notation

In elementary algebra, it\'s traditional to write $-x$ for the opposite (or additive inverse) of $x$, although one could write $0 - x$ (or perhaps $-1x$, meaning $-1 \cdot x$) instead.  This overloads the symbol $-$, with one meaning as a unary operator and another as a binary operator, related since $y - x = y + (-x)$.  However, it is *not* common to write $/x$ for the reciprocal (or multiplicative inverse) of $x$, but only to write $1/x$ (or perhaps $x^{-1}$) instead.  But it would work just as well to overload the symbol $/$, again with one meaning as a unary operator and another as a binary operator, related since $y/x = y \cdot /x$.

In a wheel, the unary operator $/$ is a fundamental part of the structure, and the binary $/$ is only defined using it.  Although it remains true that $y/x = y \cdot /x$, so that we could reasonably write $1/x$ instead of $/x$, this puts the emphasis the wrong way.  And while we could probably write $x^{-1}$ instead of $/x$, this is dangerous, since it\'s not true that $/x$ is always the multiplicative inverse of $x$ in a wheel.  (But for that matter, $-x$ is not always the additive inverse of $x$ in a wheel, even though we can still write it, and indeed define it, as $-1 \cdot x$, when this makes sense.)

If we write multiplication as juxtaposition, then $y /x$ is naturally interpreted as $y \cdot /x$, and so rather than think explicitly of the derived binary operation $/$, it\'s best to think of $y /x$ as always meaning $y \cdot /x$.  So only the unary version of $/$ really matters.  But notice that $/x y$ means $/x \cdot y$, not $/(x y)$ (which is different, and in fact equal to $/x \cdot /y$, or $/x /y$ for short).  So we are coming down on the side of $8 /2 (2 + 2) = 16$ in the great PEMDAS debate.

## Definition

A __wheel__ consists of an [[underlying set]] $W$ equipped with two binary operations written with any common notation for addition and multiplication, a unary operation written with $/$ as a prefix (called the reciprocal), and two constants $0$ and $1$, such that:

* Addition and multiplication are each associative and commutative, with $0$ and $1$ (respectively) as identities;
* The reciprocal is an [[involution]]: $//x = x$;
* Reciprocals get along with multiplication: $/(x y) = /x /y$ and $/1 = 1$;
* Multiplication sort of distributes over addition, but not quite: neither $0 x$ nor $(x + y) z$ can be simplified in general, but instead we have $(x + y) z + 0 z = x z + y z$;
* An axiom that would imply that reciprocals are multiplicative inverses if distributivity held: $(x + y z) /z = x / z + y + 0 z$;
* Special properties of $0$ that are needed since distributivity doesn\'t hold: $0 0 = 0$, $(x + 0 y) z = x z + 0 y$, and $/(x + 0 y) = /x + 0 y$; and
* Although $0 /0$ doesn\'t simplify, it\'s an [[absorbing element]] for addition: $0 /0 + x = 0 /0$.

Although these rules are somewhat odd, we need some unusual behaviour to prevent a wheel from collapsing to a triviality via $0 = 0 x = 0 /0 x = 1 x = x$.  In fact, only the last step is generally valid in a wheel.  The paradigmatic example is the wheel of fractions of a commutative ring, for which all of the axioms are easily checked, but for which simpler (invalid) statements have counterexamples.

We can generalize the quasi-distributive rule beyond two terms as
$$ (\sum_{i=1}^n x_i) y + \sum_{i=1}^{n-1} (0 y) = \sum_{i=1}^n (x_i y) .$$
That is, to multiply $y$ by the sum of $n$ terms, you need to add in $n-1$ copies of $0 y$ as well, and then the result is the sum of $n$ products.

Wheels are like [[rigs]] in that there is generally no notion of subtraction.  However, if the wheel happens to have a (necessarily unique) additive inverse of $1$, we may write it as $-1$, define $-x$ as $-1 x$, and define $y - x$ as $y + -x$.  But notice that $-x$ is *not* in general an additive inverse of $x$; rather, we have $x - x = 0 x + 0 x$ (using the modified general form of distributivity).

It is common to write $/0$ as $\infty$ and $0 \infty$ as $\bot$.

## Examples

The original example is the __wheel of real numbers__, denoted $\mathbb{R}^\odot$.  An element of $\mathbb{R}^\odot$ is an equivalence class of pairs of real numbers under a certain equivalence relation.  Morally, $(a,b)$ and $(c,d)$ are set equivalent iff $a d = b c$; however, this is not quite right, because it makes $(0,0)$ equivalent to every other pair (and so it\'s not even transitive).  So we additionally require that $(a,b) \ne (0,0)$ iff $(c,d) \ne (0,0)$, so that $(0,0)$ is actually equivalent only to itself.  We may write the equivalence class of $(a,b)$ as $a/b$, but let\'s write it as $a : b$ instead until we\'ve established what $a /b$ means in $\mathbb{R}^\odot$.  Now, we can define addition and multiplication in the usual way for fractions: $(a : b) + (c : d) \coloneqq (a d + b c) : (b d)$ and $(a : b) (c : d) \coloneqq (a c) : (b d)$.  Reciprocals are immediate: $/(a : b) \coloneqq b : a$.  The identities $0$ and $1$ in $\mathbb{R}^\odot$ are $0 : 1$ and $1 : 1$ respectively.  More generally, any real number $a$ gives us an element $a : 1$ of $\mathbb{R}^\odot$.  This function from $\mathbb{R}$ to $\mathbb{R}^\odot$ is one-to-one and respects addition and multiplication; it also respects reciprocals in that $/(a : 1) = (1/a) : 1$ if $a \ne 0$.  Now we\'re justified in writing $a /b$ for $a : b$.  We also write $\infty$ for $1 : 0$ and $\bot$ for $0 : 0$; these are the only elements of $\mathbb{R}^\odot$ that don\'t come from $\mathbb{R}$.  We have $x + \infty = \infty$ whenever $x \ne \infty, \bot$, while $\infty + \infty = \bot$ and $x + \bot = \bot$ always; and $\infty x = \infty$ whenever $x \ne 0, \bot$, while $0 \infty = \bot$ and $\bot x = \bot$ always.  In particular, $-\infty = \infty$, and $-\bot = \bot$.  Of course, $/0 = \infty$ and $/\infty = 0$, while $/\bot = \bot$.  Then $x - x = 0$ whenever $x \ne \infty, \bot$, while $\infty - \infty, \bot - \bot = 0$; and $x/x = 1$ whenever $x \ne 0, \infty, \bot$, while $0/0, \infty/\infty, \bot/\bot = \bot$.  Fortunately, $x + 0 = x$ and $1 x = x$ remain true always.  One way to think of all of this is that we are using the usual arithmetic on the version of the [[extended real numbers]] in which $\infty = -\infty$ (the [[projective line]]), but whenever the result is undefined, we take it to be $\bot$ (and any calculation involving $\bot$ stays $\bot$, as in [[domain theory]]).

## References

* Jesper Carlström (2001). _Wheels: On Division by Zero._ [Web](https://www2.math.su.se/reports/2001/11/).

[[!redirects wheel]]
[[!redirects wheels]]


wheeled graph

#Contents#
* table of contents
{:toc}

##Idea##

A **wheeled graph** in the sense of ([HRY](#HRY)) is a generalization of the notion of a [[directed pseudograph]]. What differentiates a wheeled graph from a [[directed pseudograph]] is the notion of an "exceptional cell," which should be thought of as a set of half edges which are not adjacent to any vertex.

The basic idea behind a wheeled graph is that it is a set of vertices with directed edges between them but also edges that leave and enter the graph. It can also have loops and even edges that are not adjacent to any vertex (called **exceptional edges**, see [[generalized graph]] for more). Moreover, a wheeled graph $G$ is equipped with a coloring, i.e. there is a function $Vt(G)\overset{\kappa}\to \mathcal{C}$ from the vertices of $G$ to a set of colors. Since wheeled operads have inputs and outputs, this makes them suitable for modeling [[PROPs]] and [[properads]].

The term "wheeled" refers to the fact that a [[generalized graph]] $G$ might have directed loops in it. These can be in the form of vertices with loops, closed directed paths in $G$, or "exceptional loops" that have no vertices at all (see the examples below). Sometimes, we will be interested in wheel-free graphs, which are wheeled graphs without wheels (though they are not, crucially, just [[graphs]].)

This entry relies on notation defined in [[generalized graph]].

## Definitions ##
 The following is the fifth item of Definition 2.5 of [HRY](#HRY):

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

A wheeled graph is a [[generalized graph]] equipped with a coloring, a direction and a listing (see [here](https://ncatlab.org/nlab/show/generalized+graph#properties) for definitions of these properties).

=--

## Extra Structure

Wheeled graphs can have a number of attributes. Most of the following are self-explanatory though stating them in the terminology of [[generalized graph]] as [HRY]{#HRY} does, can be tedious:

* One of these, confusingly, is being **wheel-free**. In other words, a wheeled graph which is wheel-free is a generalized graph with a coloring, a direction and a listing, but without any loops (either ordinary or exceptional) or directed paths with identical initial and terminal vertex.

* A wheeled graph is **connected** if it is a single exceptional edge, a single exceptional loop,  or has empty exceptional cell, is non-empty and between any two vertices there is an edge.

* A wheeled graph is **simply connected** if it is connected, is not an exceptional loop, and contains no cycles.

* A wheeled graph is a **unital tree** if it is simply connected and each vertex has exactly one output flag.

* A wheeled graph is a **linear graph** if it is a linear tree in which each vertex has exactly one input flag.

Using these definitions we can define the following sets:

* Define $Gr_c^{\circlearrowleft}$ to be the set of connected wheeled graphs.

* Define $Gr_c^{\uparrow}$ to be the set of connected wheel-free graphs.

* Define $Gr_{di}^{\uparrow}$ to be the set of simply connected wheel-free graphs.

* Define $UTree$ and $ULin$ to be the sets of unital trees and linear graphs, respectively.

Note that $UTree$ and $ULin$ are effectively the object sets of the category of [[trees]] and the [[simplex category]] respectively.

## Examples

1. The most elementary wheeled graph (which is also of course an ordinary [[graph]] and a [[quiver]]) is the empty wheeled graph which has the [[empty set]] as its set of flags and as such no vertices or edges.
1. Any [[quiver]] can be realized as a wheeled graph in an obvious way.
1. There is a graph $I$ with $Flag(I)=\{e_{-1},e_1\}$, $\iota(e_i)=e_i$, $\kappa(e_i)=c$, $\delta(e_i)=i$. This is the **$c$-colored exceptional edge**. It can be represented schematically by
$$
\uparrow_c.
$$
1. There is a graph $W$ which is identical to $I$ except that $\iota(f_{-1})=f_1$ and $\iota(f_1)=f_{-1}$. This is the **$c$-colored exceptional loop**. The involution $\iota$ can be thought of as "spinning" the loop. It can be represented schematically by
$$
\circlearrowleft_c.
$$

## Related concepts

* [[graph]]

* [[quiver]]

* [[generalized graph]]

* [[PROP]]

* [[properad]]

## References

* [[Philip Hackney]], [[Marcy Robertson]] and [[Donald Yau]]. _Infinity Properads and Infinity Wheeled Properads_,  Lecture Notes in Mathematics, 2147. Springer, Cham, 2015. [(arxiv version)](http://arxiv.org/pdf/1410.6716v2.pdf)
{#HRY}


wheeledgraph > corolla1


Wheeler superspace

> This entry is about the concept in [[gravity]]/[[cosmology]]. For the concept in [[supergeometry]] see at _[[superspace]]_.

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#Contents#
* table of contents
{:toc}

## Idea

In describing [[gravity]] ([[general relativity]]) on [[globally hyperbolic spacetimes]] as evolution along a given [[foliation]] of [[spacetime]] by [[spacelike]] slices, the [[Einstein equations]] describe a 1-parameter evolution in a [[configuration space (physics)|configuration space]]/[[moduli space]] of fields on each spatial slice.

For applications to [[cosmology]] this space may be drastically reduced to retain only some very large scale feature. The resulting configuration space had been called "superspaces" by [[John Wheeler]].

On this Wheeler superspace, the [[Einstein equations]] describe the [[classical mechanics]] of a configuration point moving around in superspace. Many or most approaches to [[quantum cosmology]] proceed by taking this reduced mechanical system on Wheeler superspace at face value and applying [[quantization]] to it, hoping that the resulting [[quantum mechanics]] on Wheeler superspace is a sensible approximation to full (and unknown) [[quantum gravity]], at least for purposes of studying [[cosmic structure formation]], [[cosmic inflation]] and similar issues.

## Examples

### In 2d gravity on string worldsheets
 {#In2dGravityOnStringWorldsheets}

The [[worldsheet]]-theory of [[strings]] in [[string theory]] is [[D=2 gravity]] coupled to "matter"-fields (the string's [[sigma-model]] "embedding fields"). For plain [[closed string|closed]] [[bosonic strings]] the corresponding Wheeler superspace is just the [[smooth loop space]] of the string's [[target spacetime]]: each loop in spacetime is the configuration of a closed string at a fixed value of its chosen worldsheet-"time" parameter.

For the [[type II string theory|type II]] [[superstring]] the Wheeler superspace is essentially the [[smooth loop space]] of the [[exterior bundle]] of [[target spacetime]], regarded as a [[supermanifold]].

(Beware the clash of "super"-terminology here: the Wheeler superspace of the superstring is now a [[superspace]]-[[Wheeler superspace]], a "super-superspace", where the first "super" is in the sense of [[supersymmetry]]/[[graded manifolds|graded geometry]], while the second "super" in the sense of "[[configuration space (physics)|configuration space]]".

The Wheeler-DeWitt [[quantum mechanics]] of the [[type II superstring theory|type II]] [[superstring]] on its Wheeler superspace, in this sense, is ([Witten 85, from p. 92 (32 of 39) on](supersymmetric+quantum+mechanics#Witten85)) a [[supersymmetric quantum mechanics]] of the form which for finite-dimensional leads to the relation between supersymmetry and Morse theory ([Witten 82](supersymmetric+quantum+mechanics#Witten82)): the Morse-deformed super-charge is the [[Dirac-Ramond operator]] on [[smooth loop space]] (more on this in [Schreiber 04](supersymmetric+quantum+mechanics#Schreiber04)).

## Related concepts

* [[Tomonaga-Schwinger equation]]

* [[geometrodynamics]]

* [[quantum gravity]]

* [[quantum cosmology]]

## References

Original articles:

* {#DeWitt67} [[Bryce DeWitt]], *Quantum Theory of Gravity. I. The Canonical Theory*, Phys. Rev. **160** (1967) 1113 [[doi:10.1103/PhysRev.160.1113](https://doi.org/10.1103/PhysRev.160.1113)]

The Wheeler-DeWitt equation is (5.5) in [DeWitt 1967](#DeWitt67)

Review:

* [[Domenico Giulini]], _The Superspace of Geometrodynamics_ ([arXiv:0902.3923](https://arxiv.org/abs/0902.3923))

See also:

* Wikipedia, *[Wheeler-DeWitt equation](https://en.wikipedia.org/wiki/Wheeler%E2%80%93DeWitt_equation)*

“Quantum Theory of Gravity. I. The Canonical Theory”

[[!redirects Wheeler-DeWitt equation]]
[[!redirects Wheeler-DeWitt equations]]


Wheels theorem

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#### Algebraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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=--

\tableofcontents

## Idea

The _Wheels theorem_ says that the [[quantization of 3d Chern-Simons theory|perturbative]] [[Wilson loop observable]] ([[Kontsevich integral]]) of the [[unknot]] is, as a [[universal Vassiliev invariant]] with values in [[Jacobi diagrams]], a [[series]] of wheel-shaped [[Jacobi diagrams]] with [[coefficients]] the [[modified Bernoulli numbers]].

## Related theorems

* [[Duflo isomorphism]]

* [[PBW theorem]]

* [[Rozansky-Witten Wilson loop of unknot is A-hat genus]]

## References

The full proof is due to

* [[Dror Bar-Natan]], [[Le Tu Quoc Thang]], [[Dylan Thurston]], _Two applications of elementary knot theory to Lie algebras and Vassiliev invariants_, Geom. Topol. Volume 7, Number 1 (2003), 1-31 ([euclid.gt/1513883092](https://projecteuclid.org/euclid.gt/1513883092))

The weaker version for [[Lie algebra weight systems]] was proven in:

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of  the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))

Version for [[Rozansky-Witten weight systems]]:

* [[Justin Sawon]], _Rozansky-Witten invariants of hyperkähler manifold_, Cambridge 2000 ([arXiv:math/0404360](https://arxiv.org/abs/math/0404360))

* {#RobertsWillerton} [[Justin Roberts]], [[Simon Willerton]], Section 8 of: _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

Review:

* {#Roberts01} [[Justin Roberts]], Section 8 of: _Rozansky-Witten theory_ ([arXiv:math/0112209](https://arxiv.org/abs/math/0112209))


whiskering

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# Whiskering
* table of contents
{: toc}

## Idea

In a [[2-category]], the [[horizontal composition]] of a [[2-morphism]] with [[1-morphisms]] is sometimes called _whiskering_.

Whiskering from the left with an [[equivalence]] and from the right with an inverse equivalence is a [[conjugation]] [[action]] of equivalences on [[2-morphisms]].

## Examples

An important use of whiskering is the usual definition of [[adjoint functor]]s via the [[triangle identities]]: in [[Cat]] whiskering is the [[composition]] of a [[functor]] with a [[natural transformation]] to produce a natural transformation.

If we identify a functor or 1-morphism with its [[identity natural transformation]] or [[identity 2-morphism]], then whiskering is a special case of [[horizontal composition]], and composition of 1-morphisms is a special case of whiskering.

In detail:

* If $F,G\colon C \to D$ and $H\colon D\to E$ are functors and $\eta\colon F \to G$ is a natural transformation whose coordinate at any object $A$ of $C$ is $\eta_A$, then __whiskering__ $H$ and $\eta$ yields the natural transformation $H \circ \eta\colon (H \circ F) \to (H \circ G)$ whose coordinate at $A$ is $H(\eta_A)$.
* If $F\colon C \to D$ and $G,H\colon D \to E$ are functors and $\eta\colon G\to H$ is a natural transformation whose coordinate at $A$ is $\eta_A$, then __whiskering__ $\eta$ and $F$ yields the natural transformation $\eta \circ F\colon (G \circ F) \to (H \circ F)$ whose coordinate at $A$ is $\eta_{F(A)}$.

## In dependent type theory

In [[dependent type theory]], whiskering is the type theoretic equivalent of the principle in [[set theory]] that given sets $A$, $B$, and $C$, and functions $f:A \to B$ and $g:A \to B$, if $f(x) = g(x)$ for all elements $x \in A$, then

* $h(f(x)) = h(g(x))$ for all functions $h:B \to C$ and elements $x \in A$
* $f(h(x)) = g(h(x))$ for all functions $h:C \to A$ and elements $x \in C$

Given types $A$, $B$, and $C$ and functions $f:A \to B$ and $g:A \to B$ there is a function

$$\mathrm{leftwhiskering}_{A, B, C}(f, g):\left(\prod_{x:A} f(x) =_B g(x)\right) \to \left(\prod_{h:B \to C} \prod_{x:A} h(f(x)) =_C h(g(x))\right)$$

called **left whiskering**, which is defined as the [[lambda abstraction]] of the composite of the [[function application to identities]] of function $h:B \to C$ with [[homotopy]] $H:\prod_{x:A} f(x) =_B g(x)$

$$\mathrm{leftwhiskering}_{A, B, C}(f, g)(H, h) \coloneqq \lambda x.\mathrm{ap}_h(H(x))$$

Left whiskering is frequently written simply as $h \circ H$ or $h \cdot H$.

Given types $A$, $B$, and $C$ and functions $f:A \to B$ and $g:A \to B$, there is a function

$$\mathrm{rightwhiskering}_{A, B, C}(f, g):\left(\prod_{x:A} f(x) =_B g(x)\right) \to \left(\prod_{h:C \to A} \prod_{x:C} f(h(x)) =_B g(h(x))\right)$$

called **right whiskering**, defined as the [[lambda abstraction]] of the composite of [[homotopy]] $H:\prod_{x:A} f(x) =_B g(x)$ with function $h:C \to A$

$$\mathrm{rightwhiskering}_{A, B, C}(f, g)(H, h) \coloneqq \lambda x.H(h(x))$$

Right whiskering is frequently written simply as $H \circ h$ or $H \cdot h$.

## Related concepts

* [[pasting diagram]]

* [[digraph|plane digraphs]]

## References

Whiskering (though not under this name) of [[natural transformations]] is first described in:

* {#Godement58} [[Roger Godement]], Appendix (pp. 269) of: *Topologie algébrique et theorie des faisceaux*, Actualités Sci. Ind. **1252**, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]

For textbook accounts see most of those listed at *[[category]]*, such as:

* [[Saunders MacLane]], §II.5, p. 43 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

For whiskering in [[dependent type theory]]:

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

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

For whiskering in [[category theory]]:

* [A MathOverflow question about whiskering](http://mathoverflow.net/questions/40813/what-is-the-name-for-the-composition-of-a-functor-with-a-natural-transformation/40814#40814)

* [[Peter Selinger]]: Introduction to categorical logic. [pdf](https://math.vanderbilt.edu/dept/conf/tacl2013/coursematerials/SelingerTACL20132.pdf), page 41

[[!redirects whiskering]]
[[!redirects whiskerings]]
[[!redirects whisker]]
[[!redirects whiskers]]
[[!redirects whiskered]]


whisky club

[[whiskypic.jpg:file]]


white noise

#Contents#
* table of contents
{:toc}

## Idea

A *white noise* is a kind of [[noise]]/[[stochastic process]] where all frequencies are represented with the same probability.

## References

* T. Hida, H.-H. Kuo, J. Potthoff, L. Streit, _White Noise. An infinite dimensional calculus_, Kluwer 1993.

* M. de Faria, J. Potthoff, L. Streit, _The Feynman integrand as a Hida distribution_, J. Math. Phys. __32__, 2123 (1991); [doi](http://dx.doi/org/10.1063/1.529184)

See also:

* Wikipedia [white noise](https://en.wikipedia.org/wiki/White_noise)

category: applications, probability


Whitehead

* [[Alfred North Whitehead]]

* [[George Whitehead]]

* [[John Henry Constantine Whitehead]]

category: people