Ping Xu

* [website](http://www.math.psu.edu/ping/)

## Selected writings

On [[twisted equivariant K-theory]] via [[bundle gerbes]] over [[differentiable stacks]]:

* [[Jean-Louis Tu]], [[Ping Xu]], [[Camille Laurent-Gengoux]], _Twisted K-theory of differentiable stacks_, Annales Scientifiques de l’École Normale Supérieure Volume 37, Issue 6, November–December 2004, Pages 841-910 ([arXiv:math/0306138](https://arxiv.org/abs/math/0306138), [doi:10.1016/j.ansens.2004.10.002](https://doi.org/10.1016/j.ansens.2004.10.002))

On [[fiber bundles]] and [[principal bundles]] over [[Lie groupoids]] (e.g. [[orbifolds]]):

* [[Camille Laurent-Gengoux]], [[Jean-Louis Tu]], [[Ping Xu]], _Chern-Weil map for principal bundles over groupoids_, Math. Z. 255, 451–491 (2007) ([arXiv:math/0401420](https://arxiv.org/abs/math/0401420), [doi:10.1007/s00209-006-0004-4](https://doi.org/10.1007/s00209-006-0004-4))

On the [[twisted Chern character]] in [[orbifold K-theory]]:

* [[Jean-Louis Tu]], [[Ping Xu]], _Chern character for twisted K-theory of orbifolds_, Advances in Mathematics Volume 207, Issue 2, 20 December 2006, Pages 455-483 ([arXiv:math/0505267](https://arxiv.org/abs/math/0505267), [doi:10.1016/j.aim.2005.12.001](https://doi.org/10.1016/j.aim.2005.12.001))

On the [[cohomology ring]]-structure of [[twisted equivariant K-theory]]:

* [[Jean-Louis Tu]], [[Ping Xu]], *The ring structure for equivariant twisted K-theory*, J. Reine Angew. Math. 635 (2009), 97–148 ([arXiv:math/0604160](https://arxiv.org/abs/math/0604160), [doi:10.1515/CRELLE.2009.077](https://doi.org/10.1515/CRELLE.2009.077))

category: people


pion

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[nuclear physics]], specifically in the [[chiral perturbation theory]] of [[quantum chromodynamics]], the _pion_ or _pi-meson_ ($\pi$-meson) is the [[isospin]]-triplet scalar-[[meson]] [[field (physics)|field]] in the first-[[generation of fermions]], i.e. a [[bound state]] of an [[up quark]] and a [[down quark]] (a [[light meson]]):

[[!include flavors of fundamental particles -- table]]

\linebreak

## Details

### As the Goldstone boson of chiral symmetry breaking

From the point of view of the [[quark]]-[[model (in theoretical physics)|model]] of [[nuclear physics]], the pion is the [[Goldstone boson]] corresponding to the [[spontaneous symmetry breaking]] of the "[[chiral symmetry|chiral]]"-[[symmetry group]] $SU(2)_R \times SU(2)_L$ to the [[diagonal]] [[subgroup]] $SU(2)_V$, by the [[vacuum expectation value]] $\langle q \bar q\rangle \neq 0$ of the [[quark]]-[[condensate]].

(e.g. [Machleidt-Entem 11, 2.1.3](#MachleidtEntem11))

### Plain pion field

Hence, in the sense of the [[Wigner classification]], the pion field transforms in the [[sign representation]] of the [[Lorentz group]]/[[Pin group]] (is a spacetime [[pseudoscalar]]) and in the [[adjoint representation]] of the [[isospin]] [[group]] [[SU(2)]]

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

As such, a pion [[field history]] is a [[smooth function]] from [[spacetime]] to the [[Lie algebra]] [[su(2)]]

\[
  \label{PlainPionField}
  i \vec \pi
  \;\colon\;
  \mathbb{R}^{3,1}
  \longrightarrow
  \mathfrak{su}(2)
  \,,
\]

where the vecotr notation on the left is to indicate that this is, at each spacetime point ([[event]]) $x \in \mathbb{R}^{3,1}$, an element in a [[real numbers|real]] 3-dimensional [[vector space]]

$$
  i \vec \pi(x) \;\in\; \mathfrak{su}(2) \;\simeq_{\mathbb{R}}\; \mathbb{R}^3
  \,.
$$

This means that for any choice of [[linear basis]] of $\mathfrak{su}(2)$, the pion field decomposes as three real-valued fields.

In the [[nuclear physics]]-literature the common choice is that corresponding to the [[Cartan-Weyl basis]]

$$
  \mathrm{Span}
  \big(
    \{t_+, t_-, t_0\}
  \big)
  \;\simeq_{\mathbb{R}}\;
  \mathfrak{su}(2)
$$

in terms of which the components of the pion field are hence denoted as follows

| [[pion]] field component | [[quark]] [[bound state]] |
|--------------------------|---------------------------|
|      $\pi^0$             |         $u \bar u$ or $d \bar d$ |
|      $\pi^+$             |         $u \bar d$        |
|      $\pi^-$             |         $d \bar u$       |

\linebreak

### Exponentiated pion field

Especially in [[chiral perturbation theory]], the pion field is typically represented as the [[exponentiation]] of (eq:PlainPionField) to an [[SU(2)]]-valued [[field (physics)|field]]

\[
  \label{ExponentiatedPionField}
  e^{i \vec \pi/f_\pi}
  \;\colon\;
  \mathbb{R}^{3,1}
  \longrightarrow
  SU(2)
  \,,
\]

([Witten 83, (2)](#Witten83), [Adkins-Nappi 84, (1)-(3)](#AdkinsNappi84))
nowadays called the _exponentiated pion field_ or often just _the chiral field_, for review see [Machleidt-Entem 11, (2.29)](#MachleidtEntem11), [Rho et al. 16, around (1)](#RhoEtAl16).

Here the [[physical unit|unit]] $f_\pi$ is called the _pion decay constant_.

Assuming that the pion field [[vanishing at infinity|vanishes at spatial infinity]] hence means that the exponentiated pion field is a [[map]]

$$
  e^{i \vec \pi/f_\pi}
  \;\colon\;
  \mathbb{R}^{0,1} \times (\mathbb{R}^3)^{cpt}
  \;=\;
  \mathbb{R}^{0,1} \times S^3
  \longrightarrow
  S^3
  \simeq
  SU(2)
$$

from (the [[time]]-axis [[product manifold|times]]) the [[3-sphere]] to [[SU(2)]]. The [[homotopy class]] of this [[continuous function]], an element of the ([[cohomotopy|co-]])[[homotopy group of spheres]] $\pi_3(S^3) \simeq \pi^3(S^3) \simeq \mathbb{Z}$, is the [[Skyrmion]]-number, or, in fact, the [[baryon]]-number, encoded in the [knotted stucture](On+Vortex+Atoms#RanadaTrueba01) of the pion field.

### Relation to baryon current

Explicitly, the [[baryon current]] is the [[Wess-Zumino-Witten term]] in the [[exponentiated pion field]] ([Witten 83a](#Witten83a), [Witten 83b](#Witten83b)):

$$
  \begin{aligned}
  B_{top}
  & \coloneqq \;
  Tr
  \big(
    ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} )
    \wedge
    ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} )
    \wedge
    ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} )
  \big)
  \\
  & =\;
  \big\langle
    (e^{i \vec \pi/f_\pi})^\ast(\theta)
    \wedge
    (e^{i \vec \pi/f_\pi})^\ast(\theta)
    \wedge
    (e^{i \vec \pi/f_\pi})^\ast(\theta)
  \big\rangle
  \;\;\in\;
  \Omega^3(\mathbb{R}^{3,1})
  \end{aligned}
$$

Here the expression in the first line uses the fact that [[SU(2)]] is a [[matrix group]], while the second line exporesses the same via [[pullback of differential forms|pullback]] of the [[Maurer-Cartan form]] $\theta$ from the group manifold.

### Relation to Skyrmions

The [[skyrmion]]-model (see there) realizes [[baryons]] as [[solitons]]/[[instantons]] in the exponentiated pion field.

## Related concepts

* [[omega-meson]],[[rho meson]]

* [[kaon]]

* [[B-meson]]

* [[quarkonium]]

## References

### General

Introduction and survey:

* {#MachleidtEntem11} [[Rupert Machleidt]], [[David Rodríguez Entem]], _Chiral effective field theory and nuclear forces_, Phys. Rept. 503:1-75, 2011 ([arXiv:1105.2919](https://arxiv.org/abs/1105.2919), [doi:10.1016/j.physrep.2011.02.001](https://doi.org/10.1016/j.physrep.2011.02.001))

* {#Brambilla14} Brambilla et al., Section 3.2.7 of: _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981  ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723), [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

See also

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

### Decays and interactions

On $\gamma \to \pi^0 + \pi^+ + \pi^-$:

* Ruvi Aviv, [[Anthony Zee]], _Low-Energy Theorem for $\gamma \to 3 \pi$_
Phys. Rev. D 5, 2372 (1972) ([doi:10.1103/PhysRevD.5.2372](https://doi.org/10.1103/PhysRevD.5.2372))

* {#Witten83a} [[Edward Witten]], _Global aspects of current algebra_, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 422-432 (doi:10.1016/0550-3213(83)90063-9)

* {#BDDL09}  M. Benayoun, P. David, L. DelBuono, O. Leitner, _A Global Treatment Of VMD Physics Up To The $\phi$: I. $e^+ e^-$ Annihilations, Anomalies And Vector Meson Partial Widths_, Eur. Phys. J. C65:211-245, 2010 ([arXiv:0907.4047](https://arxiv.org/abs/0907.4047))

On [[pion]]-[[nucleon]] [[interaction]]:

* E. Ruiz Arriola, J. E. Amaro, R. Navarro Perez, _Three pion nucleon coupling constants_, Modern Physics Letters A Vol. 31, No. 28, 1630027 (2016) ([arXiv:1606.02171](https://arxiv.org/abs/1606.02171))

On the [[Dalitz decay]] of the [[pion]]:

* {#Dalitz51} [[Richard Dalitz]], _On an alternative decay process for the neutral $\pi$-meson_, Proceedings of the Physical Society. Section A 64 (7), 667, 1951 ([doi:10.1088/0370-1298/64/7/115](https://doi.org/10.1088/0370-1298/64/7/115))

* {#KKN02} Karol Kampf, Marc Knecht, Jiri Novotny, _Some aspects of Dalitz decay $\pi^0 \to e^+ e^- \gamma$_,  presented at Int. Conf. Hadron Structure '02, September 2002, Slovakia ([arXiv:hep-ph/0212243](https://arxiv.org/abs/hep-ph/0212243))

* {#KKN06} Karol Kampf, Marc Knecht, Jiri Novotny, _The Dalitz decay $\pi^0 \to e^+ e^- \gamma$ revisited_, Eur. Phys. J. C46:191-217, 2006 ([arXiv:hep-ph/0510021](https://arxiv.org/abs/hep-ph/0510021))

* {#Berghauser10} Henning Berghäuser, _Investigation of the Dalitz decays and the electromagnetic form factors of the $\eta$ and $\pi^0$-meson_, 2010 ([spire:1358057](https://inspirehep.net/literature/1358057))

* {#Kunkel12} M. Kunkel, _Dalitz Decays of Pseudo-Scalar Mesons_, talk at [Light Meson Decays Workshop August 5, 2012](https://www.jlab.org/conferences/lmd2012/) ([[KunkelDalitzDecay.pdf:file]])

* Sergi González-Solís, _Single and double Dalitz decays of $\pi^0$, $\eta$ and $\eta'$ mesons_, Nuclear and Particle Physics Proceedings Volumes 258–259, January–February 2015, Pages 94-97 ([doi:10.1016/j.nuclphysbps.2015.01.021](https://doi.org/10.1016/j.nuclphysbps.2015.01.021))

* Esther Weil, Gernot Eichmann, Christian S. Fischer, Richard Williams, section III.A of: _Electromagnetic decays of the neutral pion_, Phys. Rev. D 96, 014021 (2017) ([arXiv:1704.06046](https://arxiv.org/abs/1704.06046))

### Exponentiated pion field and Skyrmions

Discussion of the exponentiated pion field ("chiral field") in [[chiral perturbation theory]] and the interpretation of its [[winding number]] as [[Skyrmion]]-number / [[baryon]]

* {#Witten83a} [[Edward Witten]], _Global aspects of current algebra_, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 422-432 (doi:10.1016/0550-3213(83)90063-9)

* {#Witten83b} [[Edward Witten]], _Current algebra, baryons, and quark confinement_, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 433-444 (doi:10.1016/0550-3213(83)90064-0)

* {#AdkinsNappi84} [[Gregory Adkins]], [[Chiara Nappi]], _Stabilization of Chiral Solitons via Vector Mesons_, Phys. Lett. 137B (1984) 251-256 ([spire:194727](http://inspirehep.net/record/194727), doi:10.1016/0370-2693(84)90239-9)

  (beware that the two copies of the text at these two sources differ!)

* {#RhoEtAl16} [[Mannque Rho]] et al., _Introduction_, In: [[Mannque Rho]] et al. (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

### Via holographic QCD

Discussion via the [[AdS/QCD correspondence]]:

* Domenec Espriu, Alisa Katanaeva, _Effects of bulk symmetry breaking on AdS/QCD predictions_ ([arXiv:2001.08723](https://arxiv.org/abs/2001.08723))

* Xuanmin Cao, Matteo Baggioli, Hui Liu, Danning Li, *Pion dynamics in a soft-wall AdS-QCD model* [[arXiv:2210.09088](https://arxiv.org/abs/2210.09088)]

[[!redirects pions]]

[[!redirects pi meson]]
[[!redirects pi mesons]]
[[!redirects pi-meson]]
[[!redirects pi-mesons]]

[[!redirects pion field]]
[[!redirects pion fields]]

[[!redirects chiral field]]
[[!redirects chiral fields]]

[[!redirects exponentiated pion field]]
[[!redirects exponentiated pion fields]]

[[!redirects π meson]]
[[!redirects π mesons]]
[[!redirects π-meson]]
[[!redirects π-mesons]]


Piotr Hajac

* [webpage](https://www.impan.pl/~pmh/)

## Selected writings

On [[parallel transport]] in [[Yang-Mills theory]]:

* [[Piotr Hajac]], _Axiomatic holonomy maps and generalized Yang-Mills moduli space_, Letters in Mathematical Physics volume 27, pages 301–309 (1993) ([doi:10.1007/BF00777377](https://doi.org/10.1007/BF00777377))

On the [[Drinfeld double]]:

* [[Piotr Hajac]], *On and around the Drinfeld double* (2009) [[pdf](http://www.impan.pl/~burgunde/WSBC09/Ddouble_Hajac.pdf), [[Hajac-DrinfeldDouble.pdf:file]]]

## Related entries

* [[Tomasz Maszczyk]]

category: people


Piotr Kucharski

* [GoogleScholar entry](https://scholar.google.com/citations?user=6n_6ZIUAAAAJ&hl=pl)

## Selected writings

Introducing the [[knots-quivers correspondence]]:

* {#KRSS19} [[Piotr Kucharski]], [[Markus Reineke]], [[Marko Stosic]], [[Piotr Sułkowski]], *Knots-quivers correspondence*, Adv. Theor. Math. Phys. **23** 7 (2019) 1849-1902 [[arXiv:1707.04017](https://arxiv.org/abs/1707.04017), [doi:10.4310/ATMP.2019.v23.n7.a4](https://doi.org/10.4310/ATMP.2019.v23.n7.a4)]

* [[Piotr Kucharski]], [[Markus Reineke]], [[Marko Stosic]], [[Piotr Sułkowski]], *BPS states, knots and quivers*, Phys. Rev. D **96** 121902 (2017) [[arXiv:1707.02991](https://arxiv.org/abs/1707.02991), [doi:10.1103/PhysRevD.96.121902](https://doi.org/10.1103/PhysRevD.96.121902)]

category: people


Piotr Podleś

[[!redirects Piotr Podles]]

__Piotr Podleś__ is a Polish mathematical physicist known for early new examples of quantum homogeneous spaces, [[Podleś sphere]]s, from

* Piotr Podleś, _Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups_, Commun.Math.Phys. 170 (1995) 1-20 [doi](https://doi.org/10.1007/BF02099436) [hep-th/9402069](https://arxiv.org/abs/hep-th/9402069)

Other works include

* [[Piotr Podleś]], [[S. L. Woronowicz]], _Quantum deformation of Lorentz group_, Comm. Math. Phys. __130__(2) (1990) 381--431 [doi](https://doi.org/10.1007/bf02473358)

category: people


Piotr Sułkowski

* [webpage](http://psulkows.fuw.edu.pl/)

## Selected writings

Introducing the [[knots-quivers correspondence]]:

* {#KRSS19} [[Piotr Kucharski]], [[Markus Reineke]], [[Marko Stosic]], [[Piotr Sułkowski]], *Knots-quivers correspondence*, Adv. Theor. Math. Phys. **23** 7 (2019) 1849-1902 [[arXiv:1707.04017](https://arxiv.org/abs/1707.04017), [doi:10.4310/ATMP.2019.v23.n7.a4](https://doi.org/10.4310/ATMP.2019.v23.n7.a4)]

* [[Piotr Kucharski]], [[Markus Reineke]], [[Marko Stosic]], [[Piotr Sułkowski]], *BPS states, knots and quivers*, Phys. Rev. D **96** 121902 (2017) [[arXiv:1707.02991](https://arxiv.org/abs/1707.02991), [doi:10.1103/PhysRevD.96.121902](https://doi.org/10.1103/PhysRevD.96.121902)]

Review:

* [[Piotr Sułkowski]], Section 5 of: *Quantum fields, strings, and physical mathematics*, PoS **390** (2021) [[arXiv:2104.03350](https://arxiv.org/abs/2104.03350), [doi:10.22323/1.390.0042](https://doi.org/10.22323/1.390.0042)]

On [[machine learning]] the detection of [[unknots]]:

* [[Sergei Gukov]], [[James Halverson]], [[Fabian Ruehle]], [[Piotr Sułkowski]], *Learning to Unknot*, Mach. Learn.: Sci. Technol. **2** (2022) 025035 [[arXiv:2010.16263](https://arxiv.org/abs/2010.16263), [doi:10.1088/2632-2153/abe91f](https://iopscience.iop.org/article/10.1088/2632-2153/abe91f)]

category: people

[[!redirects Piotr Sulkowski]]


pitch-class

# Contents
* this block creates the table of contents, leave as is
{: toc}

## Introduction

In [[music theory]], the term "pitch" is generally used to mean an individual [[frequency]], such as the pitch A4, which has a frequency of 440 Hz. The concept of pitch therefore does not take into account octave equivalence, so A4 (440 Hz) and A5 (880 Hz) are distinct pitches. On the other hand, a *pitch-class* denotes the [[equivalence relation|equivalence class]] of pitches that are octave-equivalent. Therefore A4 and A5 are members of the pitch-class A.

The need to formulate the notion of pitch-class arose with the rise of atonal music theory in the twentieth century. Due to a radical shift in musical idioms, the concepts from the past that were taken for granted for centuries were no longer adequate for music theory. The concept of pitch-class was one such notion. Rahn, in [Rahn 1980](#Rahn80), summarizes the situation as so:

> Tonal theory has been around so long and so pervasively that many of its concepts are built into our basic musical terminology. To call a particular pitch a $C\sharp$ is to accept at least two theoretical assumptions about structures of pitches. The first is the assumption of *pitch-class equivalence*: that all pitches that are called $C\sharp$, in whatever octave, have enough in common (for purposes of musical structure) that it is useful to give all of them the same name–"$C\sharp$". The second assumption of *diatonic functionality*: a $C\sharp$ cannot function as the fourth degree of an $A\flat$ major scale, even though a $C\sharp$ and a $D\flat$ (which can so function) may both be produced by hitting the same key on a piano. Of these two assumptions, that of diatonic functionality is more elaborate, more sophisticated, further along the tree of theoretical development, and more peculiarly "tonal".

> The theory of atonal music can retain the assumption of pitch-class equivalence, but must reject diatonic functionality as being too particularly "tonal". Hence there is a need for a way to notate pitches which does not incorporate diatonic functionality. For decades, theorists have been solving this problem using *integers* to notate pitches. This less specialized notation has allowed us also, as a bonus, to *construct* (rather than assume) the structures of diatonic functionality in tonal music. Tonal theory can in this sense be regarded as a special case of atonal theory, just as atonal theory is a special case (incorporating pitch-class equivalence) of  a more general theory of music.

In music theory, one therefore denotes pitches with integer notation, where 0 usually refers to $C4$, 1 to $C\sharp4$ or $D\flat4$, -12 to $C3$, and so on. One then derives the notion of pitch-class, where two pitches $p$ and $p'$, represented as integers, are equivalent iff there is an $n \in \mathbb{Z}$ such that
\[
p + 12 n = p'.
\]
This equivalence relation means that the set of pitch-classes is therefore treated as the [[cyclic group]] $\mathbb{Z}_{12}$.

It is important to note however that just because we use integers to denote pitches and pitch-classes, this does not mean that all properties of integers and cyclic groups make sense as properties of pitches/pitch-classes. For instance, 13 is a prime number, but it doesn't make sense to think of the pitch $C\sharp5$ as a prime pitch. The use of integers to denote pitches and pitch-classes is therefore a convention, which is used for the expediences of music theory.

Some aspects of the structure of the integers that also correspond to the structure of pitches/pitch-classes are the following:

* **Greater than**. For $a, b \in \mathbb{Z}$ pitches, if $a \lt b$ then the pitch denoted by $a$ sounds lower than the pitch denoted by $b$. Note that this relation does *not* make sense however for pitch-classes $x, y \in \mathbb{Z}_{12}$.

* **Interval-class**. For $w, x, y, z \in \mathbb{Z}_{12}$ pitch-classes, if $x-w = z-y$, then the pitch-class interval between $w$ and $x$ is equal to the pitch-class interval between $y$ and $z$. For instance, the pitch-class interval between $C$ (0) and $E$ (4) is $4$, as is the pitch-class interval between $F$ (5) and $A$ (9). This means that hearing a $C$ and an $E$ sounding simultaneously will have a similar harmonic "flavor" as hearing an $F$ and an $A$ simultaneously.

## References
 {#References}

* Allen Forte, _The Structure of Atonal Music_, Yale University Press, 1973.

* David Lewin, _Generalized Musical Intervals and Transformations_, Oxford University Press, USA, 2010.

* {#Mazzola12} Guerino Mazzola, _The Topos of Music: Geometric Logic of Concepts, Theory, and Performance_, Birkhäuser (2012) [[doi:10.1007/978-3-0348-8141-8](https://doi.org/10.1007/978-3-0348-8141-8)] [[web](https://www.researchgate.net/publication/257259829_The_Topos_of_Music_Geometric_Logic_of_Concepts_Theory_and_Performance)]

* Robert Morris, _Composition with Pitch-classes_, Yale University, 1987.

* {#Rahn80} John Rahn, _Basic atonal theory_, 1980.


pivotal category

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

#Contents#
* table of contents
{:toc}

## Definition

A **pivotal category** is an [[autonomous category]] equipped with a [[monoidal natural transformation|monoidal natural isomorphism]] $A \to (A^*)^*$.  Pivotal categories have also been called "sovereign categories."  This is a kind of [[category with duals]].

A right autonomous category with such an isomorphism is automatically left autonomous too, so the right/left distinction does not apply to pivotal categories. Note additionally that for the double dual to really be a functor we are assuming that every object has a distinguished dual element.

## Intuition

In an autonomous category $\mathscr{C}$, left duals and right duals have no reason to agree. That is, given an object $A\in\mathscr{C}$ we may not necessarily have $A\cong A^{**}$ (See [this MO post](https://mathoverflow.net/questions/303245/the-dual-of-a-dual-in-a-rigid-tensor-category) for a discussion). Even if we do (like in semisimple categories and braided monoidal categories) there is no a-priori reason for this isomorphism to be natural.

For this isomorphism to be natural we need a pivotal structure. Interestingly enough, if a pivotal structure exists then it will be unique up to precomposing with an element of the group $\text{Aut}^{\otimes}(\text{id}_{\mathscr{C}})$ of monoidal natural automorphisms of the identity ([Bartlett, Prop. 5.7, Lemma 6.2](#Bartlett2009)).

In the case of fusion categories, we have that $\text{Aut}^{\otimes}(\text{id}_{\mathscr{C}})\cong \text{Hom}(U_{\mathscr{C}},\mathbb{C}^{\times})$ where $U_{\mathscr{C}}$ is the universal grading group. Hence, there will be only finitely many choices of pivotal structures by a result of [Gelaki](#Gelaki2006).

## Explicit compatibility conditions

Our intuition for pivotal categories is that right duals should also be left duals in a natural/compatible way. This can be made explicit, and gives an alternative definition of pivotal. Namely, a pivotal category is a (right) rigid monoidal category equipped with morphisms

$$\text{ev}^{\text{rev}}_{A}:A^*\otimes A \to 1,$$

$$\text{coev}^{\text{rev}}_A:1\to A\otimes A^*$$

giving $A^*$ the structure of a left dual for $A$, such that $\text{ev}_A$,$\text{coev}_A$,$\text{ev}^{\text{rev}}_A$, $\text{coev}^{\text{rev}}_A$ satisfy appropriate compatibility conditions. The exact conditions feel unmotivated and clunky when written out explicitly in terms of compositions, and hence they are best stated in the language of [[string diagrams]].

\begin{proposition} Let $\mathscr{C}$ be a pivotal category, with natural isomorphism $i:\text{id}_{\mathscr{C}}\to (-)^{**}$. We graphically define the distinguished morphisms

\begin{imagefromfile}
"file_name": "pivotal-eval-coeval.png",
"width": 400,
"unit": "px"
\end{imagefromfile}

The following properties are satisfied:

* These morphisms satisfy the rigidity axioms. This is, for all $A\in \mathscr{C}$

\begin{imagefromfile}
"file_name": "rigidity-reprise.png",
"width": 300,
"unit": "px"
\end{imagefromfile}

* For all $A,B\in \mathscr{C}$,

\begin{imagefromfile}
"file_name": "something-property.png",
"width": 300,
"unit": "px"
\end{imagefromfile}

* For all $A,B\in \mathscr{C}$, $f:A\to B$,

\begin{imagefromfile}
"file_name": "morphism-duals-agree.png",
"width": 200,
"unit": "px"
\end{imagefromfile}

Conversely, any rigid category with duals satisfying these axioms will be pivotal under the functor

\begin{imagefromfile}
"file_name": "duals-to-pivotal.png",
"width": 200,
"unit": "px"
\end{imagefromfile}

This gives a bijection between pivotal categories and rigid categories with duals satisfying these axioms.

\end{proposition}
\begin{proof} See [Bartlett](#Bartlett2009) Section 5.1.
\end{proof}

## In braided monoidal categories

It is a general principle that rigidity behaves much better in [[braided monoidal categories]]. For instance, every braided rigid category will automatically have two-sided duals ([Joyal and Street](#Joyal1993), Proposition 7.2). However, there is no a-priori reason for a pivotal structure to exist. What is true is that pivotal structures can be quantified explicitly in terms of _twists_:

\begin{proposition}[Deligne] Let $\mathscr{C}$ be a braided rigid monoidal category. Given any natural isomorphism $\theta:\id_{\mathscr{C}}\to\id_{\mathscr{C}}$ such that $\theta_{A\otimes B}=\beta_{B,A}\circ \beta_{A,B}\circ (\theta_A\otimes \theta_B)$ for all $A,B\in \mathscr{C}$, the distinguished morphisms

\begin{imagefromfile}
"file_name": "co-dual.png",
"width": 300,
"unit": "px"
\end{imagefromfile}

give $\mathscr{C}$ the structure of a pivotal category. This map induces a canonical bijection between the set of pivotal structures on $\mathscr{C}$ and the set of natural isomorphism $\theta$ satisfying $\theta_{A\otimes B}=\beta_{B,A}\circ \beta_{A,B}\circ (\theta_A\otimes \theta_B)$ for all $A,B\in \mathscr{C}$.
\end{proposition}
\begin{proof} An exposition can be found in this article of [Yetter](#Yetter1992).
\end{proof}

A braided rigid monoidal category with a twist is called [[balanced monoidal category| balanced]] (not to be confused with the other unrelated notion of [[balanced category| balanced]]). This result can be read as saying that a braided rigid monoidal category is balanced if and only if it is pivotal. Morally, we have the following explanation:

>"The lemma is remarkable because the concept of a braided autonomous category does not include any assumption relating the braided structure to the autonomous structure. Moreover, the axioms for a twist depend only on the braided structure, whereas the axioms for a pivotal structure depend only on the autonomous structure. Yet, they are equivalent if $\mathscr{C}$ is braided autonomous" - Peter Selinger

Given a pivotal structure, the twist can be recovered explicitly using graphical language as follows:

\begin{imagefromfile}
"file_name": "graphical-twist.png",
"width": 300,
"unit": "px"
\end{imagefromfile}

The proof of this result is a simple untangling:

\begin{imagefromfile}
"file_name": "graphical-twist-proof.png",
"width": 300,
"unit": "px"
\end{imagefromfile}

## In fusion categories

In every fusion category, we automatically have that every object is isomorphic to its double dual. This is because fusion categories are semisimple, so $A\otimes A^{*}$ admitting a non-zero morphism into $1$ is the same as there being a $1$ in the simple object decomposition of $A\otimes A^{*}$, which in turn is the same as $1$ admitting a non-zero morphism into $A\otimes A^{*}$.

It is a deep conjecture of Etingof, Nikshych, and Ostrik that every fusion category is pivotal. This conjecture was proposed in their seminal work, [On fusion categories](#Etingof2005). What one gets a-priori, however, is a natural isomorphism between every object and its _quadruple_ dual, called the Radford isomorphism. This weaker isomorphism is already very powerful, and serves as an important tool in the theory of fusion categories.

## Related concepts

* **pivotal category

* [[rigid category]]

* [[fusion category]]

* [[spherical category]]

## References

* {#Etingof2005} [[Pavel Etingof]], [[Dmitri Nikshych]], [[Viktor Ostrik]], _On fusion categories_, Annals of Mathematics, [arXiv](https://arxiv.org/abs/math/0203060)

* {#Joyal1993} [[Andre Joyal]], [[Ross Street]], _Braided tensor categories_, Advances in Mathematics, [pdf](https://core.ac.uk/reader/82681130)

* {#Gelaki2006} [[Shlomo Gelaki]], _Nilpotent fusion categories_, Advances in Mathematics, [arXiv](https://arxiv.org/abs/math/0610726)

* {#Yetter1992} [[David Yetter]], _Framed tangles and a theorem of Deligne on braided deformations of tannakian categories_, Contemporary Mathematics, [web](http://www.ams.org/books/conm/134/)

* {#Bartlett2009} [[Bruce Bartlett]], _On unitary 2-representations of finite groups and topological quantum field theory_, PhD thesis, [arXiv](https://arxiv.org/abs/0901.3975)

[[!redirects pivotal category]]
[[!redirects pivotal categories]]
[[!redirects sovereign category]]
[[!redirects sovereign categories]]
[[!redirects pivotal structure]]
[[!redirects pivotal structures]]


pivotal monoidal category

See [Pivotal Category](https://ncatlab.org/nlab/show/pivotal+category)


pivotal symbols

The _pivotal symbols_ or _$3j$ symbols_ are a certain collection of signs attached to a [[fusion category]] arising from analyzing how invariant vectors in a $3$-particle state transform under rotation by a full revolution.

In a [[fusion category]], the nature of [[dual]]s for objects in the category can be encoded in principal $\mathbb{C}^\times$-bundle over the simple objects. Firstly, suppose one goes ahead and chooses an arbitrary dual object $V_i^*$ for every simple object $V_i$. Write
\[
 Dual(V_i \dashv V_i^*)
\]
for the set of ways that $V_i^*$ can be expressed as a right dual of $V_i$.
Notice that $Dual(V_i \dashv V_i^*)$ is a $\mathbb{C}^\times$-[[torsor]] (you can multiply the unit by a nonzero complex number, following which you are forced to multiply the counit by the reciprocal of this number if you want to obey the [[triangle identities|snake equations]]).

In a fusion category, if $V_i^*$ is a right dual of $V_i$, then it is also a left dual of $V_i$, though not in a canonical way. This gives us the principal $\mathbb{C}^\times$-bundle $L$ over the simple objects — the [[fiber]] above $V_i$ is
\[
 L_{V_i} = \left\{Dual(V_i \dashv V_i^*) \stackrel{\sim}\rightarrow Dual(V_i^* \dashv V_i) \right\},
\]
the set of isomorphisms between $Dual(V_i \dashv V_i^*)$ and $Dual(V_i^* \dashv V_i)$.

A _pivotal structure_ or _even-handed structure_ is a [[section]] of this bundle satisfying some naturality properties. In other words, it is a consistent way to turn right duals into left duals. The most stringent condition is that the invariant vectors in a tensor product of three particles must either be totally 'bosonic' or totally 'fermionic' when you rotate them through a full rotation.

That is, the endomorphism
\[
 T_{ijk} : Hom(1,V_i \otimes V_j \otimes V_k)  \rightarrow Hom(1,V_i \otimes V_j \otimes V_k)
\]
(which can be proved to be an involution) given by sending

[[turn.png:pic]]

must be equal to $\pm id$. Assuming this is satisfied (if not, there is no consistent way to turn right duals into left duals), we can call this collection of signs the __$3j$ symbols__ or the __pivotal symbols__ $\epsilon_{ijk}$ of the fusion category:
\[
  T_{ijk} = \epsilon_{ijk} id.
\]
That is to say, the $3j$ symbols are the signs arising from choosing a trivialization of the 'duality bundle' and analyzing how invariant $3$-particle tensors transform when they go through a full revolution.

It turns out that one can express the concept of a [[pivotal structure]] directly in terms of the pivotal symbols $\epsilon_{ijk}$. Namely, a pivotal structure on the category amounts to an $\epsilon$-[[twisted monoidal natural transformation]] of the identity on the category. That is to say, a collection of nonzero complex numbers $t_{i}$ satisfying
\[
 t_j t_k = \epsilon_{ijk} t_i
\]
whenever $V_i$ appears in $V_j \otimes V_k$.
If all the $3j$ symbols were equal to $1$, this would just be an ordinary [[monoidal natural transformation]] of the identity. Perhaps this phenomenon can be understood in a more enlightening way using the language of [[planar algebra|planar algebras]].

# References

* P. Etingof, D. Nikshych and V. Ostrik, [On fusion categories](http://arxiv.org/abs/math/0203060).
* T. J. Hagge and S-M Hong, [Some non-braided fusion categories of rank 3](http://arxiv.org/abs/math/0704.0208).
* B. Bartlett, [On unitary 2-representations of finite groups and topological quantum field theory](http://arxiv.org/abs/0901.3975) (chapter 6).

[[!redirects Pivotal symbols]]


Pivotal symbols > history

< [[Pivotal symbols]]
[[!redirects Pivotal symbols -- history]]


Piyush Kumar

* [webpage](http://psiepsilon.wikia.com/wiki/Piyush_Kumar)

## related $n$Lab entries

* [[string phenomenology]]

* [[Hořava-Witten theory]]

category: people


PL de Rham complex

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of the _PL de Rham complex_ ([Bousfield-Gugenheim 76](#BousfieldGugenheim76), [Sullivan 77](#Sullivan77)) is a variant of that of the [[de Rham complex]] for [[smooth manifolds]] which applies to general [[topological spaces]] and [[simplicial sets]].

The terminology "PL" for "piecewise linear" seems to have been tacitly introduced in [Bousfield-Gugenheim 76](#BousfieldGugenheim76). Beware that despite this commonly adopted terminology (e.g. [Griffith-Morgan 13](#GriffithMorgan13)), the PL de Rham complex consist of piecewise _[[polynomial differential forms|polynomial]]_ [[differential forms]], but polynomial with respect to a piecewise linear structure on the domain space/complex.

In analogy to the [[de Rham theorem]] for [[smooth manifolds]], the [[fundamental theorem of  dg-algebraic rational homotopy theory]] shows that the PL de Rham complex computes the [[rational cohomology]] (or [[real cohomology]], [[complex cohomology]]) of the given [[topological space]]/[[simplicial sets]].

Applied to a topological space that happens to carry the structure of a [[smooth manifold]], the PL de Rham complex is connected by a [[zig-zag]] of [[quasi-isomorphisms]] to the smooth [[de Rham complex]], hence both are [[isomorphism|isomorphic]] in the [[homotopy category]] of the [[model structure on connective dgc-algebras]].

## Definition

Let $k \in \{\mathbb{Q}, \mathbb{R}, \mathbb{C}\}$.

Write

\[
  \label{PolynomialDifferentialFormsOnSimplices}
  \Omega^\bullet_{polydR}
  (\Delta^\bullet)
  \;\colon\;
  \Delta^\op
  \longrightarrow
  dgcAlgebras^{\geq 0}_{k}
\]

for the [[simplicial object]] in [[dgc-algebras]] given by [[polynomial differential forms on simplices]].

+-- {: .num_defn #PLdeRhamComplex}
###### Definition
**([[PL de Rham complex]])**

The for $S \in $ [[sSet]] a [[simplicial set]], its PL de Rham complex is the
[[hom-object]] of [[simplicial objects]] from $S$ to $\Omega^\bullet_{polyDR}$ (eq:PolynomialDifferentialFormsOnSimplices), hence is the following [[end]] in [[dgcAlgebras]]:

\[
  \label{PLdERhamComplexOfSimplicialSet}
  \Omega^\bullet_{PLdR}(S)
  \;\coloneqq\;
  sSet
  \big(
    S,\,
    \Omega^\bullet_{polydR}(\Delta^\bullet)
  \big)
  \;\coloneqq\;
  \underset{
     [n] \in \Delta^{op}
  }{\int}
  \underset{
    S_n
  }{\oplus}
  \Omega^\bullet_{polydR}
  \big(
    \Delta^n
  \big)
  \,.
\]

=--

([Bousfield-Gugenheim 76, Sec. 2, p. 7](#BousfieldGugenheim76))

For $X \in $ [[Top]] a [[topological space]] its _PL de Rham complex_ is the PL de Rham complex as in (eq:PLdERhamComplexOfSimplicialSet) of its [[singular simplicial complex]]:

$$
  \Omega^\bullet_{PLdR}(X)
  \;\coloneqq\;
  \Omega^\bullet_{PLdR}
  \big(
    Sing(X)
  \big)
  \,.
$$

The [[cochain cohomology]] of the PL de Rham complex is _PL de Rham cohomology_

\[
  \label{PLdeRhamCohomology}
  H^\bullet_{PLdR}(-)
  \;\coloneqq\;
  H \Omega^\bullet_{PLdR}(-)
  \,.
\]

## Properties

### Relation to simplicial sets

+-- {: .num_prop #QuillenAdjunctionBetweenSimplicialSetsAndConnectivedgcAlgebras}
###### Proposition
**([[Quillen adjunction between simplicial sets and connective dgc-algebras]])**

The [[PL de Rham complex]]-construction (Def. \ref{PLdeRhamComplex}) is the [[left adjoint]] in a [[Quillen adjunction]] between

* the [[opposite model structure|opposite]] of the [[projective model structure on connective dgc-algebras]]

* the [[classical model structure on simplicial sets]]

$$
  \big(
    DiffGradedCommAlgebras^{\geq 0}_{k}
  \big)^{op}_{proj}
  \underoverset
    {
      \underset
        {\;\;\; exp \;\;\;}
        {\longrightarrow}
    }
    {
      \overset
        {\;\;\;\Omega^\bullet_{PLdR}\;\;\;}
        {\longleftarrow}
    }
    {\bot_{\mathrlap{Qu}}}
  SimplicialSets_{Qu}
$$

=--

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

That the [[PL de Rham complex]] [[functor]] preserves [[cofibrations]], hence sends injections of simplicial sets to surjections of dgc-algebras, is immediate from its construction.

That its [[right adjoint]] preserves fibrations, hence sends cofibrations of dgc-algebras to [[Kan fibrations]], is the statement of [Bousfield-Gugenheim 76, Lemma 8.2](#BousfieldGugenheim76).

=--

### Relation to rational cohomology
  {#RelationToRationalCohomology}

+-- {: .num_prop #PLdeRhamTheorem}
###### Proposition
**([[PL de Rham theorem]])**

Let $k$ be a [[field]] of [[characteristic zero]] (such as the [[rational numbers]], [[real numbers]] or [[complex numbers]]).

Then the evident operation of [[integration of differential forms]] over [[simplices]] induces a [[quasi-isomorphism]] between the [[PL de Rham complex]] with coefficients in $k$ (Def. \ref{PLdeRhamComplex}}) and [[cochain complex]] for [[singular cohomology]] with [[coefficients]] in $k$

$$
  \Omega^\bullet_{PLdR}(X)
  \underoverset{}{\simeq}{\longrightarrow}
  C^\bullet(X; k)
$$

and hence an [[isomorphism]] from [[PL de Rham cohomology]] (eq:PLdeRhamCohomology) to [[ordinary cohomology]] with [[coefficients]] in $k$ (such as [[rational cohomology]], [[real cohomology]], [[complex cohomology]]):

$$
  H^\bullet_{PLdR}(X)
  \underoverset{}{\simeq}{\longrightarrow}
  H^\bullet(X; k)
$$

(for $X$ any [[topological space]]).

=--

([Bousfield-Gugenheim 76, Theorem 2.2](#BousfieldGugenheim76))

### Relation to rational homotopy type

+-- {: .num_defn #NilpotententFiniteRationalHomotopyTypes}
###### Definition
**(nilpotent and finite rational homotopy types)**

Write

\[
  \label{NilpotentQFiniteHomotopyTypes}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)
\]

for the [[full subcategory]] of the [[classical homotopy category]]  ([[homotopy category]] of the [[classical model structure on simplicial sets]]) on those [[homotopy types]] $X$ which are

* [[connected topological space|connected]]: $\pi_0(X) = \ast$

* [[nilpotent space|nilpotent]]: $\pi_1(X)$ is a [[nilpotent group]]

* rational [[finite type]]: $dim_{\mathbb{Q}}\big( H^n(X;,\mathbb{Q}) \big) \lt \infty$ for all $n \in \mathbb{N}$.

and

\[
  \label{NilpotentFiniteRationalHomotopyTypes}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)
\]

for the futher [[full subcategory]] on those [[homotopy types]] that are already [[rational spaces|rational]].

Similarly, write

\[
  \label{NilpotentFiniteTypedgcAlgebras}
  Ho
  \big(
     DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}}
  \big)_{fin}^{\geq 1}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
     DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}}
  \big)
\]

for the [[full subcategory]] of the [[homotopy category]] of the [[projective model structure on connective dgc-algebras]] on those [[dgc-algebras]] $A$ which are

* connected: $H^0(A) \simeq \mathbb{Q}$

* [[finite type]]: $dim_{\mathbb{Q}}\big( H^n(A) \big) \lt \infty$ for all $n \in \mathbb{N}$.

=--

([Bousfield-Gugenheim 76, 9.2](#BousfieldGugenheim76))

+-- {: .num_pro #FundamentalTheoremOfdgAlgebraicRationalHomotopyTheory}
###### Proposition
**([[fundamental theorem of dg-algebraic rational homotopy theory]])**

The [[derived adjunction]]

$$
  Ho
  \left(
    \big(
      DiffGradedCommAlgebras^{\geq 0}_{k}
    \big)^{op}_{proj}
  \right)
  \underoverset
    {
      \underset
        {\;\;\; \mathbb{R} exp \;\;\;}
        {\longrightarrow}
    }
    {
      \overset
        {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;}
        {\longleftarrow}
    }
    {\bot}
  Ho
  \big(
    HoSimplicialSets_{Qu}
  \big)
$$

of the [[Quillen adjunction between simplicial sets and connective dgc-algebras]] (whose [[left adjoint]] is the [[PL de Rham complex]]-[[functor]]) has the following properties:

* on connected, nilpotent rationally finite homotopy types $X$ (eq:NilpotentQFiniteHomotopyTypes) the [[derived adjunction unit]] is [[rationalization]]

  $$
    \array{
    Ho
    \big(
       SimplicialSets_{Qu}
    \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil}
    &
      \overset{
      }{\longrightarrow}
    &
    Ho
    \big(
       SimplicialSets_{Qu}
    \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}
    \\
    X
    &\mapsto&
    \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X)
    }
  $$

  $$
    X
    \underoverset
      {\eta_X^{der}}
      {rationalization}
      {\longrightarrow}
    \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X)
  $$

* on the [[full subcategories]] of nilpotent and finite rational homotopy types from Def. \ref{NilpotententFiniteRationalHomotopyTypes} it restricts to an [[equivalence of categories]]:

  $$
    Ho
    \left(
      \big(
        DiffGradedCommAlgebras^{\geq 0}_{k}
      \big)^{op}_{proj}
    \right)^{\geq 1}_{fin}
    \underoverset
      {
        \underset
          {\;\;\; \mathbb{R} exp \;\;\;}
          {\longrightarrow}
      }
      {
        \overset
          {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;}
          {\longleftarrow}
      }
      {\simeq}
    Ho
    \big(
      HoSimplicialSets_{Qu}
    \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}
  $$

=--

([Bousfield-Gugenheim 76, Theorems 9.4 & 11.2](#BousfieldGugenheim76))

### Relation to smooth de Rham complex
 {#RelationToSmoothdeRhamComplex}

Write

\[
  \label{SmoothDifferentialFormsOnSimplices}
  \Omega^\bullet_{dR}
  (-)
  \;\colon\;
  \Delta^\op
  \longrightarrow
  dgcAlgebras^{\geq 0}_{\mathbb{R}}
\]

for the [[simplicial object]] in [[dgc-algebras]] given by smooth [[differential forms on simplices]].

+-- {: .num_defn #PSdeRhamComplex}
###### Definition
**(PS de Rham complex)**

The for $S \in $ [[sSet]] a [[simplicial set]], its _PS de Rham complex_ ("piecewise smooth") is the [[hom-object]] of [[simplicial objects]] from $S$ to $\Omega^\bullet_{dR}(\Delta^\bullet)$ (eq:SmoothDifferentialFormsOnSimplices), hence is the following [[end]] in [[dgcAlgebras]]:

\[
  \label{PSdERhamComplexOfSimplicialSet}
  \Omega^\bullet_{PSdR}(S)
  \;\coloneqq\;
  sSet
  \big(
    S,\,
    \Omega^\bullet_{dR}(\Delta^\bullet)
  \big)
  \;\coloneqq\;
  \underset{
     [n] \in \Delta^{op}
  }{\int}
  \underset{
    S_n
  }{\oplus}
  \Omega^\bullet_{dR}
  \big(
    \Delta^n
  \big)
  \,.
\]

=--

This receives an evident inclusion from the PL de Rham complex (eq:PSdERhamComplexOfSimplicialSet):

\[
  \label{InclusionOfPolynomialFormsIntoSmoothFormsOnSimplices}
  \Omega_{PLdR}^\bullet(-)
  \overset{
    \phantom{AA} i_{poly} \phantom{AA}
  }{\hookrightarrow}
  \Omega_{PSdR}^\bullet(-)
\]

For $X$ a [[smooth manifold]], and $S(X)$ the [[simplicial complex]] given by any smooth [[triangulation]], notice that:

* there is a [[homeomorphism]] of [[topological spaces]]

  $$
    S(X) \overset{p}{\longrightarrow} X
  $$

  which restricts to a [[diffeomorphism]] onto its image in the [[interior]] of any [[simplex]]

* there is a [[weak homotopy equivalence]] of [[simplicial sets]]

  $$
    S(X)
      \overset{ \phantom{AA} i \phantom{AA} }{\hookrightarrow}
    Sing(X)
    =
    Sing\big( \left\vert S(X)\right\vert \big)
  $$

  into the [[singular simplicial set]] of $X$ (this being the [[adjunction unit]] of the $\left\vert - \right\vert \dashv Sing$ [[Quillen equivalence]] between the [[classical model structure on simplicial sets]] and the [[classical model structure on topological spaces]], and in fact equivalently the derived adjunction unit, since $\left\vert -\right\vert$ preserves all weak equivalences, and $Sing$ those between CW-complexes, by [[Ken Brown's lemma]]).

+-- {: .num_defn #PLdeRhamComplex}
###### Definition
**([[PL de Rham complex of smooth manifold is equivalent to de Rham complex]])**

Let $X$ be a [[smooth manifold]].

We have the following [[zig-zag]] of [[dgc-algebra]] [[quasi-isomorphisms]] between the [[PL de Rham complex]] of (the [[topological space]] underlying) $X$ and the smooth [[de Rham complex]] of $X$:

$$
  \array{
    &&
    \Omega^\bullet_{PLdR}
    \big(
      S(X)
    \big)
    &&
    &&
    \Omega^\bullet_{dR}(X)
    \\
    &
      {}^{
        \mathllap{
          i^\ast
        }
      }
      \nearrow
    & &
      \searrow^{
        \mathrlap{
          i_{poly}
        }
      }
    & &
      {}^{
        \mathllap{
          p^\ast
        }
      }
      \swarrow
    \\
    \mathllap{
      \Omega^\bullet_{PLdR}(X)
      \;=\;
    }
    \Omega^\bullet_{PLdR}
    \big(
      Sing(X)
    \big)
    && &&
    \Omega^\bullet_{PSdR}
    \big(
      S(X)
    \big)
  }
$$

Here $S(X)$ is the [[simplicial complex]] corresponding to any smooth [[triangulation]] of $X$.

=--

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

For the two morphisms on the right this is [Griffith-Morgan 13, Cor. 9.9](#GriffithMorgan13).

For the morphism on the left this follows since $S(X) \hookrightarrow Sing(X)$ is a [[weak homotopy equivalence]] and since $\Omega^\bullet_{PLdR}$, being a [[left Quillen functor]] preserves weak equivalences between [[cofibrant objects]] (where every simplicial set being cofibrant), by [[Ken Brown's lemma]].

=--

## Related concepts

* [[de Rham complex]]

* [[equivariant PL de Rham complex]]

* [[rational homotopy theory]]

* [[fundamental theorem of dg-algebraic rational homotopy theory]]

## References

* {#Sullivan77} [[Dennis Sullivan]], _Infinitesimal computations in topology_, Publications Mathématiques de l'IHÉS, 47 (1977), p. 269-331 ([numdam:PMIHES_1977__47__269_0](http://www.numdam.org/item/PMIHES_1977__47__269_0/))

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))

* {#GriffithMorgan13} [[Phillip Griffiths]], [[John Morgan]], _Rational Homotopy Theory and Differential Forms_,  Progress in Mathematics Volume 16, Birkhauser (2013) ([doi:10.1007/978-1-4614-8468-4](https://doi.org/10.1007/978-1-4614-8468-4))

[[!redirects PL de Rham complexes]]

[[!redirects PL de Rham cohomology]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Preliminaries

Write

\[
  \label{PolynomialDifferentialFormsOnSimplices}
  \Omega^\bullet_{polydR}
  (\Delta^\bullet)
  \;\colon\;
  \Delta^\op
  \longrightarrow
  dgcAlgebras^{\geq 0}_{\mathbb{R}}
\]

for the [[simplicial object]] in [[dgc-algebras]] given by [[polynomial differential forms on simplices]] over the [[real numbers]] (see also at [[fundamental theorem of dgc-algebraic rational homotopy theory]] -- [change of scalars](fundamental+theorem+of+dg-algebraic+rational+homotopy+theory#ChangeOfScalars)).

+-- {: .num_defn #PLdeRhamComplex}
###### Definition
**([[PL de Rham complex]])**

The for $S \in $ [[sSet]] a [[simplicial set]], its PL de Rham complex is the
[[hom-object]] of [[simplicial objects]] from $S$ to $\Omega^\bullet_{polyDR}$ (eq:PolynomialDifferentialFormsOnSimplices), hence is the following [[end]] in [[dgcAlgebras]], here over the [[real numbers]]:

\[
  \label{PLdERhamComplexOfSimplicialSet}
  \Omega^\bullet_{PLdR}(S)
  \;\coloneqq\;
  sSet
  \big(
    S,\,
    \Omega^\bullet_{polydR}(\Delta^\bullet)
  \big)
  \;\coloneqq\;
  \underset{
     [n] \in \Delta^{op}
  }{\int}
  \underset{
    S_n
  }{\oplus}
  \Omega^\bullet_{polydR}
  \big(
    \Delta^n
  \big)
  \,.
\]

=--

([Bousfield-Gugenheim 76, Sec. 2, p. 7](#BousfieldGugenheim76))

For $X \in $ [[Top]] a [[topological space]] its _PL de Rham complex_ is the PL de Rham complex as in (eq:PLdERhamComplexOfSimplicialSet) of its [[singular simplicial complex]]:

$$
  \Omega^\bullet_{PLdR}(X)
  \;\coloneqq\;
  \Omega^\bullet_{PLdR}
  \big(
    Sing(X)
  \big)
  \,.
$$

Write

\[
  \label{SmoothDifferentialFormsOnSimplices}
  \Omega^\bullet_{dR}
  (-)
  \;\colon\;
  \Delta^\op
  \longrightarrow
  dgcAlgebras^{\geq 0}_{\mathbb{R}}
\]

for the [[simplicial object]] in [[dgc-algebras]] (over the [[real numbers]]) given by smooth [[differential forms on simplices]].

+-- {: .num_defn #PSdeRhamComplex}
###### Definition
**(PS de Rham complex)**

The for $S \in $ [[sSet]] a [[simplicial set]], its _PS de Rham complex_ ("piecewise smooth") is the [[hom-object]] of [[simplicial objects]] from $S$ to $\Omega^\bullet_{dR}(\Delta^\bullet)$ (eq:SmoothDifferentialFormsOnSimplices), hence is the following [[end]] in [[dgcAlgebras]]:

\[
  \label{PSdERhamComplexOfSimplicialSet}
  \Omega^\bullet_{PSdR}(S)
  \;\coloneqq\;
  sSet
  \big(
    S,\,
    \Omega^\bullet_{dR}(\Delta^\bullet)
  \big)
  \;\coloneqq\;
  \underset{
     [n] \in \Delta^{op}
  }{\int}
  \underset{
    S_n
  }{\oplus}
  \Omega^\bullet_{dR}
  \big(
    \Delta^n
  \big)
  \,.
\]

=--

This receives an evident inclusion from the PL de Rham complex (eq:PSdERhamComplexOfSimplicialSet):

\[
  \label{InclusionOfPolynomialFormsIntoSmoothFormsOnSimplices}
  \Omega_{PLdR}^\bullet(-)
  \overset{
    \phantom{AA} i_{poly} \phantom{AA}
  }{\hookrightarrow}
  \Omega_{PSdR}^\bullet(-)
\]

For $X$ a [[smooth manifold]], and $S(X)$ the [[simplicial complex]] given by any smooth [[triangulation]], notice that:

* there is a [[homeomorphism]] of [[topological spaces]]

  $$
    S(X) \overset{p}{\longrightarrow} X
  $$

  which restricts to a [[diffeomorphism]] onto its image in the [[interior]] of any [[simplex]]

* there is a [[weak homotopy equivalence]] of [[simplicial sets]]

  $$
    S(X)
      \overset{ \phantom{AA} i \phantom{AA} }{\hookrightarrow}
    Sing(X)
    =
    Sing\big( \left\vert S(X)\right\vert \big)
  $$

  into the [[singular simplicial set]] of $X$ (this being the [[adjunction unit]] of the $\left\vert - \right\vert \dashv Sing$ [[Quillen equivalence]] between the [[classical model structure on simplicial sets]] and the [[classical model structure on topological spaces]], and in fact equivalently the derived adjunction unit, since $\left\vert -\right\vert$ preserves all weak equivalences, and $Sing$ those between CW-complexes, by [[Ken Brown's lemma]]).

## Statement

+-- {: .num_prop #PLdeRhamComplex}
###### Proposition
**(PL de Rham complex of smooth manifold is equivalent to de Rham complex)**

Let $X$ be a [[smooth manifold]].

We have the following [[zig-zag]] of [[dgc-algebra]] [[quasi-isomorphisms]] between the [[PL de Rham complex]] of (the [[topological space]] underlying) $X$ and the smooth [[de Rham complex]] of $X$:

$$
  \array{
    &&
    \Omega^\bullet_{PLdR}
    \big(
      S(X)
    \big)
    &&
    &&
    \Omega^\bullet_{dR}(X)
    \\
    &
      {}^{
        \mathllap{
          i^\ast
        }
      }
      \nearrow
    & &
      \searrow^{
        \mathrlap{
          i_{poly}
        }
      }
    & &
      {}^{
        \mathllap{
          p^\ast
        }
      }
      \swarrow
    \\
    \mathllap{
      \Omega^\bullet_{PLdR}(X)
      \;=\;
    }
    \Omega^\bullet_{PLdR}
    \big(
      Sing(X)
    \big)
    && &&
    \Omega^\bullet_{PSdR}
    \big(
      S(X)
    \big)
  }
$$

Here $S(X)$ is the [[simplicial complex]] corresponding to any smooth [[triangulation]] of $X$.

=--

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

For the two morphisms on the right this is [Griffith-Morgan 13, Cor. 9.9](#GriffithMorgan13).

For the morphism on the left this follows since $S(X) \hookrightarrow Sing(X)$ is a [[weak homotopy equivalence]] and since $\Omega^\bullet_{PLdR}$, being a [[left Quillen functor]] preserves weak equivalences between [[cofibrant objects]] (where every simplicial set being cofibrant), by [[Ken Brown's lemma]].

=--

## References

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))

* {#GriffithMorgan13} [[Phillip Griffiths]], [[John Morgan]], _Rational Homotopy Theory and Differential Forms_,  Progress in Mathematics Volume 16, Birkhauser (2013) ([doi:10.1007/978-1-4614-8468-4](https://doi.org/10.1007/978-1-4614-8468-4))

[[!redirects the PL de Rham complex of a smooth manifold is equivalent to the de Rham complex]]


PL de Rham theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _PL de Rham theorem_ is the variant of the [[de Rham theorem]] with the smooth [[de Rham complex]] replaced by the [[PL de Rham complex]].

## Statement

+-- {: .num_prop #PLdeRhamTheorem}
###### Proposition
**([[PL de Rham theorem]])**

Let $k$ be a [[field]] of [[characteristic zero]] (such as the [[rational numbers]], [[real numbers]] or [[complex numbers]]).

Then the evident operation of [[integration of differential forms]] over [[simplices]] induces a [[quasi-isomorphism]] between the [[PL de Rham complex]] with coefficients in $k$  and [[cochain complex]] for [[singular cohomology]] with [[coefficients]] in $k$

$$
  \Omega^\bullet_{PLdR}(X)
  \underoverset{}{\simeq}{\longrightarrow}
  C^\bullet(X; k)
$$

and hence an [[isomorphism]] from [[PL de Rham cohomology]] to [[ordinary cohomology]] with [[coefficients]] in $k$ (such as [[rational cohomology]], [[real cohomology]], [[complex cohomology]]):

$$
  H^\bullet_{PLdR}(X)
  \underoverset{}{\simeq}{\longrightarrow}
  H^\bullet(X; k)
$$

(for $X$ any [[topological space]]).

=--

([Bousfield-Gugenheim 76, Theorem 2.2](#BousfieldGugenheim76))

## Lemmas

Besides the [[Poincaré lemma]] for piecewise polynomial forms, the proof uses:

\begin{lemma}\label{ExtensionLemma}
**(extension lemma)**
Given a PL form on the [[boundary of a simplex]], it extends to (hence is the [[pullback of differential forms|restriction of]]) a PL form on the full [[n-simplex]].
\end{lemma}

(e.g. [Griffiths & Morgan 2013, Lemma 9.6](#GriffithsMorgan13))

\begin{proof}  Consider [barycentric coordinates](simplex#BarycentricCoordinates) for the given [[n-simplex]]

$$
  \Delta^n
  \,\equiv\,
  \left\{
    \,
    (x^0, \cdots, x^n)
    \,\in\,
    \mathbb{R}^{n+1}
    \;\Big\vert\;
    \textstyle{\sum}_k x^k = 1
    \,,
    \forall_i \, x^i \geq 0
    \,
  \right\}
$$

such that the $i$th vertex is the point $v_i \in \Delta^n$ with coordinates

$$
  x^n(v_i)
  \;=\;
  \delta^n_i
$$

and the $i$th-face is the subset

$$
  \sigma_i\big(\Delta^n\big)
  \;=\;
  \big\{
    \,
    (x^0, \cdots, x^n)
    \,\in\,
    \Delta^n
    \;\big\vert\;
    x^i = 0
    \,
  \big\}
  \,.
$$

With these coordinate expressions consider the functions

\[
  \label{ProjectionFunctions}
  \begin{array}{rcl}
    \mathllap{
      p_i
      \;\colon\;
    }
    \Delta^n
    \setminus
    \{v_i\}
    &\longrightarrow&
    \sigma_i(\Delta^{n})
    \\
    (x^k)_{k}
    &\mapsto&
    \left(
      \frac{x^k}{1-x^i}
    \right)_{k \neq i}
    \,.
  \end{array}
\]

Observe that given a polynomial form $\alpha$ on $\sigma_i(\Delta^n)$, its [[pullback of differential forms|pullback]] along $p_i$ is a form on $\Delta^n \setminus \{v_i\}$ which is polynomial in the variables $\{x^k\}_{k \neq i}$ and in the variable $1/(1-x^i)$. Therefore there is a power $N_i \in \mathbb{N}$ such that

\[
  \label{PartialExtension}
  \widehat \alpha
  \;\coloneqq\;
  (1 - x^i)^{N_i}
  \cdot
  p_i^\ast(\alpha)
\]

is a differential form on $\Delta^n \setminus \{v_i\}$ such that

1. $\widehat \alpha$ extends to $v_i$ (by zero) to give a polynomial differential form on all of $\Delta^n$;

1. pulled back to the $i$th face, $\widehat \alpha$ coincides with $\alpha$ (there being the pullback of an identity map);

1. if $\alpha$ vanishes on the $j$-face in $\sigma(\Delta^n)$ then $\widehat{\alpha}$ vanishes on the $j$-face of $\Delta^n$ (there being a pullback of the former).

Now to complete the proof, consider a polynomial differential form $\omega$ on the [[boundary of a simplex|boundary]] $\partial \Delta^n$. We need to find an extension $\widehat \omega$ to all of $\Delta_n$.

First consider the above construction (eq:PartialExtension)
$$
  \widehat \omega_0
    \;\coloneqq\;
  \widehat {\omega \vert_{\sigma_0} }
$$
on the restriction of $\omega$ to $\sigma_0(\Delta^n)$ and notice that the difference

$$
  \omega_1
  \;\coloneqq\;
  \omega
    -
  \widehat{\omega}_0 \vert_{\partial \Delta^n}
$$

vanishes on $\sigma_0$ and coincides with $\omega$ on all other faces. Therefore consider next the above construction (eq:PartialExtension)

$$
  \widehat \omega_2
  \;\coloneqq\;
  \widehat{ \omega_1 \vert_{\sigma_1} }
$$

on this difference restricted to $\sigma_1$ and notice that the difference

$$
  \omega_2
  \;\coloneqq\;
  \omega_1
    -
  \widehat{\omega}_1 \vert_{\partial \Delta^n}
$$

vanishes on the union of faces $\sigma_0 \cup \sigma_1$ and coincides with $\omega$ on all remaining faces. Proceeding in this fashion one arrives at

$$
  \omega
  \;=\;
  \underset{
    \widehat{\omega}
  }{
    \underbrace{
      \textstyle{\sum}_k
      \widehat{\omega}_k
    }
  }
  \,
  \Big\vert_{\partial \Delta^n}
$$

and hence the term over the brace is an extension as required.
\end{proof}

\begin{remark}\label{ExtensionLemmaForPiecewiseSmoothForms}
**(extension lemma for piecewise smooth differential forms)**
  The Extension Lemma \ref{ExtensionLemma} holds also for other flavors of differential forms over simplices:

In (the proof of) [Griffiths & Morgan 2013, Cor. 9.9](#GriffithsMorgan13) this is observed for the case of smooth differential forms: Here the proof of Lemma \ref{ExtensionLemma} applies verbatim, except that the multiplication by $(1-x^i)^{N_i}$ in (eq:PartialExtension) needs to be replaced by multiplication with any [[bump function]] which vanishes in a neighbourhood of $x^i = 1$ and is unity for $x^i = 0$.

Moreover, we may observe that this same argument then also applies to differential forms on "extended simplices" (see [this Def.](shape+via+cohesive+path+∞-groupoid#SmoothExtendedSimplices))

$$
  \Delta^n_{ext}
  \,\equiv\,
  \left\{
    \,
    (x^0, \cdots, x^n)
    \,\in\,
    \mathbb{R}^{n+1}
    \;\Big\vert\;
    \textstyle{\sum}_k x^k = 1
    \,
  \right\}
  \,,
$$

where the condition $\forall_i \, x^i \geq 0$ is dropped (which plays no role in the above proof).

This is noteworthy, because it implies, with the discussion at *[[shape via cohesive path ∞-groupoid]]* and using the [[fundamental theorem of dg-algebraic rational homotopy theory]], that that rational space encoded in a [[Sullivan model]]/[[Whitehead L-infinity algebra|Whitehead $L_\infty$-algebra]] $\mathfrak{a}$ is equivalently the [[shape modality|shape]]

$$
  \esh
  \,
  \Omega^1_{dR}(-;\mathfrak{a})_{flat}
$$

of the [[smooth set]] of [[flat L-infinity algebra valued differential forms|flat $\mathfrak{a}$-valued differential forms]].
\end{remark}

## Related statements

* [[PL de Rham complex of smooth manifold is equivalent to de Rham complex]]

* [[equivariant PL de Rham theorem]]

* [[de Rham theorem]]

## References

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))

Including the variant of piecewise smooth forms:

* {#GriffithsMorgan13} [[Phillip Griffiths]], [[John Morgan]], Chapter 9 of: *Rational Homotopy Theory and Differential Forms*,  Progress in Mathematics **16**, Birkhauser (2013) [[doi:10.1007/978-1-4614-8468-4](https://doi.org/10.1007/978-1-4614-8468-4)]


place

#Contents#
* table of contents
{:toc}

## Idea

A __place__ of a [[commutative unital ring]] has different meanings in the literature:

* The most common one is an equivalence class of ([[archimedean field|archimedean]] or non-archimedean) [[absolute value]]s. (An _absolute value_ is a non-trivial multiplicative [[seminorm]].) This convention is often used in [[number theory]], where it applies to [[number field|number]] or [[function fields]].

* It can mean an equivalence class of (possibly higher-rank) [[valuations]]. This convention is sometimes used in non-archimedean [[analytic geometry]], after Zariski's work (Zariski--Riemann space).

* It can mean an equivalence class of morphism to [[fields]]. This convention is used in [[scheme]] theory, where a place is exactly the same thing as a [[prime ideal]] (easy proof).

Other notions of places can be imagined, combining the three above classical examples.

The notion of place in number theory is interesting because it is at the heart of the geometric study of [[zeta functions]]. An improvement on this notion is given by the setting of [[global analytic geometry]], which includes trivial seminorms that allow a natural geometric definition of [[adele]]s and [[idele]]s.

## Related concepts

* [[place at infinity]]

[[!redirects place]]
[[!redirects places]]


place at infinity

an archimedean [[place]]

see _[[Arakelov geometry]]_

[[!redirects places at infinity]]


Plack Collaboration

[[!redirects Plack Collaboration]]

* [webpage](http://www.rssd.esa.int/index.php?project=planck)


Plamen L. Bozhilov

* [InSpire page](https://inspirehep.net/authors/1015520)

## Selected writings

On "[[probe branes]]" in [[string theory]]:

* {#Bozhilov03} [[Plamen L. Bozhilov]], *Probe Branes Dynamics: Exact Solutions in General Backgrounds*, Nucl. Phys. B **656** (2003) 199-225 [[arXiv:hep-th/0211181](https://arxiv.org/abs/hep-th/0211181), doi:10.1016/S0550-3213(03)00119-6]

category: people

[[!redirects Plamen Bozhilov]]


planar algebra

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

#Contents#
* table of contents
{:toc}

## Idea

A _planar algebra_ $\mathcal{P}$ consists of vector spaces $P_n$ together with multilinear operations between them indexed by planar tangles -- large discs
with internal ("input") discs all connected up by non-intersecting curves
called strings. Thus a planar algebra may be thought of as made up
from generators $R_i \in P_n$ to which linear combinations of planar tangles
may be applied to obtain all elements of $\mathcal{P}$.

## References

Details are given in

* [[Vaughan Jones]], _Quadratic Tangles in Planar Algebras_ ([arXiv:1007.1158](http://arxiv.org/abs/1007.1158)),

where the author remarks

> The definition of a planar algebra has evolved a bit since the original one in [Planar algebras I](http://arxiv.org/abs/math/9909027) so we give a detailed definition which is, we hope, the ultimate one, at least for shaded planar algebras.

* [[Vaughan Jones]], _[Planar Algebra Master Class](http://www.ctqm.au.dk/events/2008/MasterClassAug/)_

> Relations will appear with [[knot theory]], [[statistical mechanics]], combinatorial [[group theory]], [[quantum groups]] and most of all, [[subfactors]]. They lead to many different algebraic structures according to the particular planar tangles chosen. Of particular interest recently have been graded algebra structures which give connections with the large $N$ limit of random matrices and free probability.

The relation to [[von Neumann algebra factors|subfactors]] and [[fusion categories]] is discussed in

* [[Scott Morrison]], _Small index subfactors, planar algebras, and fusion categories_, lectures at CT2013 ([pdf](http://tqft.net/math/2013-07-03-Macquarie.pdf))
 {#Morrison}

Plenty of discussion of planar algebras is at

* Secret Blogging Seminar, _[Planar algebra entries](http://en.wordpress.com/tag/planar-algebras/)_

[[!redirects planar algebras]]


planar graph

[[!redirects planar]]

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

#Contents#
* table of contents
{:toc}

## Definition

### Planar graphs

A **planar graph** is a [[graph]] which may be [[embedding|embedded]] into a [[surface]] of [[genus]] 0 (such as a [[sphere]] or the [[Euclidean plane]]) without crossing edges.

### Plane graphs/planar maps

Planarity is usually considered as a [[property]] of graphs, rather than as extra [[structure]]. When the structure reading is intended, this is sometimes referred to as a "[[plane graph]]" or "[[planar map]]": that is, a graph equipped with an embedding into a genus 0 surface. In this sense, a planar graph is just a graph which is isomorphic to the underlying graph of a planar map. In particular, one graph might have multiple non-isomorphic embeddings into the plane, or it may be planar while also admitting embeddings into surfaces of higher genus (e.g., $K_4$ is planar, but can also be embedded into the [[torus]]).

## Properties

### Existence of embeddings

The existence of an embedding of an arbitrary graph $G$ into a surface of genus 0 may be tested by various _planarity criteria_, such as Kuratowski's theorem ($G$ is planar iff it does not contain a subgraph that is an edge [[subdivision]] of $K_5$ or $K_{3,3}$, closely related to Wagner's theorem that this is the case iff $K_5, K_{3, 3}$ are not [[graph minors]] of $G$) or _Mac Lane's planarity criterion_ (the [[cycle space]] of $G$ has a [[basis of a vector space|basis]] such that no edge of $G$ appears in more than two basis vectors).

Related to the [[Kuratowski]] and [[Mac Lane]] characterizations is the [[matroid]] characterization of [[Hassler Whitney]]: a graph is planar iff its [[matroid|graphic matroid]] is [[matroid|dual]] to another graphic matroid.

## Related entries

* [[Euler's formula for planar graphs]]

* [[planar map]]

* [[topological map]]

## References

Some related Wikipedia articles:

* [Planar graph](https://en.wikipedia.org/wiki/Planar_graph)
* [Polyhedral graph](https://en.wikipedia.org/wiki/Polyhedral_graph)

Early articles describing different planarity criteria:

* [[Casimir Kuratowski]] (1930), "Sur le problème des courbes gauches en Topologie", _Fundamenta Mathematicae_, 15 (1):271-283. [eudml](https://eudml.org/doc/212352)
* [[Hassler Whitney]] (1932), "Non-separable and planar maps", _Transactions of the American Mathematical Society_, 34 (2): 339–362. [doi](https://doi.org/10.1090%2FS0002-9947-1932-1501641-2)
* [[Saunders Mac Lane]] (1937), "A combinatorial condition for planar graphs", _Fundamenta Mathematicae_, 28 (1): 22–32. [eudml](https://eudml.org/doc/212919)

[[!redirects planar graphs]]


planar limit

In [[gauge theory]] a _planar limit_ is a limit in which [[scattering amplitudes]] are dominated by [[Feynman diagrams]] which can be given the structure of [[planar graphs]].
In particular, this phenomenon happens in the [[large N limit]] of [[Yang-Mills theory]], as seen from [['t Hooft double line notation]].

See at [[N=4 D=4 super Yang-Mills theory]] for discussion of the main examples.

category: physics
[[!redirects planar limits]]


planar operad

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _[[operad]]_ comes in two broad flavors (apart from the choice of enriching category): _planar operads_ and _[[symmetric operads]]_. Accordingly, planar operads are also called _non-symmetric operads_. Another term is _nonpermutative operads_.

A planar operad is a collection ([[set]]/[[object]] in some enriching category) of $n$-ary operations for all $n \in \mathbb{N}$, equipped with a suitable notion of _composition_. In contrast, a _[[symmetric operad]]_ in addition carries an [[action]] of the [[symmetric group]] $\Sigma_n$ on the object on $n$-ary operations, and all structures are required to respect this action.

The notion of planar operads takes its name from the fact that the operations in a planar operad may naturally be drawn as [[planar trees]] without, in general, a relation between two trees that cannot be related by a planar deformation into each other.

Multi-coloured planar operads over [[Set]] are equivalently known as _[[multicategories]]_.

## Definition

In the context of [[(∞,1)-operads]] $\mathcal{O}$ exhibited by their [[(∞,1)-categories of operators]] $\mathcal{O}^\otimes$, a **planar $(\infty,1)$-operad** is a [[fibration of (∞,1)-operads]]

$$
  \mathcal{O}^\otimes \to Assoc^\otimes
$$

to the [[associative operad]]/[[A-∞ operad]]. ([Lurie, def. 4.1.1.6](#Lurie))

## Properties

### Relation to symmetric operads
 {#RelationToSymmetricOperads}

Planar operads embed into [[symmetric operads]] by [[slice category|slicing]] over the [[associative operad]] (regarded as a symmetric operad). For more details see at [Symmetric operad -- Relation to planar operads](symmetric+operad#RelationToPlanarOperads).

## Examples

* The operad [[Assoc]] for associative [[monoids]] is the [[terminal object]] in the category of planar $V$-operads, for choices such as $V = $  [[Set]], [[sSet]], [[Top]], etc.

## Related concepts

* [[symmetric operad]]

* [[multicategory]]

## References

Discussion in the context of the [[higher algebra]] of [[(∞,1)-operads]] is in section 4.1.1 of

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}

[[!redirects planar operads]]

[[!redirects non-symmetric operad]]
[[!redirects non-symmetric operads]]

[[!redirects nonpermutative operad]]
[[!redirects nonpermutative operads]]

[[!redirects non-permutative operad]]
[[!redirects non-permutative operads]]

[[!redirects planar (∞,1)-operad]]
[[!redirects planar (∞,1)-operads]]
[[!redirects planar (infinity,1)-operad]]
[[!redirects planar (infinity,1)-operads]]


Plancherel measure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[probability distribution]] on [[Young diagrams]].

## Definition

From an [answer](https://math.stackexchange.com/questions/846556/why-is-the-plancherel-measure-interesting/3487507#3487507) by [[Vadim Alekseev]]:

Suppose $G$ is a [[locally compact group]].
What one ultimately wants to study is (upon fixing _a_ Haar measure in the noncompact case) the left regular representation $\lambda\colon G\to U(L^2(G,\mathrm{Haar}))$. Now, general theory tells us that while it's not always possible to decompose $L^2(G)$ as a direct _sum_ of irreducible reprenentations (this already fails for $G=\mathbb{Z}$), it is always possible to decompose it as a [[direct integral]] of irreducible representations (which are parametrised by the [[unitary dual]] $\widehat G$ of $G$). Now, if $G$ is unimodular and type I,  the direct integral decomposition (with respect to _both_ left and right actions of $G$) is as follows:
$$
L^2(G) \cong \int_{\widehat G} H_\pi\,d\mu(\pi),
$$
where $H_\pi = \pi\otimes \pi^*$, and its understanding requires, in particular, to determine the measure $\mu$ on $\widehat G$ such that the above becomes an isometric isomorphism. The unique measure with this property is called the *Plancherel measure* of $G$ (associated to a given Haar measure). Equivalently, it's the unique measure such that
$$
\|f\|_2^2 = \int_{\widehat{G}} \|\pi(f)\|_{\mathrm{HS}}^2 \mathrm{d}\mu(\pi),\quad f\in L^1(G,\mathrm{Haar})\cap L^2(G,\mathrm{Haar}).
$$

From a [[MathOverflow]] [answer](https://mathoverflow.net/questions/284756/characterizing-the-plancherel-measure/284762#284762) by [[Cameron Zwarich]]:

If $G$ is a unimodular second countable Type I group, then the Plancherel measure is the unique measure $\mu$ such that
$$\|f\|_2^2 = \int_{\widehat{G}} \|\pi(f)\|_{\mathrm{HS}}^2 \mathrm{d}\mu(\pi).$$
for every $f \in \mathrm{L}^1(G) \cap \mathrm{L}^2(G)$. This appears as Theorem 18.8.2 in Dixmier's book on $C^*$-algebras.

When $G$ is not unimodular, the question becomes more complicated, because the Plancherel measure needs to be twisted by a section of a line bundle; see the paper of Duflo-Moore on the subject for the gory details. When $G$ is not second countable, I do not know of a published result; the technical details of direct integral theory are more difficult in this case and not standard. When $G$ is not Type I, the decomposition of the left regular representation into irreducibles is no longer unique, and some of the operators on the right-hand side of the formula will fail to have finite Hilbert-Schmidt norm.

The closest analogue to the definition of a [[Haar measure]] on abelian [[locally compact groups]] as a left-invariant [[Radon measure]] is the characterization of the Plancherel measure as a unique co-invariant trace (or weight) on the von Neumann algebra $\mathcal{M}$ generated by the left-regular representation of $G$. Suppose $G$ satisfies the same hypotheses as above and $\Delta : \mathcal{M} \to \mathcal{M} \overline{\otimes} \mathcal{M}$ is the comultiplication on $\mathcal{M}$ given by $\lambda(s) \mapsto \lambda(s) \otimes \lambda(s)$. Then the Plancherel trace is the unique normal semifinite trace $\tau$ on $\mathcal{M}$ such that
$$\tau((\varphi \otimes \mathrm{id}) (\Delta(a))) = \tau(a)$$
for all $a \in \mathcal{M}_\tau^+$ and $\varphi \in \mathcal{M}_*$. A similar characterization holds for the Plancherel weight of an arbitrary locally compact group, or for the Haar weight of a locally compact quantum group. For proofs, see volume 2 of Takesaki or any of the literature on von Neumann algebraic quantum groups.

## Definition for symmetric groups

For $\lambda$ a [[partition]]/[[Young diagram]], its *Plancherel probability* is (see at *[[hook length formula]]*):

$$
  p^{Pl}_\lambda
  \;\coloneqq\;
  \frac
    {
      \big(
        dim(S^{(\lambda)})
      \big)^2
    }
    {n!}
  \;=\;
  n!
  \underset{
    { 1 \leq i \leq rows(\lambda) }
    \atop
    { 1 \leq j \leq \lambda_i }
  }{\prod}
  \frac{1}{ \big(\ell hook_\lambda(i,j)\big)^2 }
  \,,
$$

where

* $S^{(\lambda)}$ denotes the [[complex numbers|complex]] [[irrep]] of the [[symmetric group]] that is labelled by $\lambda$ via the [[representation theory of the symmetric group]] (the $\lambda$th [[Specht module]])

* $\ell hook_\lambda(i,j)$ denotes the *hook length* at the $(i,j)$-box in the Young diagram $\lambda$.

## Properties
 {#Properties}

With respect to the Plancherel measure on $Part(n)$ and in the [[limit of a sequence|limit]] of large $n \to \infty$, the [[logarithm]] of $dim\big( S^{(\lambda)}\big) = \left\vert sYTableaux_\lambda \right\vert$ (see at [[hook length formula]]) is [[almost surely]] approximately constant (i.e. independent of $\lambda$) on the value

$$
  ln
  \big(
    \left\vert sYTableaux_\lambda \right\vert
  \big)
  \;\sim\;
  \tfrac{c}{2}
  \sqrt{n}
  -
  \tfrac{1}{2}\ln(n!)
$$

for some $c \in \mathbb{R}$ (numerically: $c \gt 1.8$), in that for all $\epsilon \in \mathbb{R}_+$ we have

$$
 \underset{n \to \infty}{\lim}
 p^{Pl}
 \left(
   \left\{
     \lambda \in Part(n)
     \;\left\vert\;
       \tfrac{2}{\sqrt{n}}
       ln
       \frac
         {\left\vert sYTableaux_\lambda \right\vert}
         {\sqrt{n!}}
       -
       c
       \;\lt\;
       \epsilon
     \right.
   \right\}
 \right)
 \;=\;
 1
  \,.
$$

([Vershik & Kerov 85, p. 22 (2 of 11)](#VershikKerov85))

## Related concepts

* [[Schur-Weyl measure]]

* [[hook length formula]]

* [[asymptotic representation theory]]

## References

* {#VershikKerov85} [[Anatoly Vershik]], [[Sergei Kerov]], *Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group*, Functional Analysis and Its Applications volume 19, pages 21–31 (1985) ([doi:10.1007/BF01086021](https://doi.org/10.1007/BF01086021))

* Maciej Dołęga, *Central limit theorem for random Young diagrams with respect to Jack measure* 2014 ([pdf](https://www.impan.pl/~mdolega/papers/CLT.pdf))

See also:

* Wikipedia, [Plancherel measure](https://en.wikipedia.org/wiki/Plancherel_measure)

* [[Anatoly Vershik]], *Two lectures on the asymptotic representation theory and statistics of Young diagrams*,  In: Vershik A.M., Yakubovich Y. (eds) *Asymptotic Combinatorics with Applications to Mathematical Physics* Lecture Notes in Mathematics, vol 1815. Springer 2003 ([doi:10.1007/3-540-44890-X_7](https://doi.org/10.1007/3-540-44890-X_7))

* [[G. Olshanski]], *Asymptotic representation theory*, Lecture notes 2009-2010 ([webpage](https://lpetrov.cc/art), [pdf 1](https://storage.lpetrov.cc/Olshanski_ART_course_1.pdf), [pdf 2](https://storage.lpetrov.cc/Olshanski_ART_course_2.pdf))

In relation to [[matrix models]]:

* Suvankar Dutta, Debangshu Mukherjee, Neetu, Sanhita Parihar, *A Unitary Matrix Model for q-deformed Plancherel Growth* ([arXiv:2105.09342](https://arxiv.org/abs/2105.09342))

[[!redirects Plancherel measures]]


Planck Collaboration

* [webpage](http://www.rssd.esa.int/index.php?project=planck)

  [publications](http://www.cosmos.esa.int/web/planck/publications)

## Selected writings

* {#PlanckCollaboration13} [[Planck Collaboration]], _Planck 2013 results. XXII. Constraints on inflation_ ([arXiv:1303.5082](http://arxiv.org/abs/1303.5082))

* {#BICEPKeckPlanck15} [[Planck Collaboration]], [[BICEP2]] _A Joint Analysis of BICEP2/Keck Array and Planck Data_ ([arXiv:1502.00612](http://arxiv.org/abs/1502.00612))

* {#PlanckCollaboration15} [[Planck Collaboration]], _Planck 2015, Overview of results_ ([arXiv:1502.01582](http://xxx.lanl.gov/abs/1502.01582))

* {#PlanckCollaboration15XIII} [[Planck Collaboration]], _Planck 2015 results. XIII. Cosmological parameters_ ([arXiv:1502.01589](http://xxx.lanl.gov/abs/1502.01589))

* {#PlanckCollaboration15XX} [[Planck Collaboration]], _Planck 2015 results. XX. Constraints on inflation_ ([arXiv:1502.02114](http://arxiv.org/abs/1502.02114))

* {#PlanckCollaboration18X} [[Planck Collaboration]], _Planck 2018 results. X. Constraints on inflation_ ([arXiv:1807.06211](https://arxiv.org/abs/1807.06211))

See also

* Eleonora Di Valentino, Alessandro Melchiorri, Joseph Silk, _Planck evidence for a closed Universe and a possible crisis for cosmology_, Nature Astronomy (4th November 2019) ([arXiv:1911.02087](https://arxiv.org/abs/1911.02087))

## Related $n$Lab entries

* [[cosmic structure formation]]

* [[cosmic inflation]]

  * [[Starobinsky model of cosmic inflation]]

  * [[chaotic inflation]]

* [[experiment]]

## Members

The Planck collaboration consists of $\gt 200$ members. Names linked on the $n$Lab include

* [[Jörg Rachen]]

category: people

[[!redirects Planck Collaboration]]
[[!redirects Planck collaboration]]
[[!redirects Plack Collaboration]]
[[!redirects Plack collaboration]]


Planck length

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

#Contents#
* table of contents
{:toc}

## Idea

The fundamental [[physical unit]] of [[length]].

In comparison to macorscopic [[physical units]] such as the [[meter]], the approximate value of the Planck length is $\sim 1.6 \;10^{-35}$ [[meter]].

## Definition

Two important [[physical units]] of [[length]] induced by a [[mass]] $m$ are

1. the [[Compton wavelength]]

   $$
     \ell_m \coloneqq \frac{2 \pi \hbar}{m c}
   $$

1. the _[[Schwarzschild radius]]_

   $$
     r_m \coloneqq 2 m G/c^2
   $$

where

* $c$ is the [[speed of light]];

* $\hbar$ is [[Planck's constant]];

* $G$ is the [[gravitational constant]];

Solving the [[equation]]

$$
  \array{
    & \ell_m &=& r_m
    \\
    \Leftrightarrow & 2\pi\hbar / m c &=& 2 m G / c^2
  }
$$

for $m$ yields the _[[Planck mass]]_

$$
   m_{P} \coloneqq \tfrac{1}{\sqrt{\pi}} m_{\ell = r} = \sqrt{\frac{\hbar c}{G}}
  \,.
$$

The corresponding [[Compton wavelength]] $\ell_{m_{P}}$ is given by the _[[Planck length]]_ $\ell_P$

$$
  \ell_{P} \coloneqq \tfrac{1}{2\pi} \ell_{m_P} = \sqrt{ \frac{\hbar G}{c^3} }
  \,
$$

## Related entries

* [[quantum gravity]]

* [string theory -- scales](string+theory#Scales)

* [[near-horizon geometry]]

* [[meter]], [[micrometer]], [[nanometer]], [[femtometer]]

[[!include fundamental scales -- contents]]

## References

The notion was introduced in:

* {#Planck99} [[Max Planck]], _Über irreversible Strahlungsvorgänge_, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin. 5: 440–480. pp. 478–80, 1899, ([10.1002/andp.19003060105](http://onlinelibrary.wiley.com/doi/10.1002/andp.19003060105/abstract))

See also

* Wikipedia, _[Planck length](http://en.wikipedia.org/wiki/Planck_length)_

[[!redirects Planck lengths]]


Planck mass

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

#Contents#
* table of contents
{:toc}

## Idea

The fundamental [[physical unit]] of [[mass]].

About $10^{19}$ [[GeV]].

## Definition

Two important [[physical units]] of [[length]] induced by a [[mass]] $m$ are

1. the [[Compton wavelength]]

   $$
     \ell_m \coloneqq \frac{2 \pi \hbar}{m c}
   $$

1. the _[[Schwarzschild radius]]_

   $$
     r_m \coloneqq 2 m G/c^2
   $$

where

* $c$ is the [[speed of light]];

* $\hbar$ is [[Planck's constant]];

* $G$ is the [[gravitational constant]];

Solving the [[equation]]

$$
  \array{
    & \ell_m &=& r_m
    \\
    \Leftrightarrow & 2\pi\hbar / m c &=& 2 m G / c^2
  }
$$

for $m$ yields the _[[Planck mass]]_

$$
   m_{P} \coloneqq \tfrac{1}{\sqrt{\pi}} m_{\ell = r} = \sqrt{\frac{\hbar c}{G}}
  \,.
$$

The corresponding [[Compton wavelength]] $\ell_{m_{P}}$ is given by the _[[Planck length]]_ $\ell_P$

$$
  \ell_{P} \coloneqq \tfrac{1}{2\pi} \ell_{m_P} = \sqrt{ \frac{\hbar G}{c^3} }
  \,
$$

## Related entries

[[!include fundamental scales -- contents]]

## References

* {#Planck99} [[Max Planck]], _Über irreversible Strahlungsvorgänge_, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin. 5: 440–480. pp. 478–80, 1899, ([10.1002/andp.19003060105](http://onlinelibrary.wiley.com/doi/10.1002/andp.19003060105/abstract))

See also

* Wikipedia, _[Planck mass](http://en.wikipedia.org/wiki/Planck_mass)_


Planck scale

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] the [[scale]] of the [[Planck mass]] (about $10^{19}$ [[GeV]]) or [[Planck length]] is typically called the _Planck scale_.

## Related concepts

[[length scales]] in the [[observable universe]] (from [[cosmological constant|cosmic]] scales, over [[fundamental particle]]-[[masses]] around the [[electroweak symmetry breaking]] to [[GUT scale]] and [[Planck scale]]):

> graphics grabbed from [Zupan 19](flavour+in+particle+physics#Zupan19)

\linebreak

[[!include fundamental scales -- contents]]

## References

* Wikipedia, _[Planck scale](http://en.wikipedia.org/wiki/Planck_scale)_