Picard stack

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> under construction

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## Idea

Fully generally one might call any [[Picard ∞-groupoid]] equipped with the structure of an [[∞-stack]] a _Picard ∞-stack_. But as with [[Picard groups]] themselves, this fully general concept is typically considered in the special case of [[Picard ∞-groupoids]] of [[∞-line bundles]] over a given [[space]] in [[algebraic geometry]]([[E-∞ geometry]]. That is what we discuss here: [[moduli ∞-stacks]] of [[multiplicative group]]-[[principal ∞-bundles]].

## Definition

### General abstract

For some algebraic [[site]]/[[(∞,1)-site]] such as the [[étale site]] or the [[étale (∞,1)-site]], write $\mathcal{B}$ for the [[(∞,1)-topos of (∞,1)-sheaves]] over that site. For $S\in \mathcal{B}$ any object, write $\mathcal{B}_{/S}$ for its [[slice (∞,1)-topos]].

Here $\mathcal{B}$ contains a canonical [[group object in an (∞,1)-category|group object]] $\mathbb{G}_m \in Grp(\mathcal{B})$, the absolute [[multiplicative group]] given as an [[(∞,1)-presheaf]] by the assignment which sends any [[commutative ring]]/[[E-∞ ring]] to its [[group of units]]/[[∞-group of units]]

$$
  \mathbb{G}_m \;\colon\; R \mapsto R^\times
  \,.
$$

The [[inverse image]] of $\mathbb{G}_m$ under [[base change]] along $S \to \ast$ we will still denote by $\mathbb{G}_m \in Grp(\mathcal{B}_{/S})$.

Write $\mathbf{B}\mathbb{G}_m$ for the [[delooping]] of $\mathbb{G}_m$.

For $X \in \mathcal{B}_{/S}$ any object, then morphisms

$$
  X \longrightarrow \mathbf{B}\mathbb{G}_m
$$

in $\mathcal{B}_{/S}$ modulate $\mathbb{G}_m$-[[principal ∞-bundles]] on $X$, whose canonically [[associated ∞-bundles]] are algebraic $\mathbb{G}_a$-[[∞-line bundles]]. (...) (Notice that by the [[Koszul-Malgrange theorem]] these are often better thought of as line bundles with flat holomorphic connection...)

The [[internal hom]]/[[mapping stack]]

$$
  \mathbf{Pic}(X) \coloneqq [X,\mathbf{B}\mathbb{G}_m]
  \in
  \mathcal{B}_{/\mathcal{S}}
$$

is the _Picard $\infty$-stack_ of $X$.

Unwinding the definitions, this is the [[(∞,1)-presheaf]] which sends $S'\to S$ to the [[∞-groupoid]] of [[∞-line bundle]] on the [[(∞,1)-fiber product]] with $X \to S$:

$$
  \mathbf{Pic}(X) \;\colon\; (S' \to S) \mapsto \mathcal{B}(S' \underset{S}{\times}X, \mathbf{B}\mathbb{G}_m)
  \,.
$$

In essentially this form the definition is indicated for instance in  ([Lurie 04, section 8.2](#LurieDAG)).

In good cases its [[truncated object in an (∞,1)-topos|0-truncation]] is a [[scheme]], in which case it is called the _[[Picard scheme]]_.

### More concrete realization

See at _[Picard Scheme -- Picard stack](Picard+scheme#PicardStack)_.

## Properties

### Differentiation / deformation theory

The [[Lie differentiation]] of $\tau_0 \mathbf{Pic}(X)$ is, if it exists as a [[formal scheme|formal]] [[group scheme]], the [[Artin-Mazur formal group]] $\Phi^1_X$.

## Related concepts

* [[Brauer stack]]

## References

* [[The Stacks Project]], _[The Picard stack](http://stacks.math.columbia.edu/tag/0372)_

* {#Lurie04} [[Jacob Lurie]], section 8.2 of _Derived algebraic geometry_, PhD thesis, 2004 ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG.pdf), [web](http://dspace.mit.edu/handle/1721.1/30144))

* {#Deligne74} _Lettre de Grothendieck à Deligne_, 1974 ([pdf](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/grothendieck_to_deligne_on_sinh.pdf)) (Edited by M. Künzer)

[[!redirects Picard stack]]
[[!redirects Picard stacks]]

[[!redirects Picard ∞-stack]]
[[!redirects Picard ∞-stacks]]


Picard-Fuchs equation

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## Idea

Picard-Fuchs equation is a linear [[ordinary differential equation]] of degree $2g$ for $2g$ periods of a  nonsingular projective curve of genus $g$ over a field $K$ of characteristic $0$. For curves over complex numbers it has been first derived by [[Lazarus Fuchs]]. It may be more abstractly defined using a [[Gauss-Manin connection]] which makes sense in a more general context (beyond curves and algebraic/analytic geometry).

## Literature

A description in the language of [[algebraic geometry]] is in the introduction to

* Nicholas M. Katz, _On the differential equations satisfied by period matrices_, Publications Mathématiques de l'IHÉS __35__ (1968) 71--106 [numdam](http://www.numdam.org/item/PMIHES_1968__35__71_0)
* [[Pierre Deligne]], _Equations différentielles à points singuliers réguliers_,  Lecture Notes in Math. __163__, Springer-Verlag 1970

[[mirror map|Mirror maps]] for families of Calabi-Yau are also obtained from corresponding Picard-Fuchs equations.

See also

* wikipedia [Picard-Fuchs equation](https://en.wikipedia.org/wiki/Picard%E2%80%93Fuchs_equation)

Doran has clarified in which cases the corresponding mirror map is the Hauptmoduln (j-function) for the elliptic curve:

* [[Charles Doran|Charles F. Doran]], _Picard-Fuchs uniformization: modularity of the mirror map and mirror-moonshine_, arxiv:[math.AG/9812162](https://arxiv.org/pdf/math/9812162)

category: algebraic geometry

[[!redirects Picard-Fuchs equations]]

[[!redirects Picard-Fuchs differential equation]]
[[!redirects Picard-Fuchs differential equations]]


Picard-Lefschetz theory

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## Idea

_Picard-Lefschetz theory_ is an approach which relates the study of [[singularities]] of [[holomorphic functions]] to the [[topology]] of the domain [[complex manifold]]; thus it is a certain analogue of [[Morse theory]] in [[complex geometry]]. The starting point is the __Picard-Lefschetz formula__ describing [[monodromy]] at a critical point. There is also extension to other fields (Deligne and Katz).

## References

Related $n$Lab entries include: [[monodromy]], [[locally constant sheaf]], [[Morse theory]], [[holomorphic function]], [[Riemann-Hilbert problem]], [[vanishing cycle]], [[Painlevé transcendent]], [[Milnor fiber]], [[Gauss-Manin connection]], [[Picard-Fuchs equation]], [[slope filtration]], [[Stokes phenomenon]], [[semiclassical approximation]], [[movable singularity]], [[Fuchsian equation]], [[hypergeometric function]]

* wikipedia: [Picard-Lefschetz theory](http://en.wikipedia.org/wiki/Picard%E2%80%93Lefschetz_theory)
* [[V. A. Vassiliev]], _Ramified integrals, singularities and lacunas_, Kluwer 1994, 289+xvii pp. Russian version: Ветвящиеся интегралы, Moscow 2002, 432 pp.
* P. Seidel, _Fukaya categories and Picard-Lefschetz theory_, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich, 2008. viii+326 pp. [MR2009f:53143](https://www.ams.org/mathscinet-getitem?mr=2441780), [doi](https://doi.org/10.4171/063)
* V. A. Vassiliev, _Applied Picard-Lefschetz theory_,  Mathematical Surveys and Monographs __97__. Amer. Math. Soc.  2002. xii+324 pp. [MR2003k:32043](https://www.ams.org/mathscinet-getitem?mr=1930577)
* [[V. A. Vassiliev]], _Stratified Picard-Lefschetz theory_,
Selecta Math. (N.S.) 1 (1995), no. 3, 597--621, [MR96i:32037](http://www.ams.org/mathscinet-getitem?mr=1366625), [doi](https://doi.org/10.1007/BF01589499)
* Henryk Żołądek, _The monodromy group_, Monografie Matematyczne 67, 588 pp. Birkhäuser 2006
* [[Pierre Deligne]], Nicholas Katz, _Groupes de monodromie en géométrie algébrique. II_, Springer Lect. Notes in Math. __340__, 1973, [doi](http://dx.doi.org/10.1007/BFb0060505), [MR0354657](http://www.ams.org/mathscinet-getitem?mr=0354657)

category: analysis, topology, algebraic geometry
[[!redirects Picard--Lefschetz theory]]
[[!redirects Picard-Lefschetz formula]]


Picard-Vessiot theory

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## Idea

__Picard-Vessiot theory__ is the [[Galois theory]] of homogeneous linear [[ordinary differential equations]]. A version for homogeneous linear difference equations has also been developed. Alternatively, terms differential Galois theory and difference Galois theory have also been used.

## Literature

#### Classical references

* E. R. Kolchin, _Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations_, Ann. Math. (2nd ser) __49__:1 (1948) 1-42 [jstor](https://www.jstor.org/stable/1969111) [doi](https://doi.org/10.2307/1969111)
* Charles H. Franke, _Picard-Vessiot theory of linear homogeneous difference equations_, Trans. Amer. Math. Soc. __108__:3 (1963) 491-515 [jstor](https://www.jstor.org/stable/1993595) [doi](https://doi.org/10.2307/1993595)

#### In categorical setup

Tannakian approach has been developed in

* [[Pierre Deligne]], _[[Catégories Tannakiennes]]_, [[Grothendieck Festschrift]], vol. II,  Birkhäuser Progress in Math. 87 (1990) 111-195, [pdf](https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf)
* A. Ovchinnikov, _Tannakian categories, linear differential algebraic groups, and parametrized linear differential equations_, Transformation Groups 14, 195–223 (2009) [doi](https://doi.org/10.1007/s00031-008-9042-9)

In the setup of differential abelian tensor categories over differential rings,

* [[Henri Gillet]], [[Sergey Gorchinskiy]], Alexey Ovchinnikov, _Parameterized Picard–Vessiot extensions and Atiyah extensions_, Adv. Math. __238__ (2013) 322-411 [doi](https://doi.org/10.1016/j.aim.2013.02.006)

Galois theory for differential equations is one of the examples in

* [[Francis Borceux]], [[George Janelidze]], _[[Galois Theories]]_, Cambridge Studies in Advanced Mathematics 72, Cambridge University Press, 2001.

Difference equations are viewed from topos perspective (with sporadic recourse to Galois theory) in

* Ivan Tomašić, _A topos-theoretic view of difference algebra_, [arxiv/2001.09075](https://arxiv.org/abs/2001.09075)

#### Connections to and variants in [[model theory]]

* D. Marker, _Model theory of differential fields_, In Model theory of fields, edited by Marker, Messmer, Pillay. Lecture Notes in Logic 5. ASL-Peters, 2006
* Moshe Kamensky, [[Anand Pillay]], _Interpretations and differential Galois extensions_, IMRN 24 (2016) 7390–7413 [doi](https://doi.org/10.1093/imrn/rnw019)
* Z. Chatzidakis, Anand Pillay, _Generalized Picard-Vessiot extensions and differential Galois cohomology_, 2017, [pdf](https://www3.nd.edu/~apillay/generalizedPV-PHS.zoe.pdf)

#### Hopf algebraic approach

* Mitsuhiro Takeuchi, _A Hopf algebraic approach to the Picard-Vessiot theory_, J. Algebra __122__:2 (1989) 481-509 doi

> ...Hopf algebraic approach simplifies and generalizes the theory. Among other things the generalization gives a Picard-Vessiot theory for positive characteristic, for fields with not necessarily commuting derivations, and even for fields with (a set of) higher derivations.

* K. Amano, A. Masuoka, M. Takeuchi, _Hopf algebraic approach to Picard-Vessiot theory_, Handbook of Algebra, Volume 6, (2009) 127-171 doi

[[!redirects differential Galois theory]]


Pichet Vanichchapongjaroen

* [personal page](http://www.if.nu.ac.th/personal/pichet)

## Selected writings

On the [[M5-brane]]'s [[Green-Schwarz sigma-model]]:

* Sheng-Lan Ko, [[Dmitri Sorokin]], [[Pichet Vanichchapongjaroen]], _The M5-brane action revisited_, JHEP11(2013)072 ([arXiv:1308.2231](https://arxiv.org/abs/1308.2231))

* [[Pichet Vanichchapongjaroen]], *Covariant M5-brane action with self-dual 3-form* ([arXiv:2011.14384](https://arxiv.org/abs/2011.14384))

category: people


picture changing operator

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## Idea

What is called _picture number_ in [[string theory]] is a certain [[graded object|grading]] originally seen in the [[quantization]] of the [[NSR superstring]] ([Friedan-Martinec-Shenker 86](#FriedanMartinecShenker86)).

The [[supergeometry|supergeometric]] meaning was clarified in [Belopolsky 97b](#Belopolsky97b) in terms of an extra grading on [[differential forms on supermanifolds]] induced from a choice of _integral top-form_ needed for a notion of [[integration over supermanifolds]], see [Catenacci-Grassi-Noja 18](#CatenacciGrassiNoja18)

## Related concepts

* [[superstring field theory]]

* [[integration over supermanifolds]]

## References

The appearance in the [[quantization]] of the [[NSR superstring]] is due to

* {#FriedanMartinecShenker86} [[Daniel Friedan]], [[Emil Martinec]], [[Stephen Shenker]], *Conformal invariance, supersymmetry and string theory*, Nucl. Phys. B **271** (1986) 93

A [[supergeometry|supergeometric]] interpretation of picture changing operators as part of the geometry of [[super Riemann surfaces]] is given in

* {#Belopolsky97} [[Alexander Belopolsky]], _New Geometrical Approach to Superstrings_ ([arXiv:hep-th/9703183](http://arxiv.org/abs/hep-th/9703183))

* {#Belopolsky97b} [[Alexander Belopolsky]], _Picture changing operators in supergeometry and superstring theory_ ([arXiv:hep-th/9706033](https://arxiv.org/abs/hep-th/9706033))

further amplified in

* {#Witten12} [[Edward Witten]], appendix D of _Notes On Super Riemann Surfaces And Their Moduli_ ([arXiv:1209.2459](http://arxiv.org/abs/1209.2459))

In this perspective picture number is an extra grading on [[differential forms on supermanifolds]] induced from a choice of _integral top-form_ needed to define [[integration over supermanifolds]]:

* {#CatenacciGrassiNoja18} [[Roberto Catenacci]], [[Pietro Grassi]], S. Noja, _Superstring Field Theory, Superforms and Supergeometry_, Journal of Geometry and Physics Volume 148, February 2020, 103559 ([arXiv:1807.09563](https://arxiv.org/abs/1807.09563))

  (with an eye towards [[superstring field theory]])

* [[Carlo A. Cremonini]], [[Pietro Grassi]], _Pictures from Super Chern-Simons Theory_ ([arXiv:1907.07152](https://arxiv.org/abs/1907.07152))

  (with an eye towards [[super Chern-Simons theory]])

* [[Carlo A. Cremonini]], *The Geometry of Picture Changing Operators* [[arXiv:2305.02828](https://arxiv.org/abs/2305.02828)]

Review:

* [[Pietro Grassi]], *Integral Forms and Applications*, Sestri Levante 2015 ([pdf](https://agenda.infn.it/event/8823/attachments/55101/64989/Grassi-SL.pdf), [[GrassiIntegralForms2015.pdf:file]])

See also:

* [[Ashoke Sen]], [[Edward Witten]]. *Filling the gaps with PCO’s*. JHEP 09 004 (2015) [[doi](https://doi.org/10.1007/JHEP09(2015)004), [arXiv:1504.00609](https://arxiv.org/abs/1504.00609)]

Via [[pure spinors]]:

* Andrei Mikhailov, Dennis Zavaleta, _Geometrical framework for picture changing operators in the pure spinor formalism_ ([arXiv:2003.13995](https://arxiv.org/abs/2003.13995))

[[!redirects picture changing operators]]

[[!redirects picture number]]
[[!redirects picture numbers]]


picture of mechanics

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## Idea

[[dynamics|Dynamics]] affects both [[observables]] and, [[duality between space and quantity|dually]], [[states]]; this is most well known in [[quantum mechanics]] but applies equally well to [[classical mechanics]]. The different "pictures" of mechanics differ in how the dynamics is explicitly formalized:

* In the __Heisenberg picture__, the dependence of observables on [[time]] (or more generally [[spacetime]]) is encoded, while the state is held fixed; the axiomatic formalization of this is given by the [[Haag–Kastler axioms]] of [[AQFT]].

* In the __Schrödinger picture__, states are [[propagator|propagated]] through [[time]], while observables are held fixed; the axiomatic formalization of this is given by [[cobordism category]] [[representations]] in [[FQFT]].

* The __[[Dirac (interaction) picture]]__ is a mixture of these two approaches: dynamics is split into a [[free field theory|free]] (or otherwise [[solvable system|solvable]]) part and an [[perturbation theory|interaction or perturbation]]; one of these is take to affect the states, the other the observables.

The pictures are named after those physicists ([[Werner Heisenberg]], [[Erwin Schrödinger]], and [[Paul Dirac]]) who first used or popularised these approaches to quantum physics.

## Formulation

### With global time

Let us assume a global notion of [[time]], say a [[background field|fixed background]] [[spacetime]] which is [[globally hyperbolic spacetime|globally hyperbolic]], so that it admits a [[foliation]] into [[Cauchy surfaces]], and choose a [[time] [[coordinate]] for this foliation.  The upshot of this is that each [[event]] occurs at a time $t$, and conversely we can speak of [[space]] at any time $t$ (at least within certain bounds).  Thus we may speak sensibly of either the state of the world at time $t$ or the value of some observable quantity at time $t$.

Because this is a picture of [[dynamics]], states or observables (as appropriate to the picture) will vary through time.  We therefore have a [[time evolution]] operator $U(t,t')$ between any two times $t,t'$; actually, we need consider only $U(t) \coloneqq U(t,0)$, since $U(t,t') = U(t) \circ U(t')^{-1}$.

In the Heisenberg picture, for each observable $A$, we speak of $A$ only at some time $t$, so our actual observables are of the form $A(t)$.  We write abstractly
$$ A(t) = A(0) \cdot U(t) $$
to show the evolution of the observable through time.  However, when it comes to the state of the world, we speak of a single state $\psi$ that describes the world at all times.

In the Schrödinger picture, we instead speak of $\psi(t)$, the state of the world at time $t$.  We write abstractly
$$ \psi(t) = U(t) \ast \psi(0) $$
(the [[Schrödinger equation]]) to show the evolution of the state through time.  However, when it comes to observables, we use only the observable $A$ across all times.

To see the connection between the two pictures, recall that an observable $A$ and a state $\psi$ together produce a [[probability distribution]] giving the probability that any given value of $A$ will be observed, given that the world is in state $\psi$.  (This is true throughout mechanics, although it is obscured in non-statistical classical mechanics, since the probability distributions produced by classical [[pure states]] are all [[delta measure]]s.)  Assuming that $A$ belongs to an appropriate [[algebra of observables]] and the probability measures are sufficiently nice, we may restrict attention to the [[expectation value]]s $\langle{A}\rangle_\psi$ of these distributions, since the entire distribution can be recovered from $\langle{A^n}\rangle_\psi$ as $n$ varies over [[natural numbers]].

The connection between the two pictures is then given by
$$ \langle{A \cdot U(t)}\rangle_\psi = \langle{A}\rangle_{U(t) \ast \psi} .$$

It remains to say exactly what $U(t)$ is and what the operations $\cdot$ and $\ast$ are.  Let us use the [[density matrix]] formulation of [[quantum statistical mechanics]], since classical and non-statistical mechanics may be recovered as special cases, by restricting (respectively) the allowed observables or states.  In this case, both states and observables are given by [[linear operators]] on a [[Hilbert space]] $H$, and we have
$$ \langle{A}\rangle_\psi = \tr(A \psi) $$
(using the [[trace]] operation).
Each $U(t)$ is a [[unitary operator]] on $H$ (since time evolution between Cauchy surfaces is a symmetry), and we have
$$ A \cdot U(t) = U(t)^{-1} A U(t) $$
(a [[right action]]) and
$$ U(t) \ast \psi = U(t) \psi U(t)^{-1} $$
(a [[left action]]).
We then have
$$ \langle{A \cdot U(t)}\rangle_\psi
 = tr(U(t)^{-1} A U(t) \psi)
 = tr(A U(t) \psi U(t)^{-1})
 = \langle{A}\rangle_{U(t) \ast \psi} ,$$
as desired, using the [[cyclic property of the trace]].

The time evolution operator $U(t)$ is often derived from a [[Hamiltonian]] and the formula for $A(t)$ or $\psi(t)$ is further derived from a [[differential equation]] involving this Hamiltonian.  However, this is unnecessary for the connection between the two pictures.

### Without time

If spacetime is not [[globally hyperbolic spacetime|globally hyperbolic]], then there is no time coordinate $t$, and none of the discussion above makes sense; or if we choose a coordinate $t$ and call it time regardless, then time evolution is not a symmetry and we do not have the operators $U(t)$.

In this case, the Heisenberg picture still makes sense, even though we cannot expect to calculate $A(t)$ from $A(0)$ (if it even makes sense to discuss such things).  This is easily seen in [[field theory]], where the operators called $A$ above are really of the form $A(x,y,z)$.  Then the Heisenberg picture\'s $A(t)$ is really $A(x,y,z,t)$, or simply $A(p)$ where $p$ indicates an [[event]] (a [[point]] in [[spacetime]]).  So even if the coordinates $x,y,z,t$ do not make sense, still $A(p)$ does; and even if the equations of physics cannot be thought of as describing evolution through time, still they can be thought of as describing the relationships between observables at different places in spacetime.

In contrast, the Schrödinger picture cannot be so treated.  One may be led to the contrary impression by the quantum mechanics of a single particle without any internal structure (not even [[spin]]), in which case the Hilbert space of (pure quantum-mechanical) states is naturally identified with $L^2(\mathbb{R}^3)$ and the state $\psi$ is really $\psi(x,y,z)$.  In this case, the Schrödinger picture\'s $\psi(t)$ is really $\psi(x,y,z,t)$, that is $\psi(p)$.  However, this fails in classical or statistical mechanics; and even in non-statistical quantum mechanics, it breaks down if the particle has internal structure or there is more than one particle in the world.  Then we see that the spacial coordinates $x,y,z$ generalise to the arbitrary coordinates of [[configuration space]], while $t$ remains only $t$, and there is no way to subsume it into a spacetime coordinate.

### In local field theory

from [Torre-Varadarajan 98 p.2](#TorreVaradarajan98):

The idea of evolving a quantum field from any [[Cauchy surface]] to any other seems to have originated in the mid 1940's with the work of Tomonaga [1] and
Schwinger [2] on relativisticquantum field theory. Tomonaga and Schwinger wanted an invariant generalization of the Schrödinger equation, which describes time evolution of the state of a quantum field relative to a fixed inertial reference frame. By allowing for all possible Cauchy surfaces in the
description of dynamical evolution one easily accommodates all possible notions of time for all possible inertial observers. Thus a dynamical formalism incorporating arbitrary Cauchy surfaces does allow for an invariant generalization of the Schrödinger equation. Since, the space of Cauchy surfaces is infinite-dimensional, it is impossible to describe time evolution along arbitrary surfaces by using a single time parameter.  In essence, one needs a distinct time parameter for every possible foliation of spacet
ime. As shown by Tomonaga and Schwinger, if one formulates dynamics in terms of general Cauchy surfaces, the resulting dynamical evolution equation is, formally, a functional differential equation, which is usually called the "[[Tomonaga-Schwinger equation]]. Following [3], we use the term
functional evolution to refer to the formulation of dynamical evolution in which one evolves quantities along arbitrary Cauchy surfaces.

Thus the Tomonaga-Schwinger equation appears as the analog of the Schr ̈odinger equation, when describing functional evolution. It was (and stillis) tacitly assumed that the Tomonaga-Schwinger equation defines the infinitesimal form of unitary evolution of states from one Cauchy surface to another, just as the more familiar (and mathematically more tractable) sCHRÖDINGER equation describes the infinitesimal form of unitary evolution between two hyperplanes of constant Minkowskian time. One of our principal goals in this paper is to show that this assumption is untenable.

from [[Arnold Neumaier]], [PO comment 2016](http://www.physicsoverflow.org/36936/what-picture-the-haag-ruelle-scattering-theory-formulated?show=36953#a36953).

The [[Heisenberg picture]] is the only one where relativistic quantum field theory can be rigorously defined, e.g., through the Wightman axioms. The latter imply the existence of a 4-dimensional group of translations generated by the 4-momentum vector $p$. Once one select a future-like direction to determine time, one gets from $p$ the Hamiltonian $H=p_0$, and can use it to define time-dependent states $\psi(t)$ in the usual way. Together with the spatial part of the momentum, this gives a conventional, frame-dependent Schroedinger representation of the states.

Thus the Schrödinger picture exists but is frame dependent. You can regard any operator $A$ on the Hilbert space as a Schroedinger observable and find its expectation as time changes in the usual Schroedinger way, and translate it to the Heisenberg picture in the usual way such that $A$ becomes explicitly time-dependent and the state is fixed.

However, in 4-dimensional relativistic interacting quantum field theories, fields must be smeared in space and in time in order to produce densely defined operators (rather than distributions). Thus, unlike in the nonrelativistic case, the fixed-time spatial fields $\phi(x)=\Phi(t,x)|_{t=0}$, where $x$ is 3-dimensional, are not well-defined objects.

Thus although the Schroedinger picture exists it does not represent local fields at a fixed time. The attempt to pretend it did leads to the problems mentioned in Dirac's paper. In this sense, the relativistic field theory cannot be made well-defined without transcending the Schroedinger picture.

The above is completely independent of scattering theory. In scattering theory one has to construct the asymptotic Hilbert space. Unlike the interacting Hilbert space, the asymptotic Hilbert space is a Fock space and must therefore be defined by a limiting procedure. This is what is done in the Haag-Ruelle theory. Note that because of the Lorentz invariance of the future cone, the resulting asymptotic space does not depend on the choice of the time direction, as long as it points into the future cone.

There are presentations of the Haag-Ruelle theory that are a little more in the spirit of the Heisenberg picture; see the comments at the end of p.379 of Volume 3 of Reed and Simon. One can probably work almost completely in the Heisenberg picture if one works out the algebra of asymptotic constants (in analogy with the nonrelativistic case treated in Section 3.4 of Volume 3 of Thirring), but to show that the corresponding asymptotic operators have a Fock representation one apparently needs to go through some Schroedinger-like computations.

## History

Historically, the terms 'Schrödinger picture' and 'Heisenberg picture' (at least) referred to *more* than what we discuss above; they referred to the entirety of the differences between Schrödinger\'s and Heisenberg\'s approaches to quantum mechanics.

For example, these terms included also Schrödinger\'s use of typically [[wave]]-like functions as pure states (and correspondingly [[operators]] in the higher-type-theoretic sense as observables) vs Heisenberg\'s use of infinite-dimensional [[matrix|matrices]] as observables (and correspondingly [[infinite sequences]] as pure states).  This difference was rectified by [[John von Neumann|von Neumann]]\'s application of [[Hilbert space]] to the problem, showing that (if one suitably restricts the allowed functions and sequences and also identifies equivalent functions a bit) both approaches used Hilbert space (what we would now call the infinite-dimensional  iseparable Hilbert space) as the space of pure states.

This is entirely separate from the question of whether states or observables are taken to evolve with time.  Still, there is this connection: Schrödinger evolved states, and his approach was called 'wave mechanics' after his representation for states, while Heisenberg evolved observables, and his approach was called 'matrix mechanics' after his representation for observables.

## Examples

* The [[quantization]] of the [[hydrogen atom]] in the various pictures is reviewed in [Nanni 15](#hydrogen+atom#Nanni15), chpapter 6 for the Schrödinger picture and chapter 12 for the Heisenberg picture.

## Related concepts

* [[Tomonaga-Schwinger equation]]

[[!include Isbell duality - table]]

## References

### General

See for instance

* [[Eberhard Zeidler]], sections 7.19.1--3 in _Quantum field theory. A bridge between mathematicians and physicists -- volume I_ Springer (2009) ([web](http://www.mis.mpg.de/zeidler/qft.html)).

To check conventions at least, see Wikipedia:

* [Heisenberg picture](http://en.wikipedia.org/wiki/Heisenberg_picture),
* [Schrödinger picture](http://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture).

Subtleties with the Schrödinger picture for [[field theory]] in [[spacetime]] dimension $\geq 3$ is discussed in

* {#TorreVaradarajan98} [[Charles Torre]], M. Varadarajan, _Functional Evolution of Free Quantum Fields_, Class.Quant.Grav. 16 (1999) 2651-2668 ([arXiv:hep-th/9811222](https://arxiv.org/abs/hep-th/9811222))

[[operator algebra|Operator algebraic]] formulation of [[quantum error correction]] in the Heisenberg picture:

* Cédric Bény, Achim Kempf, David W. Kribs, _Generalization of Quantum Error Correction via the Heisenberg Picture_, Phys. Rev. Lett. 98, 100502 – Published 7 March 2007 ([doi:10.1103/PhysRevLett.98.100502](https://doi.org/10.1103/PhysRevLett.98.100502), [arXiv:quant-ph/0608071](https://arxiv.org/abs/quant-ph/0608071))

* Cédric Bény, Achim Kempf, David W. Kribs, _Quantum Error Correction of Observables_, Phys. Rev. A 76, 042303 (2007) ([arXiv:0705.1574](https://arxiv.org/abs/0705.1574))

[[!include relation between algebraic and functorial field theory -- references]]

[[!redirects dynamical picture]]
[[!redirects picture of dynamics]]
[[!redirects mechanical picture]]
[[!redirects picture of mechanics]]
[[!redirects quantum mechanical picture]]
[[!redirects picture of quantum mechanics]]

[[!redirects Heisenberg picture]]
[[!redirects Schrödinger picture]]
[[!redirects Schroedinger picture]]
[[!redirects Schrodinger picture]]


PIE-limit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[2-category]]-theory, by a **PIE-limit** one means a [[strict 2-limit]] which can be constructed from

1. (P) strict [[cartesian product|products]],

1. (I) [[strict inserters]],

1. (E) [[strict equifiers]].

More precisely, the class of PIE-limits is the [[saturation of a class of weights|saturation]] of the class containing products, inserters, and equifiers.  Any PIE-limit is in particular a [[flexible limit]], and therefore also a (non-strict) [[2-limit]].

Furthermore, all [[strict pseudo-limits]] are PIE-limits, and therefore any [[strict 2-category]] which admits all PIE-limits also admits all non-strict 2-limits, although it may not have all strict 2-limits.  This is the case, for instance, for the 2-category of strict algebras and pseudo morphisms over a [[strict 2-monad]].

Some examples of PIE-limits are:

- [[comma object|Comma objects]]
- [[pseudo limit|Pseudo limits]]
- [[lax limit|Lax limits]]
- [[oplax limit|Oplax limits]]
- [[Eilenberg–Moore objects]] for [[monads]]
- [[power|Powers]]
- [[iso-inserter|Iso-inserters]]
- [[inverters|Inverters]]
- [[descent object|Descent objects]]

An intuition is that PIE-limits are those 2-dimensional limits that do not impose any equations between 1-cells. For instance, [[equalizers]] and [[pullbacks]] are not PIE-limits.

PIE-limits can also be characterized as the coalgebras for a [[pseudo morphism classifier]] [[comonad]], exhibiting them as a 2-categorical version of the notion of [[rigged limit]].

## Characterisation of PIE-weights

A PIE-limit is one whose [[weighted limit|weight]] is PIE.  [Power and Robinson](#PowerRobinson) characterised such weights $W : J^{op} \to Cat$ as those for which the induced functor $ob \circ W_0 \colon J^{op}_0 \to Cat_0 \to Set$ is [[multirepresentable]]. This holds if and only if each connected component of the [[category of elements]] of $ob \circ W_0$ has an [[initial object]].

## Related pages

* [[flexible limit]]

## References

* Blackwell, [[Max Kelly|Kelly]], and [[John Power|Power]], *Two-dimensional monad theory*, Journal of Pure and Applied Algebra 59 (1989) 1-41. doi:[10.1016/0022-4049(89)90160-6](https://doi.org/10.1016/0022-4049%2889%2990160-6)

* {#PowerRobinson} [[John Power]], [[Edmund Robinson]], *A characterization of pie limits*, Math. Proc. Cam. Phil. Soc.  **110** (1991) 33 [[doi:10.1017/S0305004100070092](https://doi.org/10.1017/S0305004100070092)]

* [[John Bourke]], *Codescent objects in 2-dimensional universal algebra*, PhD Thesis (2010), University of Sydney.

On $\lambda$-small objects in PIE-limits:

* [[Leonid Positselski]], _Notes on limits of accessible categories_, [arXiv:2310.16773](https://arxiv.org/abs/2310.16773).

[[!redirects PIE-limit]]
[[!redirects PIE limit]]
[[!redirects PIE-limits]]
[[!redirects PIE limits]]
[[!redirects pie-limit]]
[[!redirects pie limit]]
[[!redirects pie-limits]]
[[!redirects pie limits]]

[[!redirects PIE 2-limit]]
[[!redirects PIE 2-limits]]


piecewise flat spacetime

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

#Contents#
* table of contents
{:toc}

## Idea

If one allows [[pseudo-Riemannian manifolds]] with [[conical singularities]] then it makes sense to ask for [[spacetimes]] which are flat ([[isometry|isometric]] to pieces of [[Minkowski spacetime]]) away from [[stratified space|strata]] of [[positive number|positive]] [[codimension]], with all [[curvature]] concentrated in [[conical singularities]] on these lower-dimensional strata.

Such piecewise flat spacetimes have been considered as discretized models for smooth (non-singular and non-piecewise flat) [[spacetimes]] useful for [[computation]] (see [Williams-Tuckey 92](#WilliamsTuckey92)): A [[combinatorics|combinatorial]] [[functional]] on piecewise flat spacetimes, depending on the [[edge]] [[lengths]] of a metric [[simplicial complex]], was introduced in [Regge 61](#Regge61) (see also [Barrett 87](#Barrett87)) with the idea that in an appropriate [[limit of a sequence|limit]] it approaches the [[Einstein-Hilbert action]] [[functional]] on non-singular [[spacetimes]]. This has become famous as _Regge calculus_. That this [[limit of a sequence|limit]] indeed works out has been proven (only) in [Cheeger-Mueller-Schrader 84](#CheegerMuellerSchrader84), see [Cheeger 16](#Cheeger16) for review.

A variant of this perspective, but with the [[conical singularities]] constrained to be [[timelike]] as expected for "physical" singularities, has been initiated in
['t Hooft 08](#tHooft08) and worked out in some detail by [van de Meent 11](#vandeMeent11).

In both cases a more speculative motivation for considering piecewise flat spacetimes is the hope that it might help with defining [[quantum gravity]], [[non-perturbative QFT|non-perturbatively]] ([Regge-Williams 00](#ReggeWilliams00)). A direct attempt to define and compute a [[path integral quantization]] over piecewise flat spacetimes is known as "causal dynamical triangulation" (see [Ambjorn-Jurkiewicz-Loll 00](#AmbjornJurkiewiczLoll00)).

Piecewise flat spacetimes appear naturally in [[3d quantum gravity|3-dimensional gravity]], which provides much of the inspiration and motivation of various approaches.

But piecewise flat spacetimes also appear naturally as the "[[far-horizon geometry]]" ("small $N$-limit", see there) of [[BPS state|BPS]] [[black brane]] spacetimes in [[supergravity]] theories, where considerations such as at _[[M-theory on G2-manifolds]]_ suggest that the [[conical singularities]] have to be taken seriously as part of the physical model. These [[cone brane]]-singularities are necessarily time-like, as in ['t Hooft 08](#tHooft08), [van de Meent 11](#vandeMeent11), but in contrast to the assumption in general Regge calculus. Of course they are of higher dimension (and higher co-dimension) than considered in ['t Hooft 08](#tHooft08), [van de Meent 11](#vandeMeent11).

## References

### Regge calculus

* {#Regge61} [[Tullio Regge]], _General relativity without coordinates_,  Nuovo Cim (1961) 19: 558 ([doi:10.1007/BF02733251](https://doi.org/10.1007/BF02733251))

* {#CheegerMuellerSchrader84} [[Jeff Cheeger]], Werner Müller, Robert Schrader, _On the curvature of piecewise flat spaces_, Comm. Math. Phys. Volume 92, Number 3 (1984), 405-454 ([euclid:1103940867](https://projecteuclid.org/euclid.cmp/1103940867))

* {#Barrett87} [[John Barrett]], _The geometry of classical Regge calculus_, Classical and Quantum Gravity, Volume 4, Number 6, 1987 ([doi:10.1088/0264-9381/4/6/015/meta](http://iopscience.iop.org/article/10.1088/0264-9381/4/6/015/meta))

* {#WilliamsTuckey92} R. M. Williams, P. A. Tuckey, _Regge calculus: a brief review and bibliography_, Classical and Quantum Gravity, Volume 9, Number 5, 1992 ([web](http://iopscience.iop.org/article/10.1088/0264-9381/9/5/021/meta))

* {#ReggeWilliams00} [[Tullio Regge]], Ruth M. Williams, _Discrete structures in gravity_, J. Math. Phys.41:3964-3984, 2000 ([arXiv:gr-qc/0012035](https://arxiv.org/abs/gr-qc/0012035))

* {#Cheeger16} [[Jeff Cheeger]], _Curvature of piecewise flat spaces_, talk at Courant institute 2016 ([pdf](https://www.ethz.ch/content/dam/ethz/special-interest/phys/theoretical-physics/itp-dam/documents/graf/schrader/Cheeger.pdf), [[Cheeger2016.pdf:file]])

* [[Aleksandar Mikovic]], _Piecewise Flat Metrics and Quantum Gravity_ ([arXiv:2001.11439](https://arxiv.org/abs/2001.11439))

See also

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

Application to [[FRW models]] of [[cosmology]]:

* Ren Tsuda, Takanori Fujiwara, _Oscillating 4-Polytopal Universe in Regge Calculus_ ([arXiv:2011.04120](https://arxiv.org/abs/2011.04120))

### 't Hooft-van de Meent

* {#tHooft08} [[Gerard 't Hooft]], _A locally finite model for gravity_, Found. Phys. 38:733-757, 2008 ([arXiv:0804.0328](https://arxiv.org/abs/0804.0328))

* [[Maarten van de Meent]], _Piecewise Flat Gravitational Waves_,  	Class. Quant. Grav.28:075005, 2011 ([arXiv:1012.1991](https://arxiv.org/abs/1012.1991))

* [[Maarten van de Meent]], _Exact Piecewise Flat Gravitational Waves_, Class. Quantum Grav. 28 (2011) 245006 ([arXiv:1106.5380](https://arxiv.org/abs/1106.5380))

* {#vandeMeent11} [[Maarten van de Meent]], _Piecewise Flat Gravity in 3+1 dimensions_ ([arXiv:1111.6468](https://arxiv.org/abs/1111.6468))

### Causal dynamical triangulation

* J. Ambjorn, R. Loll, _Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change_, Nucl.Phys. B536 (1998) 407-434 ([arXiv:hep-th/9805108](https://arxiv.org/abs/hep-th/9805108))

* J. Ambjorn, J. Jurkiewicz, R. Loll, _Dynamically Triangulating Lorentzian Quantum Gravity_, Nucl.Phys. B610 (2001) 347-382 ([arXiv:hep-th/0105267](https://arxiv.org/abs/hep-th/0105267))

* {#AmbjornJurkiewiczLoll00} J. Ambjorn, J. Jurkiewicz, R. Loll, _A non-perturbative Lorentzian path integral for gravity_, Phys.Rev.Lett. 85 (2000) 924-927 ([arXiv:hep-th/0002050](https://arxiv.org/abs/hep-th/0002050))

* R. Loll, _Quantum Gravity from Causal Dynamical Triangulations: A Review_ ([arXiv:1905.08669](https://arxiv.org/abs/1905.08669))

* Jan Ambjorn, *Lattice Quantum Gravity: EDT and CDT* [[arXiv:2209.06555](https://arxiv.org/abs/2209.06555)]

Review:

* Dario Benedetti, *Landau Theory of Causal Dynamical Triangulations*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.11043](https://arxiv.org/abs/2212.11043)]

* J. Ambjørn, R. Loll, *Causal Dynamical Triangulations: Gateway to Nonperturbative Quantum Gravity*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.09399](https://arxiv.org/abs/2401.09399)]

[[!redirects piecewise flat spacetimes]]

[[!redirects Regge calculus]]

[[!redirects causal dynamical triangulation]]
[[!redirects causal dynamical triangulations]]


piecewise-linear manifold

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

#Contents#
* table of contents
{:toc}

## Idea

A _piecewise-linear manifold_, or _PL manifold_ for short, is a [[manifold]] whose transition functions are [[piecewise linear functions]].

## Related concepts

* [[triangulation theorem]]

## References

See also

* Wikipedia, _[Piecewise linear manifold](https://en.wikipedia.org/wiki/Piecewise_linear_manifold)_

Branched covers:

Every $n$-[[dimension|dimensional]] [[PL manifold]] is a [[branched cover]] of the [[n-sphere]]:

* {#Alexander20} [[James W. Alexander]], _Note on Riemann spaces_, Bull. Amer. Math. Soc. 26 (1920) 370–373

All [[PL manifold|PL]] [[4-manifolds]] are _simple_ branched covers of the  [[4-sphere]]:

* {#Piergallini95} [[Riccardo Piergallini]], _Four-manifolds as 4-fold branched covers of $S^4$_, Topology Volume 34, Issue 3, July 1995 (doi:10.1016/0040-9383(94)00034-I, [pdf](https://core.ac.uk/download/pdf/82379618.pdf))

* {#IoriPiergallini02} Massimiliano Iori, [[Riccardo Piergallini]], _4-manifolds as covers of the 4-sphere branched over non-singular surfaces_, Geom. Topol. 6 (2002) 393-401 ([arXiv:math/0203087](https://arxiv.org/abs/math/0203087))

The [[n-torus]] for $n \geq 3$ is not a [[cyclic group|cyclic]] branched cover of the [[n-sphere]]:

* {#HirschNeumann75} U. Hirsch, W. D. Neumann, _On cyclic branched coverings of spheres_, Math. Ann. (1975) 215: 289 ([doi:10.1007/BF01343895](https://doi.org/10.1007/BF01343895))

[[!redirects piecewise-linear manifolds]]

[[!redirects PL manifold]]
[[!redirects PL manifolds]]

[[!redirects piecewise-linear structure]]
[[!redirects piecewise-linear structures]]

[[!redirects PL structure]]
[[!redirects PL structures]]

[[!redirects PL-manifold]]
[[!redirects PL-manifolds]]
[[!redirects PL manifold]]
[[!redirects PL manifolds]]


Pieralberto Marchetti

* [institute page](http://www.dfa.unipd.it/didattica/master/comunicazione-delle-scienze/docenti/pieralberto-marchetti/)

## Selected writings

On [[anyons]] and [[braid group statistics]] via [[algebraic quantum field theory]]:

* {#FroehlichMarchetti88} [[Jürg Fröhlich]], [[Pieralberto Marchetti]], *Quantum field theory of anyons*, Lett. Math. Phys. **16** (1988) 347–358 (reprinted in [Wilczek 1990, p. 202-213](#Wilczek90)) $[$[doi:10.1007/BF00402043](https://doi.org/10.1007/BF00402043)$]$

and making explicit the role of the [[configuration space of points]]:

* {#FroehlichGabbianiMarchetti90} [[Jürg Fröhlich]], [[Fabrizio Gabbiani]], [[Pieralberto Marchetti]], *Braid statistics in three-dimensional local quantum field theory*, in: H.C. Lee (ed.) *Physics, Geometry and Topology* NATO ASI Series, **238** Springer (1990)  $[$[doi:10.1007/978-1-4615-3802-8_2](https://doi.org/10.1007/978-1-4615-3802-8_2), [[FroehlichGabbianiMarchetti-BraidStatistics.pdf:file]]$]$

category:people

[[!redirects P. Marchetti]]


Pierce spectrum

#Contents#
* table of contents
{:toc}

## Idea

Let $R$ be a commutative [[ring]]. By $E(R)$ denote the [[Boolean algebra]] of idempotents of $R$ whose meet operation is given by the multiplication of $R$. the __Pierce [[spectrum (geometry)|spectrum]]__ $\mathrm{Idl}(E(R))$ of $R$ is the [[partial order|poset]] (in fact [[locale]]) of ideals of $E(R)$.  There is a [[sheaf]] $\bar{R}$ of indecomposable rings (rings whose only idempotents are $0$ and $1$) on $\mathrm{Idl}(E(R))$, called the __Pierce sheaf__, whose ring of sections over the principal ideal $(e)$ (where $e$ is a prime [[filter]] in $E(R)$) is $R_e$.

## Related concepts

* [[spectrum (geometry)]]

## References

F. Borceux, G. Janelidze, Galois Theories, Cambridge studies in advanced mathematics, 72, Cambridge University Press 2001.

P. T. Johnstone, [[Stone Spaces]], Cambridge studies in advanced mathematics 3, Cambrdge Univ. Press 1982.


Piero Truini

* [Inspire page](https://inspirehep.net/authors/1423576)

* [institute page](https://www.difi.unige.it/it/dipartimento/persone/piero-truini)

* [institute page](https://quantumgravityresearch.org/piero-truini/)

## Selected writings

On generalizations of [[exceptional structures]], including [[E8]], [[octonions]] and the [[exceptional Jordan algebra]]:

* [[Piero Truini]], [[Michael Rios]], [[Alessio Marrani]], *The Magic Star of Exceptional Periodicity*, J. Phys.: Conf. Ser. **1194** (2019) 012106 [[arXiv:1711.07881](https://arxiv.org/abs/1711.07881), [doi:10.1088/1742-6596/1194/1/012106](https://iopscience.iop.org/article/10.1088/1742-6596/1194/1/012106)]

category: people


Pierre Ageron

Pierre Ageron is a French mathematician based at Caen. He has published research in Category theory especially on [[sketches]].  He has also written on the History and Philosophy of mathematics.

* [Home Page](https://ageron.users.lmno.cnrs.fr/)

* [Résumé of categorical work](https://ageron.users.lmno.cnrs.fr/recherche.html)

Notes on _L'Art de l'esquisse_, an unfinished book on the theory of [[sketches]]:

* _L'Art de l'esquisse_ (1) ([[L Art de l esquisse 1.xyz:file]])
* _L'Art de l'esquisse_ (2) ([[L Art de l esquisse 2.xyz:file]])
* _L'Art de l'esquisse_ (3) ([[L Art de l esquisse 3.xyz:file]])

* _Esquisse d'extension galoisienne. Esquisses presque inductives et presque projectives_, Séminaire itinérant de catégories à l'Université d'Amiens, LAMFA 18 (2001) 15-18, ([[_Esquisse d extension galoisienne.pdf:file]])

category:people


Pierre Anglès

* [webpage](http://www.icca7.ups-tlse.fr/PierreAngles.htm)

## related $n$Lab entries

* [[conformal geometry]]

category: people


Pierre Berthelot

* [webpage](http://perso.univ-rennes1.fr/pierre.berthelot/)
* [wikipedia](http://en.wikipedia.org/wiki/Pierre_Berthelot)

category:people


Pierre Boutry

* [webpage](http://igg.unistra.fr/People/boutry/)

## related $n$Lab entries

* [[Euclidean geometry]], [[synthetic geometry]]

* [[formal proof]], [[Coq]]


Pierre Cagne

* [website](http://www.normalesup.org/~cagne/)

## Selected writings

On the [[Grothendieck construction for model categories]]:

* [[Pierre Cagne]], [[Paul-André Melliès]], *On bifibrations of model categories*, Advances in Mathematics
**370** (2020) 107205 [[arXiv:1709.10484](https://arxiv.org/abs/1709.10484), [doi:10.1016/j.aim.2020.107205](https://doi.org/10.1016/j.aim.2020.107205)]

On [[symmetry]] and introducing the language of [[homotopy type theory]] for [[univalent foundations of mathematics]]:

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: *[[Symmetry]]* (2021) [[pdf](https://unimath.github.io/SymmetryBook/book.pdf)]

category: people


Pierre Cartier

**Pierre Cartier** is a French mathematician of wide interests (including also mathematical physics), ex-member of Bourbaki.

* [wikipedia](http://en.wikipedia.org/wiki/Pierre_Cartier_%28mathematician%29), [Cartier's homepage at l'IHES](http://www.ihes.fr/~cartier)

## Selected writings

* _A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry_, Bull. Amer. Math. Soc. __38__ (2001), 389-408, [pdf](http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf)

On [[Hopf algebras]]:

* [[Pierre Cartier]], _A primer on Hopf algebras_, Frontiers in number theory, physics, and geometry II, 537--615, preprint IHÉS 2006-40, 81p ([doi:10.1007/978-3-540-30308-4_12](https://doi.org/10.1007/978-3-540-30308-4_12),  [pdf](http://preprints.ihes.fr/2006/M/M-06-40.pdf),  [pdf](http://www.math.osu.edu/~kerler.2/VIGRE/InvResPres-Sp07/Cartier-IHES.pdf), [[CartierHopfAlgebras.pdf:file]])

On [[path integrals]] such as in [[quantum physics]]:

* [[Pierre Cartier]], [[Cécile DeWitt-Morette]], _Functional integration: action and symmetries_, Cambridge Monographs on Mathematical Physics (2006) [[ISBN:9780521143578](https://www.cambridge.org/ae/academic/subjects/physics/theoretical-physics-and-mathematical-physics/functional-integration-action-and-symmetries?format=PB#contentsTabAnchor)]

category: people

[[!redirects P. Cartier]]


Pierre Clairambault

* [webpage](http://www.pps.jussieu.fr/~pclairam/)

## Selected writings

On the [[categorical model of dependent types]] by [[locally cartesian closed categories]]:

* {#ClairambaultDybjer} [[Pierre Clairambault]], [[Peter Dybjer]], *The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories*, in *Typed lambda calculi and applications*, Lecture Notes in Comput. Sci. **6690** Springer (2011) [[arXiv:1112.3456](http://arxiv.org/abs/1112.3456), [doi:10.1007/978-3-642-21691-6_10](https://doi.org/10.1007/978-3-642-21691-6_10)]

category: people


Pierre Conner

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

## Selected writings

Introducing [[equivariant bordism homology theory]]:

* [[Pierre Conner]], [[Edwin Floyd]], _Differentiable periodic maps_, Bull. Amer. Math. Soc. Volume 68, Number 2 (1962), 76-86 ([euclid:bams/1183524501](https://projecteuclid.org/euclid.bams/1183524501))

* [[Pierre Conner]], [[Edwin Floyd]], _Differentiable periodic maps_, Ergebnisse der Mathematik und ihrer Grenzgebiete **33**, Springer 1964 ([doi:10.1007/978-3-662-41633-4](https://link.springer.com/book/10.1007/978-3-662-41633-4))

* [[Pierre Conner]], [[Edwin Floyd]], _Maps of Odd Period_, Annals of Mathematics Second Series, Vol. 84, No. 2 (1966), pp. 132-156 ([jstor:1970515](https://www.jstor.org/stable/1970515))

Introducing [[Conner-Floyd Chern classes]] and the [[complex cobordism theory]]-interpretation of the [[e-invariant]] via the [[Todd genus]] of complex [[manifolds with boundary]] ([[MUFr]]):

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

On [[MU]]-[[bordism theory]]:

* {#ConnerSmith69} [[Pierre Conner]], [[Larry Smith]], _On the complex bordism of finite complexes_, Publications Mathématiques de l'IHÉS, Tome 37 (1969) , pp. 117-221 ([numdam:PMIHES_1969__37__117_0](http://www.numdam.org/item/?id=PMIHES_1969__37__117_0))

On the [[torsion subgroup]] of the [[SU-bordism ring]]:

* [[Pierre Conner]], [[Edwin Floyd]], _Torsion in $SU$-bordism_, Memoirs of the American Mathematical Society 1966 ([ISBN:978-1-4704-0007-1](https://bookstore.ams.org/memo-1-60))

On [[MU]] and introducing the [[Todd character]]:

* [[Pierre Conner]], [[Larry Smith]],  _On the complex bordism of finite complexes. II_, J. Differential Geom. Volume 6, Number 2 (1971), 135-174 ([euclid:euclid.ijm/1256051760](https://projecteuclid.org/euclid.ijm/1256051760))

## Related $n$Lab entries

* [[Conner-Floyd Chern classes]]

* [[Conner-Floyd isomorphism]]

category: people


Pierre Cousin

* [Wikipedia entry](https://fr.wikipedia.org/wiki/Pierre_Cousin)

## related $n$Lab entries

* [[Heine-Borel theorem]]

category: people


Pierre de Fermat

__Pierre de Fermat__ was a 17th-century French mathematician known for:

* [[adequality]] (now a feature of [[nonstandard analysis]]),
* [[infinite descent]] (a method of proof by [[induction]]),
* the [[least time principle]] (a forerunner of the [[least action principle]]),
* [[Fermat's little theorem]] and [[Fermat's last theorem]] concerning rational points of [[Fermat curve]] (in [[number theory]], now related to arithmetic [[algebraic geometry]]),
* the first rigorous [[probability]] calculations,
* etc.

See also _[[Fermat theory]]_

category: people

[[!redirects Pierre de Fermat]]
[[!redirects Pierre Fermat]]
[[!redirects Fermat]]


Pierre Deligne

**Pierre Deligne** is a Belgian mathematician who was a student of [[Alexander Grothendieck]], then worked at [l'IHÉS](http://www.ihes.fr) and now is emeritus at [IAS](http://www.ihes.fr).

* [Wikipedia entry](http://en.wikipedia.org/wiki/Pierre_Deligne),

* [IAS page](http://www.math.ias.edu/~phares/deligne/Deligne.html),

* publication list ([pdf](https://www.ias.edu/sites/default/files/math/deligne/DelignePublications.pdf))

* list of scanned articles: [numdam](http://www.numdam.org/numdam-bin/recherche?au=Deligne,+Pierre&format=short)

* [[Luc Illusie]], _From Pierre Deligne's secret garden: looking back at some of his letters_ ([pdf](http://www.math.u-psud.fr/~illusie/Illusie-Deligne-NYC.pdf))

Deligne's main interests include [[algebraic geometry]] especially cohomology of [[algebraic varieties]], [[Hodge theory]], [[L-functions]] and [[automorphic forms]], [[Tannaka duality|Tannakian theory]], [[representation theory]] of [[algebraic groups]], and [[motives]], where he extended the conjectural picture from pure to [[mixed motives]] (Grothendieck was much earlier aware and pushing toward that extension, though not publishing about it, according to Serre and others).

Deligne has obtained the Fields medal in 1978 for a famous 1973 proof of [[Weil conjectures]].

For the sake of preparatory/foundational steps he wrote a quick amendement for the unfinished volumes of [[SGA]] in a form of practical and short (but controversial to Grothendieck) SGA $4\frac{1}{2}$. This work uses a powerful and deep yoga of [[Hodge filtrations]] discovered also by Deligne.

## Selected writings

Introducing the notion of [[algebraic stacks]] and (what came to be called) [[Deligne-Mumford stacks]], and on the example of the [[moduli space of curves]]:

* {#DeligneMumford69} [[Pierre Deligne]], [[David Mumford]], *The irreducibility of the space of curves of given genus*, Publications Mathématiques de l'IHÉS (Paris) **36**   (1969) 75-109 [[doi:10.1007/BF02684599](https://doi.org/10.1007/BF02684599), [numdam:PMIHES_1969__36__75_0](http://www.numdam.org/item?id=PMIHES_1969__36__75_0)]

On [[differential equations]] with regular singular points (and developing [[local systems]], [[twisted cohomology]], [[twisted de Rham cohomology]], [[Gauss-Manin connections]]):

* [[Pierre Deligne]], *Equations différentielles à points singuliers réguliers*, Lecture Notes  Math. **163**, Springer (1970) $[$[publications.ias:355](https://publications.ias.edu/node/355)$]$

Introducing [[Deligne cohomology]] in [[complex analytic geometry]] (by a [[chain complex]] of [[holomorphic differential forms]]) with applications to [[Hodge theory]] and [[intermediate Jacobians]]:

* {#Deligne71} [[Pierre Deligne]], _Théorie de Hodge II_ , IHES Pub. Math. (1971), no. 40, 5–57 ([pdf](http://www.math.jussieu.fr/~ai/d/Theory%20of%20Hodge%202.pdf))

On [[elliptic curves]] over general [[commutative ring|commutative]] [[ground rings]] (in [[arithmetic geometry]]):

* [[Pierre Deligne]], [[Michael Rapoport]], _Les Schémas de Modules de Courbes Elliptiques_ In: Deligne P., Kuijk W. (eds) _Modular Functions of One Variable II_, Lecture Notes in Mathematics, vol 349. Springer (1973) ([doi:10.1007/978-3-540-37855-6_4](https://doi.org/10.1007/978-3-540-37855-6_4))

On the [[Weil conjectures]]:

* _La conjecture de Weil : I_, Publications Mathématiques de l'IHÉS __43__ (1974), p. 273-307 [numdam](http://www.numdam.org/item?id=PMIHES_1974__43__273_0)

On [[rational homotopy theory]] of [[Kähler manifolds]]:

* {#DeligneGriffithsMorganSullivan75} [[Pierre Deligne]], [[Phillip Griffiths]], [[John Morgan]], [[Dennis Sullivan]], _Real homotopy theory of Kähler manifolds_, Invent Math (1975) 29: 245 ([doi:10.1007/BF01389853](https://doi.org/10.1007/BF01389853))

  (containing the statement that _[[the co-binary Sullivan differential is the dual Whitehead product]]_)

Introducing the notion of [[perverse sheaves]] (and of [[t-structures]] on [[triangulated categories]]):

* [[Alexander Beilinson]], [[Joseph Bernstein]], [[Pierre Deligne]], *Faisceaux pervers*, Astérisque **100** (1982) [[ISBN:978-2-85629-878-7](https://smf.emath.fr/publications/faisceaux-pervers), [pdf](https://publications.ias.edu/sites/default/files/Faisceaux%20pervers.pdf), [MR86g:32015](http://www.ams.org/mathscinet-getitem?mr=751966)]

On [[Tannakian categories]]:

* [[Pierre Deligne]], [[James Milne]]: *Tannakian categories*,  in: *Hodge Cycles, Motives, and Shimura Varieties*, Lecture Notes in Mathematics **900**, Springer (1982) [[doi:10.1007/978-3-540-38955-2_4](https://doi.org/10.1007/978-3-540-38955-2_4), [webpage](http://www.jmilne.org/math/xnotes/tc.html)]

On [[tensor categories]] and [[Tannaka duality]]

* {#Deligne90} [[Pierre Deligne]], section 2 of _[[Catégories Tannakiennes]]_, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp. 111-195 ([pdf](https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf))

culminating in [[Deligne's theorem on tensor categories]]:

* {#Deligne02} [[Pierre Deligne]], _Catégorie Tensorielle_, Moscow Math. Journal 2 (2002) no. 2, 227-248. ([pdf](https://www.math.ias.edu/files/deligne/Tensorielles.pdf))

Deligne led a seminar on differential systems corresponding to meromorphic connections, whose basic results were explained in a classic in this are:

* _Équations différentielles à points singuliers réguliers_, Lect. Notes in Math. __163__, Springer-Verlag (1970)

Related to this is the survey

* N. Katz, _An overview of Deligne's work on [[Riemann-Hilbert problem|Hilbert's twenty-first problem]]_, Proc. of Symp. in Pure Math., vol.28, pp. 537-557 (1976).

## Related $n$Lab entries

* [[Deligne completeness theorem]]

* [[Deligne conjecture]]

* [[Tannaka duality]]

* [[Deligne's theorem on tensor categories]]

* [[Whitehead product]],

  [[the co-binary Sullivan differential is the dual Whitehead product]]

category: people

[[!redirects Deligne]] [[!redirects P. Deligne]]


Pierre Fayet

* [Wikipedia entry](https://fr.wikipedia.org/wiki/Pierre_Fayet)

## Related $n$Lab entries

* [[supersymmetry]]

* [[supersymmetry breaking]]

category: people


Pierre Gabriel

__Pierre Gabriel__ also often as __Peter Gabriel__ (1933-2015) was a French and Swiss mathematician, professor at Zürich, was a president of Swiss Mathematical Society in 1980/1981.

* [webpage](http://www.pierre-peter-gabriel-mathematics.ch), [biography](http://www.pierre-peter-gabriel-mathematics.ch/biography/)

* The [English wikipedia](http://en.wikipedia.org/wiki/Pierre_Gabriel) gives almost nothing, but there is some bio material
at [German wikipedia](http://de.wikipedia.org/wiki/Pierre_Gabriel) and similarly at the [French wikipedia page](https://fr.wikipedia.org/wiki/Pierre_Gabriel_(mathématicien)).

Students of Gabriel include [[Bernhard Keller]].

## Selected writings

A version of Gabriel's 1960 PhD thesis has been published as *[[Des catégories abéliennes]]* in 1962. His thesis was a major breakthrough in the theory of [[localization]], and the study of [[abelian categories]], including categories of [[quasicoherent sheaves]] on schemes. In retrospective, it can be said that it was in its ideas and methods one of the starting points of modern [[noncommutative algebraic geometry]] as well.

Gabriel assisted [[Grothendieck]] in reformulating the [[pseudofunctor]] version of [[descent theory]] in invariant (property characterized way) i.e. as [[fibered categories]] which he wrote up under the guidance of Grothendieck in SGA I.6. Gabriel contributed to some other parts of SGA, namely in study of [[group schemes]] and [[formal schemes]], e.g. in SGA III.2 (Exp. 7a, P. Gabriel, Étude infinitésimale des schémas en groupe et groupes formels; Exp. 7b, P. Gabriel, Groupes formels). Soon after with Demazure writes a first tome of an unfinished but monumental work on [[algebraic group]]s which, more than EGA, emphasised [[functor of points]] view.

On [[algebraic groups]]:

* M. Demazure, P. Gabriel, _Groupes algebriques_, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970

With Zisman he introduced a general localization method in homotopy theory (see also [[calculus of fractions]] and [[Gabriel-Zisman]]):

* P. Gabriel, M. Zisman, _Calculus of fractions and homotopy theory_, Springer 1967. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35.

Introducing the notion of [[locally presentable categories]]:

* [[Pierre Gabriel]], [[Friedrich Ulmer]], *Lokal präsentierbare Kategorien*, Springer LNM **221** (1971) [[doi:10.1007/BFb0059396](https://doi.org/10.1007/BFb0059396), [MR0327863](http://www.ams.org/mathscinet-getitem?mr=0327863)]

In later part of his mathematical career, Gabriel worked on representation theory of finite-dimensional associative algebras and [[quivers]], where he found some of the basic theorems, see _[[Gabriel's theorem]]_.

## Related $n$Lab entries

* [[Gabriel filter]]

* [[Gabriel composition of filters]]

* [[Gabriel multiplication]]

* [[Gabriel-Ulmer duality]]

* Gabriel--Rosenberg [[reconstruction theorem]],

* Gabriel--Zisman [[localization]]

[[!redirects Gabriel]]
[[!redirects Peter Gabriel]]
[[!redirects Pierre Peter Gabriel]]
[[!redirects P. Gabriel]]


Pierre Grillet

Pierre Grillet is an Eleritus Professor at Tulane university. He has worked  mostly on semigroup theory, but also made important contributions to both cohomology theory and [[category theory]].

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=15000)

* [Tribute by Jonathan Leech for Pierre Grillet's 75 birthday](https://link.springer.com/article/10.1007/s00233-017-9862-9)

## Selected writings

Introducing [[Barr-exact categories]] and [[regular categories]]:

* {#BarrGrilletvonOsdool} [[Michael Barr]],  [[Pierre Grillet]], [[Donovan van Osdol]], *Exact Categories and Categories of Sheaves*,  Lec. Notes in Math. __236__, Springer 1971, ([doi:10.1007/BFb0058579](https://link.springer.com/book/10.1007/BFb0058579))

On [[semigroups]]:

* [[Pierre Grillet]], _Semigroups: An Introduction to the Structure Theory_ - Chapman & Hall/CRC Pure and Applied Mathematics 193

Showing that 'triple' (Beck) cohomology for commutative semigroups can be  described in concrete terms and related to commutative group coextensions:

* [[Pierre Grillet]], _Commutative semigroup cohomology_, Communications in Algebra, Vol. 23, Volume 23, 1995 - Issue 10.

category: people

[[!redirects Pierre A. Grillet]]
[[!redirects Pierre Antoine Grillet]]


Pierre Guillot

* [webpage](http://www-irma.u-strasbg.fr/~guillot/)

category: people


Pierre Lochak

* [personal page](https://webusers.imj-prg.fr/~pierre.lochak/)

* [Wikipedia entry](https://fr.wikipedia.org/wiki/Pierre_Lochak) (French)

## Selected writings

On relating the [[automorphism groups]] of the [[profinite completion]] of [[braid groups]] to the [[Grothendieck-Teichmüller group]]:

* [[Pierre Lochak]], [[Leila Schneps]], *The Grothendieck-Teichmüller group and automorphisms of braid groups*, in: *The Grothendieck Theory of Dessins d’Enfant*, Cambridge University Press (1994, 2011) [[pdf](https://webusers.imj-prg.fr/~leila.schneps/Fls.pdf), [doi:10.1017/CBO9780511569302](https://doi.org/10.1017/CBO9780511569302)]

category; people


Pierre Molino

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=31262)

## Selected writings

On [[G-structures]]:

* [[Pierre Molino]], _Theorie des G-Structures: Le Probleme d'Equivalence_, Lecture Notes in Mathematics, Springer (1977) ([ISBN:978-3-540-37360-5](https://www.springer.com/de/book/9783540082460))

* [[Pierre Molino]], _Sur quelques propriétés des G-structures_, J. Differential Geom. Volume 7, Number 3-4 (1972), 489-518 ([euclid:jdg/1214431168](https://projecteuclid.org/euclid.jdg/1214431168))

category: people


Pierre Ocvirk

* [personal page](http://astro.u-strasbg.fr/~ocvirk/)

## Selected writings

Simulation of [[cosmic structure formation]]/[[galaxy formation]] in feedback with [[cosmic reionization]]:

* [[Pierre Ocvirk]] et al. , _Cosmic Dawn (CoDa): the First Radiation-Hydrodynamics Simulation of Reionization and Galaxy Formation in the Local Universe_ ([arXiv:1511.00011](https://arxiv.org/abs/1511.00011))

* [[Pierre Ocvirk]], et al., _Cosmic Dawn II (CoDa II): a new radiation-hydrodynamics simulation of the self-consistent coupling of galaxy formation and reionization_ ([arXiv:1811.11192](https://arxiv.org/abs/1811.11192))

category: people


Pierre Pradic

* [ResearchGate page](https://www.researchgate.net/scientific-contributions/Pierre-Pradic-2073948059)

## Selected writings

On [[dependent linear type theory]]:

* {#KPB15} [[Neelakantan Krishnaswami]], [[Pierre Pradic]], [[Nick Benton]], *Integrating Dependent and Linear Types*,  ACM SIGPLAN Notices **50** 1 (2015)  17–30  [[pdf](https://www.cl.cam.ac.uk/~nk480/dlnl-paper.pdf),  [doi:10.1145/2775051.2676969](https://doi.org/10.1145/2775051.2676969)]

category: people


Pierre Ramond

* [webpage](http://www.phys.ufl.edu/~ramond/)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Pierre_Ramond)

## Selected writings

On the [[NSR superstring]]:

* [[Pierre Ramond]], _Dual theory for Free Fermions_, Phys Rev. D3  2415 (1971) ([doi:10.1103/PhysRevD.3.2415](https://doi.org/10.1103/PhysRevD.3.2415))

On the history of the concept of [[supersymmetry]]:

* [[Pierre Ramond]], _SuSy: The Early Years (1966-1976)_, Eur. Phys. J. C (2014) 74: 2698 ([arxiv:1401.5977](https://arxiv.org/abs/1401.5977))

On [[grand unified theory]]:

* [[Murray Gell-Mann]], [[Pierre Ramond]], Richard Slansky, _Complex Spinors and Unified Theories_, Supergravity, [[Peter van Nieuwenhuizen]] and D.Z. Freedman, eds, North Holland Publishing Co, 1979, (reprinted as [arXiv:1306.4669](http://arxiv.org/abs/1306.4669))

On historical work by [[Yoichiro Nambu]] on the [[Ising model]]:

* [[Lars Brink]], [[Pierre Ramond]], *Nambu and the Ising Model* [[arXiv:2209.01122](https://arxiv.org/abs/2209.01122)]

## Related $n$Lab entries

* [[string theory]]

* [[spinning string]], [[superstring]]

* [[Dirac-Ramond operator]]

category: people


Pierre Schapira

Pierre Schapira is a French mathematician who developed [[algebraic analysis]] as a member of [[Mikio Sato]]'s school.

* [personal page](http://people.math.jussieu.fr/~schapira)

His main collaborator is [[Masaki Kashiwara]]. Together, they developed microlocalization of sheaves and ind-sheaves, expanding Sato's groundbreaking ideas to a complete categorical toolbox to do analysis in a purely (ind)-sheaf theoretical setting.

## Selected writings

The theory of [[microlocalization]] of (ind)-sheaves was developed in the following works:

* [[Masaki Kashiwara]], [[Pierre Schapira]], _Ind-sheaves, distributions and microlocalization_, describes the program.

* [[Masaki Kashiwara]], [[Pierre Schapira]], [[Florian Ivorra]], [[Ingo Waschkies]] _Microlocalization of ind-sheaves_, gives the main results
and proofs.

* [[Masaki Kashiwara]], [[Pierre Schapira]] _Ind-sheaves_, SMF, gives a
complete account of the theory.

On [[microlocalization]] of [[abelian sheaves]] and [[Grothendieck's yoga of six operations]]:

* [[Masaki Kashiwara]], [[Pierre Schapira]], _Sheaves on manifolds_, Grundlehren **292**, Springer (1990) [[doi:10.1007/978-3-662-02661-8](https://doi.org/10.1007/978-3-662-02661-8)]

On [[category theory]] and [[sheaf and topos theory]]:

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_, Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006)

* [[Pierre Schapira]], *An Introduction to Categories and Sheaves*, lecture notes (2023) [[pdf](https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/CatShv.pdf)]

On [[Alexander Grothendieck]]'s *[[Récoltes et semailles]]*:

* [[Pierre Schapira]], *A truncated manuscript*, Inference-Review.com **7** 3 (Dec 2022) [[arXiv:2301.02898](https://arxiv.org/abs/2301.02898), [doi:10.37282/991819.22.61](https://doi.org/10.37282/991819.22.61)]

[[!redirects P. Schapira]]

category:people


Pierre Sikivie

* [webpage](https://www.phys.ufl.edu/~sikivie/)

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

## related $n$Lab entries

* [[axion]] [[fuzzy dark matter]]

category: people


Pierre Vanhove

* [personal page](https://sites.google.com/site/vanhovepierre/pierre-vanhove--en)

## Selected writings

On [[gravitational waves]] and [[perturbative quantum gravity]] form an [[S-matrix]]-perspective:

* [[Pierre Vanhove]], *S-matrix approach to general gravity and beyond* ([arXiv:2104.10148](https://arxiv.org/abs/2104.10148))

category: people


Pierre Vogel

* [webpage](http://www.math.jussieu.fr/~vogel/)

## Selected writings

On [[universal localization]] of [[group rings]]:

* [[M. Farber]], [[Pierre Vogel]], _The Cohn locaization of the free group ring_, Math. Proc.  Camb. Phil. Soc. (1992) 111, 433  ([pdf](http://www.maths.ed.ac.uk/~aar/papers/fv.pdf))

On [[topological quantum field theory]]:

* C. Blanchet, N. Habegger, [[Gregor Masbaum]], [[Pierre Vogel]], _Topological quantum field theories derived from the Kauffman bracket_, Topology Vol 34, No. 4, pp. 883-927 (1995) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/bhmv.pdf))

On [[Lie algebra weight systems]] from [[super Lie algebras]], [[knot invariants]] and [[graph cohomology]]:

* [[Pierre Vogel]], _Algebraic structures on modules of diagrams_, Journal of Pure and Applied Algebra Volume 215, Issue 6, June 2011, Pages 1292-1339 ([pdf](https://webusers.imj-prg.fr/~pierre.vogel/diagrams.pdf))

category: people

[[!redirects P. Vogel]]


Pierre-Louis Curien

* [webpage](http://www.pps.jussieu.fr/~curien/)

## Selected writings

On the [[categorical semantics]] of [[dependent type theory]] with [[function types]] in [[locally cartesian closed categories]] (see at *[[relation between category theory and type theory]]*):

* {#Curien} [[Pierre-Louis Curien]], *Substitution up to isomorphism*, Fundamenta Informaticae, **19** 1-2 (1993) 51-86 [[doi:10.5555/175469.175471](https://dl.acm.org/doi/10.5555/175469.175471)] and: Diagrammes **23** (1990) 43-66 [[numdam:DIA_1990__23__43_0](http://www.numdam.org/item/DIA_1990__23__43_0)]

* {#CurienGarnerHofmann} [[Pierre-Louis Curien]], [[Richard Garner]], [[Martin Hofmann]], *Revisiting the categorical interpretation of dependent type theory*, Theoretical Computer Science **546** 21 (2014) 99-119 [[doi:10.1016/j.tcs.2014.03.003](https://doi.org/10.1016/j.tcs.2014.03.003), [[CurienGarnerHofmann.pdf:file]]]

## Related pages

* [[PPS]]

category: people


Pierre-Loïc Méliot

* [personal page](https://www.imo.universite-paris-saclay.fr/~meliot/)

## Selected writings

On the [[Schur-Weyl measure]]:

* [[Pierre-Loïc Méliot]], *Kerov’s central limit theorem for Schur-Weyl and Gelfand measures*, [DMTCS proc.](https://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/) AO, 2011, 669–680 ([pdf](https://www.imo.universite-paris-saclay.fr/~meliot/files/kerovclt.pdf))

* {#Meliot10a} [[Pierre-Loïc Méliot]], _Kerov's central limit theorem for Schur-Weyl measures of parameter 1/2_ ([arXiv:1009.4034](https://arxiv.org/abs/1009.4034))

* {#Meliot10b} [[Pierre-Loïc Méliot]], _Asymptotics of the Gelfand models of the symmetric groups_ ([arXiv:1009.4047](https://arxiv.org/abs/1009.4047))

category: people


Pierre-Yves Strub

## Publications

* Cédric Fournet, Markulf Kohlweiss, [[Pierre-Yves Strub]], *Modular code-based cryptographic verification*, CCS '11: Proceedings of the 18th ACM conference on Computer and communications securityOctober 2011 Pages 341–350 ([doi:10.1145/2046707.2046746](https://doi.org/10.1145/2046707.2046746))

* [[Sophie Bernard]], [[Cyril Cohen]], [[Assia Mahboubi]], [[Pierre-Yves Strub]], *Unsolvability of the Quintic Formalized in Dependent Type Theory*, ITP 2021 - 12th International Conference on Interactive Theorem Proving, Jun 2021, Rome / Virtual, France ([hal:hal-03136002](https://hal.inria.fr/hal-03136002))

category: people


Pieter Belmans

* [webpage](https://www.uantwerpen.be/en/staff/pieter-belmans/)

category: people


Pieter Blue

* [webpage](https://www.maths.ed.ac.uk/~pblue/)

## Selected writings

On a rigorous [[proof]] of [[moduli stabilization]] of vacuum [[spacetimes]], refuting [Penrose 03](moduli+stabilization#Penrose03) (for $\geq 10$-dimensions with a [[Killing spinor]] on the [[compact topological space|compact]] [[fiber]] and for [[Schwarzschild spacetime|Schwarzschild]]-asymptotics):

* {#AndersonBlueWyattYau20} [[Lars Andersson]], [[Pieter Blue]], [[Zoe Wyatt]], [[Shing-Tung Yau]], _Global stability of spacetimes with supersymmetric compactifications_ ([arXiv:2006.00824](https://arxiv.org/abs/2006.00824))

category: people


Pieter Cuijpers

* [webpage](https://www.tue.nl/en/research/researchers/pieter-cuijpers/)

## selected writings

* {#Cuipers13a} [[Pieter Cuijpers]], _Prefix Orders as a General Model of Dynamics_, Proceedings of the [9th International Workshop on Developments in Computational Models (DCM)](http://www.dcm-workshop.org.uk/2013/) p. 25–29 ([pdf](http://www.dcm-workshop.org.uk/2013/dcm-2013-pre-proc.pdf#page=30))

* {#Cuipers13b} [[Pieter Cuijpers]], _The Categorical Limit of a Sequence of Dynamical Systems_, [EPTCS 120 : Proceedings EXPRESS/SOS 2013](https://www.win.tue.nl/expresssos2013/), p. 78-92, 2013 ([arXiv:1307.7445](https://arxiv.org/abs/1307.7445), [doi:10.4204/EPTCS.120.7]())

## related $n$Lab entries

* [[prefix order]]

category: people


Pieter Hofstra

* [website](https://mysite.science.uottawa.ca/phofstra/)

### Selected writings

* [[Pieter Hofstra]], [[Philip Scott]], _Aspects of categorical recursion theory_, ([arXiv:2001.05778](https://arxiv.org/abs/2001.05778))

### Related entries

* [[recursive mathematics]]

category: people


Pietro Antonio Grassi

* [institute page](https://upobook.uniupo.it/pietro.grassi)

* [personal page](http://fisicateorica.mfn.unipmn.it/people/pietroantoniograssi.html)

## Selected writings

On the [[D'Auria-Fre formulation of supergravity]], [[pure spinors]] and the [[supermembrane]]

* [[Pietro Fré]], [[Pietro Antonio Grassi]],
_Pure spinors, free differential algebras, and the supermembrane_, Nucl. Phys.  B 763:1-34, 2007 ([arXiv:hep-th/0606171] (http://www.arxiv.org/abs/hep-th/0606171))

On the [[D'Auria-Fre formulation of supergravity]], [[rheonomy]] and [[pure spinors]]:

* [[Pietro Fré]], [[Pietro Antonio Grassi]], _Free differential algebras, rheonomy, and pure spinors_ ([arXiv:0801.3076] (http://www.arxiv.org/abs/0801.3076))

Discussion of the [[Hodge star operator]] on [[supermanifolds]] (in terms of [[picture changing operators]] and integral top-forms for [[integration over supermanifolds]]):

* [[Leonardo Castellani]], [[Roberto Catenacci]], [[Pietro Antonio Grassi]], _Hodge Dualities on Supermanifolds_, Nuclear Physics B Volume 899, October 2015, Pages 570-593 ([arXiv:1507.01421](https://arxiv.org/abs/1507.01421))

and generally in terms of [[Berezin integration]]:

* [[Leonardo Castellani]], [[Roberto Catenacci]], [[Pietro Antonio Grassi]], _The Hodge Operator Revisited_ ([arXiv:1511.05105](https://arxiv.org/abs/1511.05105))

On [[D=7 supergravity]]:

* [[Pietro Fré]], [[Pietro Antonio Grassi]], [[Lucrezia Ravera]], M. Trigiante, _Minimal $D=7$ Supergravity and the supersymmetry of Arnold-Beltrami Flux branes_, JHEP 1606 (2016) 018 ([arXiv:1511.06245](https://arxiv.org/abs/1511.06245))

On [[supersymmetric quantum mechanics]] in the perspective of [[supergeometry]] ([[integration over supermanifolds]], [[picture changing operators]]):

* [[Leonardo Castellani]], [[Roberto Catenacci]], [[Pietro Grassi]], _Super Quantum Mechanics in the Integral Form Formalism_, Ann. Henri Poincaré (2018) 19: 1385 ([arXiv:1706.04704](https://arxiv.org/abs/1706.04704))

On [[superstring field theory]], [[supergeometry]], [[integration over supermanifolds]] of [[differential forms on supermanifolds]] and on [[picture changing operators]]:

* [[Pietro Grassi]], *Integral Forms and Applications*, Sestri Levante 2015 ([pdf](https://agenda.infn.it/event/8823/attachments/55101/64989/Grassi-SL.pdf), [[GrassiIntegralForms2015.pdf:file]])

* {#CatenacciGrassiNoja18} [[Roberto Catenacci]], [[Pietro Grassi]], [[Simone Noja]], _Superstring Field Theory, Superforms and Supergeometry_, Journal of Geometry and Physics Volume 148, February 2020, 103559 ([arXiv:1807.09563](https://arxiv.org/abs/1807.09563))

On [[integration over supermanifolds]] and [[picture changing operators]] in view of [[super Chern-Simons theory]]:

* [[C. A. Cremonini]], [[Pietro Grassi]], _Pictures from Super Chern-Simons Theory_ ([arXiv:1907.07152](https://arxiv.org/abs/1907.07152))

On super-[[Wilson lines]] via [[integration over supermanifolds]]:

* [[C. A. Cremonini]], [[Pietro Grassi]], S. Penati, _Supersymmetric Wilson Loops via Integral Forms_ ([arXiv:2003.01729](https://arxiv.org/abs/2003.01729))

On [[Lie algebra cohomology]] of [[super Lie algebras]] (see also the [[Green-Schwarz action functional|brane scan]]) in relation to [[integration over supermanifolds|integrable forms]] of coset [[supermanifolds]]:

* [[Roberto Catenacci]], [[C. A. Cremonini]], [[Pietro Antonio Grassi]], [[Simone Noja]], _Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces_ ([arXiv:2012.05246](https://arxiv.org/abs/2012.05246))

* [[C. A. Cremonini]], [[Pietro Antonio Grassi]], *Cohomology of Lie Superalgebras: Forms, Pseudoforms, and Integral Forms* ([arXiv:2106.11786](https://arxiv.org/abs/2106.11786))

On [[self-dual field theory]] via [[supergeometry]]:

* [[C. A. Cremonini]], [[Pietro Grassi]], _Self-Dual Forms in Supergeometry I: The Chiral Boson_ ([arXiv:2012.10243](https://arxiv.org/abs/2012.10243))

Relating the [[D'Auria-Fré formulation of supergravity]] to [[pure spinor]]-formalism:

* [[Pietro Antonio Grassi]], *Remarks on the Integral Form of $D=11$ Supergravity* [[arXiv:2304.01743](https://arxiv.org/abs/2304.01743)]

On the [[self-dual higher gauge field]] on the [[M5-brane]] lifted to [[superspace]] in the [[D'Auria-Fré formulation of supergravity]]:

* {#ACDGMNRT22} [[Laura Andrianopoli]], [[C. A. Cremonini]], [[Riccardo D'Auria]], [[Pietro A. Grassi]], R. Matrecano, R. Noris, [[Lucrezia Ravera]], [[Mario Trigiante]], *M5-brane in the superspace approach*, Phys. Rev. D **106** 2 (2022) 026010 [[arXiv:2206.06388](https://arxiv.org/abs/2206.06388), [doi:10.1103/PhysRevD.106.026010](https://doi.org/10.1103/PhysRevD.106.026010)]

category: people

[[!redirects Pietro Grassi]]
[[!redirects Pietro A. Grassi]]


Pietro Fré

* [website](http://personalpages.to.infn.it/~fre/)

## Selected writings

The original articles on the [[D'Auria-Fre formulation of supergravity]]:

* {#DAuriaFre80} [[Riccardo D'Auria]], [[Pietro Fré]] [[Tullio Regge]], _Graded Lie algebra, cohomology and supergravity_, Riv. Nuov. Cim. 3, fasc. 12 (1980) ([spire](http://inspirehep.net/record/156191))

* {#DAuriaFre82}  [[Riccardo D'Auria]], [[Pietro Fré]], _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140 (doi:10.1016/0550-3213(82)90376-5,  [[GeometricSupergravityErrata.pdf:file]])

* {#CFGPN83} [[Leonardo Castellani]], [[Pietro Fré]], F. Giani, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], *Gauging of $d = 11$ supergravity?*, Annals of Physics **146** 1 (1983) 35-77 [[spire:11998](https://inspirehep.net/literature/11998), doi:10.1016/0003-4916(83)90052-0]

* [[Riccardo D'Auria]], [[Pietro Fré]], [[Paul Townsend]],  *Invariance of actions, rheonomy and the new minimal $N=1$ supergravity in the group manifold approach*, Ann. Phys. 155 (1984) 423-446 [[cds:143990](https://cds.cern.ch/record/143990), doi:10.1016/0003-4916(84)90007-1, [pdf](https://cds.cern.ch/record/143990/files/198305027.pdf)]

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

On [[Freund-Rubin compactifications]] of [[11d supergravity]]:

* [[Riccardo D'Auria]], [[Pietro Fre]], [[Peter van Nieuwenhuizen]], _Symmetry Breaking in $d=11$ Supergravity on the Parallelized Seven Sphere_, Phys. Lett. B 122 (1983) 225 ([spire:181634](https://inspirehep.net/literature/181634), doi:10.1016/0370-2693(83)90689-5)

On the work of [[Tullio Regge]] ([[Regge calculus]], [[D'Auria-Fre formulation of supergravity]], ...):

* [[Leonardo Castellani]],  [[Anna Ceresole]], [[Riccardo D'Auria]], [[Pietro Fré]] (eds.): _Tullio Regge: An Eclectic Genius_, World Scientific 2019 ([doi:10.1142/11643](https://doi.org/10.1142/11643))

On [[D=7 supergravity]]:

* [[Pietro Fré]], [[Pietro Antonio Grassi]], [[Lucrezia Ravera]], M. Trigiante, _Minimal $D=7$ Supergravity and the supersymmetry of Arnold-Beltrami Flux branes_, JHEP 1606 (2016) 018 ([arXiv:1511.06245](https://arxiv.org/abs/1511.06245))

On [[N=1 D=4 super Yang-Mills theory]] via [[D3 branes]] at resolved [[orbifold singularities]] and invoking a generalized [[McKay correspondence]]:

* [[Pietro G. Fré]], *Lectures on resolutions à la Kronheimer of orbifold singularities, McKay quivers for Gauge Theories on D3 branes, and the issue of Ricci flat metrics on the resolved three-folds* [[arXiv:2308.14022](https://arxiv.org/abs/2308.14022)]

On [[special geometry]]

* [[Pietro Fré]], _Lectures on special Kähler geometry and electric-magnetic duality rotations_, Nucl. Phys. B Proc. Suppl. __45__:2-3, pp. 59--114 (1996) ([hep-th/9512043](https://arxiv.org/abs/hep-th/9512043) doi)

Textbook on [[gravity]]:

* [[Pietro Fré]], *Gravity, a Geometrical Course*, Volume 1: *Development of the Theory and Basic Physical Applications*, Spinger (2013) [[doi:10.1007/978-94-007-5361-7](https://doi.org/10.1007/978-94-007-5361-7)]

and specifically on [[black holes]], [[cosmology]] and the ([[D'Auria-Fré formulation of supergravity|D'Auria-Fré formulation of]]) [[supergravity]]:

* [[Pietro Fré]], *Gravity, a Geometrical Course*, Volume 2: *Black Holes, Cosmology and Introduction to Supergravity*, Springer (2013) [[doi:10.1007/978-94-007-5443-0](https://doi.org/10.1007/978-94-007-5443-0)]

## Related entries

* [[D'Auria-Fre formulation of supergravity]]

* [[gauged supergravity]]

* [[brane]]

* [[M2-brane]], [[AdS-CFT]]

* [[7d supergravity]]

* [[M-theory super Lie algebra]]

* [[11-dimensional supergravity]]

* [[12-dimensional supergravity]]

category: people

[[!redirects Pietro Fre]]

[[!redirects Pietro G. Fré]]


Pietro Polesello

* [website](http://www.math.unipd.it/~pietro/index_ing.html)

## related $n$Lab entries

* [[locally constant ∞-stack]]

* [[homotopy group of an ∞-stack]]

* [[local system]]


Pilar Carrasco

Pilar Carrasco Carrasco is a Spanish mathematician and expert on homotopical and homological algebra, especially with regard to [[crossed modules]] and their higher dimensional analogues. In particular she is an expert on simplicial groups and their structure, and showed a version of the Dold-Kan correspondence between simplicial groups and [[hypercrossed complex]]es.

Her web page in the Departamento de Álgebra, at Granada is [here](http://www.ugr.es/~mcarrasc/).

##Related entries

* [[hypercrossed complex]]

* [[Moore complex]]

##Ph.D. thesis

* Pilar Carrasco, 1987, _Complejos Hipercruzados, Cohomologia y Extensiones_, Ph.D. thesis, Universidad de Granada.

## Selected writings

* Pilar Carrasco and [[Antonio Cegarra|A. M. Cegarra]], _Group-theoretic Algebraic Models for Homotopy Types_, J. Pure Appl. Alg., 75, (1991), 195 – 235.

* P. Carrasco and [[T. Porter]], _Coproduct of 2-crossed modules. Applications to a definition of a tensor product for 2-crossed complexes_, Collectanea Mathematica, (DOI) 10.1007/s13348-015-0156-9.

On [[non-abelian cohomology]] with [[coefficients]] in [[braided 2-groups]] and [[symmetric 2-groups]]:

* [[Manuel Bullejos]], [[Pilar Carrasco]], [[Antonio M. Cegarra]], *Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane's classification theorem*, Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 163 - 189 ([doi:10.1017/S0305004100071498](https://doi.org/10.1017/S0305004100071498))

category:people

[[!redirects P. Carrasco]]
[[!redirects Carrasco]]


Piljin Yi

* [webpage](http://newton.kias.re.kr/~piljin/public_html/frameset.htm)

* [Institute page at KIAS](http://www.kias.re.kr/sub03/sub03_02_02_01_01.jsp?empid=MTAwNTc=)

## Selected writings

On [[hadrons]], in particular [[baryons]] as [[Skyrmions]]. via [[holographic QCD]]:

* Deog Ki Hong, Mannque Rho, Ho-Ung Yee, [[Piljin Yi]], _Chiral Dynamics of Baryons from String Theory_, Phys. Rev. D76:061901, 2007 ([arXiv:hep-th/0701276](https://arxiv.org/abs/hep-th/0701276))

* {#Yi09} [[Piljin Yi]], _Holographic Baryons_ ([arXiv:0902.4515](https://arxiv.org/abs/0902.4515), [doi:10.1142/9789814280709_0016](https://www.worldscientific.com/doi/abs/10.1142/9789814280709_0016)), Chapter 16 in: [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

* {#Yi11} [[Piljin Yi]], _Precision Holographic Baryons_, AIP Conference Proceedings 1388, 106 (2011) ([arXiv:1103.1684](https://arxiv.org/abs/1103.1684), [doi:10.1063/1.3647358](https://aip.scitation.org/doi/abs/10.1063/1.3647358))

* {#Yi13} [[Piljin Yi]], _Two Approaches to Holographic Baryons/Nuclei_, Few-Body Syst (2013) 54: 77. ([doi:10.1007/s00601-012-0373-7](https://doi.org/10.1007/s00601-012-0373-7))

Introducing the [[nuclear matrix model]] for [[baryons]]/[[nucleons]] in the [[Witten-Sakai-Sugimoto model]]:

* {#HashimotoIizukaYi10} [[Koji Hashimoto]], [[Norihiro Iizuka]], [[Piljin Yi]], _A Matrix Model for Baryons and Nuclear Forces_, JHEP 1010:003, 2010 ([arXiv:1003.4988](https://arxiv.org/abs/1003.4988))

category: people


pillowcase orbifold

+-- {: .rightHandSide}
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###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The _pillowcase orbifold_ is the 2-dimensional [[flat orbifold|flat]] ([[complex orbifold|complex]]) [[orbifold]] which is the [[global quotient orbifold|global quotient]] of the [[torus]] $\mathbb{C}/\mathbb{Z}[i]$ by the [[reflection]] [[involution]] $z \mapsto -z$:

$$
  Pillow
  \;\coloneqq \;
  (\mathbb{C}/\mathbb{Z}[i])\sslash_{refl} \mathbb{Z}_2
$$

## Properties

### Coarse underlying space

The [[Weierstrass elliptic function]] $\wp$, regarded as a [[holomorphic function]] with values in the [[Riemann sphere]] $\mathbb{C}P^1$, exhibits the coarse underlying topological space of the [[pillowcase orbifold]] as the [[2-sphere]]:

$$
  \wp \;\colon\;
  \mathbb{C}/\mathbb{Z}[i]
  \longrightarrow
  (\mathbb{C}/\mathbb{Z}[i])/_{refl} \mathbb{Z}_2
  \overset{homeo}{\longrightarrow}
  \mathbb{C}P^1
$$

(e.g [Mukase 04, end of 1.4](Weierstrass+elliptic+function#Mukase04), [Eskin-Okounkov 05, p. 1](#EskinOkounkov05), [Goujard-Moeller 18, 2.3](#GoujardMoeller18))

## References

* {#EskinOkounkov05} [[Alex Eskin]], [[Andrei Okounkov]], _Pillowcases and quasimodular forms_, In: Ginzburg V. (ed.) _Algebraic Geometry and Number Theory_, Progress in Mathematics, vol 253. Birkhäuser 2006
 ([arXiv:math/0505545](https://arxiv.org/abs/math/0505545), [doi:10.1007/978-0-8176-4532-8_1](https://doi.org/10.1007/978-0-8176-4532-8_1))

* {#GoujardMoeller18} Elise Goujard, Martin Moeller, _Pillowcase covers: Counting Feynman-like graphs associated with quadratic differentials_ ([arXiv:1809.05016](https://arxiv.org/abs/1809.05016))

[[!redirects pillowcase orbifolds]]

[[!redirects pillow orbifold]]
[[!redirects pillow orbifolds]]


pin group

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* table of contents
{:toc}

## Idea

A _$Pin$-group_ is a double [[covering|cover]] of an [[orthogonal group]]. Its restriction along the inclusion of the [[special orthogonal group]] is a [[Spin group]]. Hence a $Pin$-group is "like the corresponding Spin group, but including [[reflections]]".

## Definition

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

A _quadratic vector space_ $(V, \langle -,-\rangle)$ is a [[vector space]] $V$ over finite [[dimension]] over a [[field]] $k$ of [[characteristic]] 0, and equipped with a symmetric [[bilinear form]] $\langle -,-\rangle \colon V \otimes V \to k$.

=--

Conventions as in ([Varadarajan 04, section 5.3](#Varadarajan04)).

We write $q\colon v \mapsto \langle v ,v \rangle$ for the corresponding [[quadratic form]].

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

The _[[Clifford algebra]]_ $CL(V,q)$ of a quadratic vector space, def. \ref{QuadraticVectorSpace}, is the [[associative algebra]] over $k$ which is the [[quotient]]

$$
  Cl(V,q)
    \coloneqq
  T(V)/I(V,q)
$$

of the [[tensor algebra]] of $V$ by the ideal generated by the elements $v \otimes v - q(v)$.

=--

Since the [[tensor algebra]] $T(V)$ is naturally $\mathbb{Z}$-graded, the Clifford algebra $Cl(V,q)$ is naturally $\mathbb{Z}/2\mathbb{Z}$-graded.

Let $(\mathbb{R}^n, q = {\vert -\vert})$ be the $n$-dimensional [[Cartesian space]] with its canonical [[scalar product]]. Write $Cl^\mathbb{C}(\mathbb{R}^n)$ for the [[complexification]] of its [[Clifford algebra]].

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

There exists a unique [[complex number|complex]] [[representation]]

$$
  Cl^{\mathbb{C}}(\mathbb{R}^n) \longrightarrow End(\Delta_n)
$$

of the algebra $Cl^\mathbb{C}(\mathbb{R}^n)$ of smallest [[dimension]]

$$
  dim_{\mathbb{C}}(\Delta_n) = 2^{[n/2]}
  \,.
$$

=--

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

The [[Pin group]] $Pin(V;q)$ of a quadratic vector space, def. \ref{QuadraticVectorSpace}, is the [[subgroup]] of the [[group of units]] in the [[Clifford algebra]] $Cl(V,q)$

$$
  Pin(V,q) \hookrightarrow GL_1(Cl(V,q))
$$

on those elements which are multiples $v_1 \cdots v_{n}$ of elements $v_i \in V$ with $q(v_i) = 1$.

The [[Spin group]] $Spin(V,q)$ is the further [[subgroup]] of $Pin(V;q)$ on those elements which are even number multiples $v_1 \cdots v_{2k}$ of elements $v_i \in V$ with $q(v_i) = 1$.

Specifically, "the" Spin group is

$$
  Spin(n) \coloneqq Spin(\mathbb{R}^n)
  \,,
$$

where we understand the standard [[quadratic form]] on $\mathbb{R}^n$ for either global sign

$$
  \array{
    \mathbb{R}^n &\overset{q_{\pm}}{\longrightarrow}& \mathbb{R}
    \\
    \vec x &\mapsto& \pm \underset{i}{\sum} (x^i)^2
  }
$$

The corresponding two $Pin$-groups are denoted

$$
  Pin_\pm(n)
  \;\coloneqq\;
  Pin\big( \mathbb{R}^n, q_\pm\big)
$$

=--

## Examples

* [[Pin(2)]]
* [[Pin(5)]]

## Related concepts

[[!include table of orthogonal groups and related]]

[[!include low dimensional rotation groups -- table]]

* [[pseudoscalar representation]], [[pseudovector representation]]

## References

A standard textbook account is

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], chapter I, section 2 of _[[Spin geometry]]_, Princeton University Press (1989)

See also

* {#Varadarajan04} [[Veeravalli Varadarajan]], section 7 of _[[Supersymmetry for mathematicians]]: An introduction_, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)

The following article discusses which of the Pin groups are in fact compatible with [[general relativity]]

* Bas Janssens, _The Pin Groups in General Relativity_ ([arXiv:1709.02742](https://arxiv.org/abs/1709.02742))

[[!redirects pin groups]]

[[!redirects Pin group]]
[[!redirects Pin groups]]


Pin(2)

+-- {: .rightHandSide}
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### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[Pin group]] in [[dimension]] 2.

## Properties

Let $\mathbb{H} \simeq \mathbb{C} \oplus j \mathbb{C}$ be the [[quaternions]] realized as the [[Cayley-Dickson double]] of the [[complex numbers]], and identify the [[circle group]]

$$
  SO(2) \simeq S\big( \mathbb{C}\big) \hookrightarrow \mathbb{H}
$$

with the [[unit circle]] in $\mathbb{C} \hookrightarrow \mathbb{H}$ this way, with group structure now thought of as given by multiplication of [[quaternions]]. Then the [[Pin group]] $Pin_-(2)$ is [[isomorphism|isomorphic]] to the [[subgroup]] of the [[group of units]] $\mathbb{H}^\times$ of the [[quaternions]] which consists of this copy of [[SO(2)]], together with the multiplies of the imaginary quaternion $j$ with this copy:

$$
  Pin_-(2)
  \;\simeq\;
  S\big( \mathbb{C}\big)
   \;\cup\;
  j \cdot S\big( \mathbb{C}\big)
  \;\subset\;
  \mathbb{H}^\times
  \,.
$$

This is in fact inside the [[unit sphere]] $S(\mathbb{H}) \simeq $ [[Spin(3)]].

By the classification of the [[finite subgroups of Spin(3)]] this means that the restriction of $Pin_-(2)$ to the inclusion of the [[dihedral group]] $D_{2n}$ into the [[orthogonal group]] $O(2)$ is the [[binary dihedral group]] $2 D_{2n}$:

$$
  \array{
    2 D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Pin_-(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    \\
    D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    O(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)
  }
$$

### Appearance in Seiberg-Witten theory and Floer homology

On a [[spin structure|spin]] [[4-manifold]], the $U(1)$-symmetry of the [[Seiberg-Witten equations]] enhances to a $Pin(2)$-symmetry ([Furuta 01](#Furuta01)). Because of this, $Pin(2)$-equivariance appears in [[Seiberg-Witten theory]] and [[Floer homology]]. See the references below.

## Related concepts

[[!include low dimensional rotation groups -- table]]

## References

$Pin(2)$-[[equivariant homotopy theory]]/[[equivariant cohomology theory]]

application to [[Seiberg-Witten theory]] and [[Floer homology]]:

* {#Furuta01} Mikio Furuta, [_Monopole equation and the $11/8$-conjecture_](https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2001/0008/0003/MRL-2001-0008-0003-a005.pdf) Mathematical Research Letters, Volume 8, Number 3, (2001).

* Ciprian Manolescu, _$Pin(2)$-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture_ ([arXiv:1303.2354](https://arxiv.org/abs/1303.2354))

* Matthew Stoffregen, _$Pin(2)$-equivariant Seiberg-Witten Floer homology of Seifert fibrations_ ([arXiv:1505.03234](https://arxiv.org/abs/1505.03234))

* Matthew Stoffregen, _A Remark on $Pin(2)$-Equivariant Floer Homology_,     Michigan Math. J. Volume 66, Issue 4 (2017), 867-884 ([euclid:1508810818](https://projecteuclid.org/euclid.mmj/1508810818))

$Pin(2)$-[[equivariant K-theory|equivariant]] [[KO-theory]]:

* Jianfeng Lin, _Pin(2)-equivariant KO-theory and intersection forms of spin four-manifolds_, Algebr. Geom. Topol. 15 (2015) 863-902 ([arXiv:1401.3264](https://arxiv.org/abs/1401.3264))

[[!redirects Pin(2)-group]]
[[!redirects Pin(2) group]]

[[!redirects pin(2)]]
[[!redirects pin(2)-group]]
[[!redirects pin(2) group]]


Pin(3)

+-- {: .rightHandSide}
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### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[pin group]] in [[dimension]] 3.

## Properties

$Pin_+(3) \simeq SO(3) \times C_4$ and $Pin_-(3) \simeq SU(2) \times C_2$.

## Related concepts

[[!include low dimensional rotation groups -- table]]

## References


Pin(4)

+-- {: .rightHandSide}
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#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[pin group]] in [[dimension]] 4.

## Related concepts

[[!include low dimensional rotation groups -- table]]


Pin(5)

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[pin group]] in [[dimension]] 5.

## Properties

### Exceptional isomorphisms

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

The [[exceptional isomorphism]] [[Spin(5)]] $\simeq$ [[Sp(2)]] ([this Prop.](Spin5#ExceptionalIsoToSp2)) generalizes to

$$
  Pin^\pm(5)
  \;\simeq\;
  Sp(2)
    \sqcup
  \omega Sp(2)
  \phantom{AAA}
  \omega^2 = \pm e
$$

where $\omega \in Z\big( Pin^+(5)\big)$ is an element in the [[center of a group|center]] which, for $Pin^+(5)$, squares to the the [[neutral element]] $e$ (corresponding to the [[Clifford algebra]] element $+1$) or, for $Pin^-(5)$, to $-e$ (the Clifford algebra element $-1$).

=--

(e.g. [Varlamov 99, Theorem 5](#Varlamov99))

## Related concepts

[[!include low dimensional rotation groups -- table]]

## References

* {#Varlamov99} Vadim V. Varlamov, _Fundamental Automorphisms of Clifford Algebras and an Extension of Dabrowski Pin Groups_, Hadronic J. 22 (1999) 497-533 ([arXiv:math-ph/9904038v2](https://arxiv.org/abs/math-ph/9904038v2))


Ping Gao

* [institute page](https://physics.mit.edu/faculty/ping-gao/)

* [GoogleScholar page](https://scholar.google.com/citations?user=jGTQByMAAAAJ&hl=en)

## Selected writings

Argument that the image of *[[wormhole]] traversal* under [[AdS-CFT duality]] is *[[quantum teleportation]]*:

* [[Ping Gao]], [[Daniel Louis Jafferis]], [[Aron C. Wall]], *Traversable Wormholes via a Double Trace Deformation*, Journal of High Energy Physics **2017** 151 (2017) [[arXiv:1608.05687](https://arxiv.org/abs/1608.05687), doi:10.1007/JHEP12(2017)151]

category: people