Poisson tensor

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## Definition

For $(X, \{-,-\})$ a [[Poisson manifold]], there is a unique [[bivector]] (skew-symmetric rank (2,0)-[[tensor field]]) $\pi \in \wedge^2 \Gamma(T X)$ such that for all functions $f,g \in C^\infty(X)$ the [[Poisson bracket]] is given by the [[Schouten bracket]] $[-,-]$ as

$$
  \{f,g\} = [[\pi, f],g]
  \,.
$$

This $\pi$ is called the **Poisson tensor** or **Poisson bivector** of $(X, \{-,-\})$.

Every bivector $\pi$ such that $[\pi, \pi] = 0$ in the [[Schouten bracket]] arises this way.

## In terms of Lie algebroids

The Poisson tensor constitutes the [[anchor map]] of the [[Poisson Lie algebroid]] $\mathfrak{P}$ which corresponds to the [[Poisson manifold]].

Regarded as an element in the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{P}) \simeq (\wedge^\bullet \Gamma(T X), [\pi,-])$,  the Poisson tensor also constitutes the canonical [[Lie algebroid cocycle]] on $\mathfrak{p}$ which is in [[transgression]] with the canonical [[invariant polynomial]] on $\mathfrak{P}$, the one that exhibits $\mathfrak{P}$ as a [[symplectic Lie n-algebroid|symplectic Lie 1-algebroid]].

## Related concepts

* [[deformation quantization]]

  * [[Moyal quantization]]

[[!redirects Poisson tensors]]
[[!redirects Poisson bivector]]
[[!redirects Poisson bivectors]]


Poisson vertex algebra

A _Poisson vertex algebra_ is a unital commutative associative algebra with a derivation and a $\lambda$-bracket; it has to satisfy the axioms of a Lie conformal algebra; and the $\lambda$-bracket and the multiplication form a Poisson algebra (i.e. satisfy the Leibniz rule). It is a Poisson analogue of a conformal Lie algebra.

* Haisheng Li, _Vertex algebras and vertex Poisson algebras_, Commun. Contemp. Math. [http://arxiv.org/abs/math/0209310](http://arxiv.org/abs/math/0209310)
* A. De Sole, [[V. G. Kac]], M.; Wakimoto, _On classification of Poisson vertex algebras_, Transform. Groups __15__ (2010), no. 4, 883–907 [MR2012a:17051](http://www.ams.org/mathscinet-getitem?mr=2753261) [doi](http://dx.doi.org/10.1007/s00031-010-9110-9)
* B. Bakalov, A. De Sole, _Non-linear Lie conformal algebras with three generators_,
Selecta Math. (N.S.) 14 (2009), no. 2, 163–198 [dpo](http://dx.doi.org/10.1007/s00029-008-0058-8) [MR2009m:17024](http://www.ams.org/mathscinet-getitem?mr=2480713)
* Alberto De Sole, Victor G. Kac, Daniele Valeri, _Classical W-algebras and Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras_, [arxiv/1207.6286](http://arxiv.org/abs/1207.6286)
* Gaywalee Yamskulna, _Vertex Poisson algebras associated with Courant algebroids and their deformations; I_, [math.QA/0509122](http://arxiv.org/abs/math/0509122)
* A. De Sole, _Poisson vertex algebras in the theory of Hamiltonian equations_, Talk at Algebraic Lie Theory, Newton Inst. U. Cambridge 2009, [video](http://sms.cam.ac.uk/media/538800)

Non-local Poisson vertex algebras are studied with applications to the study of integrability of Hamiltonian PDE-s in

* Alberto De Sole, [[Victor Kac|Victor G. Kac]], _Non-local Hamiltonian structures and applications to the theory of integrable systems I_, [arxiv/1210.1688](http://arxiv.org/abs/1210.1688); _II_, [arxiv/1211.2391](http://arxiv.org/abs/1211.2391); merged version, [arxiv/1302.0148](http://arxiv.org/abs/1302.0148)
* Aliaa Barakat, Alberto De Sole, Victor G. Kac, _Poisson vertex algebras in the theory of Hamiltonian equations_, [arxiv/0907.1275](http://arxiv.org/abs/0907.1275)

[[!redirects Poisson vertex algebras]]
[[!redirects vertex Poisson algebra]]


Poisson-Lie T-duality

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## Idea

What has come to be called _nonabelian T-duality_ ([de la Ossa-Quevedo 92](#OssaQuevedo92)) or _Poisson-Lie T-Duality_ (due to [Klimcik-Ševera 95](#KlimcikSevera95), [von Unge 02](#vonUnge02)) is a generalization of [[T-duality]] from [[fiber bundles]] equipped with an [[abelian group]] of [[isometries]] ([[torus]] [[bundles]]) to those with a [[nonabelian group]] of isometries.

Poisson-Lie T-duality may also be made manifest at the level of [[type II supergravity ]] in the framework of [[double field theory]] on [[Lie group|group manifolds]]. Using this framework both the NS/NS sector and the R/R sector are captured, and this allows to derive the transformation of the [[RR fields]] for full Poisson-Lie T-duality ([Hassler 17](#Hassler17)).

## As a partial duality of string theory

While ordinary abelian [[T-duality]] is supposedly a full [[duality in string theory]], in particular in that it is an equivalence on the [[string perturbation series]] to all orders of the squared [[string length]]/[[Regge slope]] $\alpha'$ and the [[string coupling constant]] $g_s$, it has apparently been shown by [[Martin Roček]] (citation?) that there are topological [[obstructions]] at higher [[genus of a surface|genus]] for non-abelian T-duality, letting it break down in higher orders of $g_s$; and already a genus-0 ([[tree level]]) it apparently breaks down for the [[open string]] (i.e. on punctured [[disks]]) at some order of $\alpha'$.

But see [Hassler 20](#Hassler20), [Borsato-Wulff 20](#BorsatoWulff20).

## Related $n$Lab entries

* [[topological T-duality]]

* [[spherical T-duality]]

## References

The original articles are

* {#OssaQuevedo92} Xenia C. de la Ossa, Fernando Quevedo, _Duality Symmetries from Non--Abelian Isometries in String Theories_, Nucl.Phys. B403 (1993) 377-394 ([hep-th/9210021](http://xxx.lanl.gov/abs/hep-th/9210021))

* {#KlimcikSevera95} [[Ctirad Klimcik]], [[Pavol Ševera]], _Dual non-Abelian duality and the Drinfeld double_, Physics Letters B, Volume 351, Issue 4, 1 June 1995, Pages 455-462 (doi:10.1016/0370-2693(95)00451-P)

* {#vonUnge02} [[Rikard von Unge]], _Poisson-Lie T-plurality_,  Journal of High Energy Physics, Volume 2002, JHEP07 (2002) ([arXiv:hep-th/0205245](https://arxiv.org/abs/hep-th/0205245))

Review includes

* I. Petr, _From Buscher Duality to Poisson‐Lie T‐Plurality on Supermanifolds_, AIP Conference Proceedings 1307, 119 (2010) (doi:10.1063/1.3527407)

* [[Konstadinos Sfetsos]], _Recent developments in non-Abelian T-duality in string theory_, Fortschr. Phys., Special Issue: Proceedings of the “Schools and Workshops on Elementary Particle Physics and Gravity” (CORFU 2010), 29 August – 12 September 2010, Corfu (Greece) Volume59, Issue11‐12 ([arXiv:1105.0537](https://arxiv.org/abs/1105.0537))

Relation to [[T-folds]]:

* Mark Bugden, _Non-abelian T-folds_ ([arXiv:1901.03782](https://arxiv.org/abs/1901.03782))

* Ladislav Hlavatý, Ivo Petr, _T-folds as Poisson-Lie plurals_ ([arXiv:2004.08387](https://arxiv.org/abs/2004.08387))

Discussion of the duality at the level of [[type II supergravity]] [[equations of motion]] is (using [[Riemannian geometry]] of [[Courant algebroids]]) due to

* [[Branislav Jurco]], [[Jan Vysoky]], _Poisson-Lie T-duality of String Effective Actions: A New Approach to the Dilaton Puzzle_, Journal of Geometry and Physics Volume 130, August 2018, Pages 1-26 ([arXiv:1708.04079](https://arxiv.org/abs/1708.04079))

* [[Pavol Ševera]], [[Fridrich Valach]], _Courant algebroids, Poisson-Lie T-duality, and type II supergravities_ ([arXiv:1810.07763](https://arxiv.org/abs/1810.07763))

and in relation to [[double field theory]]:

* {#Hassler17} [[Falk Hassler]], _Poisson-Lie T-Duality in Double Field Theory_ ([arXiv:1707.08624](https://arxiv.org/abs/1707.08624))

Discussion within a broader picture of [[duality in physics|dual]] [[higher gauge theories]], including 4d [[electric-magnetic duality]]:

* [[Ján Pulmann]], [[Pavol Ševera]], [[Fridrich Valach]], _A non-abelian duality for (higher) gauge theories_ ([arXiv:1909.06151](https://arxiv.org/abs/1909.06151))

Discussion of non-abelian T-duality from a comprehensive picture of [[higher differential geometry]], relating [[Kaluza-Klein compactification]] on [[principal ∞-bundles]] to [[double field theory]], [[T-folds]], [[type II geometry]], [[exceptional geometry]], etc.:

* [[Luigi Alfonsi]], Section 4 of _Global Double Field Theory is Higher Kaluza-Klein Theory_ ([arXiv:1912.07089](https://arxiv.org/abs/1912.07089))

See also

* {#FraserManolopoulosSfetsos18} Benjo Fraser, Dimitrios Manolopoulos, [[Konstantinos Sfetsos]], _Non-Abelian T-duality and Modular Invariance_ ([arXiv:1805.03657](https://arxiv.org/abs/1805.03657))

* Francesco Bascone, Franco Pezzella, Patrizia Vitale, _Poisson-Lie T-Duality of WZW Model via Current Algebra Deformation_ ([arXiv:2004.12858](https://arxiv.org/abs/2004.12858))

* Falk Hassler, Thomas B. Rochais
_$O(D,D)$-covariant two-loop β-functions and Poisson-Lie T-duality_ ([arXiv:2011.15130](https://arxiv.org/abs/2011.15130))

Discussion in [[cosmology]]:

* Ladislav Hlavatý, Ivo Petr, _Poisson-Lie plurals of Bianchi cosmologies and Generalized Supergravity Equations_ ([arxiv:1910.08436](https://arxiv.org/abs/1910.08436))

Generalization to [[U-duality]] in [[exceptional generalized geometry]]:

* [[Emanuel Malek]], [[Daniel C. Thompson]], _Energy Physics - Theory Poisson-Lie U-duality in Exceptional Field Theory_, JHEP 0420 (2020) 058 [[arxiv:1911.07833](https://arxiv.org/abs/1911.07833)]

* [[Chris D. A. Blair]], [[Daniel C. Thompson]], Sofia Zhidkova, _Exploring Exceptional Drinfeld Geometries_, JHEP 09 (2020) 151 [[arxiv:2006.12452](https://arxiv.org/abs/2006.12452)]

In the context of the  [[BMN matrix model]]:

* [[Yolanda Lozano]], [[Carlos Nunez]], Salomon Zacarias, _BMN Vacua, Superstars and Non-Abelian T-duality_, JHEP 09 (2017) 008 ([arXiv:1703.00417](https://arxiv.org/abs/1703.00417))

Discussion of $\alpha'$-corrections:

* {#Hassler20} [[Falk Hassler]], Thomas Rochais, _$\alpha'$-corrected Poisson-Lie T-duality_ ([arXiv:2007.07897](https://arxiv.org/abs/2007.07897))

* {#BorsatoWulff20} Riccardo Borsato, Linus Wulff, _Quantum correction to Poisson-Lie and non-abelian T-duality_ ([arXiv:2007.07902](https://arxiv.org/abs/2007.07902))

Application to [[AdS-CFT duality]]:

* Ali Eghbali, Reza Naderi, Adel Rezaei-Aghdam, *Non-Abelian T-duality of AdSd≤3 families by Poisson-Lie T-duality* ([arXiv:2111.07700](https://arxiv.org/abs/2111.07700))

See also:

* Alex S. Arvanitakis, Chris D. A. Blair, Daniel C. Thompson, *A QP perspective on topology change in Poisson-Lie T-duality* ([arXiv:2110.08179](https://arxiv.org/abs/2110.08179))

[[!redirects nonabelian T-duality]]

[[!redirects non-abelian T-duality]]


Pokman Cheung

* [webpage](http://maths.dept.shef.ac.uk/maths/staff_info.php?id=459)

category: people


Pol Vanhaecke

* [webpage](http://wwwmathlabo.univ-poitiers.fr/~vanhaeck/)

* [Wikipedia entry](https://de.wikipedia.org/wiki/Pol_Vanhaecke)

## related $n$Lab entries

* [[polynomial Poisson algebra]]

* [[formal deformation quantization]]

category: people


polar coordinates

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#### Geometry
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#Contents#
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## Idea

A _polar coordinate system_ for [[Cartesian space]] $\mathbb{R}^{n+1}$ is a [[coordinate system]] adapted to the decomposition of the [[complement]] $\mathbb{R}^{n+1} \setminus \{0\}$ of the origin as a [[Cartesian product]] of the [[unit sphere|unit]] [[n-sphere]] times the [[positive real numbers]] inside the [[real line]]:

$$
  \mathbb{R}^{n+1} \setminus \{0\}
  \;\simeq\;
  S^n \times \mathbb{R}_{\gt 0}
  \,.
$$

If $dvol_{S^n} \in \Omega^n(S^n)$ denotes the standard [[volume form]] on the [[unit sphere|unit]] [[n-sphere]], then the standard [[volume form]] $dvol_{\mathbb{R}^{n+1}}$ of [[Cartesian space]] in polar coordinates is

$$
  dvol_{\mathbb{R}^{n+1}}
  \;=\;
  r^n d r \wedge dvol_{S^n}
  \,,
$$

where $r \colon \mathbb{R}_{\gt 0} \to \mathbb{R}$ denotes the canonical [[coordinate function]] along the [[radius|radial]] direction.

If a [[smooth function]] $\mathbb{R}^{n+1} \to \mathbb{R}$ depends at most on the [[radius]] coordinate $r$ and the [[angle]] $\theta$ of [[vectors]] in $\mathbb{R}^{n+1}$ to any fixed line through the origin, then

$$
  \label{VolumeElementForFunctionDependingOnRadiusandTheta}
  \begin{aligned}
    f(\vec x) dvol_{\mathbb{R}^{n+1}}
      & =
    f(r,\theta)
   \,\, (r \sin(\theta))^{n-1} dvol_{S^{n-1}} \,\wedge r d\theta\, \wedge d r
  \end{aligned}
$$

> sign correct?

## Related concepts

* [[coordinates]]

## References

See also

* Wikipedia, _[Polar coordinate system](http://en.wikipedia.org/wiki/Polar_coordinate_system)_

* Wikipedia, _[Spherical coordinare system](https://en.wikipedia.org/wiki/Spherical_coordinate_system)_

[[!redirects polar coordinate system]]
[[!redirects polar coordinate systems]]

[[!redirects spherical coordinate]]
[[!redirects spherical coordinates]]
[[!redirects spherical coordinate systems]]
[[!redirects spherical coordinate system]]
[[!redirects spherical coordinate systems]]


polar decomposition

#Contents#
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## Idea ##
This page is about the polar decomposition of [[bounded operators]] on [[Hilbert spaces]]. Any complex number $z$ has a representation as $z = r e^{i \phi}$ with $r \in \mathbb{R}, r \ge 0$ being the absolute value of $z$ and the complex number $e^{i \phi}$ of norm $1$ being the modulus, or the complex sign, of $z$. The polar decomposition of a bounded operator is a generalization of this representation.

## Definition ##
Let $\mathcal{H}$ be a [[Hilbert space]] and $S_1, S_2$ be closed linear subspaces.

+-- {: .un_def}
###### Definition
An unitary isomorphism
$$
   U: S_1 \to S_2
$$
is called a **partial isometry** with **initial space** $S_1$ and **final space** or range $S_2$
=--

Let $T$ be a bounded operator on $\mathcal{H}$

+-- {: .un_def}
###### Definitions
The positive operator
$$
   |T| := (T^*T)^{\frac{1}{2}}
$$
is called the **modulus** of T.
=--

## The Theorem ##
For every bounded operator $T$ on $\mathcal{H}$ there exists a unique partial isometry $U$ such that

1. U has initial space $\overline{R(|T|)}$ and range $\overline{R(T)}$

2. $T = U |T| = U (T^*T)^{\frac{1}{2}}$

## Properties ##
We have stated the theorem for the [[operator algebra]] $\mathcal{B}(\mathcal{H})$ only, for a general [[C-star algebra]] $C$ it need not hold because the partial isometry $U$ need not be contained in $C$.

This is true however for every [[von Neumann algebra]].

## Examples ##

## References ##
Most textbooks about operators on Hilbert spaces mention the polar decomposition, for example it can be found in the beginning of

* Kadison, Ringrose: _Fundamentals of the Theory of Operator Algebras_ , volume 2, _Advanced Theory_

* wikipedia [polar decomposition](http://en.wikipedia.org/wiki/Polar_decomposition)
[[!redirects polar decompositions]]


polar homology

Polar homology is defined in

* Boris Khesin, Alexei Rosly, _Polar homology_, [pdf](http://www.math.toronto.edu/khesin/papers/polarhom.pdf)


polarity in type theory

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### Context
#### Type theory
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\tableofcontents

## Idea

In [[type theory]], a type may be classified by its "polarity", either [[positive type|positive]] or [[negative type|negative]], according to whether its _[[term constructors]]_ or its _[[term eliminators]]_ are regarded as primary, respectively. The idea is due to [[Jean-Marc Andreoli]] and [[Jean-Yves Girard]].

[[positive type|Positive types]] are [[inductive type|inductively]] defined by their [[term introduction]] rules and correspond, under [[categorical semantics]], to [[objects]] defined as covariant [[representable functors]] ($C \to Set$) such as [[colimits]] and [[left adjoints]].
On the other hand, [[negative types]] are [[coinductive type|coinductively]] defined by their [[term elimination]] rules and correspond under [[categorical semantics]] to [[objects]] defined as [[contravariant functor|contravariant]] [[representable functors]] ($C^{op} \to Set$) such as [[limits]] and [[right adjoints]].

Note that polarity is not so much a property of the [[types]] themselves, but rather of their presentation by introduction and elimination rules: It may happen that a positive and a negative type are [[equivalence in type theory|equivalent]] (this is the case, for example, for the [[product type]] in [[simply typed lambda calculus]]).
In terms of [[categorical semantics]] this is simply the fact that a single [[object]] may exhibit multiple [[universal properties]].

Generally, in [[computational effect|effectful]] languages, the [[positive type|positive types]] are better behaved in [[call-by-value]] evaluation strategies and [[negative type|negative types]] are better behaved in [[call-by-name]] and other [[lazy evaluation]] strategies.
A [[category theory|category-theoretic]] explanation of this fact is that [[call-by-value]] languages can be modeled by the [[Kleisli category]] of a [[monad]] $T$ on a category $C$. Usually, the category $C$ will have both [[limits]] (negatives) and [[colimits]] (positives), however the canonical functor $F \colon C \to C_T$ is a [[left adjoint]] so it only generally preserves the colimits (positives). Dually, call-by-name languages can be modeled by the Kleisli category of a [[comonad]] $W$ and the canonical functor $U \colon C \to C_W$ is a [[right adjoint]], and [[right adjoints preserve limits|so]] [[preserved limit|preserves]] [[limits]] (negatives).

## Related concepts

* [[positive type]]

* [[negative type]]

* [[focusing]]

* [[representability determines functoriality]]

## References

* [[Noam Zeilberger]], _The Logical Basis of Evaluation Order and Pattern-Matching_, ([pdf](http://noamz.org/thesis.pdf))

* [[Robert Harper]], _Polarity in Type Theory_, ([blog post](https://existentialtype.wordpress.com/2012/08/25/polarity-in-type-theory/))


polarization

> This entry is about polarization of [[phase spaces]] (or of any [[symplectic manifold]]) into canonical "position" coordinates and [[canonical momenta]]. Different concepts of a similar name include the _[[polarization identity]]_ (such as in an [[inner product space]] or a [[Jordan algebra]]) or _[[wave polarization]]_ (such as polarized [[light]]). On the other hand, the concept of _[[polarized algebraic variety]]_ is closely related.

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### Context
#### Geometric quantization
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[[!include geometric quantization - contents]]
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#### Symplectic geometry
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[[!include symplectic geometry - contents]]
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# Contents
* table of contents
{: toc}

## Idea

For a [[symplectic manifold]] $(X, \omega)$ regarded as the [[phase space]] of a [[physical system]], a choice of _polarization_ is, locally, a choice of decomposition of the [[coordinates]] on $X$ into "[[canonical coordinates]]" and "[[canonical momenta]]".

The archtypical example is that where $X = T^* \Sigma$ is the [[cotangent bundle]] of a [[manifold]] $\Sigma$. In this case the canonical [[canonical coordinates]] are those parameterizing $\Sigma$ itself, while the canonical [[canonical momenta]] are coordinates on each [[fiber]] of the cotangent bundle.

But for general symplectic manifolds there is no such canonical choice of coordinates and momenta. Moreover, in general there is not even a global notion of canonical momenta. Instead, a choice of (real) polarization is a [[foliation]] of phase space by [[Lagrangian submanifolds]] and then

* the "[[canonical coordinates]]" are coordinates on the corresponding [[leaf space]] (parameterizing the leaves);

* the "[[canonical momenta]]" are coordinates along each [[leaf]]. If there is no _typical leaf_ then this is not a globally defined notion, only the polarization itself is.

Locally this is a choice of [[coordinate patch]] $\phi : \mathbb{R}^{2n} \to X$ such that the [[symplectic form]] takes the form

$$
  \phi^* \omega = \sum_{i = 1}^n \mathbf{d} q^i \wedge \mathbf{d}p_i
$$

where the $\{q^i : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\}$ are the canonical coordinates on the first $\mathbb{R}^n$-factor of the [[Cartesian space]] $\mathbb{R}^{2n}$, and where $\{p_o : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\}$ are canonical coordinates  on the second $\mathbb{R}^n$-factor.

## Definition

The traditional notion of polarization applies to a [[symplectic manifold]].

* [Of a symplectic manifold](OfASymplecticManifold)

Symplectic manifold are the lowest step in a tower of notions in [[higher symplectic geometry]] which proceeds with [[n-plectic geometry]] for all $n$ and manifolds refined to [[smooth infinity-groupoids]]. The next simplest cases in this tower are [[symplectic Lie n-algebroids]], which for $n=1$ are [[Poisson Lie algebroids]] and for $n = 2$ are [[Courant Lie 2-algebroids]]:

* [Of a Poisson Lie algebroid](#OfAPoissonLieAlgebroid)

* [Of a Courant Lie 2-algebroid](#OfACourantLie2Algebroid)

* [Of a symplectic Lie n-algebroid](#OfASymplecticLienAlgebroid)

### Of a symplectic manifold
 {#OfASymplecticManifold}

Let $(X, \omega)$ be a [[symplectic manifold]].

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

A **real polarization** of $(X, \omega)$ is a [[foliation]] by [[Lagrangian submanifolds]].

=--

For instance ([Weinstein, p. 9](#Weinstein)).

More generally

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

A **polarization** of $(X,\omega)$ is a choice of involutive [[Lagrangian subspace|Lagrangian subbundle]] $\mathcal{P} \hookrightarrow T_{\mathbb{C}} X$ of of the [[complexification|complexified]] [[tangent bundle]] of $X$.

=--

For instance ([Bates-Weinstein, def. 7.4](#BatesWeinstein))

### Of a Poisson Lie algebroid
 {#OfAPoissonLieAlgebroid}

A [[Poisson Lie algebroid]] $\mathfrak{P}$ is a [[symplectic Lie n-algebroid]] for $n = 1$. Regarding its [[Chevalley-Eilenberg algebra]] as the algebra of functions on a [[dg-manifold]], that dg-manifold carries a graded [[symplectic form]] $\omega$. One can then say

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

A dg-[[Lagrangian submanifold]] of $(\mathfrak{P}, \omega)$ is also called a **$\Lambda$-structure**. ([Ševera, section 4](#Severa)).

Hence we might say **real polarization** of $(\mathfrak{P}, \omega)$ is a foliation by dg-Lagrangian submanifolds.

=--

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

For $(X, \pi)$ the [[Poisson manifold]] underlying a [[Poisson Lie algebroid]] $(\mathfrak{P}, \omega)$, a dg-Lagrangian submanifold of $(\mathfrak{P}, \omega)$ corresponds to a [[coisotropic submanifold]] of $(X, \pi)$.

=--

([Ševera, section 4](#Severa))

+-- {: .num_remark}
###### Remark

The dg-Lagrangian submanifolds also correspond to [[branes]] in the [[Poisson sigma-model]] (see there) on $(\mathfrak{P}, \omega)$.

=--

### Of a Courant Lie 2-algebroid
 {#OfACourantLie2Algebroid}

A [[Courant Lie algebroid]] $\mathfrak{C}$ is a [[symplectic Lie n-algebroid]] for $n = 2$. Regarding its [[Chevalley-Eilenberg algebra]] as the algebra of functions on a [[dg-manifold]], that dg-manifold carries a graded [[symplectic form]] $\omega$. One can then say

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

A dg-[[Lagrangian submanifold]] of $(\mathfrak{C}, \omega)$ is also called a **$\Lambda$-structure**. ([Ševera, section 4](#Severa)).

Hence we might say **real polarization** of $(\mathfrak{C}, \omega)$ is a foliation by dg-Lagrangian submanifolds.

=--

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

The dg-Lagrangian submanifolds of a Courant Lie 2-algebroid $(\mathfrak{C}, \omega)$ correspond to [[Dirac structures]] on $(\mathfrak{C}, \omega)$.

=--

([Ševera, section 4](#Severa))

### Of a symplectic Lie $n$-algebroid
 {#OfASymplecticLienAlgebroid}

??

### Of an $n$-plectic smooth $\infty$-groupoid

A simple notion of a real polarization for [[2-plectic manifolds]] is
considered within the context of higher geometric quantization in
[Rogers, Chap. 7](#Rogers).

## Examples

### Real polarizations

(...)

### Kähler polarizations
 {#KählerPolarization}

If the  [[symplectic manifold]] $(X,\omega)$ lifts to the structure of a [[Kähler manifold]] $(X, J, g)$, hence with [[Riemannian metric]] $g(-,-) = \omega(-,I(-))$, then the holomorphic/antiholomorphic decomposition induced by the [[complex manifold]] structure is a polarization of $(X,\omega)$. Polarizations of this form are therefore called **[[Kähler polarizations]]**.

### Convex polarizations

* [[p-convex polarization]]

## Related concepts

### Quantum states and wave functions

Upon ([[geometric quantization|geometric]]) [[quantization]] of the physical system described by the [[symplectic manifold]] $(X, \omega)$ a [[quantum state]] is supposed to be a function on $X$ -- or rather a [[section]] of a [[prequantum line bundle]] which is a "wave-function that only depends on the canonical coordinates", not on the canonical momenta. In terms of polarizations this is formalized by saying that a [[space of states (in geometric quantization)|quantum state]] is a section which is [[covariant derivative|covariant constant]] along the [[leaves]] of the polarization (along the "momentum direction").

### Bohr-Sommerfeld leaves

After a choice of [[prequantum line bundle]] $\nabla$ lifting $\omega$, a **[[Bohr-Sommerfeld leaf]]** of a (real) polarization is a [[leaf]] on which the prequantum line bundle is not just flat, but also trivializable as a [[circle bundle]] with connection.

### Liouville integrability

If a polarization on an $2n$-dimensional symplectic manifold is generated from $n$ [[Hamiltonian vector fields]] whose [[Hamiltonians]] commute with each other under the [[Poisson bracket]] (and one of them is regarded as that generating time evolution of a mechanical system) then one speaks of a [[Liouville integrable system]].

### Refinement to higher geometric quantization

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

### Refinement to higher quantization by push-forward

[[!include orientations in higher quantization - table]]

### Other

* [[canonical transformation]]

* [[polarized algebraic variety]]

## References

Lecture notes include

* [[Alan Weinstein]], _Lectures on Symplectic manifolds_ Lecture 2 _Lagrangian splittings, real and complex polarizations, Kähler manifolds_, CBMS Regional Conference Series in Mathematics, AMS (1977)
 {#Weinstein}

* Sean Bates, [[Alan Weinstein]], _Lectures on the geometry of quantization_, [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)
 {#BatesWeinstein}

* Kristin Shaw, _An introduction to polarizations_ ([pdf](http://www.math.toronto.edu/karshon/grad/2006-07/polarizations.pdf))

and section 4 and 5 of

* [[Matthias Blau]], _Symplectic Geometry and Geometric Quantization_ ([[BlauGeometricQuantization.pdf:file]])
 {#Blau}

or section 5 of

* A. Echeverria-Enriquez, M.C. Munoz-Lecanda, N. Roman-Roy, C. Victoria-Monge, _Mathematical Foundations of Geometric Quantization_ Extracta Math. 13 (1998) 135-238 ([arXiv:math-ph/9904008](http://arxiv.org/abs/math-ph/9904008))

or

* Yuichi Nohara, _Independence of Polarization
in Geometric Quantization_ ([pdf](http://geoquant2007.mi.ras.ru/nohara.pdf))

Lagrangian submanifolds of [[L-infinity algebroids]] are considered in

* {#Severa} [[Pavol Ševera]], _Some title containing the words "homotopy" and "symplectic", e.g. this one_ ([arXiv:0105080](http://arxiv.org/abs/math/0105080))

In the case that the polarization integrates to the [[action]] of a [[Lie group]] $G$ one may think of passing to polarized sections as equivlent to passing to $G$-[[gauge equivalence|gauge equivalence classes]]. This point of view is highlighted in

* [[Gabriel Catren]], _On the Relation Between Gauge and Phase Symmetries_, Foundations of Physics 44 (12):1317-1335 (2014) ([web](http://philpapers.org/rec/CATOTR))

A candidate for polarizations for higher geometric quantization in $n$-plectic geometry is discussed in Chapter 7 of

* {#Rogers} [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068)).

[[!redirects polarizations]]

[[!redirects canonical coordinate]]
[[!redirects canonical coordinates]]

[[!redirects real polarization]]
[[!redirects real polarizations]]

[[!redirects complex polarization]]
[[!redirects complex polarizations]]

[[!redirects canonical position]]
[[!redirects canonical positions]]


polarization identity

> This entry is about the notion in [[linear algebra]] relating bilinear and quadratic forms. For the notion in [[symplectic geometry]] see at _[[polarization]]_. For polarization of light, see _[[wave polarization]]_.

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

# The polarization identity
* table of contents
{: toc}

## Idea

Any [[symmetric bilinear form]] $(-) \cdot (-)$ defines a [[quadratic form]] $(-)^2$.  The _polarization identity_ reconstructs the bilinear form from the quadratic form.  More generally, starting from any [[bilinear form]], the polarization identity reconstructs its symmetrization.  A slight variation applies this also to [[sesquilinear forms]], which don\'t require any kind of symmetry to work.  The whole business actually applies to [[bilinear maps]], not just forms (that is, taking arbitrary values, not just values in the [[base field]] or some other [[line]]).  Some kind of linearity is crucial: polarization doesn\'t work without addition (and subtraction); we must also be able to divide by $2$ for the best results.  It is also possible to generalize from quadratic forms to higher-order [[homogenous form]]s if we can divide by larger integers.

## Statements

Let $R$ be a [[commutative ring]].  Let $V$ and $W$ be $R$-[[modules]], and let $m\colon V \times V \to W$ be a [[bilinear map]]; that is, we have an $R$-module [[homomorphism]] $V \otimes V \to W$.  Let $Q\colon V \to W$ be the [[quadratic map]] given by $Q(x) = m(x,x)$; this is *not* an $R$-module homomorphism.

Then we have:

* the _[[parallelogram law]]_: $2 Q(x) + 2 Q(y) = Q(x + y) + Q(x - y)$, and
* the __polarization identity__: $2 m(x,y) + 2 m(y,x) = Q(x + y) - Q(x - y)$.

Writing $Q(x)$ as $x^2$ and $m(x,y)$ as $x y$, these read:

* $2 x^2 + 2 y^2 = (x + y)^2 + (x - y)^2$,
* $2 x y + 2 y x = (x + y)^2 - (x - y)^2$.

The polarization identity also has these alternative forms:

* $x y + y x = (x + y)^2 - x^2 - y^2$,
* $x y + y x = x^2 + y^2 - (x - y)^2$;

these may derived by adding or subtracting the parallelogram law and the first polarization identity assuming that multiplication by $2$ is cancellable in $W$; they can also both be proved directly without any such assumption.

The most general form of such expressions is
$$ k x y + k y x = \sum_i c_i (a_i x + b_i y)^2 $$
for any [[finite sequence]] of triples $(a,b,c)$ from $R$ such that
$$ \sum_i a_i ^ 2 c_i = \sum_i b_i ^ 2 c_i = 0 ,$$
if
$$ k = \sum_i a_i b_i c_i .$$
In the three specific versions of the polarization identity listed above, the values of $a$, $b$, and $c$ are all always $0$, $1$, or $-1$, but there is no reason to restrict ourselves to this or even assume that they are integers (or even rational numbers or more generally members of the [[prime field]] of $R$).  (If we want to allow $R$ to be a [[rig]], then we can use a version of this with sums on both sides of the equation, but that seems like [[centipede mathematics]] since I don\'t know any application of that case.)

Now suppose that $m$ is [[symmetric bilinear map|symmetric]], so that $x y = y x$.  And suppose that multiplication by $2$ is (not merely cancellable but also) invertible in $W$.  (If $1/2 \in R$, then that is more than sufficient.)  Then the polarization identities read:

* $x y = \frac{1}{4} (x + y)^2 - \frac{1}{4} (x - y)^2$,
* $x y = \frac{1}{2} (x + y)^2 - \frac{1}{2} x^2 - \frac{1}{2} y^2$,
* $x y = \frac{1}{2} x^2 + \frac{1}{2} y^2 - \frac{1}{2} (x - y)^2$.

That is, we may recover $m$ from $Q$ (in any of these ways).  The second one is probably the most widely used.  Regardless of the original symmetry of $m$, we may recover its symmetrization $x \circ y = (x y + y x)/2$ (the [[Jordan product]]) in any these ways.  The most general formula for $x \circ y$ is
$$ x \circ y
= \frac{\sum_i c_i (a_i x + b_i y)^2}
{\sum_i 2 a_i b_i c_i} $$
when the sums of $a^2 c$ and $b^2 c$ are zero (as above) and the sum in the denominator is invertible.  (In that case, one could also normalize the $c$s so that the denominator becomes $1$.)  Notice that $2$ must be invertible to have any hope of recovering $m$ or its symmetrization; but we could recover $x y + y x$ without that.

We can also go the other direction: given a quadratic map $Q$, if $2$ is invertible, then any polarization identity defines a symmetric bilinear map $m$; these all agree iff $Q$ obeys the parallelogram law, and then $Q$ may be recovered from this $m$ once more.

If $R$ is a $*$-[[star-ring|ring]], then $m$ could be [[sesquilinear map|sesquilinear]] instead of bilinear.  Then $Q$ would satisfy $Q(t x) = t^* t Q(x)$ (which I don\'t know a term for, maybe 'sesqui-quadratic'? or using whatever\'s Latin for 3/4?) instead of $Q(t x) = t^2 Q(x)$ as for a quadratic map.  Since everything is still bilinear or quadratic over the [[integers]], the parallelogram identity still follows, as do the polarization identities in which only integers appear and before assuming symmetry.  Then *regardless* of whether $m$ is symmetric (or conjugate-symmetric or anything else), we can recover $m$ from $Q$ if $R$ has (in addition to $1/2$) an [[imaginary unit]]: an element $i$ such that $i + i^* = 0$ and $i^* i = 1$.  Specifically,
$$ m(x,y) = \frac{1}{4} Q(x + y) - \frac{1}{4} Q(x - y) - \frac{1}{4} i Q(x + i y) + \frac{1}{4} i Q(x - i y) $$
when $m$ is conjugate-linear in the first variable and linear in the second.  (Swap the signs on the two terms involving $i$ if $m$ is linear in the first variable and conjugate-linear in the second.)  The more general version of this is
$$ m(x,y)
= \frac{\sum_i c_i Q(a_i x + b_i y)}
{\sum_i a_i^* b_i c_i} $$
when
$$ \sum_i a_i^* a_i c_i = \sum_i b_i^* b_i c^i = \sum_i a_i b_i^* c_i = 0 $$
and the denominator is invertible.  (Notice that the last two of these conditions are contradictory if the involution is trivial; or rather, they imply that $R$ is the [[trivial ring]].)

## Examples

This is best known in the case of bilinear and quadratic *forms*, where $W$ is the [[ground ring]] $R$.  Here, $m$ is an [[inner product]], making $V$ into an [[inner product space]], and $Q$ is (the square of) the norm, making $V$ into a [[normed space]].  Instead of abbreviating $m(x,y)$ as $x y$, we might abbreviate it as $\langle x, y \rangle$ (or any other common notation for an inner product) and $Q(x)$ as $\|x\|^2$.  (In the general algebraic context that we are assuming here, $Q$ may not have a [[square root]], much less a canonical 'principal' square root to give meaning to $\|x\|$ alone.  Even assuming that $R$ is the field $\mathbb{R}$ of [[real numbers]], that $Q(x) \geq 0$ is an additional assumption, the [[positive-definite|positive (semi)-definiteness]] of the inner product.)

Besides forms, the general theory also applies to [[commutative algebras]], where $W$ is $V$.  Actually, there is no need for $m$ to be associative; although one rarely studies commutative but [[non-associative algebras]], we have an exception with [[Jordan algebras]].  Although the Jordan identity is simpler to express in terms of the multiplication operation (as usual), the application to [[quantum mechanics]] may be more easily motivated through the squaring operation (since the square of an [[observable]] has a more obvious meaning than the Jordan product of two observables), and the polarization identities allow us to recover multiplication from squaring (and subtraction and division by $2$).

When $V = W$ is a $*$-[[*-algebra|algebra]] over $R$, then although the usual multiplication is bilinear, we can also define a (nonassociative) sesquilinear operator $m$ by explicitly taking the adjoint of one argument: $m(x,y) \coloneqq x^* y$.  When $R$ is a complete [[normed field]] and $V = W$ is a $C^*$-[[C-star algebra|algebra]], the resulting 3/4-quadratic operator $Q$ must have a principal square root, but now we write $Q(x) \coloneqq x^* x$ as $|x|^2$ rather than $\|x\|^2$, which is taken.  (However, $|{-}|$ is not an [[absolute value]], since it\'s not multiplicative, only submultiplicative.)

## Higher order

Let $R, V, W$ be as before.  Let $p$ be a [[natural number]], and let $Q\colon V \to W$ be homogeneous of degree $p$; that is,
$$ Q(t v) = t^p Q(v) .$$
We wish to turn $Q$ into a symmetric [[multilinear map]] $m$ of rank $p$ (so $m\colon \Sym_p V \to W$ is linear) as follows:

* If $p = 0$, $m() = Q(0)$;
* If $p = 1$, $m(v_1) = Q(v_1)$;
* If $p = 2$, $m(v_1, v_2) = Q(v_1 + v_2) - Q(v_1) - Q(v_2)$;
* If $p = 3$, $m(v_1, v_2, v_3) = Q(v_1 + v_2 + v_3) - Q(v_1 + v_2) - Q(v_1 + v_3) - Q(v_2 + v_3) + Q(v_1) + Q(v_2) + Q(v_3)$;
* If $p = 4$, $m(v_1, v_2, v_3, v_4) = Q(v_1 + v_2 + v_3 + v_4) - Q(v_1 + v_2 + v_3) - Q(v_1 + v_2 + v_4) - Q(v_1 + v_3 + v_4) - Q(v_2 + v_3 + v_4) + Q(v_1 + v_2) + Q(v_1 + v_3) + Q(v_1 + v_4) + Q(v_2 + v_3) + Q(v_2 + v_4) + Q(v_3 + v_4) - Q(v_1) - Q(v_2) - Q(v_3) - Q(v_4)$;
* etc.

(Morally, the expressions on the right-hand side of each item above should end with $\pm Q(0)$, but $Q(0) = 0$ for any homogeneous map $Q$ of positive degree (since $Q(0) = Q(0 v) = 0^p Q(v) = 0 Q(v) = 0$ when $p \gt 0$), so it is convenient to leave off this term except for $p = 0$.  The $p = 0$ case is instead $Q(0) = Q(0 v) = 0^p Q(v) = 1 Q(v) = Q(v)$, so a homogeneous map of degree $0$ is simply a constant map, and we are defining $m$ to be this constant.)

So defined, $m$ is manifestly symmetric, but it might *not* be multilinear just because $Q$ is homogeneous (for $p \gt 0$); instead, we *define* a __homogeneous polynomial__ of degree $p$ from $V$ to $W$ be such a homogeneous map $Q$ such that $m$ *is* multilinear.  Then a __polynomial__ from $V$ to $W$ is a sum of homogeneous polynomials of various degrees.  Note that if $V$ is the [[free module]] $R^n$ and $W$ is $R$, then this is the usual notion of a polynomial function on $n$ variables over $R$.  (But different [[polynomials]] can give rise to the same polynomial function; see there for examples.)

To get the same $m$ as in earlier sections of this text, the expressions here must be divided by the [[factorial]] $p!$.  Of course, this only works if $p!$ is invertible in $R$; even if $R$ is a [[field]], we need its [[characteristic]] to be $0$ (or at least greater than $p$ if we are only interested in certain values of $p$).  The expressions above work regardless of this to define what a homogeneous polynomial is.  However, if you wish to recover $Q$ from $m$, then you do need to divide by the factorial; otherwise, the only rule that holds in general is that
$$ m(v,\ldots,v) = p!\, Q(v) .$$

## Related concepts

* [[cubical structure on a line bundle]]

[[!redirects polarization identity]]
[[!redirects polarization identities]]
[[!redirects polarisation identity]]
[[!redirects polarisation identities]]


polarized algebraic variety

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Geometric quantization
+-- {: .hide}
[[!include geometric quantization - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _polarization_ on an [[algebraic variety]] is a choice of [[line bundle]] on it whose [[Chern class]] is analogous to the class represented by a [[Kähler form]] in [[complex analytic geometry]].

In other words, the concept of polarized algebraic variety is the generalization of that of _[[Kähler polarization]]_ from [[symplectic geometry]]/[[complex geometry]] to more general [[algebraic geometry]]. In fact it is the generalization of the concept of a [[holomorphic line bundle|holomorphic]] [[prequantum line bundle]] compatible with a [[Kähler polarization]].

(Notice however the reversion of the logic: in [[symplectic geometry]] the symplectic form is given and then a [[complex structure]] is chosen to match it, whereas here in [[algebraic geometry]] the analog of the complex structure exists beforehand, given by the algebraic structure, and now a polarization conversely asks for the analog of a compatible symplectic form (and for its [[prequantization]])).

Therefore a polarized variety together with a choice of [[Theta characteristic]] on it is the algebraic geometry version of a polarized [[phase space]] with [[metaplectic correction]]. See at _[[geometric quantization]]_ for more on this.

## Definition

A _polarization_ of an [[algebraic variety]] $X$ over an [[algebraically closed field]] $k$ is a choice of [[ample sheaf|ample]] element in its [[Néron-Severi group]] $Pic_X/Pic_X^0$. In other words, a polarization is the choice of an [[equivalence class]] of an [[ample sheaf|ample]] algebraic [[line bundle]] over $X$ (e.g. [[holomorphic line bundles]] if $k$ is the [[complex numbers]]) where two such are regarded as equivalent if they differ by tensoring with one whose underlying topological [[Chern class]] is trivial.

The polarization is called _principal_ if it is of [[degree of a coherent sheaf|degree]] 1.

## Examples

### Jacobians and higher Jacobians

The [[Jacobian variety]] of an algebraic variety is principally polarized by the [[theta divisor]].

See also [this MO discussion](http://mathoverflow.net/q/95288/381).

More generally the higher [[intermediate Jacobians]] with their Weil complex structure are polarized. On the other hand, when equipped with the  Griffith complex structure they carry a [[p-convex polarization]] which is [[symplectomorphism|symplectomorphic]] but not [[biholomorphism|biholomorphic]] to the Weil polarization.

## Related concepts

* [[p-convex polarization]]

* [[Prym-Tyurin variety]]

## References

* {#Voisin02} [[Claire Voisin]], section 7 of _[[Hodge theory and Complex algebraic geometry]] I,II_,  Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

* [[eom]], _[Polarized algebraic variety](http://www.encyclopediaofmath.org/index.php/Polarized_algebraic_variety)_

* Jeroen Sijsling, _What is... a polarization?_ [pdf](http://pub.math.leidenuniv.nl/~strengtc/cm/polariz.pdf)

Polarization specifically of [[Jacobian varieties]] is discussed for instance in

* {#Polsihchuk03} [[Alexander Polishchuk]], section 17 of _Abelian varieties, Theta functions and the Fourier transform_, Cambridge University Press (2003) ([pdf](http://math1.unice.fr/~beauvill/pubs/poli.pdf))

[[!redirects polarized algebraic varieties]]

[[!redirects polarized variety]]
[[!redirects polarized varieties]]

[[!redirects principally polarized algebraic variety]]
[[!redirects principally polarized algebraic varieties]]
[[!redirects principally polarized variety]]
[[!redirects principally polarized varieties]]


Polchinski's flow equation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]] via the method of [[effective quantum field theories]] what is called _Wilsonian RG_ (remark \ref{GroupoidOfEFTs} below) and specifically _Polchinski's flow equation_ (prop. \ref{FlowEquationPolchinski} below) is a characterization of the ([[infinitesimal]]) dependence of [[relative effective actions]] $S_{eff,\Lambda}$ ("effective potentials") on the choice of [[UV cutoff]]-scale $\Lambda$.

Solving Polchinki's flow equation with a choice of initial conditions may be used to choose a [[renormalization|("re"-)normalization]] of an [[interacting field theory|interacting]] [[perturbative QFT]].

This is related to, but conceptually different from, the _[[renormalization group flow]] via [[beta functions]]_ in the sense of [[Gell-Mann-Low renormalization cocycles]].

## Statement

+-- {: .num_remark #GroupoidOfEFTs}
###### Remark
**(Wilsonian [[groupoid]] of [[effective quantum field theories]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to [this def.](S-matrix#VacuumFree)) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum ([this def.](effective+quantum+field+theory#CutoffsUVForPerturbativeQFT)).

Then the [[relative effective actions]] $\mathcal{S}_{eff,\Lambda, \Lambda_0}$ ([this def.](efffective+QFT#EffectiveActionRelative)) satisfy

$$
  S_{eff, \Lambda', \Lambda_0}
  \;=\;
  \left(
    \mathcal{S}_{\Lambda'}^{-1}
    \circ
    \mathcal{S}_\Lambda
  \right)
  \left(
    S_{eff, \Lambda, \Lambda_0}
  \right)
  \phantom{AAA}
  \text{for}
  \,
  \Lambda,\Lambda' \in [0,\infty)
  \,,\,
  \Lambda_0 \in [0,\infty) \sqcup \{\infty\}
  \,.
$$

This is similar to a [[group]] of UV-cutoff scale-transformations. But since the [[composition]] operations are only sensible when the UV-cutoff labels match, as shown, it is really a [[groupoid]] [[groupoid action|action]].

This is often called the _Wilsonian RG_, following ([Wilson 71](#Wilson71)).

=--

We now consider the [[infinitesimal]] version of this "flow":

+-- {: .num_prop #FlowEquationPolchinski}
###### Proposition
**([[Polchinski's flow equation]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to [this def.](S-matrix#VacuumFree)), let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum  (def. \ref{CutoffsUVForPerturbativeQFT}), such that $\Lambda \mapsto \Delta_{F,\Lambda}$ is [[differentiable function|differentiable]].

Then for _every_ choice of [[UV regularization]] $\mathcal{S}_\infty$ ([this prop.](effective+QFT#UVRegularization)) the
corresponding [[relative effective actions]] $S_{eff,\Lambda}$ ([this def.](effective+QFT#EffectiveActionRelative)) satisfy the following [[differential equation]]:

$$
  \frac{d}{d \Lambda}
  S_{eff,\Lambda}
  \;=\;
  -
  \frac{1}{2} \frac{1}{i \hbar}
  \frac{d}{d \Lambda'}
  \left(
    S_{eff,\Lambda}
      \star_{F,\Lambda'}
    S_{eff,\Lambda}
  \right)\vert_{\Lambda' = \Lambda}
  \,,
$$

where on the right we have the [[star product]] induced by $\Delta_{F,\Lambda'}$ ([this def.](star+product#PropagatorStarProduct)).

=--

This goes back to ([Polchinski 84, (27)](#Polchinski84)). The rigorous formulation and proof is due to ([Brunetti-Dütsch-Fredenhagen 09, prop. 5.2](#BrunettiDuetschFredenhagen09), [Dütsch 10, theorem 2](#Duetsch10)).

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

First observe that for any [[polynomial observable]] $O \in PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$ we have

$$
  \begin{aligned}
  &
  \frac{1}{(k+2)!}
  \frac{d}{d \Lambda}
  (
    \underset{
      k+2 \, \text{factors}
    }{
    \underbrace{
      O
        \star_{F,\Lambda}
        \cdots
        \star_{F,\Lambda}
      O
    }
    }
  )
  \\
  & =
  \frac{1}{(k+2)!}
  \frac{d}{d \Lambda}
  \left(
    prod
    \circ
    \exp\left(
      \hbar
      \underset{1 \leq i \lt j \leq k}{\sum}
      \left\langle
         \Delta_{F,\Lambda}
         ,
         \frac{\delta}{\delta \mathbf{\Phi}_i}
         \frac{\delta}{\delta \mathbf{\Phi}_j}
      \right\rangle
    \right)
    (
      \underset{
        k + 2 \, \text{factors}
      }{
      \underbrace{
        O \otimes \cdots \otimes O
      }
      }
    )
  \right)
  \\
  & =
  \underset{
    = \frac{1}{2} \frac{1}{k!}
  }{
  \underbrace{
    \frac{1}{(k+2)!}
    \left(
      k + 2
      \atop
      2
    \right)
  }}
  \left(
    \frac{d}{d \Lambda}
    O
     \star_{F,\Lambda}
    O
  \right)
    \star_{F,\Lambda}
  \underset{
    k \, \text{factors}
  }{
  \underbrace{
  O
    \star_{F,\Lambda}
    \cdots
    \star_{F,\Lambda}
  O
  }
  }
  \end{aligned}
$$

Here $\frac{\delta}{\delta \mathbf{\Phi}_i}$ denotes the functional derivative of the $i$th tensor factor of $O$, and the binomial coefficient counts the number of ways that an unordered pair of distinct labels of tensor factors may be chosen from a total of $k+2$ tensor factors, where we use that the [[star product]] $\star_{F,\Lambda}$ is commutative (by symmetry of $\Delta_{F,\Lambda}$) and associative (by [this prop.](star+product#AssociativeAndUnitalStarProduct)).

With this and the defining equality $\mathcal{S}_\Lambda(S_{eff,\Lambda}) =  \mathcal{S}(g S_{int} + j A)$ ([this equation](effective+QFT#eq:RelativeEffectiveActionRelativeToInfinity)) we compute as follows:

$$
  \begin{aligned}
    0
    & =
    \frac{d}{d \Lambda} \mathcal{S}(g S_{int} + j A)
    \\
    & =
    \frac{d}{d \Lambda}
    \mathcal{S}_\Lambda(S_{eff,\Lambda})
    \\
    & =
    \left(
      \frac{1}{i \hbar}
      \frac{d}{d \Lambda} S_{eff,\Lambda}
    \right)
      \star_{F,\Lambda}
    \mathcal{S}_\Lambda(S_{eff,\Lambda})
    +
    \left(
      \frac{d}{d \Lambda}
      \mathcal{S}_{\Lambda}
   \right)
   \left( S_{eff, \Lambda} \right)
    \\
    & =
    \left(
      \frac{1}{i \hbar}
      \frac{d}{d \Lambda} S_{eff,\Lambda}
    \right)
      \star_{F,\Lambda}
    \mathcal{S}_\Lambda(S_{eff,\Lambda})
    \;+\;
    \frac{1}{2}
    \frac{d}{d \Lambda'}
    \left(
      \frac{1}{i \hbar} S_{eff,\Lambda}
        \star_{F,\Lambda'}
      \frac{1}{i \hbar} S_{eff, \Lambda}
    \right)
    \vert_{\Lambda' = \Lambda}
      \star_{F,\Lambda}
    \mathcal{S}_\Lambda
    \left(
      S_{eff, \Lambda}
    \right)
  \end{aligned}
$$

Acting on this equation with the multiplicative inverse $(-) \star_{F,\Lambda} \mathcal{S}_\Lambda( - S_{eff,\Lambda} )$ (using that $\star_{F,\Lambda}$ is a commutative product, so that exponentials behave as usual) this yields the claimed equation.

=--

## Related concepts

* [[renormalization group flow]]

* [[gauge coupling unification]]

## References

The idea of [[effective quantum field theory]] was promoted in

* {#Wilson71} [[Kenneth Wilson]], _Renormalization group and critical phenomena_ , I., Physical review B 4(9) (1971).

The flow equation in its original form is due to

* {#Polchinski84} [[Joseph Polchinski]], equation (27) in _Renormalization and effective Lagrangians_ , Nuclear Phys. B B231, 1984 ([pdf](http://max2.physics.sunysb.edu/~rastelli/2016/Polchinski.pdf))

The rigorous formulation and proof in [[causal perturbation theory]]/[[perturbative AQFT]] is due to

* {#BrunettiDuetschFredenhagen09} [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], prop. 5.2 of _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](https://arxiv.org/abs/0901.2038))

* {#Duetsch10} [[Michael Dütsch]], theorem 2 in _Connection between the renormalization groups of Stückelberg-Petermann and Wilson_, Confluentes Mathematici, Vol. 4, No. 1 (2012) 12400014 ([arXiv:1012.5604](https://arxiv.org/abs/1012.5604))

reviewed in

* {#Duetsch18} [[Michael Dütsch]], section 3.8 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

[[!redirects Wilsonian RG]]
[[!redirects Wilsonian RGs]]

[[!redirects Wilsonian RG flow]]
[[!redirects Wilsonian RG flows]]

[[!redirects Polchinski's flow equation]]
[[!redirects Polchinski's flow equations]]

[[!redirects Polchinski flow equation]]
[[!redirects Polchinski flow equations]]


pole

## Related concepts

* [[singularity]]

## References

* Wikipedia, _[Pole (complex analysis)](http://en.wikipedia.org/wiki/Pole_(complex_analysis))_

[[!redirects poles]]

[[!redirects simple pole]]
[[!redirects simple poles]]


Polish space

# Contents
* table of contents
{: toc}

## Introduction

A **Polish space** is a [[topological space]] that's [[homeomorphism|homeomorphic]] to a [[separable space|separable]] [[complete space|complete]] [[metric space]]. Every [[second countable space|second countable]] [[locally compact space|locally compact]] [[Hausdorff space]] is a Polish space, among others.

Polish spaces, or the [[measurable spaces]] that they define ([[standard Borel spaces]]) provide a useful framework for doing [[measure theory]].  As with any topological space, we can take a Polish space and regard it as a [[measurable space]] with its [[sigma-algebra]] of [[Borel set]]s.  Then, there is a very nice classification of Polish spaces up to measurable bijection: there is one for each [[countable set|countable]] [[cardinality]], one whose cardinality is that of the [[continuum]], and no others.

Why are Polish spaces 'not very big'?  In other words, why are there none with cardinality exceeding the continuum?  As with any separable metric space, it's because any Polish space has a countable dense subset and you can write any point as a limit of a sequence of points in this subset.  So, you only need a [[sequence]] of integers to specify any point in a Polish space. More sharply, see Lemma \ref{baire} below.

+-- {: .num_example}
###### Example
The primordial example (and in practice, one of the most convenient) is [[set-theoretic Baire space|Baire space]] $B$, viz. the space of irrational real numbers between $0$ and $1$, with the [[subspace topology]] inherited from the [[real number|real line]]. This is obviously not complete with respect to the metric induced from the real line, but it is homeomorphic to the product space $\mathbb{N}^\mathbb{N}$ via regular [[continued fraction]] expansions, and the latter is metrizable by a complete metric where the distance between two sequences $a = (a_1, a_2, \ldots)$ and $b = (b_1, b_2, \ldots)$ of positive integers is given by the formula $d(a, b) = 1/2^n$ where $n$ is the least integer such that $a_n \neq b_n$. A countable dense subset is given by continued fractions that eventually repeat (quadratic surds).
=--

The convenience of Baire space is attested to by the fact that in [[descriptive set theory]], a "real number" is often taken to mean just a point of Baire space. See also Theorem \ref{Borel} below.

## Properties

Much of the material here is adapted from [Marker](#Marker), who in turn cites the text by Kechris as a main source.

* A countable product $X = \prod_n X_n$ of Polish spaces is Polish. Indeed, each $X_n$ is metrizable by a complete metric $d_n$ that takes values in $[0, 1]$, and then we may define another such complete metric on $X$ by
$$d(f, g) = \sum_n \frac1{2^{n+1}} d_n(f(n), g(n)).$$
The metric topology on $X$ coincides with the product topology.

In particular, the [[Hilbert cube]] $[0, 1]^\mathbb{N}$ is Polish.

* A subspace of a Polish space $X$ is Polish iff it is a [[G-delta set]] of $X$. See [Marker](#Marker), Theorem 1.33.

### "Universal" Polish spaces

+-- {: .num_prop #hilbert}
###### Proposition
Any Polish space is homeomorphic to a subspace of the Hilbert cube.
=--

In fact, this is true of any [[separable space|separable metrizable space]]. Note that mere separability does not suffice, as there are separable spaces that are not first-countable, such as the Stone-Čech compactification of $\N$, and hence cannot be subspaces of any metrizable space. What distinguishes Polish spaces is that they are, up to homeomorphism, precisely the _$G_\delta$ subsets_ of the Hilbert cube.

+-- {: .num_lemma #baire}
###### Lemma
Every [[inhabited]] Polish space $X$ admits a continuous surjection from Baire space.
=--

+-- {: .proof}
###### Proof (sketch)
Construct by induction a collection of closed sets (balls) $C_s$ indexed over _finite_ sequences $s$ of positive integers, with the following properties:

* For the empty sequence $e$, $C_e = X$;

* For nonempty sequences $s$, $diam(C_s) \leq 1/n$ where $n = length(s)$;

* Letting $(s, k)$ denote the extension of $s$ obtained by appending to $s$ the final element $k$,
$$C_s = \bigcup_{k=0}^\infty C_{(s, k)};$$

* If $t$ extends $s$, then $center(C_t) \in C_s$.

Then define $f: B \to X$ where for each infinite sequence $a \in \mathbb{N}^\mathbb{N}$,

$$f(a) = \bigcap \{C_s: a\; extends\; s\}.$$

One may check that $f$ is continuous and surjective.
=--

### Cantor-Bendixson rank

Let $C \subseteq X$ be a closed subset of a Polish space $X$. The following operation traces back to Cantor's work in Fourier analysis, which in turn led to his study of [[countable ordinal|countable ordinals]] and [[ordinal]] analysis.

For a subset $A \subseteq X$, recall that $x \in A$ is a [[limit point]] if $x \in Cl(A \setminus \{x\})$. A point $x \in A$ that is not a limit point of $A$ is called an _isolated point_ of $A$. Clearly each isolated point is open relative to $A$, as is therefore the set of isolated points.

+-- {: .num_defn}
###### Definition
The **Cantor-Bendixson derivative** of $C$ is the set $C' \subseteq C$ of limit points relative to $C$. For each ordinal $\alpha$ the iterated derivative $C^\alpha$ is defined by recursion: $C^0 = C$, $C^{\alpha + 1} = (C^\alpha)'$, and $C^\alpha = \bigcap_{\beta \lt \alpha} C^\beta$ if $\alpha$ is a limit ordinal.
=--

Since $\beta \lt \alpha$ implies $C^\beta \supseteq C^\alpha$, it is clear that there is a least ordinal $\alpha$ for which $C^\alpha = C^{\alpha + 1}$. This ordinal is called the _Cantor-Bendixson rank_ of $C$. A _[[perfect set]]_ is a closed set $C$ such that $C = C'$. (Some people insist that a perfect set also be nonempty; we do not.)

+-- {: .num_prop #cantor}
###### Proposition
For each nonempty perfect set $P$ in a Polish space $X$, there is a continuous injection $i: \mathbf{2}^\mathbb{N} \to P$ from [[Cantor space]]. In particular, the cardinality of $P$ is the [[continuum]] $c$.
=--

+-- {: .proof}
###### Proof
Let $T$ be the complete infinite binary tree, whose infinite paths from the root correspond to points in Cantor space $\mathbf{2}^\mathbb{N}$. To each node $s$ in $T$ (a finite sequence of $0$'s and $1$'s) we construct by induction an open set $U_s$ with the following properties:

* $U_e = X$ for the empty sequence $e$,

* $\widebar{U_t} \subseteq U_s$ if $s$ is an initial segment of $t$,

* For the two children $(s, 0)$ and $(s, 1)$ of $s$, the sets $U_{(s, 0)}$ and $U_{(s, 1)}$ are disjoint,

* $diam(U_s) \leq 1/n$ where $n$ is the length of (nonempty) $s$,

* $U_s \cap P \neq \emptyset$ for each $s$.

Indeed, if $U_s$ has been constructed, and given $x \in U_s \cap P$, there are (at least!) two points $x_0, x_1 \in U_s \cap P$ since $P$ is perfect. We can easily find disjoint neighborhoods $U_{(s, 0)}, U_{(s, 1)}$ of these points respectively with the required properties.

Then for each path $p = (a_0, a_1, \ldots) \in \mathbf{2}^\mathbb{N}$, define

$$i(p) = \bigcap_{s \preceq p} U_s$$

where $s \preceq p$ means $s$ is an initial segment of $p$. The intersection consists of a single point because it equals the intersection of a decreasing chain of closed sets with shrinking diameter (thus closing in on a limit of a [[Cauchy sequence]]). The map $i$ is clearly injective by the disjointness of open sets of children, and it is easy to see $i$ is continuous.
=--

+-- {: .num_theorem}
###### Theorem
For $C$ a closed subset of a Polish space $X$, the Cantor-Bendixson rank is a countable ordinal $\alpha$. The complement $C \setminus C^\alpha$ is at most countable.
=--

+-- {: .proof}
###### Proof
Let $U_i$ be a countable basis of $X$. For each $\beta \lt \alpha$, each point $x \in C^\beta \setminus C^{\beta + 1}$ is an isolated point, so we can find an $U_{i(x)}$ in the basis such that $U_{i(x)} \cap (C^\beta \setminus C^{\beta + 1}) = \{x\}$. It is then clear that $x \mapsto i(x)$ is injective, so each $C^\beta \setminus C^{\beta + 1}$ is countable. Similarly, whenever $C^\beta \setminus C^{\beta + 1}$ is nonempty, we can find a basis element $U_{j(\beta)}$ that isolates one of its points (say $x$), and this same $U_j$ cannot isolate any point of an earlier $C^\gamma \setminus C^{\gamma + 1}$ since $x$ is a limit point of $C^\gamma$. It follows that $\beta \mapsto j(\beta)$ is an injective map, so that $\alpha$ must be a countable ordinal, and the collection $F \coloneqq C \setminus C^\alpha = \bigcup_{\beta \lt \alpha} C^\beta \setminus C^{\beta + 1}$ is (at most) countable.
=--

+-- {: .num_cor #CH}
###### Corollary
A closed set in a Polish space is the disjoint union $P \cup F$ of a perfect set $P$ and a finite or countable set $F$. Hence an infinite closed subset in a Polish space has cardinality either $\aleph_0$ or the continuum $c = 2^{\aleph_0}$ ([[continuum hypothesis]] for closed sets).
=--

### Borel isomorphism between Polish spaces

Recall that a function $f: X \to Y$ between topological spaces is _Borel_ if $f^{-1}(V)$ is a Borel set in $X$ for every open $V$ in $Y$. Topological spaces and Borel functions form a category.

+-- {: .num_lemma}
###### Lemma
If $A, B$ are countably infinite $T_1$-[[separation axiom|spaces]], then any bijection $f: A \to B$ is a Borel isomorphism.
=--

For, the inverse image $f^{-1}(U)$ of an open set, being countable, is an $F_\sigma$ set. In particular, any two denumerable Polish spaces are Borel isomorphic.

+-- {: .num_lemma}
###### Lemma
The unit interval $[0, 1]$ and Cantor space $2^\mathbb{N}$ are Borel isomorphic.
=--

+-- {: .proof}
###### Proof
Let $E \subseteq \mathbf{2}^\mathbb{N}$ be the set of $(0, 1)$-sequences that are eventually constant. Then

$$f(a_1, a_2, \ldots) = \sum_n \frac{a_n}{2^n}$$

maps $2^\mathbb{N} \setminus E$ homeomorphically onto the space of non-(dyadic rational) numbers in $[0, 1]$. Pick any bijection $g: E \to \{dyadic\; rationals\}$. Then the union of $f$ and $g$ defines a Borel isomorphism $h: \mathbf{2}^\mathbb{N} \to [0, 1]$.
=--

+-- {: .num_prop #subset}
###### Proposition
Any Polish space $X$ is Borel isomorphic to a Borel subset of Cantor space.
=--

+-- {: .proof}
###### Proof
There is a sequence of maps

$$X \hookrightarrow [0, 1]^\mathbb{N} \stackrel{h^\mathbb{N}}{\to} (\mathbf{2}^{\mathbb{N}})^\mathbb{N} \stackrel{j}{\to} \mathbf{2}^\mathbb{N}$$

where the first map is an inclusion of a $G_\delta$ set, the second is induced from a Borel isomorphism $h$, and the third is a homeomorphism.
=--

+-- {: .num_theorem #Borel}
###### Theorem
Any two Polish spaces of the same cardinality are Borel isomorphic.
=--

+-- {: .proof}
###### Proof
It suffices to prove this for uncountable Polish spaces (which have continuum cardinality, as we saw in Corollary \ref{CH}). We show that any such $X$ is Borel isomorphic to Cantor space. By Proposition \ref{subset}, we have an inclusion $i: X \to \mathbf{2}^{\mathbb{N}}$ that maps $X$ Borel isomorphically onto its image, and by Proposition \ref{cantor}, we have an inclusion $j: \mathbf{2}^{\mathbb{N}} \to X$ that maps Cantor space Borel isomorphically (even homeomorphically) onto its image. The rest is just a matter of checking that at least one of the proofs of the [[Cantor-Schroeder-Bernstein theorem]] applies to this Borel context.

Indeed, as explained [here](https://ncatlab.org/nlab/show/Cantor-Schroeder-Bernstein+theorem#alt), we may consider

$$S = \bigcap_{n \geq 0} (\neg \exists_j \neg \exists_i)^n(X)$$

Each of the direct image maps $\exists_i$ and $\exists_j$ takes Borel sets to Borel sets, since their inverses $i^{-1}, j^{-1}$ are Borel functions on their domains. So each of the iterates $(\neg \exists_j \neg \exists_i)^n(X)$ is a Borel set and so is their countable intersection $S$. The set $S$ is a fixed point of $\neg \exists_j \neg \exists_i$ and so we construct a Borel isomorphism $h: X \to \mathbf{2}^{\mathbb{N}}$ as

$$\array{
x & \mapsto & i(x) & if\; x \in S \\
 & \mapsto & j^{-1}(x) & if\; x \in \neg S
}$$

following the proof of the [[Cantor-Schroeder-Bernstein theorem]].
=--

For another proof, see theorem 3.1.1 of [Berberian](#Ber).

## Further examples

* The classical $L^p$-[[Lebesgue space|spaces]] for $p \lt \infty$ are Polish spaces.

* For a metric space $X$, let $K(X)$ be the set of nonempty [[compact space|compact]] subsets of $X$, equipped with the [[Hausdorff metric]]. If $X$ is a separable complete metric space, then so is $K(X)$.

* A [[local compactum|locally compact Hausdorff space]] is Polish iff it is second-countable.

* If $X, Y$ are Polish and $X$ is locally compact, then the [[exponentiable space|exponential]] $Y^X$ (the space of continuous maps with the compact-open topology) is also Polish. (See for example A.10 [here](http://www.math.tamu.edu/~kerr/book/appendixA.pdf).)

* If $X$ is Polish, then the space $P(X)$ of Borel probability measures $\mu$ equipped with the Prokhorov metric is also Polish. Definitions: for $x \in X$ and $A \subset X$ nonempty, put $d(x, A) = \inf \{d(x, a): a \in A\}$, and for $\alpha \gt 0$ define $A_\alpha = \{x \in X: d(x, A) \lt \alpha\}$ and $\emptyset_\alpha = \emptyset$. Define the *Prokhorov metric* $d_P$ by
$$d_P(\mu, \nu) = \inf \{\alpha \gt 0: \forall_{Borel\; A} \mu(A) \lt \nu(A_\alpha) + \alpha \; and \; \nu(A) \lt \mu(A_\alpha) + \alpha\}.$$
Then the map $u_X: X \to P(X)$ sending $x$ to the [[Dirac measure]] $\delta_x$ is continuous. If $x_n$ is a countable dense set of $X$, then the set of rational convex combinations $\sum_{i=1}^n q_n \delta_{x_n}$ (with $0 \leq q_n \leq 1$ and $\sum_i q_i = 1$) is a countable dense subset of $P(X)$. More at [[Giry monad]].

* Spaces of [[structure in model theory|structures]] and [[models]] (in the [[model theory]] sense), and spaces of $n$-[[n-type (model theory)|types]] (again in the model theory sense), quite often provide examples of Polish spaces. For example, if $L$ is a countable language (a countable [[signature (in logic)|signature]]), then the collection of possible $L$-structures $M$ on the countable universe $\mathbb{N}$, topologized by taking as basic opens
$$U_\phi = \{M \in Struct(L): M \models \phi\}$$
where $\phi$ is a quantifier-free sentence, is a Polish space homeomorphic to the product space
$$\prod_{relations\; R} \mathbf{2}^{\mathbb{N}^{arity(R)}} \times \prod_{functions\; f} \mathbb{N}^{\mathbb{N}^{arity(f)}}$$
(taking constants to be functions of arity $0$ in the signature).

As an example of the last principle, we have a kind of continuum hypothesis for substructures:

+-- {: .num_prop}
###### Proposition
Let $X$ be a countable structure of a language; then the number of substructures of $X$ is either countable or the continuum.
=--

+-- {: .proof}
###### Proof
A subset of $X$ is specified by its [[characteristic function]] $\chi \in 2^X$, where $2^X$ is regarded as a Polish space. In order for the subset *not* to support a substructure, then must be some function symbol $f$ of the language, say of arity $n$, and elements $a_1, \ldots, a_n$ such that $\chi(a_i) = 1$ for all $i$ and $\chi(f_X(a_1, \ldots, a_n)) = 0$. The basic open

$$U_{f; a_1, \ldots, a_n} \coloneqq \{\omega \in 2^X: \; \omega(a_1) = 1, \ldots, \omega(a_n) = 1, \; \omega(f_X(a_1, \ldots, a_n)) = 0\}$$

would thus contain $\chi$ and also exclude any substructure; we conclude that the collection of substructures forms a closed subset $C$ of $2^X$. Then the Cantor-Bendixson theorem (i.e., Corollary \ref{CH}) shows that ${|C|}$ is either countable or the continuum.
=--

##References##

* [Polish spaces](http://golem.ph.utexas.edu/category/2008/08/polish_spaces.html), blog discussion, $n$-category café, 2008-

* David Marker, _Descriptive Set Theory_, UIC Course Notes (Fall 2002) ([pdf](http://homepages.math.uic.edu/~marker/math512/dst.pdf))
{#Marker}

* S.K. Berberian, _Borel spaces_, ([pdf](https://www.ma.utexas.edu/mp_arc/c/02/02-156.pdf))
{#Ber}

* A.S. Kechris, _Classical descriptive set theory_, Springer-Verlag (1994).

Internal groupoids in Polish spaces are considered in

* Martino Lupini, _Polish groupoids and functorial complexity_, [arxiv/1407.6671](http://arxiv.org/abs/1407.6671)

[[!redirects Polish spaces]]


poly-morphism

> This page is about the notion of poly-morphism considered by [[Shinichi Mochizuki]]. For the concept in [[computer science]] see [[polymorphism]]. Or see [[universe polymorphism]] for the concept in [[type theory]].

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

#Contents#
* table of contents
{:toc}

## Idea

For a [[category]] $C$, a **poly-morphism** is a collection of [[morphisms]] with common [[source]] and [[target]], considered as a morphism in a new category $C^{poly}$ with the same [[objects]], but with [[hom-sets]] $C^{poly}(a,b) \coloneqq P(C(a,b))$, the [[power set]] of the original hom-sets.

These play a substantive rôle in [[Shinichi Mochizuki]]'s [[inter-universal Teichmüller theory]] ([Mochizuki 12, section 0](#Mochizuki12)). Here we put these into the broader context of [[change of enriching categories]] along [[lax monoidal functors]] $F$ as the special case that $F$ is a [[power set]]-functor, or an [[G-set|equivariant]] version thereof.

## Abstract approach

In the following, fix a [[monoidal category]] $V$ and a $V$-[[enriched category]]

$$
  C \in V Cat
  \,.
$$

Examples to keep in mind for $V$ are the category [[Set]] of sets considered as a [[cartesian monoidal category]], or the category [[G-set]] for a given [[group]] $G$, again with the cartesian monoidal structure.

Recall:

+-- {: .num_defn #ChangeOfEnrichingCategory}
###### Definition
**([[change of enriching category]] from $V$ to itself)**

For $F\colon V\to V$ a [[lax monoidal functor]] from $V$ to itself, the image $F_\ast(C)$ of $C$ under the corresponding [[change of enriching category]] is the $V$-category with the same [[objects]] as $C$, and with $V$-[[hom objects]] given by

$$
  F_\ast(C)(a,b) \coloneqq F(C(a,b))
  \,.
$$

The [[composition]]-, [[unit]]-, [[associator]]- and [[unitor]]-[[morphisms]] of $F_\ast(C)$ are the [[images]] of those of $C$ under $P$ suitably composed with the structure morphisms of the lax monoidal functor $P$.

=--

For the case of poly-morphisms the relevant [[lax monoidal functors]] are [[power set]]-functors

+-- {: .num_example #PowerSetFunctors}
###### Examples
**([[lax monoidal functor|lax monoidal]] [[power set]]-[[functors]])

* For $V=$[[Set]] equipped with its [[cartesian monoidal category]] [[structure]], $C$ an arbitrary [[locally small category]], take $F \coloneqq P \;\colon\; Set \to Set$ to be the [[covariant functor|covariant]] [[power set]] functor. This is lax monoidal since every [[pair]] of [[subsets]] $U \subseteq S$ and $V\subseteq T$ gives a subset $U\times V \subseteq S\times T$.

* For $G$ a group, $V=$[[GSet]] with the [[cartesian monoidal category]] [[structure]], $C$ a [[enriched category|category enriched in]] [[G-sets]], let $F \coloneqq P\;\colon\; GSet \to GSet$ also be the covariant [[power set]] functor, but for $S$ a $G$-set, equip $P(S)$ with the $G$-[[action]]
$$
  U\mapsto g\cdot U
  =
  \{g\cdot x \in S \mid x\in U\}
  \,.
$$

=--

+-- {: .num_defn #PolyMorphisms}
###### Definition
**(poly-morphisms)**

For $C$ a plain category and $F \coloneqq P \colon Set \to Set$ the [[power set]]-functor (Def. \ref{PowerSetFunctors}), we write

$$
  C^{poly}
  \;\coloneqq\;
   P_\ast(C)
$$

for the image of $C$ under the [[change of enriching category]] (Def. \ref{ChangeOfEnrichingCategory}) along $P$.

We say that the [[morphisms]] of $C^{poly}$ are the **poly-morphisms** of $C$.

Explicitly, this means [[composition]] is defined to be
$$
C^{poly}(a,b) \otimes C^{poly}(b,c) = P(C(a,b)) \otimes P(C(b,c)) \to P(C(a,b)\otimes B(b,c)) \to P(C(a,c)) = C^{poly}(a,c)
$$
and the [[unit]] map is
$$
  I \to P(I) \to P(C(a,a)) = C^{poly}(a,a)
  \,.
$$

A **poly-isomorphism** of $C$ is defined to be a poly-morphism of the [[core]] of $C$, hence a morphism of $(Core(C))^{poly}$.
Hence a poly-isomorphism is a collection of _[[invertible morphisms]]_ of $C$.
(Note that these are **not** in general the [[isomorphisms]] in $C^{poly}$: all the isomorphisms in $C^{poly}$ are actual isomorphisms in $C$.)

=--

## References

The above construction for enrichment in plain sets is considered (without the above [[category theory|category-theoretic]] formulation) in:

* {#Mochizuki12} [[Shinichi Mochizuki]], section 0 of _Inter-universal Teichmüller theory I, Construction of Hodge theaters_ (2012) ([pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf))

The construction for enrichment in [[posets]] is considered in

* Alveen Chand, [[Ittay Weiss]], _An ordered framework for partial multivalued functors_, Computer Science and Engineering (APWC on CSE), 2015 2nd Asia-Pacific World Congress on Computer Science ([arXiv:1511.00746](https://arxiv.org/abs/1511.00746))

[[!redirects poly-morphisms]]


Polyakov action

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

#Contents#
* table of contents
{:toc}

## Idea

The _Polyakov action functional_ or _energy functional_ is a standard [[kinetic action|kinetic]] [[action functional]] for [[sigma models]] with [[worldvolume]] $(\Sigma,g)$ and [[target spacetime]] $(X,\mu)$ ([[pseudo-Riemannian manifold|pseudo]])[[Riemannian manifolds]]. Its [[critical points]] are the [[harmonic maps]] from $\Sigma$ to $X$

With a suitable "[[worldvolume]] [[cosmological constant]]" added and with the worldvolume metric "integrated out", then the Polyakov action is [[classical field theory|classically]] equivalent to the [[Nambu-Goto action]] functional, which is simply the "proper volume" function for images of $\Sigma$ in $Y$. But as opposed to the Nambu-Goto action the Polyakov action is quadratic in derivatives. Therefore it is lends itself better to [[perturbation theory]] of [[scattering amplitudes]] -- where the kinetic contributions have to be [[Gaussian integrals]] -- such as in [[worldline formalism]] for [[quantum field theory]] as well as in [[perturbative string theory]].

On the other hand, the [[Nambu-Goto action]] lends itself better to generalizations such as the [[Dirac-Born-Infeld action]] for [[D-branes]].

## Definition

Let

* $p \in \mathbb{N}$ (for [[p-brane]] dynamics),

* $(X,\mu)$ a [[pseudo-Riemannian manifold]] ([[target spacetime]]),

* $\Sigma$ a [[compact topological space|compact]] [[smooth manifold]] ([[manifold with boundary|with boundary]]) of dimension $(p+1)$ ([[worldvolume]]),

* $[\Sigma,X]$ the [[diffeological space]] of [[smooth functions]] $\Sigma \to X$,

* $Met(\Sigma)$ the [[moduli space]] of ([[pseudo-Riemannian metrics|pseudo]])[[Riemannian metrics]] $g$ on $\Sigma$.

The Polyakov action is the [[smooth function]]

$$
  S_{Pol}
  \;\colon\;
  [\Sigma,X]\times Met(\Sigma)
  \longrightarrow
  \mathbb{R}
$$

which on a map $\phi \colon \Sigma \to X$ and a metric $g$ on $\Sigma$ is given by the [[integral]]

$$
  S_{Pol}(\phi,g)
  \coloneqq
  \int_\Sigma
    \Vert d\phi\Vert^2
  dvol_g
$$

where the [[derivative]] $d \phi \in \Gamma(T^\ast \Sigma\otimes \phi^\ast T X)$, and where the [[norm]] $\Vert - \Vert$ is given jointly by the metrics $g$ and $\mu$ of $\Sigma$ and $X$.

When both $\Sigma$ and $X$ are covered by single [[coordinate charts]] $(\sigma^a)_{a = 0}^p$ and $(x^\mu)$, then this reads

$$
  S_{Pol}(\phi,g)
  =
  \int_{\Sigma}
  \sqrt{det g}
  \left(
     g^{a b} \eta_{\mu \nu} (\partial_a \phi^\mu) (\partial_b \phi^\nu)
  \right)
  d \sigma^0 \cdots d \sigma^p
$$

with a sum over repeated indices understood. Here $det g$ denotes the [[absolute value]] of the [[determinant]] of $(g_{a b})$ (often written $-det g$ in the pseudo-Riemannian case.)

## Properties

### Relation to Nambu-Goto action
  {#RelationToNambuGotoAction}

The Polyakov action with a suitable worldvolume cosmological constant term added is classically equivalent to the [[Nambu-Goto action]] (e.g. [Nieto 01, section 2](#Nieto01)).

This [[shell|on-shell]] equivalence is exhibited by the smooth function

$$
  \tilde S
  \;\colon\;
  [\Sigma,X] \times Met(\Sigma)\times \Gamma(T \Sigma \otimes T \Sigma \otimes Dens \Sigma)
  \longrightarrow \mathbb{R}
$$

which on a triple $(\phi,g,\Lambda)$ is given by

$$
  \tilde S(\phi,g,\Lambda)
  \coloneqq
  \int_\Sigma
  \left(
    dvol(g) + \Lambda( \vert d \phi \vert^2 - g)
  \right)
  \,,
$$

where now $\vert d\phi \vert^2$ denotes the square norm only with respect to the metric on $X$.

Because, on the one hand, the [[equations of motion]] induced by $\tilde S$ for variation of $\Lambda$ are

$$
  \vert d \phi \vert^2 - g = 0
$$

and substituting that constraint back into $\tilde S$ gives the [[Nambu-Goto action]]. On the other hand, the equations of motion induced by $\tilde S$ for variation of $g$ are

$$
  \tfrac{1}{2}g^{-1}dvol(g) - \Lambda = 0
$$

and substituting that back into $\tilde S$ gives

$$
  \tfrac{1}{2}\int_\Sigma
  \left(
    \Vert d \phi\Vert^2
    +
    (p-1)
  \right)
  dvol(g)
$$

which is the Polyakov action with "[[cosmological constant]]" $(p-1)$.

(So the case where this cosmological constant correction disappears is $p = 1$ corresponding to the [[string]].)

## Related concepts

* [[minimal surface]]

* [[bosonic string]]

* [[Liouville theory]]

* [[Nambu-Goto action]]

## References

The Polyakov action  is named after

* [[Alexander Polyakov]], *Quantum geometry of bosonic strings*, Phys. Lett. B **103** (1981) 207-210 [doi:10.1016/0370-2693(81)90743-7, [pdf](http://qft.itp.ac.ru/polyakov-2.pdf)]

where it is discussed for the [[bosonic string]] and related to [[Liouville theory]], and

* [[Alexander Polyakov]], *Quantum geometry of fermionic strings*, Physics Letters B **103** 3 (1981) 211-213 [doi:10.1016/0370-2693(81)90744-9]

where it is generalized to the [[superstring]], but in fact (cf. [Polyakov (2008), p. 3](Alexander+Polyakov#Polyakov08)) it originates already in discussion of the [[spinning string]] in:

* [[Stanley Deser]], [[Bruno Zumino]], *A complete action for the spinning string*, Physics Letters B **65** 4 (1976) 369-373 [doi:10.1016/0370-2693(76)90245-8[pdf](http://cdsweb.cern.ch/record/201648/files/CM-P00061857.pdf?version=1)]

* [[Lars Brink]], [[Paolo Di Vecchia]], [[Paul Howe]], _A locally supersymmetric and reparametrization invariant action for the spinning string_ , Physics Letters B **65** Issue 5, 20 (1976) 471-474 [doi:10.1016/0370-2693(76)90445-7]

Further early discussion of the Polyakov action:

* [[Jean-Loup Gervais]], [[André Neveu]], *The dual string spectrum in Polyakov's quantization (I)*, Nuclear Physics B **199** 1 (1982) 59-76 [doi:10.1016/0550-3213(82)90566-1]

* [[Jean-Loup Gervais]], [[André Neveu]], *Dual string spectrum in Polyakov's quantization (II). Mode separation*, Nuclear Physics B **209** 1 (1982) 125-145 [doi:10.1016/0550-3213(82)90105-5]

* {#Nag96} Subhashis Nag, *Mathematics in and out of String Theory*, Topology and Teichmüller Spaces (1996) 187-220 [[arXiv:alg-geom/9512010](https://arxiv.org/abs/alg-geom/9512010), [doi:10.1142/9789814503921_0011](https://doi.org/10.1142/9789814503921_0011)]

and further discussion of the related [[Liouville theory]]:

* [[Colin Guillarmou]], [[Rémi Rhodes]], [[Vincent Vargas]], *Polyakov's formulation of 2d bosonic string theory*, Publ. Math. IHES **130** (2019) 111–185 [[arXiv:1607.08467](https://arxiv.org/abs/1607.08467), [doi:10.1007/s10240-019-00109-6](https://doi.org/10.1007/s10240-019-00109-6)]

Detailed discussion of the relation to the [[Nambu-Goto action]] and the [[Dirac-Born-Infeld action]]

* {#Nieto01} J. A. Nieto, _Remarks on Weyl invariant p-branes and Dp-branes_ ([arXiv:hep-th/0110227](http://arxiv.org/abs/hep-th/0110227))

Review:

* [[Igor Dolgachev]], pp, 43, 51 in: *Introduction to string theory for mathematicians*, lecture at *[1999 Summer on Algebraic Geometry and Physics](https://people.sissa.it/~bruzzo/sagp99/sagp.html)* (1999) [[pdf](https://dept.math.lsa.umich.edu/~idolga/stringy.pdf), [[Dolgachev-IntroductionStringTheory.pdf:file]]]

and mathematical details:

* [[Jürgen Jost]], *Bosonic Strings: A Mathematical Treatment*, AMS/IP Stud. Adv. Math. **21** (2001) [[ISBBN:978-0-8218-4336-9](https://bookstore.ams.org/view?ProductCode=AMSIP/21.S.B), [spire:1388134](https://inspirehep.net/literature/1388134)]

Discussion of the Polyakov action and [[Nambu-Goto action]] on [[worldsheets]] with [[manifold with boundary|boundary]] (i.e. in the generality of [[open strings]]) and cast in [[BV-BRST formalism]]:

* {#MartinoliSchiavina22} S. Martinoli, [[Michele Schiavina]], *BV analysis of Polyakov and Nambu–Goto theories with boundary*,  Lett. Math. Phys. **112** 35 (2022) [[doi:10.1007/s11005-022-01526-1](https://doi.org/10.1007/s11005-022-01526-1), [arXiv:2106.02983](https://arxiv.org/abs/2106.02983)]

See also

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

[[!redirects Polyakov action functional]]

[[!redirects energy functional]]


Polyakov gauge-string duality

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

#Contents#
* table of contents
{:toc}

[[!include Polyakov gauge-string duality -- section]]

## References

[[!include Polyakov gauge-string duality -- references]]

[[!redirects gauge-string duality]]
[[!redirects gauge-string dualities]]

[[!redirects gauge/string duality]]
[[!redirects gauge/string dualities]]


Polyakov gauge-string duality -- references

### Polyakov gauge/string duality
 {#PolyakovGaugeStringDualityReferences}

Key ideas underlying what is now known as *[[AdS-CFT correspondence|holographic]] [[duality in string theory|duality in]] [[string theory]]* and specifically as *[[AdS-QCD correspondence|holographic QCD]]* (see notably also at *[[holographic light front QCD]]*) were preconceived by [[Alexander Polyakov]] (cf. historical remarks in [Polyakov (2008)](#Polyakov08)) under the name *[[gauge/string duality]]* (cf. historical review in [Polyakov (2008)](#Polyakov08)), in efforts to understand [[confinement|confined]] [[QCD]] (the [[mass gap problem]]) by regarding [[color charge|color]]-[[flux tubes]] ([[Wilson lines]]) between [[quarks]] as dynamical [[strings]]:

Early suggestion that [[confinement|confined]] [[QCD]] is described by regarding the [[color charge|color]]-[[flux tubes]] as [[string]]-like dynamical degrees of freedoms:

* {#KogutSusskind74} [[John Kogut]], [[Leonard Susskind]], *Vacuum polarization and the absence of free quarks in four dimensions*, Phys. Rev. D **9** (1974) 3501-3512 $[$[doi:10.1103/PhysRevD.9.3501](https://doi.org/10.1103/PhysRevD.9.3501)$]$

* {#Wilson74} [[Kenneth G. Wilson]], *Confinement of quarks*, Phys. Rev. D **10** (1974) 2445 $[$[doi:10.1103/PhysRevD.10.2445](https://doi.org/10.1103/PhysRevD.10.2445)$]$

  > (argument in [[lattice gauge theory]])

* {#KogutSusskind75} [[John Kogut]], [[Leonard Susskind]], *Hamiltonian formulation of Wilson's lattice gauge theories*, Phys. Rev. D **11** (1975) 395 $[$[doi:10.1103/PhysRevD.11.395](https://doi.org/10.1103/PhysRevD.11.395)$]$

  > "The gauge-invariant configuration space consists of a collection of strings with quarks at their ends. The strings are lines of non-Abelian electric flux. In the strong coupling limit the dynamics is best described in terms of these strings. Quark confinement is a result of the inability to break a string without producing a pair. $[$...$]$"

  > "The confining mechanism is the appearance of one dimensional electric flux tubes which must link separated quarks. The appropriate description of the strongly coupled limit consists of a theory of interacting, propagating strings. $[$...$]$"

  > "This picture of the strongly coupled Yang-Mills theory in terms of a collection of stringlike flux lines is the central result of our analysis. It should be compared with the phenomenological use of stringlike degrees of freedom which has been widely used in describing hadrons."

* {#Polyakov79} [[Alexander Polyakov]], *String representations and hidden symmetries for gauge fields*, Physics Letters B **82** 2 (1979) 247-250 $[$doi:10.1016/0370-2693(79)90747-0$]$

* {#Polyakov80} [[Alexander Polyakov]], *Gauge fields as rings of glue*, Nuclear Physics B **164** (1980) 171-188 $[$doi:10.1016/0550-3213(80)90507-6$]$

  > "The basic idea is that gauge fields can be considered as chiral fields, defined on the space of all possible contours (the [[free loop space|loop space]]). The origin of the idea lies in the expectation that, in the confining phase of a gauge theory, [[closed strings]] should play the role of elementary excitations."

* {#MakeenkoMigdal81} [[Yuri Makeenko]], [[Alexander A. Migdal]], *Quantum chromodynamics as dynamics of loops*, Nuclear Physics B **188** 2 (1981) 269-316 $[$doi:10.1016/0550-3213(81)90258-3$]$

  > "So the [[world sheet]] of [[string]] should be interpreted as the color magnetic dipole sheet. The string itself should be interpreted as the electric [[flux tube]] in the [[monopole]] plasma."

* {#Polyakov87} [[Alexander Polyakov]], *Gauge Fields and Strings*, Routledge, Taylor and Francis (1987, 2021) $[$[doi:10.1201/9780203755082](https://doi.org/10.1201/9780203755082), [oapen:20.500.12657/50871](https://library.oapen.org/handle/20.500.12657/50871)$]$

> $[$[old personal page](https://phy.princeton.edu/people/alexander-polyakov)$]$: "My main interests this year $[$1993?$]$ were directed towards [[string theory]] of [[quark]] [[confinement]]. The problem is to find the string Lagrangian for the Faraday's "[[flux tube|lines of force]]",which would reproduce perturbative corrections from the [[Yang-Mills theory]] to the Coulomb law at small distances and would give permanent [[confinement]] of quarks at large distances."

Cf. also

* [[Alexei Morozov]], p. 6 of: *String theory: what is it?*, Sov. Phys. Usp. **35** (1992) 671-714 \[doi:10.1070/PU1992v035n08ABEH002255\]

Early suggestion, due to the [[Liouville theory|Liouville field]] seen in the [[quantization]] of the [[bosonic string]] via the [[Polyakov action]],

* {#Polyakov81} [[Alexander Polyakov]], *Quantum geometry of bosonic strings*, Phys. Lett. B **103** (1981) 207-210 \[doi:10.1016/0370-2693(81)90743-7, [pdf](http://qft.itp.ac.ru/polyakov-2.pdf)\]

that such [[flux tubes]] regarded as [[confinement|confining]] [[strings]] are to be thought of a probing higher dimensional spacetime, exhibiting a [[holographic principle]] in which actual spacetime appears as a [[intersecting brane model|brane]]:

* {#Polyakov98} [[Alexander Polyakov]], *String Theory and Quark Confinement*, talk at [Strings'97](https://inspirehep.net/conferences/970718), Nucl. Phys. Proc. Suppl. **68** (1998) 1-8 $[$[arXiv:hep-th/9711002](https://arxiv.org/abs/hep-th/9711002), doi:10.1016/S0920-5632(98)00135-2$]$

  > "In other words the open string flies in the $(x,\phi)$-space keeping its feet (its ends) on the ground $(\phi=0)$."

* {#Polyakov99} [[Alexander Polyakov]], *The wall of the cave*, Int. J. Mod. Phys. A **14** (1999) 645-658 $[$[arXiv:hep-th/9809057](https://arxiv.org/abs/hep-th/9809057), [doi:10.1142/S0217751X99000324](https://doi.org/10.1142/S0217751X99000324)$]$

  > "We add new arguments that the Yang-Mills theories must be described by the non-critical strings in the five dimensional curved space. The physical meaning of the fifth dimension is that of the [[renormalization]] scale represented by the [[Liouville field]]."

eventually culminating in the formulation of the dictionary for the [[AdS-CFT correspondence]]:

* {#GubserKlebanovPolyakov98} [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], *Gauge theory correlators from non-critical string theory*, Physics Letters B **428** 105-114 (1998) $[$[hep-th/9802109](http://arxiv.org/abs/hep-th/9802109), doi:10.1016/S0370-2693(98)00377-3$]$

> "Relations between gauge fields and strings present an old, fascinating and unanswered question. The full answer to this question is of great importance for theoretical physics. It will provide us with a theory of quark confinement by explaining the dynamics of color-electric fluxes."

* {#PolyakovRychkov00} [[Alexander Polyakov]], [[Vyacheslav Rychkov]] *Gauge fields-strings duality and the loop equation*, Nucl. Phys. B **581** (2000) 116-134 $[$[arXiv:hep-th/0002106](https://arxiv.org/abs/hep-th/0002106), doi:10.1016/S0550-3213(00)00183-8$]$

and the suggestion of finding the [[AdS-QCD correspondence|string-QCD correspondence]]:

* {#Polyakov02} [[Alexander Polyakov]], _Gauge Fields and Space-Time_, Int. J. Mod. Phys. A **17** S1 (2002) 119-136 $[$[arXiv:hep-th/0110196](https://arxiv.org/abs/hep-th/0110196), [doi:10.1142/S0217751X02013071](https://doi.org/10.1142/S0217751X02013071)$]$

> "in the strong coupling limit of a [[lattice gauge theory]] the elementary excitations are represented by [[closed strings]] formed by the color-electric fluxes. In the presence of quarks these strings open up and end on the quarks, thus guaranteeing quark confinement. Moreover, in the $SU(N)$ gauge theory the strings interaction is weak at large $N$. This fact makes it reasonable to expect that also in the physically interesting continuous limit (not accessible by the strong coupling approximation) the best description of the theory should involve the [[flux tubes|flux lines]] (strings) and not fields, thus returning us from Maxwell to Faraday. In other words it is natural to expect an exact duality between gauge fields and strings. The challenge is to build a precise theory on the string side of this duality."

Historical reminiscences:

* {#Polyakov05} [[Alexander Polyakov]], *Confinement and Liberation*, in [[Gerardus ’t Hooft]] (ed.) *50 Years of Yang-Mills Theory* (2005) 311-329 $[$[arXiv:hep-th/0407209](https://arxiv.org/abs/hep-th/0407209), [doi:10.1142/9789812567147_0013](https://doi.org/10.1142/9789812567147_0013), [doi:10.1142/5601](https://doi.org/10.1142/5601)$]$

> "Already in 1974, in his [famous large $N$ paper](large+N+limit#tHooft74), 't Hooft already tried to find the string-gauge connections. His idea was that the lines of Feynman’s diagrams become dense in a certain sense and could be described as a 2d surface. This is, however, very different from the picture of strings as [[flux tube|flux lines]]. Interestingly, even now people often don’t distinguish between these approaches. In fact, for the usual amplitudes Feynman’s diagrams don’t become dense and the flux lines picture is an appropriate one. However there are cases in which t’Hooft’s mechanism is really working."

* {#Polyakov07} [[Alexander M. Polyakov]],  §1 in: *Beyond Space-Time*, in *The Quantum Structure of Space and Time*, Proceedings of the 23rd Solvay Conference on Physics, World Scientific (2007) $[$[arXiv:hep-th/0602011](https://arxiv.org/abs/hep-th/0602011), [pdf](http://www.solvayinstitutes.be/pdf/Proceedings_Physics/23rd_Solvay_Physics.pdf)$]$

* {#Polyakov08} [[Alexander M. Polyakov]], *From Quarks to Strings* $[$[arXiv:0812.0183](https://arxiv.org/abs/0812.0183)$]$

  published as *Quarks, strings and beyond*, section 44 in: [[Paolo Di Vecchia]] et al. (ed.), *The Birth of String Theory*, Cambridge University Press (2012) 544-551 $[$[doi:10.1017/CBO9780511977725.048](https://doi.org/10.1017/CBO9780511977725.048)$]$

  > "By the end of ’77 it was clear to me that I needed a new strategy $[$for understanding [[confinement]]$]$ and I became convinced that the way to go was the [[gauge/string duality]]. $[$...$]$"

  > "Classically the string is infinitely thin and has only transverse oscillations. But when I quantized it there was a surprise -- an extra, longitudinal mode, which appears due to the quantum "thickening" of the string. This new field is called the [[Liouville field|Liouville mode]]. $[$...$]$"

  > "I kept thinking about [[gauge/string duality|gauge/strings dualities]]. Soon after the [[Liouville field|Liouville mode]] was discovered it became clear to many people including myself that its natural interpretation is that random surfaces in 4d are described by the strings flying in 5d with the Liouville field playing the role of the fifth dimension. The precise meaning of this statement is that the wave function of the general string state depends on the four center of mass coordinates and also on the fifth, the Liouville one. In the case of minimal models this extra dimension is related to the matrix eigenvalues and the resulting space is flat."

  > "Since this 5d space must contain the flat 4d subspace in which the gauge theory resides, the natural ansatz for the metric is just the Friedman universe with a certain warp factor. This factor must be determined from the conditions of conformal symmetry on the world sheet. Its dependence on the Liouville mode must be related to the renormalization group flow. As a result we arrive at a fascinating picture -- our 4d world is a projection of a more fundamental 5d string theory. $[$...$]$"

  > "At this point I was certain that I have found the right language for the [[gauge/string duality]]. I attended various conferences, telling people that it is possible to describe gauge theories by solving Einstein-like equations (coming from the conformal symmetry on the world sheet) in five dimensions. The impact of my talks was close to zero. That was not unusual and didn’t bother me much. What really caused me to delay the publication ([Polyakov 1998](#Polyakov98)) for a couple of years was my inability to derive the asymptotic freedom from my equations. At this point I should have noticed the paper of [Klebanov 1997](https://arxiv.org/abs/hep-th/9702076) in which he related [[D3 branes]] described by the supersymmetric Yang Mills theory to the same object described by supergravity. Unfortunately I wrongly thought that the paper is related to matrix theory and I was skeptical about this subject. As a result I have missed this paper which would provide me with a nice special case of my program. This special case was presented little later in full generality by Juan Maldacena ([Maldacena 1997](AdS-CFT+correspondence#Maldacena97a)) and his work opened the flood gates."

A detailed monograph:

* [[N. D. Hari Dass]], *Strings to Strings -- Yang-Mills Flux Tubes, QCD Strings and Effective String Theories*, Lecture Notes in Physics **1018**, Springer (2024) \[doi:10.1007/978-3-031-35358-1\]

  Ch. 23:

   *Effective String Theories (EST) of Yang-Mills Flux Tubes* \[doi:10.1007/978-3-031-35358-1_22, arXiv:2312.10629\]