Noether's theorem

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#Contents#
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## Idea
 {#Idea}

What is commonly called _Noether's theorem_ or _Noether's first theorem_ is a [[theorem]] due to [[Emmy Noether]] ([Noether 1918](#Noether1918)) which makes precise and asserts that to every continuous [[symmetry of the Lagrangian]] [[physical system]] ([[prequantum field theory]]) there is naturally associated a [[conservation law]] stating the conservation of a [[charge]] ([[conserved current]]) when the [[equations of motion]] hold.

For instance the [[time]]-translation invariance of a [[physical system]] equivalently means that the quantity of [[energy]] is conserved, and the [[spacetime|space]]-[[translation group|translation]] invariant of a physical system means that [[momentum]] is preserved.

The original and still most common formulation of the theorem is in terms of [[variational calculus]] applied to a [[local action functional]]. A modern version of this formulated properly in terms of the [[variational bicomplex]] we discuss below in

* _[Lagrangian version -- In terms of the variational bicomplex](#InTermsOfTheVariationalBicomplex)_

There is another formulation of the same physical content, but using the formalism of [[symplectic geometry]] of [[phase spaces]]. In this formulation of [[physics]] the relation between [[symmetries of the Lagrangian]] and [[charges]]/[[conserved currents]] happens to be built deep into the formalism in terms of [[Hamiltonian flows]] generated by the [[Poisson bracket]] with a [[Hamiltonian]] function. Accordingly, in this powerful formalism Noether's theorem becomes almost a tautology. This we discuss in

* _[Hamiltonian/symplectic version -- In terms of moment maps](#HamiltonianNoetherTheorem)_.

## Lagrangian version
 {#InTermsOfTheVariationalBicomplex}

Here we formulate Noether's theorem for [[local action functional]] in terms of the [[variational bicomplex]] and the [[covariant phase space]].

### Simple schematic idea
 {#LagrangianVersionSimpleSchematicIdea}

Before coming to the precise and general formulation, we indicate here schematically the simple idea which underlies Noether's first theorem (in its original Lagrangian version).

Consider a [[local Lagrangian]] $L$ and assume for simplicity
that it depends only on the first [[derivatives]] $\nabla \phi$ (the [[gradient]]) of the [[field (physics)|fields]] $\phi$, hence

$$
  L \colon \phi \mapsto L(\phi, \nabla \phi)
  \,.
$$

(We write $\nabla \cdot (-)$ in the following for the [[divergence]].)

Then the [[variational derivative]] of $L$ by the fields is

$$
  \begin{aligned}
    \delta L
    & =
    \left(\frac{\delta}{\delta \phi} L\right)  \delta \phi
    +
    \left(\frac{\delta}{\delta \nabla \phi} L\right) \cdot \nabla \delta \phi
    \\
    & =
    \left(
      \frac{\delta}{\delta \phi} L
      -
      \nabla \cdot \left(\frac{\delta}{\delta \nabla\phi}L\right)
    \right)
    \cdot (\delta \phi)
    +
    \nabla \cdot \left( \left(\frac{\delta }{\delta \nabla \phi} L\right)
    \delta \phi
   \right)
  \end{aligned}
  \,,
$$

where in the second step the total derivative was introduced via the [[product rule]] of [[differentiation]] $f (\nabla g) = -(\nabla f) g + \nabla (f g)$.

From this law for the variation of the Lagrangian, one derives both the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] as well as Noether's theorem by making different assumptions and setting different terms to zero:

1. Demanding that the variation $\delta \phi$ vanishes on some [[boundary]] of [[spacetime]] implies that the rightmost term in the above equation disappears in the variation $\delta S = \delta \int L$ of the [[action functional]] (by the [[Stokes theorem]]) and hence demanding that $\delta S = 0$ under variation that vanishes on the boundary is equivalent to demanding the [[Euler-Lagrange equation]]

   $$
      \frac{\delta}{\delta \phi} L
      -
      \nabla \cdot \left(\frac{\delta}{\delta \nabla\phi}L\right)
    = 0
    \,.
   $$

1. On the other hand, assuming that for given $\delta \phi$ the variation $\delta L $ vanishes when these equations of motion hold -- hence assuming that $\delta \phi$ is an _[[on-shell]] [[symmetry of the Lagrangian|symmetry]]_ of $L$ -- is equivalent to assuming that the above expression is zero even without the left term, hence that

  $$
    \nabla \cdot
    \left(\left(\frac{\delta }{\delta \nabla\phi} L\right) \delta \phi\right)
     = 0
    \,.
  $$

  This is the statement of _Noether's theorem_. The object

  $$
    p_\phi \delta \phi
    \coloneqq
    \left(\frac{\delta }{\delta \nabla \phi} L\right) \delta \phi
  $$

  (here $p_\phi$ is the [[canonical momentum]] of the [[field (physics)|field]] $\phi$) is called the _[[Noether current]]_ and the above says that this is ([[on-shell]]) a  [[conserved current]] precisely if $\delta \phi$ is a [[symmetry of the Lagrangian]].

This is at least the way that Noether's theorem has been introduced and is often considered. But this formulation is more restrictive than is natural. Namely it is unnatural to demand of a symmetry that it leaves the Lagrangian entirely invariant, $\delta L = 0$:

{#WeakSymmetrySchematicIdea} More generally for the symmetry to be a symmetry of the [[action functional]] $\int L$ over a [[closed manifold]] it is sufficient that the Lagrangian changes by a [[divergence]], $\delta L = \nabla \cdot \sigma$, for some term $\sigma$.

(This is really a sign of a [[higher gauge symmetry]], where the symmetry holds only up to a [[homotopy]] $\sigma$. It happens for instance for the gauge-coupling term in the [[Wess-Zumino-Witten model]] because the WZW term is not strictly invariant under [[gauge transformations]], but instead transforms by a total derivative. See at _[conserved current -- In higher prequantum geometry](conserved+current#InHigherPrequantumGeometry)_).

In this more general case the above [[conservation law]] induced by the "weak" symmetry becomes

$$
  \nabla \cdot
  \left(
    p_\phi \delta \phi - \sigma
  \right)
  = 0
  \,.
$$

(This may be regarded as the [[Legendre transform]] of $\sigma$.)

### Formulation via the variational bicomplex
 {#LagrangianVersionFormalContext}

Let $X$ be a [[spacetime]] of [[dimension]] $n$, $E \to X$ a [[field bundle]], $Jet(E) \to X$ its [[jet bundle]] and write

$$
  \Omega^{\bullet,\bullet}(Jet(E)), (d = d_H + d_V)
$$

the corresponding [[variational bicomplex]] with $d_V$ being the vertical and $d_H$ the horizontal [[de Rham differential]].

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

For $\mathbf{L} \in \Omega^{n,0}(Jet(E))$ a [[local Lagrangian]] we have a unique decomposition of its [[de Rham differential]]

$$
  d L = d_V L = \mathbf{E} - d_H \Theta
$$

such that $\mathbf{E}$ is a [[source form]] -- the [[Euler-Lagrange form]] of $\mathbf{L}$ -- and for some $\Theta \in \Omega^{n-1,0}(j_\infty E)$.

=--

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

The _dynamical [[shell]]_ $\mathcal{E} \hookrightarrow Jet(E)$ is the [[zero locus]] of $\mathbf{E}$ together with its differential consequences.

The [[covariant phase space]] of the Lagrangian is the [[zero locus]]

$$
  \{\phi \in \Gamma(E) | \mathbf{E}(j_\infty \phi) = 0\}
$$

that solves the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]].

For $\Sigma \subset X$ any [[compact topological space|compact]] $(n-1)$-dimensional [[submanifold]],

$$
  \delta \theta \coloneqq \delta \int_\Sigma \Theta
$$

is the [[presymplectic structure]] on [[covariant phase space]].

=--

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

An **infinitesimal variational [[symmetry of the Lagrangian|symmetry]]** of $\mathbf{L}$ is a [[vertical vector field]] $v$ such that

$$
  \mathcal{L}_v \mathbf{L} = d_H \sigma_v
$$

(with $\mathcal{L}_v$ denoting the [[Lie derivative]]) for some

$$
  \sigma_v \in \Omega^{n-1,0}(Jet(E))
  \,.
$$

=--

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

By [[Cartan's magic formula]] and since $v$ is assumed vertical while $L$ is horizontal, this is equivalent to

$$
  \iota_{v} d_V \mathbf{L} = d_H \sigma_v
  \,.
$$

=--

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

A **[[conserved current]]** is an element

$$
  j \in \Omega^{n-1, 0}(Jet(E))
$$

which is horizontally closed on the dynamical shell

$$
  (d_H\, j)|_{\mathcal{E}} = 0
  \,.
$$

=--

With the above notions and notation, **Noether's theorem** states:

+-- {: .num_theorem #NoetherTheoremViaVariationalBicomplex}
###### Theorem

For $v \in T_v(j_\infty E)$ an infinitesimal variational symmetry according to def. \ref{Symmetry}, then

$$
  j_v \coloneqq \sigma_v - \iota_v \Theta
$$

is a [[conserved current]], def. \ref{ConservedCurrent}.

=--

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

By prop. \ref{VariationOfTheLagrangian} and def. \ref{Symmetry} we have

$$
  \begin{aligned}
    d_H (\sigma_v - \iota_v \Theta)
    & =
    \iota_v d_V \mathbf{L} + \iota_v d_H \Theta
    \\
    & = \iota_v \mathbf{E}
  \end{aligned}
  \,.
$$

=--

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

A symmetry of the [[Lepage form]]

$$
  \rho \coloneqq \mathbf{L} + \Theta
  \,.
$$

would be defined with the full differential $d = d_V + d_H$:

$$
  \mathcal{L}_v \rho = d (\sigma_v + \kappa_v)
  \,,
$$

where $\sigma$ is horizontal and $\kappa$ is vertical.
Decomposing the result into horizontal and vertical components,
then for vertical $v$ this is equivalent to the pair of equations

$$
  \left\{
    \array{
       \iota_v \mathbf{E} & = d_H (\sigma_v - \iota_v \Theta)
       \\
       \iota_v \omega & = d_H \kappa_v + d_V \sigma_v
    }
  \right.
$$

The first one expresses the conserved current corresponding to $v$ as in
theorem \ref{NoetherTheoremViaVariationalBicomplex}, the second constrains $v$ to be a [[Hamiltonian vector field]] with respect to the presymplectic current.

=--

## Hamiltonian/symplectic version -- In terms of moment maps
 {#HamiltonianNoetherTheorem}

### In traditional symplectic geometry

In [[symplectic geometry]] the analog of Noether's theorem is the statement that the [[moment map]] of a [[Hamiltonian action]] which preserves a given time evolution is itself conserved by this time evolution.

Souriau called this _the symplectic Noether theorem_, sometimes it is called the _Hamiltonian Noether theorem_. A review is for instance in ([Butterfield 06](#Butterfield06)).

Let $(X,\omega)$ be a [[symplectic manifold]] and let $\mathbb{R} \to \mathfrak{Poisson}(X,\omega)$ be a [[Hamiltonian action]] with [[Hamiltonian]] $H \in C^\infty(X)$, thought of as the time evolution of a [[physical system]] with [[phase space]] $(X,\omega)$.

Then let $G$ be a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$ and let $\mathfrak{g} \to \mathfrak{Poisson}(X,\omega)$ be a [[Hamiltonian action]] with [[Hamiltonian]]/[[moment map]]
$\Phi \in C^\infty(X,\mathfrak{g}^\ast)$. We say this preserves the (time evolution-)Hamiltonian $H$ if for all $\xi \in \mathfrak{g}$ the [[Poisson bracket]] between the two vanishes,

$$
  \delta_\xi H \coloneqq  \{\Phi(\xi), H\} = 0
  \,.
$$

In this situation now the statement of Noether's theorem is that the generators $\Phi(\xi)$ of the symmetry are preserved by the time evolution

$$
  \frac{d}{d t} \Phi^\xi = 0
  \,.
$$

In this symplectic formulation this is immediate, because

$$
  \frac{d}{d t}\Phi^\xi = \{H,\Phi^\xi\} = - \{\Phi^\xi, H\} = 0
  \,,
$$

by the above assumtion that $H$ is preserved. Hence the "Hamiltonian Noether theorem" is all captured already by the very notion of [[Hamiltonian action]] and the  statement that the [[Poisson bracket]] is skew-symmetric (is a [[Lie algebra]] bracket).

Specifically, if one has a global [[polarization]] of $(X,\omega)$ with [[canonical coordinates]] $\{q^i\}$ and [[canonical momenta]] $\{p_i\}$ and if the symmetry action is on the canonical coordinates (on configuration space), then for $v_\xi$ the [[vector field]] corresponding to the generator $\xi$ the moment map is

$$
  \Phi^\xi = p_i (v_\xi)^i
  \,.
$$

The term on the right is in the form in which the conserved quantity obtained from Noether's theorem is traditionally written
(using that, given a [[Lagrangian]] $L$, we have $p_i = \frac{\delta L}{\delta (\dot q^i)}$).

## Examples

### Energy-momentum current

* [[energy-momentum tensor]] -- see at _[[geometry of physics -- A first idea of quantum field theory]]_ [this example](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#ScalarFieldEnergyMomentum)

### Dirac current

* [[Dirac current]] -- see at [[geometry of physics -- A first idea of quantum field theory]] [this example](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#DiracCurrent)

## Related concepts

* [[conservation law]]

* [[conserved current]]

* [[charge]]

* [[moment map]]

## References
 {#References}

The original article is

* {#Noether1918} [[Emmy Noether]], _Invariante Variationsprobleme_ Nachrichten der Königlichen Gesellschaft der Wissenschaften zu  Göttingen, Math. Phys. Kl, 235 (1918). ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN00250510X))

Lecture notes include

* _Chapter 7, Noether's theorem_ ([pdf](http://www-physics.ucsd.edu/students/courses/fall2010/physics110a/LECTURES/CH07.pdf))

* Máximo Bañados, Ignacio A. Reyes, _A short review on Noether's theorems, gauge symmetries and boundary terms, for students_ ([arXiv:1601.03616](http://arxiv.org/abs/1601.03616))

* _[A first idea of quantum field theory -- Symmetries](geometry+of+physics+--+perturbative+quantum+field+theory#Symmetries)_ ([PF Insights version](https://www.physicsforums.com/insights/newideaofquantumfieldtheory-symmetries/))

A comprehensive exposition of both the Lagrangian and the Hamiltonian version of the theorem is in

* {#Butterfield06} [[Jeremy Butterfield]], _On symmetry and conserved quantities in classical mechanics_, in _Physical Theory and its Interpretation_,
The Western Ontario Series in Philosophy of Science Volume 72, 2006, pp 43-100(2006) ([[ButterfieldNoether.pdf:file]])

Textbook accounts include

* [[Yvette Kosmann-Schwarzbach]], _Les théorèmes de Noether:
invariance et lois de conservation au XXe siècle : avec une traduction de l'article original, "Invariante Variationsprobleme"_, Editions Ecole Polytechnique, 2004 ([pdf](http://www.math.cornell.edu/~templier/junior/The-Noether-theorems.pdf))

* {#VinogradovKrasilshchik99} [[Alexandre Vinogradov]], I. S. Krasilshchik (eds.) _Symmetries and Conservation Laws for Differential Equations of Mathematical Physics_, vol. 182 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1999. ([pdf](ftp://softbank.iust.ac.ir/MathBooks/V/Vinogradov%20-%20Symmetries%20and%20Conservation%20Laws%20for%20Differential%20equations%20of%20mathematical%20physics.pdf))

For 1-parameter groups of symmetries in classical mechanics, the formulation and the proof of [[Noether's theorem]] can be found in the monograph

* [[Vladimir Arnold]], _Mathematical methods of classical mechanics_

For more general case see for instance the books by [[Peter Olver]].

The Hamiltonian Noether theorem is also reviewed in a broader context of mathematical physics as theorem 7.3.2 in

* [[Frédéric Paugam]], _Towards the mathematics of quantum field theory_ ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/Towards-the-maths-of-QFT.pdf))

Discussion in the context of the [[variational bicomplex]] includes

* {#BarnichBrandt01} [[Glenn Barnich]], [[Friedemann Brandt]], _Covariant theory of asymptotic symmetries, conservation laws and central charges_, Nucl.Phys.B633:3-82,2002 ([arXiv:hep-th/0111246](http://arxiv.org/abs/hep-th/0111246))

Discussion of a generalization to [[discrete groups]] of symmetries includes

* Anthony Ashton, _Conservation laws and non-Lie symmetries for linear PDEs_, Journal of Non-linear Mathematical Physics, 2013 ([web](http://www.tandfonline.com/doi/pdf/10.2991/jnmp.2008.15.3.5))

The example of conserved currents in [[Chern-Simons theory]] is discussed around (5.381) on p. 925 of

* Vladimir Ivancevic, Tijana Ivancevi, _Applied differential geometry: a modern introduction_

and also in

* M. Francaviglia, M. Palese, E. Winterroth, _Locally variational invariant field equations and global currents: Chern-Simons theories_, Communications in Matheamtical Physics 20 (2012) ([pdf](http://cm.osu.cz/sites/default/files/contents/20-1/cm020-2012-1_13-22.pdf))

and for [[higher dimensional Chern-Simons theory]] in

* G.Giachetta, L.Mangiarotti, G.Sardanashvily, _Noether conservation laws in higher-dimensional Chern-Simons theory_, Mod. Phys. Lett. A18 (2003) 2645-2651 ([arXiv:math-ph/0310067](http://arxiv.org/abs/math-ph/0310067))

A formalization of Noether's theorem in [[cohesive homotopy type theory]] is discussed in sections  "2.7 Noether symmetries and equivariant structure" and "3.2 Local observables, conserved currents and higher Poisson bracket homotopy Lie algebras" of

* {#Schreiber13} [[Urs Schreiber]], _[[schreiber:Classical field theory via Cohesive homotopy types]]_ (2013)

A formalization of invariance of Lagrangians in [[parametricity|parametric]] [[dependent type theory]] is discussed in

* [[Robert Atkey]], _From Parametricity to Conservation Laws, via Noether's Theorem_, talk at _[Principles of Programming Languages (POPL) 2014](http://popl.mpi-sws.org/2014/index.html)_  ([pdf article](http://bentnib.org/conservation-laws.pdf), [web slides](http://bentnib.org/posts/2014-01-29-popl-slides.html), [pdf slides](http://bentnib.org/docs/conservation-laws-20140124.pdf))

Discussion in the convenient context of [[smooth sets]]:

* {#GiotopoulosSati23} [[Grigorios Giotopoulos]], [[Hisham Sati]], §6 in: *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

[[!redirects Noether's theorems]]

[[!redirects Noether theorem]]
[[!redirects Noether theorems]]

[[!redirects symplectic Noether theorem]]
[[!redirects symplectic Noether theorems]]

[[!redirects symplectic Noether's theorem]]
[[!redirects symplectic Noether's theorems]]

[[!redirects Hamiltonian Noether theorem]]
[[!redirects Hamiltonian Noether theorems]]

[[!redirects Hamiltonian Noether's theorem]]
[[!redirects Hamiltonian Noether's theorems]]


Noetherian bimodule

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## Idea

A *Noetherian bimodule* is a bimodule which satisfies the [[ascending chain condition]] on its subbimodules.

## Definition

Given [[rings]] $R$ and $S$ and an $R$-$S$-[[bimodule]] $B$, let $\mathrm{Mono}(B)$ be the [[category]] whose [[objects]] are $R$-$S$-[[subbimodules]] of $B$ and whose [[morphisms]] are $R$-$S$-[[bimodule monomorphisms]]. An ascending chain of $R$-$S$-subbimodules is a [[direct sequence]] of $R$-$S$-subbimodules in $\mathrm{Mono}(B)$, a [[sequence]] of $R$-$S$-subbimodules $A:\mathbb{N} \to \mathrm{Mono}(B)$ with the following [[dependent sequence]] of $R$-$S$-bimodule monomorphisms: for [[natural number]] $n \in \mathbb{N}$, a dependent $R$-$S$-bimodule monomorphism $i_n:A_n \hookrightarrow A_{n+1}$.

An $R$-$S$-[[bimodule]] $B$ is **Noetherian** if it satisfies the [[ascending chain condition]] on its subbimodules: for every ascending chain of $R$-$S$-subbimodules $(A, i_n)$ of $B$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the $R$-$S$-bimodule monomorphism $i_n:A_n \hookrightarrow A_{n+1}$ is an $R$-$S$-[[bimodule isomorphism]].

## Examples

A [[ring]] $R$ is [[Noetherian ring|Noetherian]] if it is Noetherian as a $R$-$R$-[[bimodule]] with respect to its canonical bimodule structure, with its [[left action]] $\alpha_L:R \times R \to R$ and [[right action]] $\alpha_R:R \times R \to R$ defined as its multiplicative binary operation and its [[biaction]] $\alpha:R \times R \times R \to R$ defined as its ternary product:
$$\alpha_L(a, b) \coloneqq a \cdot b$$
$$\alpha_R(a, b) \coloneqq a \cdot b$$
$$\alpha(a, b, c) \coloneqq a \cdot b \cdot c$$

## See also

* [[bimodule]]

* [[Artinian bimodule]]

* [[Noetherian ring]]

[[!redirects Noetherian bimodules]]


noetherian category

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## Definition

A [[category]] $C$ is __noetherian__ if it is [[svelte category|svelte]] and every object in $C$ is a [[noetherian object]].

An [[abelian category]] is called __locally noetherian__ if it satisfies axiom (AB5) and has a small generating family of noetherian objects.

The [[full subcategory]] of noetherian objects in any locally noetherian abelian category is itself a noetherian abelian category. A category $R Mod$ of [[modules]] over a [[noetherian ring|noetherian]] [[commutative unital ring]] $R$, and more generally, the category $Qcoh(X)$ of [[quasicoherent sheaf|quasicoherent sheaves]] of $\mathcal{O}$-modules over any [[noetherian scheme]] $X$, is a locally noetherian abelian category; moreover the full subcategory of noetherian objects in $Qcoh(X)$ in this case is the category of [[coherent sheaf|coherent sheaves]] of $\mathcal{O}$-modules over $X$.

A [[filtered colimit]] of [[injective objects]] in any locally noetherian abelian category is injective.

## Related concepts

* [[Noetherian ring]]

* [[Noetherian poset]]

[[!redirects locally noetherian category]]
[[!redirects locally noetherian abelian category]]

[[!redirects noetherian categories]]

[[!redirects Noetherian category]]
[[!redirects Noetherian categories]]


Noetherian module

#Contents#
* table of contents
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## Idea

In the presence of the [[axiom of choice]], a [[module]] is _Noetherian_ if every submodule is [[finite number|finitely]] [[generators and relations|generated]].

## Related concepts

* [[Noetherian ring]]

* [[Noetherian poset]]

## References

* Wikipedia, _[Noetherian module](http://en.wikipedia.org/wiki/Noetherian_module)_

[[!redirects Noetherian modules]]

[[!redirects noetherian module]]
[[!redirects noetherian modules]]


noetherian object

An object $X$ in a category $C$ is __noetherian__ if each ascending chain of [[subobject]]s of $X$ is stationary (= only finitely many inclusions in the chain are not isomorphisms in $C$).

Cf. [[noetherian ring]], [[noetherian topological space]], [[noetherian category]].


Noetherian poset

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## Definition

A [[poset]] is _Noetherian_ if every ascending chain of [[ideals]] ([[sieves]]) stabilizes, and it is _co-Notherian_ if every ascending chain of ideals in the opposite poset stabilizes.

## References

* bourwiki, _[Noetherian poset](http://bourwiki.org/wiki/Noetherian_poset)_

## Related concepts

* [[Noetherian ring]]


noetherian ring

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## Idea

A Noetherian (or often, as below, noetherian) [[ring]] (or [[rng]]) is one where it is possible to do [[induction]] over its ideals, because the ordering of ideals by reverse inclusion is [[well-founded relation|well-founded]].

## Definition

### Noetherian rings

Every [[ring]] $R$ has a canonical $R$-$R$-[[bimodule]] structure, with [[left action]] $\alpha_L:R \times R \to R$ and [[right action]] $\alpha_R:R \times R \to R$ defined as the multiplicative binary operation on $R$ and [[biaction]] $\alpha:R \times R \times R \to R$ defined as the ternary product on $R$:
$$\alpha_L(a, b) \coloneqq a \cdot b$$
$$\alpha_R(a, b) \coloneqq a \cdot b$$
$$\alpha(a, b, c) \coloneqq a \cdot b \cdot c$$

Let $\mathrm{TwoSidedIdeals}(R)$ be the [[category of two-sided ideals]] in $R$, whose objects are [[two-sided ideals]] $I$ in $R$, [[subbimodules|sub-$R$-$R$-bimodules]] of $R$ with respect to the canonical bimodule structure on $R$, and whose [[morphisms]] are $R$-$R$-[[bimodule monomorphisms]].

An ascending chain of two-sided ideals in $R$ is a [[direct sequence]] of two-sided ideals in $R$, a [[sequence]] of two-sided ideals $A:\mathbb{N} \to \mathrm{TwoSidedIdeals}(R)$ with the following [[dependent sequence]] of $R$-$R$-bimodule monomorphisms: for natural number $n \in \mathbb{N}$, a dependent $R$-$R$-bimodule monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$.

A [[ring]] $R$ is **Noetherian** if it satisfies the [[ascending chain condition]] on its two-sided ideals: for every ascending chain of two-sided ideals $(A, i_n)$ in $R$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the $R$-$R$-bimodule monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$ is an $R$-$R$-[[bimodule isomorphism]].

### Left Noetherian rings

Let $\mathrm{LeftIdeals}(R)$ be the category of [[left ideals]] in $R$, whose objects are [[left ideals]] $I$ in $R$, sub-left-$R$-modules of $R$ with respect to the canonical [[left module]] structure $(-)\cdot(-):R \times R \to R$ on $R$, and whose [[morphisms]] are left $R$-module monomorphisms.

An ascending chain of left ideals in $R$ is a [[direct sequence]] of left ideals in $R$, a [[sequence]] of left ideals $A:\mathbb{N} \to \mathrm{LeftIdeals}(R)$ with the following [[dependent sequence]] of left $R$-module monomorphisms: for natural number $n \in \mathbb{N}$, a dependent left $R$-module monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$.

A [[ring]] $R$ is **left Noetherian** if it satisfies the [[ascending chain condition]] on its left ideals: for every ascending chain of left ideals $(A, i_n)$ in $R$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the left $R$-module monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$ is an left $R$-module isomorphism.

### Right Noetherian rings

Let $\mathrm{RightIdeals}(R)$ be the category of [[right ideals]] in $R$, whose objects are [[right ideals]] $I$ in $R$, sub-right-$R$-modules of $R$ with respect to the canonical [[right module]] structure $(-)\cdot(-):R \times R \to R$ on $R$, and whose [[morphisms]] are right $R$-module monomorphisms.

An ascending chain of right ideals in $R$ is a [[direct sequence]] of right ideals in $R$, a [[sequence]] of right ideals $A:\mathbb{N} \to \mathrm{RightIdeals}(R)$ with the following [[dependent sequence]] of right $R$-module monomorphisms: for natural number $n \in \mathbb{N}$, a dependent right $R$-module monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$.

A [[ring]] $R$ is **right Noetherian** if it satisfies the [[ascending chain condition]] on its right ideals: for every ascending chain of right ideals $(A, i_n)$ in $R$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the right $R$-module monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$ is an right $R$-module isomorphism.

## Examples

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

Every [[field]] is a noetherian ring.

=--

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

Every [[principal ideal domain]] is a noetherian ring.

=--

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

For $R$ a [[Noetherian ring]] (e.g. a [[field]] by example \ref{FieldIsNoetherianRing}) then

1. the [[polynomial algebra]]  $R[X_1, \cdots, X_n]$

1. the [[formal power series algebra]] $R[ [ X_1, \cdots, X_n ] ]$

over R in a [[finite number]] $n$ of [[coordinates]] are Noetherian.

=--

## Properties

Spectra of noetherian rings are glued together to define [[noetherian scheme|locally noetherian schemes]].

### General

One of the best-known properties is the Hilbert basis theorem. Let $R$ be a (unital) ring.

+-- {: .num_theorem}
###### Theorem
**(Hilbert)**
If $R$ is left Noetherian, then so is the [[polynomial|polynomial algebra]] $R[x]$. (Similarly if "right" is substituted for "left".)
=--

+-- {: .proof}
###### Proof
(We adapt the proof from [Wikipedia](https://en.wikipedia.org/wiki/Hilbert%27s_basis_theorem#First_Proof).) Suppose $I$ is a left ideal of $R[x]$ that is not finitely generated. Using the [[axiom of dependent choice]], there is a [[sequence]] of polynomials $f_n \in I$ such that the left ideals $I_n \coloneqq (f_0, \ldots, f_{n-1})$ form a strictly increasing chain and $f_n \in I \setminus I_n$ is chosen to have degree as small as possible. Putting $d_n \coloneqq \deg(f_n)$, we have $d_0 \leq d_1 \leq \ldots$. Let $a_n$ be the leading coefficient of $f_n$. The left ideal $(a_0, a_1, \ldots)$ of $R$ is finitely generated; say $(a_0, \ldots, a_{k-1})$ generates. Thus we may write

$$\label{kill} a_k = \sum_{i=0}^{k-1} r_i a_i$$

The polynomial $g = \sum_{i=0}^{k-1} r_i x^{d_k - d_i} f_i$ belongs to $I_k$, so $f_k - g$ belongs to $I \setminus I_k$. Also $g$ has degree $d_k$ or less, and therefore so does $f_k - g$. But notice that the coefficient of $x^{d_k}$ in $f_k - g$ is zero, by (eq:kill). So in fact $f_k - g$ has degree less than $d_k$, contradicting how $f_k$ was chosen.

=--

### A homological characterization

+-- {: .num_theorem}
###### Theorem
For a unital ring $R$ the following are equivalent:

1. $R$ is left Noetherian
1. Any small direct sum of injective left $R$-modules is injective.
1. $\operatorname{Ext}^k_R(A, \cdot)$ commutes with small direct sums for any finitely generated $A$.
=--

Direct sums here can be replaced by filtered colimits.

+-- {: .proof}
###### Proof
$1 \Rightarrow 2$: assume that $R$ is Noetherian and $I_\alpha$ are injective modules. In order to verify that $I := \bigoplus_\alpha I_\alpha$ is injective it is enough to show that for any ideal $\mathfrak{j}$ any morphism of left modules $f : \mathfrak{j} \to I$ factors through $\mathfrak{j} \to R$. Since $R$ is Notherian, $\mathfrak{j}$ is finitely generated, so the image of $f$ lies in a finite sum $I_{\alpha_1} \oplus \dots \oplus I_{\alpha_n}$. Thus an extension to $R$ exists by the injectivity of each $I_{\alpha_k}$.

$2 \Rightarrow 1$: if $R$ is not left Noetherian then there is a sequence of left ideals $\mathfrak{j}_1 \subsetneq \mathfrak{j}_2 \subsetneq \dots$. Take $\mathfrak{j} := \bigcup_k \mathfrak{j}_k$. The obvious map $j \to \prod_k (\mathfrak{j} / \mathfrak{j}_k)$ factors through $\bigoplus_k (\mathfrak{j} / \mathfrak{j}_k)$, since any element lies in all but finitely many $\mathfrak{j}_k$. Now take any injective $I_k$ with $0 \to \mathfrak{j} / \mathfrak{j}_k \to I_k$. The map $\mathfrak{j} \to \bigoplus_k I_k$ cannot extend to the whole $R$, since otherwise its image would be contained in a sum of finitely many $I_k$. Therefore, $\bigoplus_k I_k$ is not injective.

$2 \Rightarrow 3$: $\operatorname{Ext}^k_R(A, \bigoplus_\alpha X_\alpha)$ can be computed by taking an injective resolution of $\bigoplus_\alpha X_\alpha$. Since direct sums of injective modules are assumed to be injective, we can take a direct sum of injective resolutions of each $X_\alpha$. It remains to note that Hom out of a finitely generated module commutes with arbitrary direct sums.

$3 \Rightarrow 2$: Follows from the fact that $I$ is injective iff $\operatorname{Ext}^1_R(R / \mathfrak{i}, I) = 0$ for any ideal $\mathfrak{i}$.
=--

## Noetherian and Artinian rings

A dual condition is artinian: an __[[artinian ring]]__ is a ring satisfying the descending chain condition on ideals. The symmetry is severely broken if one considers unital rings: for example every unital artinian ring is noetherian; artinian rings are intuitively much smaller than generic noetherian rings.

## Related concepts

* [[Noetherian module]]

* [[Noetherian bimodule]]

* [[Noetherian poset]]

* [[Noetherian E-∞ ring]]

## References

Introduced by [[Emmy Noether]] in

* [[Emmy Noether]], _Idealtheorie in Ringbereichen_, Mathematische Annalen 83:1 (1921), 24–66.  [doi:10.1007/bf01464225](https://doi.org/10.1007/bf01464225).

* [wikipedia](http://en.wikipedia.org/wiki/Noetherian_ring), _[[noetherian object]]_

[[!redirects noetherian ring]]
[[!redirects noetherian]]
[[!redirects noetherian rings]]
[[!redirects Noetherian ring]]
[[!redirects Noetherian rings]]


noetherian scheme

#Contents#
* table of contents
{:toc}

## Definition

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

An [[algebraic scheme]] is __locally noetherian__ if it can be covered by a family $U_\alpha$ of open subsets of the form $U_\alpha = Spec R_\alpha$ where $R_\alpha$ are [[noetherian rings]]. A scheme is __noetherian__ if it is locally noetherian and [[quasicompact scheme|quasicompact]].

=--

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

If $R$ is a commutative [[noetherian ring]], then every [[localization]] of the form $R[f^{-1}]$ where $f\in R$ is also  noetherian. If a scheme $X$ is locally noetherian this implies that there is a base of topology of $X$ consisting of spectra of noetherian rings. In particular _every_ affine subscheme in $X$ has a basis of topology of spectra of noetherian rings.
=--

## Properties

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

An affine scheme is a [[spectrum of a commutative ring|spectrum]] of a [[noetherian ring]] precisely if it is a locally noetherian scheme.

=--

+-- {: .num_cor}
###### Corollary

Every affine subscheme of a locally noetherian scheme is the spectrum of a noetherian ring. A scheme is noetherian if _and only if_ it has a finite cover by spectra of noetherian rings.

=--

## References

* Springer eom: _[Noetherian scheme](http://eom.springer.de/n/n066860.htm)_

category: algebraic geometry
[[!redirects locally noetherian scheme]]
[[!redirects locally noetherian schemes]]
[[!redirects locally Noetherian scheme]]
[[!redirects locally Noetherian schemes]]
[[!redirects noetherian schemes]]
[[!redirects Noetherian scheme]]


noetherian topological space

A [[topological space]] is __noetherian__ if it satisfies the _ascending chain condition_ for inclusions of open sets. Equivalently, it satisfies the _descending chain condition_ for inclusions of closed sets.

Typical examples are the underlying topological spaces of (classical) quasiprojective varieties over a field $k$ in Zariski topology. The fact that the affine space $\mathbb{A}^n_k$ is noetherian is a consequence of the [[Hilbert basis theorem]].

Every noetherian topological space is a closed union of a finitely many [[irreducible topological spaces]] (its irreducible components).

[[!redirects noetherian space]]


noise

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[white noise]]

* [[signal processing]]

* [[error correction code]]

* [[topological data analysis]]

In [[quantum computation]]:

* [[quantum noise]]

* [[quantum error correction]]

## References

* See also:

* Wikipedia, *Noise_(signal_processing)*


nominal set

## Idea

A nominal set is a formalization of the idea of a set whose elements can have "variables" or "atoms" in them. Nominal sets are used in [[computer science]] to formalize notions of bound variables and [[alpha-equivalence]].

## Definition

A nominal set is an object of the [[Schanuel topos]].
One concrete description of this topos is that it is the [[category of G-sets]] for the group of permutations of the natural numbers topologized as a subspace of the [[Baire space of sequences]]. The continuity can be described as the finite support property below.

A nominal set is a set $X$ with an action of the group of permutations of natural numbers such that each element has finite support: for each $x \in X$ there is some finite subset $supp(x) \subseteq N$ such that if $\pi$ fixes $supp(x)$ then $\pi \cdot x = x$.

## References

Related $n$Lab entries include [[Schanuel topos]],

A survey:

* Murdoch J. Gabbay, _Foundations of nominal techniques: logic and semantics of variables in abstract syntax_. Bulletin of Symbolic Logic 17:2, June 2011, 161-229

* Wikipedia, [Nominal techniques](http://en.wikipedia.org/wiki/Nominal_techniques)

In computer science, nominal sets were introduced in

* Murdoch James Gabbay, _A theory of inductive definitions with alpha-equivalence_, PhD thesis, Cambridge 2001, [pdf](http://www.gabbay.org.uk/papers/thesis.pdf)

See also

* [[Andrew Pitts|A. M. Pitts]], _Nominal sets_, Cambridge University Press 2013

* Daniela Petrisan, _Algebra and topology over nominal sets_, PhD thesis, Leicester 2010, [pdf](https://www.irif.fr/~petrisan/thesis.pdf)

* M. J. Gabbay, [[Andrew Pitts|A. M. Pitts]], _A NEW approach to abstract syntax with variable binding_, Formal Aspects of Computing **13** (2002) pp.341-363. ([draft](http://www.cl.cam.ac.uk/~amp12/papers/newaas/newaas-jv.pdf))

[[!redirects nominal sets]]


non-abelian de Rham cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $X$ a [[smooth manifold]], the traditional [[coboundary]]-relation which defines the ordinary [[de Rham cohomology]]-classes $[\omega] \in H^n{dR}(X)$ of [[closed differential forms|closed]] [[differential n-forms]] $\omega \in \Omega_{dR}^n(X)_{clsd}$

$$
  [\omega] = [\omega']
  \;\;\;\;
     \Leftrightarrow
  \;\;\;\;
  \underset{
    \alpha \in \Omega_{dR}^{n-1}(X)
  }{\exists}
  \omega' = \mathrm{d}\alpha+ \omega
$$

is equivalent to the [[concordance]]-relation [[FSS20, Prop. 6.4](#FSS20)]

\[
  \label{OrdinaryDeRhamCoboundaryAsConcordance}
  [\omega] = [\omega']
  \;\;\;\;
     \Leftrightarrow
  \;\;\;\;
  \underset{
    \widehat{\omega} \in \Omega_{dR}^{n1}(X \times [0,1])_{clsd}
  }{\exists}
  \;
  \left\{
  \begin{array}{l}
    \omega = \widehat\omega\vert_{0}
    \,,
    \\
    \omega' = \widehat\omega\vert_{1}
    \,.
  \end{array}
  \right.
\]

But the latter [[concordance]]-relation immediately generalizes to [[flat L-infinity algebra valued differential forms|flat $L_\infty$-algebra valued differential forms]]

$$
  \Omega_{dR}\big(
    X;
    \mathfrak{a}
  \big)_{clsd}
  \;\;
  \coloneqq
  \;\;
  Hom_{dgAlg}\big(
    CE(\mathfrak{a})
    ,\,
    \Omega^\bullet_{dR}(X)
  \big)
$$

with [[coefficients]] in any $L_\infty$-algebra $\mathfrak{a}$, which reduces to the ordinary case for $\mathfrak{a} \equiv b^{n-1} \mathbb{R}$ the [[line Lie n-algebra|line Lie $n$-algebra]].

Therefore it makes sense to define [[FSS20, Def. 6.3](#FSS20)]:

\begin{definition}
The *non-abelian de Rham cohomology* of a smooth manifold $X$ with [[coefficients]] in a [[L-infinity algebra|$L_\infty$-algebra]] $\mathfrak{a}$ is the [[set]] of [[concordance]] [[equivalence class|classes]] of [[flat L-infinity algebra valued differential forms|flat $\mathfrak{a}$-valued differential forms]] on $X$:

\[
  \label{NonabelianDeRhamCohomology}
  H_{dR}\big(
    X
    ;\,
    \mathfrak{a}
  \big)
  \;\;
  \coloneqq
  \;\;
  \Omega_{dR}\big(
    X
    ;\,
    \mathfrak{a}
  \big) _{clsd}
  \big/
  \mathrm{concordance}
  \,.
\]
\end{definition}

## Examples

\begin{example}
  With (eq:OrdinaryDeRhamCoboundaryAsConcordance) it follows that the ordinary [[de Rham cohomology]] in degree $n$ is equivalently non-abelian de Rham cohomology with [[coefficients]] in the [[line Lie n-algebra]] $b^{n-1}\mathfrak{u}(1)$:
$$
  H_{dR}\big(
    X
    ;\,
    b^{n-1}\mathfrak{u}(1)
  \big)
  \;\;
  \simeq
  \;\;
  H^n_{dR}\big(
    X
    ;\,
    b^{n-1}\mathbb{R}
  \big)
  \,.
$$
\end{example}

\begin{example}\label{TotalFluxInHigherGaugeTheories}
  In [[higher gauge theories]] [of Maxwell-type](higher+gauge+field#HigherGaugeTheoryOfMaxwellType), nonabelian de Rham cohomology of a [[Cauchy surface]] with coefficients in an [[L-infinity algebra]] characteristic of the theory's [[Gauss law]] reflects the *total [[flux]]* of the [[higher gauge fields]].

See at *[[geometry of physics -- flux quantization]]* the section *[Total flux in Nonabelian de Rham cohomology](geometry+of+physics+--+flux+quantization#TotalFluxInNonabelianDeRhamCohomology)*.
\end{example}

## Properties

### Recipient of non-abelian character map

For $\mathcal{A}$ (the [[homotopy type]] of) a [[topological space]] which is [[nilpotent topological space|nilpotent]] (for instance: [[simply connected topological space|simply connected]]) and of rational [[finite type]] (all its [[rational cohomology]]-groups are [[finite-dimensional vector spaces|finite-dimensional $\mathbb{Q}$-vector spaces]]) one may regard the [[homotopy classes]] of [[maps]] into $\mathcal{A}$ as the [[nonabelian cohomology]] classified by $\mathcal{A}$ (the non-abelian cohomology in degree=1 with coefficients in the [[loop space]] [[infinity-group|$\infty$-group]] $\Omega \mathcal{A}$ ):

\[
  \label{NonabelianCohomology}
  H\big(
    X
    ;\,
    \mathcal{A}
  \big)
  \;\;
  \coloneqq
  \;\;
  \pi_0
  \,
  Maps\big(
    X
    ,\,
    \mathcal{A}
  \big)
  \;\;
  \simeq
  \;\;
  \pi_0
  \,
  Maps\big(
    X
    ,\,
    B \Omega \mathcal{A}
  \big)
  \;\;
  \equiv
  \;\;
  H^1\big(
    X
    ;\,
    \Omega \mathcal{A}
  \big)
  \,.
\]

For example, in the case that

$$
  \mathcal{A} \,\equiv\, K(n,A)
$$

is an [[Eilenberg-MacLane space]] for a [[discrete group|discrete]] [[abelian group]] $A$, this reduces to [[ordinary cohomology]]:

$$
  H\big(
    X
    ;\,
    K(n,A)
  \big)
  \;\;
  \simeq
  \;\;
  H^n(X;\, A)
  \,,
$$

or if

$$
  \mathcal{A}
  \;\equiv\;
  KU_0
  \,\simeq\,
  B U \times \mathbb{Z}
$$

is the [[classifying space]] [[KU]]$_0$ for complex [[topological K-theory]], then this reduces to to complex topological K-theory:

$$
  H\big(
    X
    ;\,
    KU_0
  \big)
  \;\;
  \simeq
  \;\;
  K(X)
  \,.
$$

Generally, if $\mathcal{E}$ is an [[Omega-spectrum]] of spaces, then

$$
  H\big(
    X
    ;\,
    E_n
  \big)
  \;\;
  \simeq
  \;\;
  E^n(X)
$$

coincides with the [[Whitehead-generalized cohomology|Whitehead-generalized $E$-cohomology]].

Now the [[rationalization]]-[[unit of an adjunction|unit]] $\eta^{\mathbb{Q}}_{\mathcal{A}} \,\colon\, \mathcal{A} \to \mathcal{A}_{\mathbb{Q}}$ followed by suitable [[extension of scalars]] along $\mathbb{Q} \to \mathbb{R}$ induces [[cohomology operations]] in the non-abelian cohomology (eq:NonabelianCohomology), to what may be called non-abelian [[rational cohomology]], and non-abelian [[real cohomology]] with coefficients in $\mathcal{A}$

\[
  \label{TheNonabelianCharacterMap}
  H\big(
    -;\,
    \mathcal{A}
  \big)
  \longrightarrow
  H\big(
    -;\,
    \mathcal{A}_{\mathbb{Q}}
  \big)
  \longrightarrow
  H\big(
    -;\,
    \mathcal{A}_{\mathbb{R}}
  \big)
  \;\;
  \simeq
  \;\;
  H_{dR}\big(
    -
    ;\,
    \mathfrak{l}\mathcal{A}
  \big)
  \,,
\]

and, shown on the right, a non-abelian version of the [[de Rham theorem]] --- given essentially by the [[fundamental theorem of dg-algebraic rational homotopy theory]] --- identifies this non-abelian real cohomology with coefficients in $\mathcal{A}$ with the non-abelian de Rham cohomology (eq:NonabelianDeRhamCohomology) with coefficients in the real-[[Whitehead L-infinity algebra|Whitehead $L_\infty$-algebra]] $\mathfrak{l}\mathcal{A}$ of $\mathcal{A}$.

For the case that $\mathcal{A} = KU_0$ the [[cohomology operation]] (eq:TheNonabelianCharacterMap) coincides with the [[Chern character]] on complex [[topological K-theory]], and generally for $\mathcal{A} = \mathcal{E}_n$ a term in an [[Omega-spectrum]] it coincides with  the [[Chern-Dold character]] map on [[Whitehead-generalized cohomology]] ([Prop. 7.2](#FSS20)).

Therefore, it makes sense to refer to (eq:TheNonabelianCharacterMap) generally as the *character map on nonabelian cohomology* taking values in non-abelian de Rham cohomology ([FSS20, Part IV](#FSS20)).

\begin{imagefromfile}
    "file_name": "NonabCharacter-240123.jpg",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

## Related entries

* [[non-abelian cohomology]]

* [[differential non-abelian cohomology]]

## References

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Part III of: _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalized]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

[[!redirects nonabelian de Rham cohomology]]

[[!redirects nonabelian de Rham theorem]]
[[!redirects nonabelian de Rham theorems]]

[[!redirects non-abelian de Rham theorem]]
[[!redirects non-abelian de Rham theorems]]


non-abelian group

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

#Contents#
* table of contents
{:toc}

## Idea

A [[group]] which is not an [[abelian group]].

## Related concepts

* [[noncommutative ring]]

* [[nonabelian cohomology]]

* [[non-abelian T-duality]]

[[!redirects non-abelian groups]]

[[!redirects nonabelian group]]
[[!redirects nonabelian groups]]


Non-Archimedean Analysis

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
=--
=--
This page collects material related to the textbook

* [[Siegfried Bosch]], [[Ulrich Güntzer]], [[Reinhold Remmert]],

  _Non-Archimedean Analysis -- A systematic approach to rigid analytic geometry_,

  Grundlehren der Mathem. Wissen. __261__, 1984

  ([MR0746961](http://www.ams.org/mathscinet-getitem?mr=0746961), [pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

on [[rigid analytic geometry]].

#Contents#
* table of contents
{:toc}

## Part A. Linear Uktrametric Analysis and Valuation Theory

### Chapter 1. Norms and valuations

#### 1.1 Semi-normed and normed groups

* [[normed group]], [[triangle inequality]]

#### 1.2 Semi-normed and normed rings

* [[normed ring]]

#### 1.4 Strictly convergent power series

#### 1.5 Non-archimedean valuations

* [[valuation]], [[absolute value]]

## Part B. Affinoid algebras

* [[affinoid algebra]]

### Chapter 3. Extensions of norms and valuations

#### 3.7 Banach algebras

### Chapter 5. Strictly convergent power series

### Chapter 6. Affinoid algebras and Finiteness theorems

* [[affinoid algebra]]

## Part C. Rigid analytic geometry

* [[rigid analytic geometry]]

### Chapter 9. Rigid analytic varieties

#### 9.1. Grothendieck topologies

* [[G-topology]]

#### 9.7. An application to elliptic curves

* [[elliptic curve]]

* [[Tate curve]]

category: reference


non-archimedean analytic geometry

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

#Contents#
* table of contents
{:toc}

## Idea

Non-archimedean geometry is ([[algebraic geometry|algebraic]]) [[geometry]] over [[non-archimedean fields]].

While the concrete results are quite different, the basic formalism of algebraic [[schemes]] and [[formal schemes]] over a non-archimedean field $K$ is the special case of the standard formalism over any field. The "correct" [[analytic geometry]] over [[non-archimedean fields]], however, is _not_ a straightforward analogue of the [[complex analytic geometry|complex analytic]] case.

As Tate noticed, the sheaf of $K$-valued functions which can be locally written as converging power series over the affine space $K^n$ is too big  (too many analytic functions) due to the fact that the underlying topological space is totally disconnected. Also there are very few $K$-analytic manifolds. This naive approach paralleling the [[complex analytic geometry]] is called by Tate _wobbly $K$-analytic varieties_ and, apart from the case of non-archimedean _local_ fields it is of little use. For this reason Tate introduced a better $K$-algebra of analytic functions, locally takes its maximal spectrum and made a Grothendieck topology which takes into account just a certain _smaller_ set of open covers; this topology is viewed as _rigidified_, hence the varieties based on gluing in this approach is called [[rigid analytic geometry]]. Raynaud has shown how some classes of rigid $K$-varieties can be realized as generic fibers of formal schemes over the ring of integers of $K$; this is called a _formal model_ of a rigid variety. Different formal models are birationally equivalent, more precisely they are related via _admissible blow-ups_. Later more sophisticated approaches appeared:

* [[Berkovich analytic space]]s, which take prime spectra together with an extension of the abolsute value to the residue field
* Huber's [[adic space]]s which take into account also valuations of higher rank
* Fujiwara-Kato's approach ('rigid geometry') which takes a limit over all formal models; this has spirit of [[birational geometry]]
* [[global analytic geometry]].

## Literature

For literature on specific approaches see the $n$Lab entries [[Berkovich analytic space]], [[adic space]], [[global analytic geometry]], [[rigid analytic geometry]], [[Huber space]], [[perfectoid space]].

A nice survey is in

* {#LeStum12} [[Bernard Le Stum]], _One century of $p$-adic geometry -- From Hensel to Berkovich and beyond_, talk notes, June 2012 ([pdf](http://www-irma.u-strasbg.fr/IMG/pdf/NotesCoursLeStum.pdf))

A standard textbook on [[rigid analytic geometry]] is

* {#BoschGuntzerRemmert84} S. Bosch, U. Güntzer, [[Reinhold Remmert]], _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

For comparison see

* [[Brian Conrad]], _Several approaches to non-archimedean geometry_, lectures at Arizona winter school 2007, [pdf](http://www.wstein.org/swc/aws/07/ConradNotes11Mar.pdf)

* Kiran S. Kedlaya, _Reified valuations and (re)adic spectra _, [arxiv/1309.0574](http://arxiv.org/abs/1309.0574)

* Kazuhiro Fujiwara, Fumiharu Kato, _Foundations of Rigid Geometry I_, [arxiv/1308.4734](http://arxiv.org/abs/1308.4734)

* [[Oren Ben-Bassat]], [[Kobi Kremnizer]], _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))

* {#BoschGuntzerRemmert84} S. Bosch, U. Güntzer, [[Reinhold Remmert]], _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

category: geometry

[[!redirects non-Archimedean analytic geometry]]
[[!redirects nonarchimedean analytic geometry]]
[[!redirects non-archimedean geometry]]


non-associative geometric algebra

+-- {: .rightHandSide}
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### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Super-Algebra and Super-Geometry
+-- {: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A non-associative version of a [[geometric algebra]].

## Definition

Given a [[commutative ring]] $R$, a **non-associative geometric algebra** is an [[N-graded module|$\mathbb{N}$-graded]] $R$-[[module]] $A$ with a [[bilinear function]] $(-)(-):A \times A \to A$ and a ring [[isomorphism]] $j:\langle A \rangle_0 \cong R$ such that

* for natural numbers $m:\mathbb{N}$ and $n:\mathbb{N}$, the product of every $m$-multivector and $n$-multivector is an $(m+n)$-multivector: for all $a \in \vert A \vert_m$ and $b \in \vert A \vert_n$, there exists $c \in \vert A \vert_{m+n}$ such that $a b = c$

* the product of every $1$-vector with itself is a $0$-vector: for all $a \in \langle A \rangle_1$ there exists $c \in \langle A \rangle_0$ such that $a^2 = c$.

Due to the ring isomorphism $j$ and the linearity of the grade projection operation, the algebra is [[unital magma|unital]].

## See also

* [[geometric algebra]]

* [[N-graded module]]


non-canonical isomorphism

# Non-canonical isomorphisms

## Idea

Often in [[category theory]], requirements that certain things "be isomorphic" are more precisely stated as saying that a certain "canonical" morphism is invertible.  However, in certain situations, it happens that the existence of a non-canonical isomorphism automatically *entails* the "correct" stronger property that the canonical morphism is invertible.

For now, see the references for details.

## References

* {#Lack09} [[Stephen Lack]], _Non-canonical isomorphisms_, ([arXiv:0912.2126](http://arxiv.org/abs/0912.2126)).

* [[Fernando Lucatelli Nunes]], *Pseudoalgebras and non-canonical isomorphisms*, [arxiv](https://arxiv.org/abs/1711.02051)

[[!redirects non-canonical isomorphisms]]


non-commutative analytic space

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###Context###
#### Geometry
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[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A notion of (Berkovich) [[analytic space]] in the context of [[noncommutative geometry]].

## Related concepts

* [[p-adic number]]

* [[non-archimedean field]]

## References

* [[Yan Soibelman]], _On non-commutative analytic spaces over non-archimedean fields_, in [[Anton Kapustin]], [[Maximilian Kreuzer]], [[Karl-Georg Schlesinger]], _Homological mirror symmetry_, lecture notes in physics 757, Springer (2009) ([ps](http://www.math.ksu.edu/~soibel/lapkin-spaces-2.ps))

[[!redirects non-commutative analytic spaces]]

[[!redirects noncommutative analytic space]]
[[!redirects noncommutative analytic spaces]]


[[!redirects Non-commutative global analytic index theory]]

+-- {: .rightHandSide}
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###Context###
#### Index theory
+-- {: .hide}
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#### Analytic geometry
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[[!include analytic geometry -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## The motivation for the development of non-commutative index theory

There has been various non-commutative index theorems, such as for example, the [[Connes-Moscovisci index theorem]], and the [[Kashiwara-Schapira index theorem]], the [[Arthur-Selberg trace formula]] and the [[Lefschetz trace formula]]. All these index type results should be treated in a uniform setting. This ideas was present in the litterature since a long time (Dold maybe; see the references at the end of this section).

It is not yet clear how one may give an optimal setting for a non-commutative global analytic index theory, but here is what one may try to do: using $\infty$-stacks (e.g., group actions) and formal stacks (foliations), one may try to treat Connes-Moscovisci formulae in a more geometric way. However, if one looks at the $\mathcal{D}$-module setting, for example, one sees a deeply non-commutative situation. This means that the following generalized geometry may be use to define a very general notion of trace and Chern character:

1. First, one may try to work with an $\infty$-topos $X$ together with a sheaf of associative algebras $\mathcal{A}$. The ideas here extend the work of Kashiwara-Schapira and someone else that will be cited later on: one will use the category $BiMod_f(\mathcal{A})$ of modules on $\mathcal{A}$ with good finiteness properties (e.g., coherent good in the case of $\mathcal{D}$-modules; perfect for $\mathcal{O}$-modules on a stack; coherent for $\mathcal{O}$-modules on a usual space, etc...). This category is equipped with two monoidal structure. Let's use the first (left) one, that is given by
$$
(\mathcal{M},\mathcal{N})\mapsto \mathcal{M}\circ\mathcal{N}:=\mathcal{M}\otimes_\mathcal{A}\mathcal{N}.
$$
Remark that one may define the notion of direct sum of $\mathcal{A}$-bimodule, denoted $\oplus$. We thus actually have a bimonoidal category
$$(BiMod_f(\mathcal{A}),\oplus,\circ)$$
that is not symmetric in $\circ$ but symmetric in $\oplus$. This is thus a categorification of an associative ring. One may easily define the notion of a seminorm $|\cdot|$ on such an associative categorical ring $(\Ac,\oplus,\circ)$, following the approach explained in [[generalized global analytic geometry]], and also define a Berkovich spectrum $\mathcal{M}(\mathcal{A},\oplus,\circ)$. This will give a topological space or a $G$-topological space (or an $\infty$-topos, in the étale topology situation) together with a sheaf $U\mapsto \mathcal{A}(U)$ of seminormed associative categorical rings. Remark that to every multiplicative seminorm on $\mathcal{A}$, one may associate a prime ideal in $\mathcal{A}$. We are thus in a conceptually abstract situation that is very close to the Toen-Vezzosi approach to the Chern character (the monoidal structure is not symmetric).

2. The definition of a categorical trace for objects of $\mathcal{A}$ acting on $\mathcal{A}$ by the left tensor product $\circ$ may be given if $\mathcal{A}$ is equipped with a kind of rigidity structure. In the $\mathcal{D}$-module setting, if we work with bimodules such as $\mathcal{M}\boxtimes D\mathcal{M}$, Kashiwara and Schapira use the notion of trace kernel to define the corresponding class in Hochschild homology. One may try to give a similar construction using rigidity, e.g., the fact that a natural dual for the above bimodule is simply $D\mathcal{M}\omega\boxtimes \omega\mathcal{M}$. The diagonal $\Delta$ seems to play an important role here.

Now a natural constraint on this situation if we look at the Kashiwara and Schapira results is to use the associated Hochschild or negative cyclic homology to define a trace: one must not suppose that $(\mathcal{A},\circ)$ is a rigid monoidal category, because this is not true in the example of elliptic pairs (Atiyah-Singer) (the corresponding trace kernel is given by tensor product of a $\mathcal{D}$-module kernel with a constructible kernel). One should only suppose something weaker, related to the fact that one wants a trace to be defined on cyclic and/or Hochschild cohomology of the situation (a global invariant on $X$, that has a meaning in the Atiyah-Singer situation).

To be continued.

## Possible references

Connes-Moscovisci

Kashiwara-Schapira

Selberg

Arthur

Ramados-Tang-Tsen Hochschild-Lefschetz class for $\mathcal{D}$-modules

PoNing Chen, Vasiliy Dolgushev: A Simple Algebraic Proof of the Algebraic Index Theorem


non-commutative Witt vectors

## References

* [[Lars Hesselholt]], Witt vectors of non-commutative rings and topological cyclic homology, [pdf](http://www.math.uiuc.edu/K-theory/0135/derived.pdf)

see also [[Hochschild-Witt complex]]


non-connective algebraic K-theory

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#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Motivic cohomology
+--{: .hide}
[[!include motivic cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Non-connective algebraic K-theory_, sometimes called _Bass K-theory_, is a variant of [[algebraic K-theory]] with non-trivial [[homotopy groups]]/[[cohomology groups]] in negative degrees.

This universally arises as the hom-spectra in the [[(∞,1)-category of noncommutative motives]].

A review and further discussion is in ([Blumberg-Gepner-Tabuada 10, section 9](BlumbergGepnerTabuada10)).

## Related concepts

[[!include noncommutative motives - table]]

## References

* [[R. W. Thomason]], Thomas Trobaugh, _Higher algebraic K-theory of schemes and of derived categories_, The Grothendieck Festschrift, 1990, 247-435.

The characterization of non-connective algebraic K-theory via [[noncommutative motives]] is due to

* [[Denis-Charles Cisinski]], [[Gonçalo Tabuada]], _Non connective K-theory via universal invariants_. Compositio
Mathematica 147 (2011), 1281–1320 ([arXiv:0903.3717](http://arxiv.org/abs/0903.3717))

and further expanded on in section 9 of

* [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], _A universal characterization of higher algebraic K-theory_,  Geometry and Topology ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282))
 {#BlumbergGepnerTabuada10}

[[!redirects nonconnective algebraic K-theory]]
[[!redirects Bass K-theory]]


Non-Euclidean geometry

#Contents#
* table of contents
{:toc}

## Idea

Generally, [[geometry]] which is not [[Euclidean geometry]]. Specifically [[hyperbolic geometry]] and [[elliptic geometry]] violating the [[parallel postulate]].

## Related concepts

* [[synthetic geometry]]

## References

* Wikipedia, _[Non-Euclidean geometry](http://en.wikipedia.org/wiki/Non-Euclidean_geometry)_

[[!redirects Non-Euclidean geometries]]

[[!redirects non-Euclidean geometry]]
[[!redirects non-Euclidean geometries]]

[[!redirects noneuclidean geometry]]
[[!redirects noneuclidean geometries]]


non-geometric string vacuum

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#### Vacua
+-- {: .hide}
[[!include vacua -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In general, a [[perturbative string theory vacuum]] is defined by a [[worldsheet]] [[2d SCFT]], regarded as the [[formal dual]] ("[[2-spectral triple]]") of the corresponding emergent [[target spacetime]].

Some such [[2d SCFTs]] arise as [[sigma-models]] from actual [[differential geometry|differential geometric]] [[pseudo-Riemannian manifold|pseudo-Riemannian]] [[spacetime]] [[smooth manifolds|manifolds]] with differential geometric [[background fields]] on them. These are hence called the _geometric vacua_ or _geometric backgrounds_.

All other vacua are then called _non-geometric backgrounds_. These are defined _only_ [[Isbell duality|Isbell-dually]] in terms of [[vertex operator algebra]] on the [[worldsheet]]. For instance a [[symmetry]] of a non-geometric background is dually an [[automorphism of a vertex operator algebra]].

In the point-particle limit, these non-geometric string vacua correspond to [[spectral triples]] that are defined purely algebraically, such as the non-geometric fiber spaces in the [[Connes-Lott model]].

## Examples

### General

* [[rational 2d CFT]]

  * [[Gepner model]]

* [[non-commutative geometry]] (as in [[Connes-Lott models]])

### Type II superstring vacua

* [[type II geometry]]

  * [[T-folds]]

  * backgrounds of the [[exotic branes]]

* [[exceptional generalized geometry]]

### Heterotic string vacua
 {#ExamplesHeteroticString}

Strictly speaking, the [[perturbative string theory vacua]] of  [[heterotic string theory]] are _all_ non-geometric, even if the apparent [[target spacetime]] is plain [[Minkowski spacetime]]:

The [[2d SCFT]] defining the [[heterotic string]] is the [[direct sum]] of two rather different chiral halves:

1. the holomorphic part, say, is one chiral half of a [[superstring]] [[sigma model]] on 10d [[Minkowski spacetime]]

2. the antiholomorphic part is one chiral half of the [[bosonic string]] on a 26d-dimensional [[Minkowski spacetime]] that is [[KK-compactification|compactified]] on the [[Leech lattice]].

While the [[direct sum]] of these two [[super vertex operator algebras]] is again a consistent [[super vertex operator algebra]], in this combination this is not a [[sigma-model]] anymore, even if its two halves do arise in (two distinct!) sigma-models.

Analogous statements hold for most [[rational CFT]]-constructions of [[perturbative string theory vacua]]. Even if parts of these algebras arise as chiral half of [[sigma-models]] (notably [[WZW models]]), the way they are put together chirally yields a non-geometric construction that defines a non-geometric [[perturbative string theory vacuum]].

## Related concepts

[[!include Isbell duality - table]]

## References

We list references that use the "non-geometric"-terminology. But notice that all [[rational 2d CFT]]/[[Gepner model]] compactifications discussed in the 1990s are "non-geometric", but the term wasn't around then. Hence see the references at _[[Gepner model]]_ for more.

### General

* Alex Flournoy, Brian Wecht, Brook Williams, _Constructing Nongeometric Vacua in String Theory_, Nucl.Phys.B706:127-149, 2005 ([arXiv:hep-th/0404217](https://arxiv.org/abs/hep-th/0404217))

* [[Katrin Becker]], [[Melanie Becker]], [[Cumrun Vafa]], [[Johannes Walcher]], _Moduli Stabilization in Non-Geometric Backgrounds_,  Nucl. Phys. B **770** (2007) 1-46 [[arXiv:hep-th/0611001](https://arxiv.org/abs/hep-th/0611001)]

* Erik Plauschinn, _Non-geometric backgrounds in string theory_, j.physrep.2018.12.002 ([arXiv:1811.11203](https://arxiv.org/abs/1811.11203))

* Dan Israel, _Mirrored K3 automorphisms and non-geometric compactifications_ 2018 ([pdf](http://www2.yukawa.kyoto-u.ac.jp/~nfst2018/Slide/Israel.pdf))

### Heterotic string

Non-geometric [[heterotic string]] vacua

* [[Jock McOrist]], [[David Morrison]], [[Savdeep Sethi]], _Geometries, Non-Geometries, and Fluxes_ ([arXiv:1004.5447](https://arxiv.org/abs/1004.5447))

* Anamaría Font, Christoph Mayrhofer, _Non-Geometric Vacua of the $SO(32).\mathbb{Z}_2$ Heterotic String and Little String Theories_ ([arXiv:1708.05428](https://arxiv.org/abs/1708.05428))

* Anamaría Font, Iñaki García-Etxebarria, [[Dieter Lüst]], Stefano Massai, Christoph Mayrhofer, _Heterotic T-fects, 6D SCFTs, and F-Theory_, JHEP08(2016)175 ([arXiv:1603.09361](https://arxiv.org/abs/1603.09361))

* Iñaki García-Etxebarria, [[Dieter Lüst]], Stefano Massai, Christoph Mayrhofer, _Ubiquity of non-geometry in heterotic compactifications_, JHEP03(2017)046 ([arXiv:1611.10291](https://arxiv.org/abs/1611.10291))

### Flux compactifications

Non-geometric [[flux compactification]]:

* Anamaria Font, Adolfo Guarino, Jesus M. Moreno, _Algebras and non-geometric flux vacua_, JHEP 0812:050, 2008 ([arXiv:0809.3748](https://arxiv.org/abs/0809.3748))

* David Andriot, Andre Betz, _Supersymmetry with non-geometric fluxes, or a β-twist in Generalized Geometry and Dirac operator_ ([arXiv:1411.6640](https://arxiv.org/abs/1411.6640))

*  Stefano Risoli, _On non-geometric string vacua_, 2016 ([pdf](https://www.phys.uniroma1.it/fisica/sites/default/files/allegati/Risoli_0.pdf))

### T-Folds

[[T-folds]]:

* [[Chris Hull]], _A Geometry for Non-Geometric String Backgrounds_ ([arXiv:hep-th/0406102](https://arxiv.org/abs/hep-th/0406102))

* [[Chris Hull]], _Global Aspects of T-Duality, Gauged Sigma Models and T-Folds_ ([arXiv:hep-th/0604178](http://arxiv.org/abs/hep-th/0604178))

* Aaron Bergman, Daniel Robbins, _Ramond-Ramond Fields, Cohomology and Non-Geometric Fluxes_ ([arXiv:0710.5158](http://arxiv.org/abs/0710.5158))

* [[Dieter Lüst]], Stefano Massai, Valentí Vall Camell, _The monodromy of T-folds and T-fects_ ([arXiv:1508.01193](http://arxiv.org/abs/1508.01193))

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

* Yoan Gautier, [[Chris Hull]], Dan Israël, _Heterotic/type II Duality and Non-Geometric Compactifications_ ([arXiv:1906.02165](https://arxiv.org/abs/1906.02165))

[[!redirects non-geometric string vacua]]

[[!redirects non-geometric string theory vacuum]]
[[!redirects non-geometric string theory vacua]]

[[!redirects non-geometric vacuum]]
[[!redirects non-geometric vacua]]

[[!redirects non-geometric background]]
[[!redirects non-geometric backgrounds]]

[[!redirects non-geometric solution]]
[[!redirects non-geometric solutions]]

[[!redirects non-geometry]]


non-Hausdorff manifold

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### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
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#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

It may superficially seem that every [[locally Euclidean space]] is a [[Hausdorff topological space]], since [[Euclidean space]] is. While the analogous conclusion is in fact true for genuine "local" properties of Euclidean space (such as the $T_1$-[[separation axiom]], [[sober topological space|sobriety]] and [[locally compact topological space|local compactness]], see [this prop.](topological+manifold#LocalPropertiesOfLocallyEuclideanSpace)), it is actually wrong for the "non-local" $T_2$= Hausdorff-conditon: There are [[locally Euclidean spaces]] which are not [[Hausdorff topological spaces]], such as the [[line with two origins]].

Since a ([[paracompact topological space|paracompact]]/[[sigma-compact]]) [[Hausdorff topological space|Hausdorff]] [[locally Euclidean space]] is called a _[[topological manifold]]_, (paracompact/sigma-compact) non-Hausdorff locally Euclidean spaces are sometimes referred to as "non-Hausdorff manifolds" (an instance of the [[red herring principle]]).

## Examples

The usual example is the [[line with two origins]]: the [[real line]] with the point $0$ 'doubled'.  Explicitly, this is a [[quotient space]] of $\mathbb{R} \times \{a,b\}$ (given the [[product topology]] and with $\{a,b\}$ given the [[discrete topology]]) by the [[equivalence relation]] generated by identifying $(x,a)$ with $(y,b)$ iff $x = y \ne 0$.

## References

* Mathieu Baillif, Alexandre Gabard, _Manifolds: Hausdorffness versus homogeneity_ ([arXiv:0609098](http://arxiv.org/abs/math.GN/0609098))

* Steven L. Kent, Roy A. Mimna, and Jamal K. Tartir, _A Note on Topological Properties of  Non-Hausdorff Manifolds_, ([web](http://www.hindawi.com/journals/ijmms/2009/891785/))

[[!redirects non-Hausdorff manifold]]
[[!redirects non-Hausdorff manifolds]]
[[!redirects nonHausdorff manifold]]
[[!redirects nonHausdorff manifolds]]

[[!redirects non-Hausdorff locally Euclidean space]]
[[!redirects non-Hausdorff locally Euclidean spaces]]


non-Hausdorff topological space

+-- {: .rightHandSide}
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#Contents#
* table of contents
{:toc}

## Idea

A _non-Hausdorff topological space_ is a [[topological space]] which is not a _[[Hausdorff topological space]]_.

## Examples

Since the [[full subcategory]] of [[Hausdorff topological spaces]] inside the [[category of topological spaces]] is [[reflective subcategory|reflective]], all [[limits]] in [[Top]] of [[diagrams]] of Hausdorff spaces are again Hausdorff spaces, but for [[colimits]] this fails in general ("non-Hausdorff quotients", the colimit in the subcategory is given by applying [[Hausdorff reflection]] to the colimit formed in [[Top]]).

Hence examples of non-Hausdorff spaces generically arise from forming [[quotient topological spaces]] of Hausdorff spaces:

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

Consider the [[real line]] $\mathbb{R}$ regarded
as the 1-dimensional [[Euclidean space]] with its [[metric topology]]
and consider the [[equivalence relation]] $\sim$ on $\mathbb{R}$ which identifies two [[real numbers]] if they
differ by a [[rational number]]:

$$
  \left(
    x \sim y
  \right)
    \;\Leftrightarrow\;
  \left(
    \underset{p/q \in \mathbb{Q} \subset \mathbb{R}}{\exists}
      x = y + p/q
  \right)
  \,.
$$

Then the [[quotient topological space]]

$$
  \mathbb{R}/\mathbb{Q}
    \;\coloneqq\;
  \mathbb{R}/\sim
$$

is a [[codiscrete topological space]], hence in particular a
not a Hausdorff space.

=--

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

The [[line with two origins]].

=--

[[!redirects non-Hausdorff topological spaces]]


non-Lagrangian field theory

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#### Quantum systems
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#### Algebraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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#### Functorial Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

A [[field theory]] is called *non-Lagrangian* if it can not be described using a [[Lagrangian density]]. This is in contrast to *[[Lagrangian field theories]]*.

## Examples

\begin{example}
A basic example of non-Lagrangian field theories used to be [[self-dual higher gauge theories]]: Superficially, these would be supposed to have higher [[Maxwell theory]]-type Lagrangian densities of the form $H_{2k+1} \wedge \star H_{2k+1}$, but if the self-duality constraint $\star H_{2k+1} = H_{2k+1}$ is imposed beforehand, then this expression vanishes identically instead of inducing the expected [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]].

This issue concerns already the [[Green-Schwarz sigma model]] for the single (abelian) [[M5-brane]] and even more so the would-be theory of coincident [[M5-branes]], involving a [[D=6 N=(2,0) SCFT]] with a nonabelian ([[higher gauge theory|higher]]) [[gauge field]].

However, more complicated Lagrangians have been found ([Pasti, Sorokin & Tonin 1997](M5-brane#PST97), [Sen 2020](#self-dual+higher+gauge+theory#Sen20)) whose [[equations of motion]] do reproduce at least the abelian self-dual fields, and at least if one is willing to disregard some decoupled [[auxiliary fields]], illustrating that the question of whether or not a "field theory" is or is not Lagrangian may require further specification to be well-defined.
\end{example}

## Literature

* S.L. Lyakhovich, [[A.A. Sharapov]], _Quantizing non-Lagrangian gauge theories: an augmentation method_, JHEP 0701:047 (2007) arXiv:[hep-th/0612086](https://arxiv.org/abs/hep-th/0612086)

* P. C. Argyres, M. R. Plesser, N. Seiberg, E. Witten, _New N=2 superconformal field theories in four-dimensions_, Nucl. Phys. B461 (1996) 71--84 arXiv:[hep-th/9511154](https://arxiv.org/abs/hep-th/9511154)

category: physics

[[!redirects non-Lagrangian field theories]]

[[!redirects non-lagrangian field theory]]
[[!redirects non-lagrangian field theories]]

[[!redirects non-Lagrangean field theory]]
[[!redirects non-Lagrangean field theories]]


non-Lorentzian type II string theory

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#### String theory
+-- {: .hide}
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#### Super-Geometry
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[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What one usually refers to as [[type II string theory]] is defined on [[Lorentzian geometry]]. It is thought that analogues of Type II string theory (and, more generally, of the other theories appearing in [[string theory]], such as [[M-theory]]) may be defined on spaces with [[pseudo-Riemannian metric|signature]] other than Lorentz.

In [Hull (1998)](#Hul98duality) it is argued that there exists a web of Type II theories related by **space-** or **time-like** [[T-duality]], as well as [[S-duality]], summarized in:

\begin{imagefromfile}
        "file_name": "nonlorentziantypesii.PNG",
"width": 500
    \end{imagefromfile}

where in particular "IIA/B" refers to [[type II string theory]] and "$IIA^*/B^*$" to [[type II* string theory]]. These theories are described in terms of their spacetime [[action functional]]. For the Type IIA's:
$$
S_{IIA} = \int d^{10}x\sqrt{\vert g\vert} (e^{-2\Phi}(R+4(\partial\Phi)^2 )-(\pm e^{-2\Phi}H^2\pm G^2_2 \pm G^2_4))+\cdots
$$

\begin{imagefromfile}
        "file_name": "typeiias.PNG",
"width": 500
    \end{imagefromfile}
while for the Type IIB's:
$$
S_{IIA} = \int d^{10}x\sqrt{\vert g\vert} (e^{-2\Phi}(R+4(\partial\Phi)^2 )-(\pm e^{-2\Phi}H^2\pm G^2_1 \pm G^2_3\pm G^2_5 ))+\cdots
$$

\begin{imagefromfile}
        "file_name": "typeiibs.PNG",
"width": 500
    \end{imagefromfile}

## References

General:

* [[Christopher Hull]]. *Timelike T-duality, de Sitter space, large N gauge theories and topological field theory*, JHEP 07 (1998) 021 [[hep-th/9806146](https://arxiv.org/abs/hep-th/9806146)\]

* {#Hul98duality} [[Christopher Hull]]. *Duality and the signature of space-time*, JHEP 9811 (1998) 017 [[hep-th/9807127](https://arxiv.org/abs/hep-th/9807127)\]

* [[Christopher Hull]] and [[Ramzi Khuri]], *Branes, times and dualities*, Nucl.Phys. B536 (1998) 219 [[hep-th/9808069](https://arxiv.org/abs/hep-th/9808069)\]

* [[Christopher Hull]]. *Duality and Strings, Space and Time*. (1999) [[hep-th/9911080](https://arxiv.org/abs/hep-th/9911080)\]

* [[Hitoshi Nishino]], [[Jim Gates]]. *The $*$-Report*. (1999). [[hep-th/9908136](https://arxiv.org/abs/hep-th/9908136)\]

On the relation between the [[super Lie algebra|super Lie algebras]] of II and $II^\ast$ and the [[orthosymplectic super Lie algebra]] $\mathfrak{osp}(1\vert 32)$:

* [[Eric Bergshoeff]], [[Antoine Van Proeyen]]. *The many faces of $OSp(1\vert 32)$*. (2000). [[hep-th/0003261](https://arxiv.org/abs/hep-th/0003261)\]

* [[Eric Bergshoeff]], [[Antoine Van Proeyen]]. *The unifying superalgebra $OSp(1\vert 32)$*. (2000). [[hep-th/0010194](https://arxiv.org/abs/hep-th/0010194)\]

The supposed branes in $II^\ast$, [[E-branes]], are discussed in:

* Somdatta Bhattacharya, Sudipta Mukherji, Shibaji Roy. *On the effective action of a space-like brane*. (2003). [[hep-th/0308069](https://arxiv.org/abs/hep-th/0308069)\]

That the boundary state for the Dirichlet [[S-brane]] constructed is a solution of type $II^\ast$ rather than type II string theory is in:

* Bruno Durin, [[Boris Pioline]]. *Aspects of Dirichlet S-branes*. (2005) [[hep-th/0507059](https://arxiv.org/abs/hep-th/0507059)\]

The suggestion that $II^\ast$ and its would-be [[M-theory]] lift may play a role in interpreting M-theory as defined on the base space of an [[octonionic projective space]] $\mathbb{O}P^2$ bundle with structure group the [[exceptional Lie group]] $F_4$ is in Remark 5, Section 2.3 of:

* [[Hisham Sati]]. *On the geometry of the supermultiplet in M-theory*. (2009). [[arXiv:0909.4737](https://arxiv.org/abs/0909.4737)\]

On the relation of twisted $\mathcal{N}=2$ algebra, [Para-quaternionic Kähler manifolds](https://ncatlab.org/nlab/show/para-quaternionic+structure), and compactifications of $II^\ast$ on [[Calabi-Yau variety|Calabi-Yau's]]:

* Maxime Médevielle, [[Thomas Mohaupt]], and G. Pope. *Type-II Calabi-Yau compactifications, T-duality and special geometry in general spacetime signature*. J. High Energ. Phys. 2022, 48 (2022). [[doi:10.1007/JHEP02\(2022\)048](https://doi.org/10.1007/JHEP02(2022)048)\]


non-perturbative effect

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#Contents#
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## Idea

An effect in [[non-perturbative quantum field theory]] that cannot be seen in [[perturbative quantum field theory]] is called a _non-perturbative effect_.

More in detail, [[theory (physics)|theories]] with [[instanton]] [[field (physics)|field]] configurations (such as in [[Yang-Mills theory]], hence in [[QCD]] and [[QED]]) or [[branes]] (such as in [[string theory]]), etc., are expected to have [[observables]] which as [[functions]] of the [[coupling constant]] $g$ are [[transseries]] of the form

\[
  \label{TypicalTransseries}
  Z(g)
  =
  \sum_n a_n g^n
  +
  e^{-A/g}
  \sum_n a_n^{(1)} g^n
  +
  e^{-B/g^2}
  \sum_n b_n^{(1)} g^n
  \,,
\]

where the first sum is the [[Feynman perturbation series]] itself and where the terms with a [[analytic function|non-analytic]] dependence of the form $\exp(-A/g)$ or $\exp(-A/g^2)$ are the contributions of the [[instantons]]. Since all the [[derivatives]] of the functions $g \mapsto e^{-1/g}$ or $g \mapsto e^{-1/g^2}$ vanish at [[coupling constant]] $g = 0$, the [[Taylor series]] of this part of the observable does not appear in [[perturbative QFT]], even though it is present. Therefore this is called a _non-perturbative effect_.

Related is _[[resurgence theory]]_. See also at _[perturbation theory -- Divergence/convergence](perturbation+theory#DivergenceConvergence)_ for more.

{#Ubiquituous} Notice that, while non-perturbative effects are generally the least understood, they are not exotic but ubiquitous: Essentially the entire parameter space of a generic "[[theory (physics)|physical theory]]" is non-perturbative. In fact, the non-perturbative region of parameter space is the [[complement]] of an [[infinitesimally thickened point]] (labeled "perturbative" in the following graphics):

## Examples

### Schwinger effect

* [[Schwinger effect]]

### Confinement and the mass gap

A central example of a non-perturbative effect is [[confinement]] (hence the "[[mass gap problem]]") in [[Yang-Mills theory]] at low [[temperature]]. Perturbation theory is not suited to explain this (e.g [Espiru 94, section 7](http://arxiv.org/abs/hep-ph/9410287)).

### Quark-gluon plasma

At the other extreme of high [[temperature]] QCD, also the [[quark-gluon plasma]], while now [[confinement|deconfined]] is thought to be strongly coupled.

### Hadronic physics in flavour and $g-2$ anomalies
 {#HadronicPhysics}

Non-perturbative effects in [[hadron]]-physics affects the
discussion of possible beyond-[[standard model of particle physics|standard model]] physics as seen in

1. [[flavour anomalies]] ([Nierste 18, 26/59](#Nierste18))

1. [[muon anomalous magnetic moment]] ([Jegerlehner 18b, section 2](#Jegerlehner18b))

### Worldsheet and brane instantons in string/M-theory
 {#WorldsheetAndBraneInstantons}

Consider the [[M-theory]] [[scales]]

* $\ell_P$, the [[Planck length]] in 11-dimensions;

* $R_{10}$ the [[length]] (circumference) of the $S^1_{10}$ [[circle]] [[fiber]] for [[KK-compactification]] to 10 dimensions

and the [[string theory]] [[scales]]

* $\ell_s$, the [[string length scale]];

* $g_{s}$, the [[string coupling constant]] of [[perturbative string theory]].

Then under the [[duality between M-theory and type IIA string theory]] these scales are related as follows:

$$
  \ell_P \;=\; g_{st}^{1/3} \ell_s
  \,,
  \phantom{AAA}
  R_{10} \;=\; g_{st} \ell_s
$$

equivalently

$$
  \ell_s \;=\; R_{10}/ g_{st}
  \,,
  \phantom{AAA}
  \ell_P \;=\; g_{st}^{-2/3} R_{10}
$$

equivalently

$$
  g_{st} \;=\; (R_{10}/\ell_P)^{3/2}
  \,,
  \phantom{AAA}
  \ell_s \;=\; \ell_P (R_{10}/\ell_P)^{-1/2}
  \,.
$$

Hence a [[membrane instanton]], which on a 3-[[cycle]] $C_3$ gives a contribution

$$
  \exp\left(
    - \frac{ vol(C_3) }{ \ell^3_P }
  \right)
$$

becomes

1. if the cycle [[wrapped brane|wraps]], $C_3 = C_2 \cup S^1_{10}$, a [[worldsheet instanton]]

   $$
     \exp\left( - \frac{ vol(C_3) }{ \ell_P^3 }  \right)
     \;=\;
     \exp\left( - \frac{ R_{10} vol(C_2) }{ g_{st} \ell_s^3 }  \right)
     \;=\;
     \exp\left( - \frac{ vol(C_2) }{ \ell_s^2 }  \right)
   $$

1. the cycle does not [[wrapped brane|wrap]], a spacetime instanton contribution, specifically a [[D2-brane instanton]]

   $$
     \exp\left( - \frac{ vol(C_3) }{ \ell_P^3 }  \right)
     \;=\;
     \exp\left( - \frac{ vol(C_3)/\ell_s^3 }{ g_{st} }  \right)
   $$

(This unification of the two different [[non-perturbative effects]] in [[perturbative string theory]]
([[worldsheet instantons]] and spacetime [[instantons]]), to a single type of effect ([[membrane instanton]])
 in [[M-theory]] was maybe first made explicit in [Becker-Becker-Strominger 95](#BeckerBeckerStrominger95). Brief review includes [Marino 15, sections 1.2 and 1.3](#Marino15)).

## Related concepts

* [[non-perturbative quantum field theory]]

* [[strict deformation quantization]]

* [[transseries]], [[resurgence theory]]

* [[fiber bundles in physics]]

* [[instanton]], [[brane]], [[renormalon]]

  * [[membrane instanton]]

* [[lattice gauge theory]]

* [perturbation theory -- convergence/divergence](perturbation+theory#DivergenceConvergence)

* [[higher curvature correction]]

* [[interacting vacuum]]

* [[proton spin crisis]]

* [[AdS/CFT correspondence]], [[AdS/QCD correspondence]]

* [string theory FAQ -- Isn't it fatal that the string perturbation series does not converge?](string+theory+FAQ#NonConvergenceOfPerturbationSeries)

## References

### In field theory

* {#BakulevShirkov10} Alexander P. Bakulev, [[Dmitry Shirkov]], _Inevitability and Importance of Non-Perturbative Elements in Quantum Field Theory_, Proceedings of the 6th Mathematical Physics Meeting, Sept. 14--23, 2010, Belgrade, Serbia, pp. 27--54 ([arXiv:1102.2380](https://arxiv.org/abs/1102.2380), [ISBN 978-86-82441-30-4](http://www.proceedings.com/22646.html), [book webpage](http://www.mphys6.ipb.ac.rs/proceedings6.htm), [book TOC pdf](http://toc.proceedings.com/22646webtoc.pdf))

Discussion of traditional [[algebraic quantum field theory]] as being about non-perturbative effects (and therefore lacking any interesting models in 3+1 dimensions, so far...):

* {#Strocchi13} [[Franco Strocchi]], _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

Genral introduction and toy examples (e.g. [[phi^4-theory]] or anharmonic oscillator):

* {#FloryHellingSluka12} [[Mario Flory]], [[Robert C. Helling]], Constantin Sluka, Section 2 of: *How I Learned to Stop Worrying and Love QFT* ([arXiv:1201.2714](https://arxiv.org/abs/1201.2714))

* [Mariño 12, section 2, section 3.1](#Marino12)

Discussion for [[phi^4 theory]] is in

* {#Serone18} Marco Serone, from 2:46 on in _A look at $\phi^4_2$ using perturbation theory_, January 2018 ([recording](https://www.youtube.com/watch?v=J4nxvY1rOhI))

Discussion for [[QCD]] includes

* [[Marcos Mariño]], _Instantons and Large $N$ -- An introduction to non-perturbative methods in QFT_ ([pdf](http://laces.web.cern.ch/laces/LACES10/notes/instlargen.pdf))

and in ([[super Yang-Mills theory|super]]-)[[Yang-Mills theory]] and [[string theory]] is in

* {#Marino12} [[Marcos Mariño]], _Lectures on non-perturbative effects in large N gauge theories, matrix models and strings_, Fortschritte der Physik 62.5‐6 (2014): 455-540 ([arXiv:1206.6272](http://arxiv.org/abs/1206.6272))

* {#Marino15} [[Marcos Mariño]], _Non-perturbative effects in string theory and AdS/CFT_, [Spring School on Superstring Theory and Related Topics 2015](http://indico.ictp.it/event/a14251/)  ([[MarinoNonPerturbative.pdf:file]], [pdf](http://indico.ictp.it/event/a14251/session/89/contribution/401/material/0/0.pdf), [recording](https://youtu.be/5hw6l-JSpc4?list=PLzYFBWfshN8eqXZFYg2jBZSii7lHaeyXJ))

In [[QCD]] ([[hadrodynamics]]) via [[operator product expansion]]:

* [[Mikhail Shifman]], *OPE-based Methods in Nonperturbative QCD*, Eur. Phys. J. C (2022) [[arXiv:2208.10600](https://arxiv.org/abs/2208.10600)]

### In phenomenology

In [[flavour anomalies]]:

* {#Nierste18} Ulrich Nierste, _Flavour Anomalies: Phenomenology and BSM Interpretations_, 2018 ([pdf](https://indico.desy.de/indico/event/18498/session/5/contribution/52/material/slides/0.pdf))

* {#Jegerlehner18b} [[Fred Jegerlehner]], _The Role of Mesons in Muon $g-2$_ ([arXiv:1809.07413](https://arxiv.org/abs/1809.07413))

### In cosmology

In [[cosmology]]:

* Anna Ijjas, Frans Pretorius, [[Paul Steinhardt]], _Stability and the gauge problem in non-perturbative cosmology_,  Journal of Cosmology and Astroparticle Physics, Volume 2019, January 2019 ([arXiv:1809.07010](https://arxiv.org/abs/1809.07010), [doi:10.1088/1475-7516/2019/01/015](https://iopscience.iop.org/article/10.1088/1475-7516/2019/01/015))

### In string theory

The form of the contribution of non-perturbative effects in [[string theory]] was originally observed in

* [[Stephen Shenker]], _The Strength of nonperturbative effects in string theory_, presented at the Cargese Workshop on Random Surfaces, Quantum Gravity and Strings, Cargese, France, May 1990 ([spire](http://inspirehep.net/record/298523?ln=en))

The interpretation via [[D-branes]] of non-perturbative effects in the string [[coupling constant]] is due to

* {#Polchinski94} [[Joseph Polchinski]], _Combinatorics of boundaries in string theory_, Phys. Rev. D 50, 6041 (1994) ([arXiv:hep-th/9407031](https://arxiv.org/abs/hep-th/9407031))

The identification of non-perturbative contributions in [[string theory]] with brane contributions is due to

* {#BeckerBeckerStrominger95} [[Katrin Becker]], [[Melanie Becker]], [[Andrew Strominger]], _Five-branes, membranes and nonperturbative string theory_, Nucl. Phys. B 456, 130 (1995) ([hep-th/9507158](http://arxiv.org/abs/hep-th/9507158))

Review includes

* [Marino 15](#Marino15)

Reviews specifically in [[type II string theory]] include

* Hugo Looyestijn, _Non-perturbative effects in type IIA string theory_, Master Thesis 2006 ([pdf](http://testweb.science.uu.nl/ITF/teaching/2006/Looyestijn.pdf))

* [[Angel Uranga]], _Non-perturbative effects and D-brane instanton resummation in string theory_ ([[UrangaNonPerturbativeBrane.pdf:file]], [pdf](https://indico.cern.ch/event/75810/contributions/1250733/attachments/1050849/1498253/uranga.pdf))

### In M-Theory

The relation of non-perturbative effects in string theory to [[M-theory]] goes back to

* {#Witten95} [[Edward Witten]], section 2.2 of _[[String Theory Dynamics In Various Dimensions]]_, Nucl.Phys.B443:85-126,1995 ([arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124))

In [Becker-Becker-Strominger 95](#BeckerBeckerStrominger95) it was realized that the [[worldsheet instantons]] and [[D-brane instantons]] of string theory unify to [[membrane instantons]] (see [Marino 15, section 1.3](#Marino15))

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