normed algebra

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## Definition

A __normed algebra__ $A$ over a [[field]] $k$ of [[real numbers|real]] or [[numbers|complex numbers]] is a [[normed vector space]] equipped with an [[associative algebra]] structure, such that the algebra multiplication is [[continuous function|continuous]] with respect to the [[norm]], i.e. such that there is a [[positive number|positive]] [[real number]] $C\gt 0$ such that

$$
 {\vert a b \vert}
 \;\leq\;
  C {\vert a \vert} {\vert b \vert}
$$

for all $a,b\in A$. One can rescale the norm to another norm to get $C = 1$ ([[absolute value]]). A normed algebra whose underlying normed space is [[complete space|complete]] is called a __[[Banach algebra]]__.

A normed algebra with $C = 1$ is equivalently a [[normed division algebra]]. See there for more.

## Related concepts

* [[normed division algebra]]

[[!include analytic geometry ingredients -- table]]

## References

* M. A. Naimark, _Normed rings_
* R. Kadison, _Introduction to operator algebras_
* A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988
* Marco Thill, _Introduction to normed $\ast$-algebras and their representations_, Kassel Univ. Press, [pdf](http://www.uni-kassel.de/upress/online/frei/978-3-933146-82-3.volltext.frei.pdf)
* [[I. M. Gel'fand]], M. A. Naimark, _Normed rings with involutions and their representations_, Izv. Akad. Nauk SSSR Ser. Mat., 12:5 (1948), 445–480, Russian orig. [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=3092&volume=12&year=1948&issue=5&fpage=445&what=fullt&option_lang=eng)

[[!redirects normed algebra]]
[[!redirects normed algebras]]


normed division algebra

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# Normed division algebras
* table of contents
{: toc}

## Idea

A normed division algebra is a [[not-necessarily associative algebra]] over the real numbers that is:

1. [[unital]] (there is an element $1$ such that $1a = a = a1$ for all $a$),

1. a [[division algebra]] $\big( \text{i.e.}\, (a \cdot b = 0) \Rightarrow (a = 0 \,\text{ or }\, b = 0) \big)$, and

1. a [[normed algebra]] such that $\|a b\| = \|a\| \|b\|$ for all $a,b$.

It turns out (by [[Hurwitz' theorem]]) that over the [[real numbers]] there are precisely only four finite-dimensional normed division algebras up to [[isomorphism]]: the algebras of

1. [[real numbers]],

1. [[complex numbers]],

1. [[quaternions]],

1. [[octonions]].

In this sense real normed division algebras may be thought of as a natural generalization of the [[real numbers]] and the [[complex numbers]].

Moreover, if one regards the [[real numbers]] as a [[star-algebra]] with trivial [[anti-involution]], then each step in the above sequence is given by applying the [[Cayley-Dickson construction]]. (While the process of applying the [[Cayley-Dickson construction]] continues, next with the [[sedenions]], these and the following are no longer [[division algebras]].)

This classification of real normed division algebras is closely related to various other systems of [[universal exceptionalism|exceptional]] structures in [[mathematics]] and [[physics]]:

* The _[[Hopf invariant one theorem]]_ says that the only [[continuous functions]] between [[spheres]] of the form $S^{2n-1}\to S^n$ whose [[Hopf invariant]] is equal to 1 are the [[Hopf constructions]] on the four real normed division algebras, namely

   1. the [[real Hopf fibration]];

   1. the [[complex Hopf fibration]];

   1. the [[quaternionic Hopf fibration]];

   1. the [[octonionic Hopf fibration]].

* Patterns related to [[Majorana spinors]] in [[spin geometry]] are intimately related to the four normed division algebras, and, induced by this, so is the classification of _[[supersymmetry]]_ in the form of [[super Poincaré Lie algebras]] and [[super Minkowski spacetimes]] (which are built from these real spin representations). For more on this see at _[[supersymmetry and division algebras]]_.

(Moreover, apparently these two items are not unrelated, see [[schreiber:Equivariant cohomology of M2/M5-branes|here]].)

## Definition

A __normed division algebra__ is

* a [[division algebra]]

* that is also a [[Banach algebra]]

* such that ${\|x y\|} = {\|x\|} {\|y\|}$

While the [[norm]] in a [[Banach algebra]] is in general only submultiplicative (${\|x y\|} \leq {\|x\|} {\|y\|}$), the norm in a normed division algebra must obey the stronger condition ${\|x y\|} = {\|x\|} {\|y\|}$.  Accordingly, this norm is considered to be an [[absolute value]]
and often written ${|{-}|}$ instead of ${\|{-}\|}$.  There is also a converse: if the norm on a Banach algebra is multiplicative (including ${\|1\|} = 1$), then it must be a division algebra.  While the usual definition of a 'normed division algebra' does not include the [[complete space|completeness]] condition of a Banach algebra, in fact
([UrbanikWright60](#UrbanikWright60)) the only examples have finite [[dimension]] and are therefore complete, hence Banach algebras.

Any normed division algebras is in particular a [[composition algebra]].

## Properties

### Classification
 {#Classification}

Over the [[complex numbers]],  [[generalized the|the]] only normed division algebra is the algebra of complex numbers themselves.

The [[Hurwitz theorem]] says that over the [[real numbers]] there are, up to [[isomorphism]], exactly four finite-dimensonal normed division algebras :

*  $\mathbb{R}$, the algebra of [[real numbers]],
*  $\mathbb{C}$, the algebra of [[complex numbers]],
*  $\mathbb{H}$, the algebra of [[quaternions]],
*  $\mathbb{O}$, the algebra of [[octonions]].

In fact these are also exactly the real [[alternative algebra|alternative]] division algebras:

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

The only [[division algebras]] over the [[real numbers]] which are also [[alternative algebras]] are the [[real numbers]] themselves, the [[complex numbers]], the [[quaternions]] and the [[octonions]].

=--

([Zorn 30](#Zorn30)).

Each of these is produced from the previous one by the [[Cayley–Dickson construction]]; when applied to $\mathbb{O}$, this construction produces the algebra of [[sedenions]], which do *not* form a division algebra.

The Cayley--Dickson construction applies to an algebra with [[involution]]; by the abstract nonsense of that construction, we can see that the four normed division algebras above have these properties:

*  $\mathbb{R}$ is [[associative algebra|associative]], [[commutative algebra|commutative]], and with trivial involution,
*  $\mathbb{C}$ is associative and commutative but has nontrivial involution,
*  $\mathbb{H}$ is associative but noncommutative and with nontrivial involution,
*  $\mathbb{O}$ is neither associative, commutative, nor with trivial involution.

However, these algebras do all have some useful algebraic properties; in particular, they are all [[alternative algebra|alternative]] (a weak version of associativity).  They are also all [[composition algebra]]s.

A __[[normed field]]__ is a commutative normed division algebra; it follows from the preceding that the only normed fields over $\mathbb{R}$ are $\mathbb{R}$ and $\mathbb{C}$ (e.g. [Tornheim 52](#Tornheim52)).

It is in fact true that all _unital_ normed division algebras over $\mathbb{R}$ are already finite dimensional, by ([Urbanik-Wright 1960](#UrbanikWright60)) (the authors give a reference on a non-unital infinite-dimensional normed division algebra). Hence the [[Hurwitz theorem]] together with [Urbanik-Wright 1960](#UrbanikWright60) says that the above four exhaust all real normed division algebras.

### Automorphisms
 {#Automorphisms}

The [[automorphism groups]] of the real normed division algebras, as [[normed algebras]], are

* $Aut(\mathbb{R}) = 1$, the [[trivial group]]

* $Aut(\mathbb{C}) = \mathbb{Z}/2$ the [[group of order 2]], acting by [[complex conjugation]];

* $Aut(\mathbb{H}) = SO(3)$, the [[special orthogonal group]] acting via its canonical representation  on the 3-dimensional space of imaginary quaternions;

* $Aut(\mathbb{O}) = G_2$, the [[exceptional Lie group]] [[G2]].

Incidentally, there is a sense in which this sequence of groups continues, with the [[infinity-group]] [[G3]] (the [[Dwyer-Wilkerson H-space]]):

| $n=$ | 0 | 1 | 2 | 3 | 4 |
|--|--|--|--|--|--|
| $DI(n)=$ | [[trivial group|1]] |  [[Z/2]] | [[SO(3)]] | [[G2]] | [[G3]] |
|          | = Aut(R) | [= Aut(C)](complex+number#AutomorphismsOfComplexNumbersIsZ2)        | [= Aut(H)](quaternion#AutomorphismsOfQUatrnionsAlgebraIsSO3)       |  [= Aut(O)](octonion#AutomorphismsOfOctonionAlgebraIsG2)  |

### Relation to H-space structures on sphere (Hopf invariant one)

The [[Hopf invariant one theorem]] says that the [[spheres]] carrying [[H-space]] structure are precisely the unit spheres in one of the normed division algebras

<img src="http://ncatlab.org/nlab/files/AdamsHopfInvariantProofFlow.jpg" width="700">

([Adams 60](Hopf+invariant+one#Adams60))

### Magic square

The [[Freudenthal magic square]] is a special square array of [[Lie algebras]]/[[Lie groups]] labeled by pairs of real normed division algebras and including all the [[exceptional Lie groups]] except [[G2]].

## Related concepts

* [[Hopf invariant one]]

* [[Cayley–Dickson construction]]

[[!include exceptional spinors and division algebras -- table]]

see [[division algebra and supersymmetry]]

[[!include normed division algebra Riemannian geometry -- table]]

## References

The classification of real division composition algebras is originally due ([[Hurwitz theorem]]) to

* {#Hurwitz1898} [[Adolf Hurwitz]], _Über die Composition der quadratischen Formen von beliebig vielen Variabeln_, Nachr. Ges. Wiss. Göttingen (1898) 309–316

The alternative classification as real [[alternative algebra|alternative]] division algebras is due to

* {#Zorn30} [[Max Zorn]], _Theorie der alternativen Ringe_, Abhandlungen des Mathematischen Seminars der Universität Hamburg 8 (1930), 123-147

General discussion includes includes

* {#Tornheim52} Leonard Tornheim, _Normed fields over the real and complex fields_, Michigan Math. J. Volume 1, Issue 1 (1952), 61-68. ([Euclid](http://projecteuclid.org/euclid.mmj/1028989727))

* Silvio Aurora, _On normed rings with monotone multiplication_, Pacific J. Math.
Volume 33, Number 1 (1970), 15-20 ([JSTOR](http://projecteuclid.org/euclid.pjm/1102977236))

The result about removing the assumption of finite-dimensionality from unital normed division algebras appears in:

* {#UrbanikWright60} Kazimierz Urbanik and Fred B. Wright, _Absolute-valued algebras_, Proc. Amer. Math. Soc. **11** (1960), 861-866, doi:[10.1090/S0002-9939-1960-0120264-6](https://doi.org/10.1090/S0002-9939-1960-0120264-6)

Exposition with emphasis on the [[octonions]] is in

* [[John Baez]], _[Normed Division Algebras](http://math.ucr.edu/home/baez/octonions/node2.html)_

* [[John Baez]], [This Week's Finds --- Week 59](http://math.ucr.edu/home/baez/week59.html)

Discussion of [[Riemannian geometry]] and [[special holonomy]] modeled on the different normed division algebras is in

* {#Leung02} [[Naichung Conan Leung]], _Riemannian Geometry Over Different Normed Division Algebras_, J. Differential Geom. Volume 61, Number 2 (2002), 289-333. ([euclid](http://projecteuclid.org/euclid.jdg/1090351387))

[[!redirects normed division algebra]]
[[!redirects normed division algebras]]

[[!redirects real normed division algebra]]
[[!redirects real normed division algebras]]


normed division algebra Riemannian geometry -- table

| $\;$[[normed division algebra]]$\;$ | $\;\mathbb{A}\;$ | $\;$Riemannian $\mathbb{A}$-manifolds$\;$ | $\;$special Riemannian $\mathbb{A}$-manifolds$\;$ |
|--|--------------|-----------------------------------|------------------|
| $\;$[[real numbers]]$\;$ | $\;\mathbb{R}\;$ | $\;$[[Riemannian manifold]]$\;$ | $\;$[[orientation|oriented]] [[Riemannian manifold]]$\;$ |
| $\;$[[complex numbers]]$\;$ | $\;\mathbb{C}\;$ | $\;$[[Kähler manifold]]$\;$ | $\;$[[Calabi-Yau manifold]]$\;$ |
| $\;$[[quaternions]]$\;$ | $\;\mathbb{H}\;$ | $\;$[[quaternion-Kähler manifold]]$\;$ | $\;$[[hyperkähler manifold]]$\;$ |
| $\;$[[octonions]]$\;$ | $\;\mathbb{O}\;$ | $\;$[[Spin(7)-manifold]]$\;$ | $\;$[[G2-manifold]]$\;$ |

([Leung 02](special+holonomy#Leung02))


normed field

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# Contents
* table of contents
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## Idea

A _normed field_ $K$ is a [[normed ring]] whose underlying [[ring]] is a [[field]].

If the product preserves the [[norm]] strictly (so that one has a _multiplicative norm_  or [[absolute value]] in that for all $f,g \in K$ we have the [[equality]] ${\vert f \cdot g\vert} = {\vert f\vert} \cdot {\vert g\vert}$ instead of just the [[inequality]] ${\vert f \cdot g\vert} \leq {\vert f\vert} \cdot {\vert g\vert}$) then one speaks of a _[[valued field]]_  (e.g. [Berkovich 09, def. 1.1.1](#Berkovich09)).

If the underlying [[normed group]] is a [[complete topological space]] then one speaks of a _[[complete field|complete normed field]]_.

## Examples

* Every [[field]] carries the trivial norm (which is [[non-archimedean field|non-archimedean]]), whose value is always $1$ (except that the norm of $0$ is $0$) and is complete with respect to this norm.  (In [[constructive mathematics]], either the field must be a [[discrete field]] or the norm must be allowed to take values in the [[lower real numbers]].)

* The field $\mathbb{R}$ of [[real numbers]] and the field $\mathbb{C}$ of [[complex numbers]], with their usual [[absolute value]] as the norm, are complete [[archimedean field|archimedian]] normed fields.

* For each [[prime number]] $p$, the field $\mathbb{Q}_p$ of $p$-[[adic numbers]] is a complete non-archimedean normed field with respect to the [[p-adic valuation]].

* The field $\mathbb{Q}$ of [[rational numbers]], with any of the norms in the two previous entries, is an incomplete normed field whose completion is $\mathbb{R}$ or $\mathbb{Q}_p$.

## Properties

### Relation to algebraic closure
 {#RelationToAlgebraicClosure}

The norm of a [[non-archimedean field]] extends uniquely to its [[algebraic closure]] and the completion of that with respect to this norm is still algebraically closed ([Bosch-Guntzer-Remmert 84, prop. 3.4.1.3](#BoschGuntzerRemmert84), [Berkovich 09, fact 1.1.4](#Berkovich09)).

For example the [[p-adic complex numbers]] $\mathbb{C}_p$ arise this way from the [[p-adic rational numbers]] $\mathbb{Q}_p$.

## Related concepts

* [[normed division algebra]]

[[!include analytic geometry ingredients -- table]]

## References

* {#Tornheim52} Leonard Tornheim, _Normed fields over the real and complex fields_, Michigan Math. J. Volume 1, Issue 1 (1952), 61-68. ([Euclid](http://projecteuclid.org/euclid.mmj/1028989727))

* {#BoschGuntzerRemmert84} S. Bosch, U. Guntzer, and R. Remmert, _Non-Archimedean analysis_, Grundlehren der Mathematischen Wissenschaften, vol. 261, Springer-Verlag, Berlin, 1984.

* {#Berkovich09} [[Vladimir Berkovich]], _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))

[[!redirects normed field]]
[[!redirects normed fields]]

[[!redirects complete normed field]]
[[!redirects complete normed fields]]


normed group

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# Contents
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{: toc}

## Idea

A **normed group** is to a [[group]] what a [[normed vector space]] is to a [[vector space]].  It consists of a group together with a length function (a [[norm]]) and, as for normed vector spaces, gives rise to a [[metric space]].

A [[complete space|complete]] normed group is a _[[complete normed group]]_.

## Definition

### Normed groups

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

A **normed group** is a pair $(G,\rho)$ where $G$ is a [[group]] and $\rho \colon G \to [0,\infty) \subset \mathbb{R}$ is a [[function]], the _norm_, satisfying the following conditions:

1. $\rho(g) = 0$ if and only if $g$ is the identity element in $G$,
2. $\rho(g^{-1}) = \rho(g)$,
3. $\rho(g h) \le \rho(g) + \rho(h)$.

=--

+-- {: .num_defn #homomorphisms}
###### Definitions

There are a few different senses of [[homomorphism]] between normed groups; the first is usually taken as the default but the second fits in better with [[normed rings]] and the obvious notion of [[isomorphism]] as two structures\' being 'the same'.

A __bounded homomorphism__ $(G_1,\rho_1)\to (G_2,\rho_2)$ of normed groups, def. \ref{NormedGroup}, is a group homomorphism $f \colon G_1 \to G_2$
of the underlying groups such that there is $C \in \mathbb{R}_{\geq 0} $
such that for all $g\in G_1$ we have $\rho_2(f(g)) \leq C\cdot\rho_1(g)$.

A __short homomorphism__ $(G_1,\rho_1)\to (G_2,\rho_2)$ of normed groups is a group homomorphism $f \colon G_1 \to G_2$
of the underlying groups such that for all $g\in G_1$ we have $\rho_2(f(g)) \leq \cdot\rho_1(g)$.
=--

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

If $G$ is a [[vector space]] (viewed as an [[abelian group]]) the conditions on $\rho$ in def. \ref{NormedGroup} almost correspond to the [[axioms]] for a [[norm]] in the context of a [[normed vector space]].  The difference is that homogeneity is only assumed for $-1$ instead of for all elements of the coefficient field.

=--

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

A norm on a group in def. \ref{NormedGroup}
defines two [[metrics]]:

$$
d_L(g,h) = \rho(g^{-1} h), \qquad d_R(g,h) = \rho(g h^{-1})
$$

The former is left invariant, the latter right invariant.

=--

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

A normed group is not necessarily a [[topological group]], see ([Bingham-Ostaszweszki](#BinghamOstaszweszki)).

=--

### Normed groupoids

The definition can be extended to [[groupoids]].

+-- {: .num_defn }
###### Definition
A **normed groupoid** is a pair $(G,\rho)$ where $G = (G_1,G_2)$ is a [[groupoid]] and $\rho \colon G_2 \to [0,\infty)$ is a function on the arrows of $G$ satisfying the conditions:

1. $\rho(g) = 0$ if and only if $g$ is an identity arrow,
2. $\rho(g) = \rho(g^{-1})$,
3. if $g h$ exists then $\rho(g h) \le \rho(g) + \rho(h)$.
=--

From a normed groupoid we do not just get a single metric space.  Rather we get one metric space for each object.  For $x \in G_1$ the underlying set of the corresponding metric space is the set of all arrows with source $x$.  The metric is then $d_x(g,h) = \rho(g h^{-1})$.  An arrow from $x$ to $y$ induces an [[isometry]] by right translation.

This reverses: from a [[metric space]], say $X$, we get a normed groupoid by taking the trivial groupoid on $X$.  An arrow in this groupoid is simply a pair $(x,y)$ of elements, whence we define the norm on $G \coloneqq X \times X$ by $\rho(x,y) = d_X(x,y)$.

## Related concepts

* [[bornological group]]

* [[Banach space]]

* [[normed ring]]

[[!include analytic geometry ingredients -- table]]

## References

* {#BinghamOstaszweszki} N. Bingham, A. Ostaszewski, _Normed groups: dichotomy and duality_ (, [pdf](http://www.maths.lse.ac.uk/Personal/adam/DM656NormedGroups.pdf), [pdf](http://www.maths.lse.ac.uk/Personal/adam/BOst12-rev.pdf))

[[!redirects normed group]]
[[!redirects normed groups]]


normed ring

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# Contents
* table of contents
{: toc}

## Idea

A _normed ring_ is a [[ring]] compatibly equipped with a [[norm]] on the underlying [[abelian group]].

If this is suitably [[complete topological space|complete]] with respect to the norm, then a normed ring is called a _[[Banach ring]]_. A normed ring which is a [[field]] is, naturally, called a [[normed field]], and if the norm is multiplicative it is also called a [[valued field]].

The [[Berkovich spectrum]] of a normed ring $R$ is the set of multiplicative [[seminorms]] on $R$ that are bounded by the norm on $R$.

## Definition

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

A _normed commutative ring_ is a [[commutative ring]] $R$ equipped with a [[function]]

$$
  {\vert -\vert} \;\colon\; R \longrightarrow \mathbb{R}_{\geq 0}
$$

to the non-negative [[real numbers]] such that for all $f,g \in R$

1. ${\vert f \vert} = 0$ precisely if $f = 0$;

2. ${\vert f + g \vert} \leq {\vert f \vert}+ {\vert g \vert}$ ([[triangle identity]])

3. ${\vert f \cdot g\vert} \leq {\vert f \vert\cdot {\vert g \vert}}$.

=--

e.g ([Berkovich 09, def. 1.2.1](#Berkovich09))

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

One might also define a normed ring to be a [[commutative monoid]] [[internalization|internal]] to the [[monoidal category]] $NGrp$ of [[normed groups]].  If the morphisms in $NGrp$ are taken to be the [[short map|short]] group homomorphisms and the [[projective cross norm]] is used on the tensor product, then this reproduces the definition above.  If (as is often seen) the morphisms are generalized to [[bounded map|bounded]] group homomrophisms, then this generalizes the third clause in def. \ref{NormedCommutativeRing}
to

* there is $C \in \mathbb{R}_{\gt 0}$ such that for all $f,g \in R$

  $$
    {\vert f \cdot g\vert} \leq C \cdot {\vert f \vert\cdot {\vert g \vert}}
  $$

see e.g. ([Bassat-Kremnitzer 13, remark 6.32](#BassatKremnitzer13))
=--

## Examples

A [[normed field]] is of course in particular a [[normed ring]].

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

For $R$ a normed commutative ring, then for each $n \in \mathbb{N}$ the [[matrix algebra]] $Mat_n(R)$ becomes a normed ring with norm

$$
 {\vert A\vert} \coloneqq max_{1 \leq i,j \leq n}({\vert A_{i,j}\vert})
 \,.
$$

Notice that even if $R$ if the norm on $R$ is multiplicative (is an [[absolute value]]) that on $Mat_n(R)$ is not in general. If $R$ is a [[Banach ring]], then so is $Mat_n(R)$.
=--

(e.g. [Jarden 11](#Jarden11)).

## Related concepts

* [[division ring]]

[[!include analytic geometry ingredients -- table]]

## References

* {#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))

* Naoki Kimura, _A note on normed ring_, Kodai Math. Sem. Rep.
Volume 1, Number 3-4 (1949), 23-24. ([Euclid](http://projecteuclid.org/euclid.kmj/1138833472))

* {#Naimark59} [[Mark Naimark]],  _Normed Rings_, Groningen, Netherlands: P. Noordhoff N. V., 1959.

* {#Jarden11} M. Jarden, _Normed rings_, chapter 2 of  _Algebraic patching_, Springer Monographs in Mathematics, 2011 ([[JardenNormedRings.pdf:file]])

* {#Berkovich09} [[Vladimir Berkovich]], _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))

* {#BassatKremnitzer13} [[Oren Ben-Bassat]], [[Kobi Kremnizer]], section 6.5 of _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))

* ProofWiki, _[Norm (Division ring)](https://proofwiki.org/wiki/Definition:Norm_(Division_Ring))_

For more see the references at _[[Banach ring]]_.

[[!redirects normed ring]]
[[!redirects normed rings]]


Northwestern TFT Conference 2009

This page contains material related to the event

* Northwestern University, Chicago: **Topological Field Theories**

* Workshop: May 18-22, 2009

* Conference: May 25-29, 2009

* Organizers: Kevin Costello, Ezra Getzler, and Paul Goerss

* [website](http://www.math.northwestern.edu/~pgoerss/tftemphasis/)

  * [program](http://www.math.northwestern.edu/~pgoerss/tftemphasis/workshop.pdf)

  * [titles and abstract](http://www.math.northwestern.edu/~pgoerss/tftemphasis/abstracts.pdf)

#Workshop lectures#

In addition to the notes provided by Alex Hoffnung below, there are also notes being posted by Evan Jenkins and Gabriel C. Drummond-Cole:

* Evan Jenkins [talk notes](http://www.math.uchicago.edu/~ejenkins/nwtft.html)

* Gabriel C. Drummond-Cole [talk notes](http://www.math.sunysb.edu/~blafard/notes/)

## David Ben-Zvi on   Topological Field Theory, Loop Spaces, and Representation Theory

* [[David Ben-Zvi]], [[DBZLecture1.ps:file]], talk notes by [[Alex Hoffnung]]

* [[David Ben-Zvi]], [[DBZLecture2.ps:file]], talk notes by [[Alex Hoffnung]]

* [[David Ben-Zvi]], [[DBZLecture3.ps:file]], talk notes by [[Alex Hoffnung]]

* [[David Ben-Zvi]], [[DBZLecture4.ps:file]], talk notes by [[Alex Hoffnung]]

The above instant talk notes have been refined into typeset lecture notes:

* Orit Davidovich, Alex Hoffnung, _David Ben-Zvi, Workshop on TFTs at Northwestern_

  * Lecture 1 ([blog](http://golem.ph.utexas.edu/category/2009/07/benzvis_lectures_on_topologica.html#more), [pdf](http://math.ucr.edu/~alex/Ncafe_postings/DBZ_lecture1.pdf))

### related $n$Lab entries ###

* [[On the Classification of Topological Field Theories]]

  * [[Dijkgraaf-Witten theory]]

* [[geometric infinity-function theory]]

  * [[geometric function theory]]

##Jacob Lurie on Topological Quantum Field Theories ##

* [[Jacob Lurie]], [[LurieLecture1.ps:file]], talk notes by [[Alex Hoffnung]]

* [[Jacob Lurie]], [[LurieLecture2.ps:file]], talk notes by [[Alex Hoffnung]]

* [[Jacob Lurie]], [[LurieLecture3.ps:file]], talk notes by [[Alex Hoffnung]]

* [[Jacob Lurie]], [[LurieLecture4.ps:file]], talk notes by [[Alex Hoffnung]]

### Related $n$Lab entries ###

* [[On the Classification of Topological Field Theories]]

##Bertrand Toën on Secondary K-theory ##

* [[Bertrand Toën]], [[ToenLecture1.ps:file]], talk notes by [[Alex Hoffnung]]

* [[Bertrand Toën]], [[ToenLecture2.ps:file]], talk notes by [[Alex Hoffnung]]

* [[Bertrand Toën]], [[ToenLecture3.ps:file]], talk notes by [[Alex Hoffnung]]

* [[Bertrand Toën]], [[ToenLecture4.ps:file]], talk notes by [[Alex Hoffnung]]

##Orit Davidovich on Modular Tensor Categories ##

* [[Orit Davidovich]], [[Davidovich.ps:file]], talk notes by [[Alex Hoffnung]]

### related $n$Lab entries ###

* [[semisimple category]]

* [[fusion category]]

* [[modular tensor category]]

##John Francis##

* [[John Francis]], [[Francis.ps:file]], talk notes by [[Alex Hoffnung]]

##Christoph Wockel##

* [[Christoph Wockel]], [[Wockel.pdf:file]], talk summary by [[Christoph Wockel]]

##Gabriel Drummond-Cole##

* [[Gabriel Drummond-Cole]]: “Gravity and homotopy BV”

#Conference Talks#

##Julie Bergner##

* [[Julie Bergner]], [[bergner.pdf:file]], talk notes by [[Christoph Wockel]]

##Kevin Costello##

* [[Kevin Costello]], [[costello.pdf:file]], talk notes by [[Christoph Wockel]]

### related $n$Lab entries ###

* [[factorization algebra]]

  * [[vertex operator algebra]]

* [[BV theory]]

## Chris Douglas##

* [[Chris Douglas]], [[douglas.pdf:file]], talk notes by [[Christoph Wockel]]

## Liang Kong ##

* [[Liang Kong]], [[kong.pdf:file]], talk notes by [[Christoph Wockel]]

## Jacob Lurie ##

* [[Jacob Lurie]], [[lurie.pdf:file]], talk notes by [[Christoph Wockel]]

#Further links#

* $n$Café [blog entry](http://golem.ph.utexas.edu/category/2009/05/tft_at_northwestern.html)

* [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]

category: reference


Noson Yanofsky

* [Webpage](http://www.sci.brooklyn.cuny.edu/~noson/)

category:people


Not used

deleted


Notation > history


Note on Formatting > history


Notes on geometric Langlands

This entry is about the collection of notes

* [[Dennis Gaitsgory]], _Notes on geometric Langlands_, [web](http://www.math.harvard.edu/~gaitsgde/GL/)

on the [[categorical geometric Langlands conjecture]], which is formulated as an equivalence of [[stable (infinity,1)-categories]] of

* [[D-modules]] on the [[derived stack]] of [[G-bundles]] on a [[curve]] $X$,

and

* [[quasi-coherent sheaves]] on the [[derived stack]] of $G^\vee$-equivariant [[local systems]] on $X$,

where $X$ is a smooth complete [[curve]] over a [[field]] of [[characteristic zero]], and $G$ is a [[reductive group]] and $G^\vee$ is its [[Langlands dual]].  (More precisely, the latter category should be replaced by some appropriate category of [[ind-coherent sheaves]].)

## See also

* [[D. Gaitsgory]], [[N. Rozenblyum]], _[[A study in derived algebraic geometry]]_.

* [[D. Gaitsgory]] et. al., _Geometric representation theory_, graduate seminar, Fall 2009--Spring 2010, [web](http://www.math.harvard.edu/~gaitsgde/grad_2009/).

## Contents

### Foundational stuff

* [[Dennis Gaitsgory]], _Generalities on DG categories_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/textDG.pdf)

Studies some aspects of the [[symmetric monoidal (infinity,1)-category]] of [[dg-categories]], including [[colimits in an (infinity,1)-category|colimits]], [[limits in an (infinity,1)-category|limits]], and [[dualizable objects]] in it, and [[categories of modules]] over it.

* [[N. Rozenblyum]], _Filtered colimits of categories_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/colimits.pdf).

An explicit description of [[filtered colimits]] in the [[(infinity,1)-category of (infinity,1)-categories]].

* [[Dennis Gaitsgory]], _Functors given by kernels, adjunctions and duality_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/kernels.pdf).

* [[Dennis Gaitsgory]], _Stacks_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/Stackstext.pdf)

A concise review of [[derived stacks]] and [[derived schemes]] in [[derived algebraic geometry]] (the version based on coconnective [[commutative dg-algebras]]).

* [[Dennis Gaitsgory]], _Quasi-coherent sheaves on stacks_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/QCohtext.pdf)

A concise review of [[stable (infinity,1)-categories]] of [[quasi-coherent sheaves]] and [[perfect complexes]] on [[derived stacks]].

* [[Dennis Gaitsgory]], _Sheaves of categories and the notion of 1-affineness_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/shvsofcat.pdf).

* [[Dennis Gaitsgory]], _Ind-coherent sheaves_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/IndCohtext.pdf).

On the [[stable (infinity,1)-category]] of [[ind-coherent sheaves]] on a [[derived stack]], and the [[six operations]] in this setting.

* [[Dennis Gaitsgory]], [[N. Rozenblyum]], _DG Indschemes_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/IndSchemestext.pdf).

Develops the theory of [[ind-schemes]] in [[derived algebraic geometry]].

* [[Dennis Gaitsgory]], [[N. Rozenblyum]], _Crystals and D-modules_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/Crystalstext.pdf).

Studies [[crystals]] and [[D-modules]] in [[derived algebraic geometry]].

* [[V. Drinfeld]], [[Dennis Gaitsgory]], _On some finiteness questions for algebraic stacks_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/Finiteness.pdf).

### ...

rest of the contents to be filled in


Notes on homotopical algebra

This page contains pointers to the text

* [[Zhen Lin Low]]

  _Notes on homotopical algebra_

  ([pdf (15-11-10)](http://zll22.user.srcf.net/writing/homotopical-algebra/2015-11-10-Main.pdf))

collecting fundamental material in [[simplicial homotopy theory]], [[(∞,1)-category theory]] and [[(∞,1)-topos theory]].

category: reference


nothing

+-- {: .rightHandSide}
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### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

According to _[[Science of Logic]]_, the [[unity of opposites|opposite]] of [[being]].

| |  | [[Dasein]] | |  |
|--|--|--|--|--|
| [[becoming]] : | [[nothing]] | $\;\;\;\dashv$ | [[being]] | : [[ceasing]] |

## Related concepts

* [[empty type]], [[empty set]]

* [[Aufhebung]]

* [[vacuum]]

## References

* [[Georg Hegel]], _[[Science of Logic]]_

* Stanford Encyclopedia of Philosophy, _[Nothingness](http://plato.stanford.edu/entries/nothingness/#WhyTheSomRatThaNot)_

[[!redirects nothingness]]

[[!redirects Nichts]]


notions of dimension -- contents

**notion of [[dimension]]**

* [[homotopy dimension]]

* [[cohomology dimension]] ([[virtual cohomological dimension|virtual]])

* [[covering dimension]]

* [[Heyting dimension]]

* [[dimension of a manifold]]

* [[dimension of a cell complex]]

* [[dimension of a separable metric space]]

* small [[inductive dimension]] of a [[regular space]]

* large [[inductive dimension]] of a [[normal space]]


notions of pullback -- contents

**Notions of pullback:**

* [[pullback]], [[fiber product]] ([[limit]] over a [[cospan]])

* [[wide pullback]]

* [[lax pullback]], [[comma object]] ([[lax limit]] over a cospan)

* [[(∞,1)-pullback]], [[homotopy pullback]], ([[(∞,1)-limit]] over a cospan)

* [[base change]], [[context extension]]

* [[pullback bundle]]

* [[contravariant functor]]

  * [[pullback in cohomology]], [[d-invariant]]

  * [[pullback of differential forms]]

  * [[pullback of a distribution]]


notions of subcategory

* **[[subcategory]]**

  * [[full subcategory|full]], [[wide subcategory|wide]], [[dense subcategory|dense]], [[replete subcategory|replete]]

  * [[reflective subcategory|reflective]], [[coreflective subcategory|coreflective]], [[bireflective subcategory|bireflective]]

  * [[abelian subcategory|abelian]], [[Serre subcategory|Serre]] [[topologizing subcategory|topologizing]]

  * [[internal subcategory]]

  * [[dense sub-site]]

* [[sub-2-category]]

  * [[full sub-2-category]]

  * [[locally full sub-2-category]]

  * [[reflective sub-2-category]]

* [[localizing subcategory]]

* [[sub-(∞,1)-category]] ([[sub-(infinity,1)-category - internal formulation|internal formulation]])

  * [[reflective sub-(∞,1)-category]] ([[reflective sub-(∞,1)-category - internal formulation|internal formulation]])

  * [[reflective product-preserving sub-(∞,1)-category]] ([[reflective product-preserving sub-(∞,1)-category - internal formulation|internal formulation]])

* [[thick subcategory]] (of a [[triangulated category]])

[Edit this sidebar](/nlab/edit/notions+of+subcategory)


notions of type

**[[type]]**, [[type theory]]

* [[propositions as types]], [[propositions as some types]]

* [[inductive type]], [[W-type]]

* [[empty type]], [[unit type]]

* [[pointed type]]

**[[dependent type]]**, [[dependent type theory]], [[Martin-Löf dependent type theory]]

* [[dependent sum type]], [[dependent product type]]

**[[homotopy type]]**, [[homotopy type theory]]

* [[identity type]], [[heterogeneous identity type]]

* [[homotopy n-type]], [[connected type]]

* [[higher inductive type]]

* [[geometric homotopy type]], [[cohesive homotopy type]]


nous

[[!redirects Nous]]

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

#Contents#
* table of contents
{:toc}

## Idea

In _([Hegel, Science of Logic, §54](Science+of+Logic#54))_ it reads:

> [[Anaxagoras]] is praised as the man who first declared that Nous, thought, is the principle of the world, that the essence of the world is to be defined as thought. In so doing he laid the foundation for an intellectual view of the universe, the pure form of which must be logic. What we are dealing with in logic is not a thinking about something which exists independently as a base for our thinking and apart from it, nor forms which are supposed to provide mere signs or distinguishing marks of truth; on the contrary, the necessary forms and self-consciousness of thought are the content and the ultimate truth itself.

In  [[Georg Hegel]], _[[Lectures on the History of Philosophy]]_, Vol 1 a similar remark is made about the ideas of [[Heraclitus]], see at _[[logos (in philosophy)]]_.

According to ([Hegel, Phenomenology of the Spirit, Vorrede §55](Phenomenology+of+Spirit#Vorrede55)) this is _[[Dasein]]_ :

> Das [[Dasein]] ist Qualität, sich selbst gleiche Bestimmtheit oder bestimmte Einfachheit, bestimmter Gedanke; dies ist der Verstand des Daseins. Dadurch ist es Nus, als für welchen Anaxagoras zuerst das Wesen erkannte.

> Existence is Quality, self-identical determinateness, or determinate simplicity, determinate thought: this is existence from the point of view of Understanding. On this account it is Nous, as Anaxagoras first thought reality to be.

According to ([Hegel, Philosophy of the Spirit -- The soul](Science+of+Logic#PS311)) this nous is

> Spirit as the abstract soul of nature is simple, sidereal, and terrestrial life. Is is the nous of the ancients, the simple, unconscious thought, which (a) as this general essence is the inner idea and would have its reality in the underlying externality of nature. But since it, as soul, is immediate substance, its existence is the particularisation of its natural being, an immediate and natural determinacy, which has its presupposed reality in the individual earth.

## Related concepts

* [[logos (in philosophy)]]

* [[道德經]]

* [[metaphysics]]

* [[hermeticism]]

## References

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


Novalis

Georg Philipp Friedrich Freiherr von Hardenberg (Novalis) (May 2, 1772 – March 25, 1801), was a German Romantic poet, philosopher, author, scientific polymath, and salt-mine manager. He is distinguished from the other German Romantics in that he is the only one who was academically qualified and professionally trained in the sciences. He undertook a grand philosophico-poetic-scientific encyclopedia project - the _Allgemeine Brouillon_ - and encapsulated this project in narrative form in _The Disciples at Sais_, among other things. As evinced by the mathematical fragments below, mathematics held much importance in his work.

* [Wikipedia entry](http://en.wikipedia.org/wiki/Novalis)
* [Stanford Encyclopedia of Philosophy entry](http://plato.stanford.edu/entries/novalis/)

### Mathematical Fragments

* "In the end, mathematics is only common, simple philosophy, and philoso- phy, is higher mathematics in general. In particular, higher mathematics connects common mathematics with the system of mathematics, while the latter borders on the philosophy of mathemat- ics—or philosophical mathematics, just as systematic science is generally always the precursor and boundary of a higher degree of science—of the philosophical degree.222 (Degrees of scientific character. The highest degree of scientific character would be termed philosophy). The philosophical degree again divides into 3 parts and immediately—passes over into the higher series, or into the higher de- gree of the philosophy of philosophy, and so on. (Just as the man of nature passes over into the common and complex human being, so too pure science into the common and higher. Higher science is the transition to a system, just as the scholar, or complex man is the transition to the systematist)."

* "(Has philosophy originated from the contemplation of mathematics?) Philosophy is universal—or higher mathematics—the animating principle of mathematics—poetical mathematics—Or the substance, if mathematics is the form. Mathematics is merely objective philosophy—formal philosophy—and so- called philosophy—is merely subjective philosophy or mathematics—real philoso- phy. By combining them in a manner analogous to that of the combination of chemistry and mechanics—there arises substantial—synthetic—philosophy—or mathematics, or physics. Contrasted with philosophy, physics is mathematics— while contrasted with mathematics—it is philosophy"

* "All the universal sciences—e.g. physics and mathematics, etc., really resemble philosophy in one respect—they are Proteusses—universal substances—indications etc."

* "The mathematical method is the essence of mathematics. Whoever fully understands this method, is a mathematician. As the scientific method in general it is extremely interesting, and perhaps supplies us with the most accurate model for the classification of knowledge, or for the faculty of experience. Axioms and postulates denote the theoretical (a.) and practical (b.) cognitive faculty as such. Problems denote the desire. Solution and proof, the analytic (ad a.) and synthetic (ad b.) ability. Explanations and corollaries also have their significance. This reveals that our desire for knowledge is the intelligence's desire for life, a play of intellectual forces."

* "Mathematics is genuine science—because it contains created knowledge—the products of its own spiritual activity—and because it methodically inspires. It is art, because it has fashioned inspired procedures into rules—because it teaches one to be a genius—and because it replaces Nature with reason. Higher mathematics is concerned with the spirit of quantities—with their political principle—with the world of quantities"

* "All sciences should become mathematics. Up to now, mathematics has merely been the first and simplest expression or revelation of true scientific spirit."

* "The external is the common. The internal, is the particular./ The inte- gration is much more difficult than the differentiation. In relation to physics and philosophy./ The science that joins and puts both into contact with one another—that instructs in deriving the particular from the common, and the inverse, as well as with the external and internal aspects—this science is the connecting—and higher science. If the first is quantative mathematics, and the second qualitative mathe- matics, then the third is relative mathematics—which appear in four systems of elements and in a single universal system."

* "The highest life is mathematics."

* "Pure mathematics is religion."

* "Pure mathematics is the intuition of the intellect as a universe. Genuine mathematics is the actual element of the magician … In music it appears formally, as revelation - as creative idealism"

### Rondas en Sais

[[Fernando Zalamea]] edited a Spanish language volume which updates Novalis in light of contemporary mathematics, culture, and philosophy:

* _Rondas en Sais. Ensayos sobre matemáticas y cultura contemporánea_ introduce algunos desarrollos profundos en matemáticas modernas (1830-1950: Galois, Riemann, Peirce, Florenski) y contemporáneas (1950-hoy: Grothendieck, Connes, Lawvere, Shelah, Zilber), para luego reflexionar sobre las transformaciones que esos avances han producido y pueden llegar a producir en el ámbito general de la cultura, aquí explorada a través de diversas vertientes (filosofía, literatura, cine, arte). En homenaje a Los Discípulos de Sais de Novalis, y continuando con la simbiosis de filosofía natural y la especulativa presente en su clarividente borrador general, Rondas en Sais reúne ensayos expresamente preparados para esta ocasión por reconocidos especialistas del mundo hispánico en historia y filosofía de las matemáticas. Con ello se registra un estado de la cuestión por primera vez a nivel internacional y se plantean problemáticas a desarrollar en el futuro próximo. El pensamiento matemático avanzado ha explorado con sumo detalle algunas fuerzas directrices tránsitos fronterizos, contaminaciones estructurales, deformaciones conceptuales, fluxiones pláticas, procesos reflexivos, por ejemplo que han permeado la cultura en el último medio siglo, y que han sido rara vez estudiadas desde la riqueza de los trasfondos matemáticos subyacentes. Rondas en Sais pretende cubrir en parte esa ausencia y ayudar a hacer comprender la matemática como un pensamiento dinámico, imprescindible, parte integral de la cultura como un todo.

### References

* Dalia Nassar (ed), _[The Relevance of Romanticism: Essays on German Romantic Philosophy](http://books.google.com/books?id=V6rFAgAAQBAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q=wood&f=false)_ Oxford University Press (2014)

  * [[Paul Redding]], "[Mathematics, Computation, Language and Poetry: The Novalis Paradox](http://paulredding.net/Novalis%20Paradox-final.pdf)", essay 12

  * [[David W. Wood]], "The "Mathematical" Wissenschaftslehre: On a Late Fichtean Reflection of Novalis", essay 14

* [[Fernando Zalamea]], _[Singularidades, Ramificatión, y Continuidad: Un Encuentro entre Riemann, Beethoven, y Novalis](http://www.virtual.unal.edu.co/revistas/bolmat/revista/volumen11/parte1/art2.pdf)_

NDPR review of [[David W. Wood]]'s translation of the _Allgemeine Brouillon_, and [[David W. Wood]]'s associated introductory essay on Novalis & the _Allgemeine Brouillon_ :

* [https://ndpr.nd.edu/news/23728-notes-for-a-romantic-encyclopaedia-das-allgemeine-brouillon/](https://ndpr.nd.edu/news/23728-notes-for-a-romantic-encyclopaedia-das-allgemeine-brouillon/)

* [https://www.sunypress.edu/pdf/61378.pdf](https://www.sunypress.edu/pdf/61378.pdf)

free digital copy of _The Disciples at Sais and other fragments_:

* [https://archive.org/details/disciplesatsais00nova](https://archive.org/details/disciplesatsais00nova)

general introduction to German Romanticism:

* interview with Philosopher Dalia Nassar: [http://www.3ammagazine.com/3am/the-jena-romantics-et-al/](http://www.3ammagazine.com/3am/the-jena-romantics-et-al/)

category: people


Novikov conjecture

## Idea

A special case of the [[Baum-Connes conjecture]].

## References

Named after _[[Sergei Novikov]]_

* Wikipedia, _[Novikov conjecture](http://en.wikipedia.org/wiki/Novikov_conjecture)_


Novikov field

# Novikov fields

* table of contents
{: toc}

## Idea

A *Novikov field* (or Novikov ring) is a field of generalized [[formal power series]] that allows non-integer exponents.  In contrast to a [[Hahn series]] field, in a Novikov field the series are only $\omega$-long rather than transfinitely long.  This requires a "left-finiteness" condition on the exponents to ensure closure under multiplication.

## Definition

There are many variant definitions of Novikov fields in the literature, see e.g. [this question](https://mathoverflow.net/q/13203).  Here we give a reasonable-seeming general and abstract definition.

Let $k$ be a [[commutative ring]] and $G$ a [[linear order|linearly]] [[ordered abelian group]].

+--{: .un_defn}
###### Definition
The **Novikov ring** of $k$ with value group $G$ is the ring of functions $f:G\to k$ such that for any $y\in G$ the set $\{ x\in G \mid x \lt y \wedge f(x)\neq 0\}$ is finite.  Such functions are added pointwise, and multiplied by the formula
$$ (f\cdot g)(z) = \sum_{x+y=z} f(x) \cdot g(y). $$
=--

Left-finiteness of the support of $f$ and $g$ implies that the above sum is finite.  Specifically, if $g\neq 0$ then since $G$ is totally ordered, there is a least $y_0$ such that $g(y_0)\neq 0$.  Then the set $\{ x\mid x \le z-y_0 \wedge f(x)\neq 0\}$ is finite, and hence so is its subset $\{ x \mid \exists y. x+y = z \wedge f(x)\neq 0 \wedge g(y)\neq 0 \}$.  Note that this depends on the fact that $G$ is totally ordered and a group; a partially ordered monoid would not suffice.

Notationally, we write such a function as $\sum_{x\in G} f(x)\, t^x$ for $t$ a formal variable.  If $k$ is a [[field]], then so is the Novikov ring.

## Examples

* When $G=\mathbb{R}$, the Novikov field is sometimes called the "universal Novikov field".

* When $G=\mathbb{Q}$ and $k =\mathbb{R}$, the Novikov field is known as the **Levi-Civita field**.

## Characterizations

The Novikov field embeds into the [[Hahn series]] field $k[[t^G]]$.  It can (probably) be characterized therein as

* The set of Hahn series with order type $\omega$ that converge to themselves in the valuation topology.

* The closure of the field $k(t^G)$ of [[generalized rational functions]] inside the Hahn series field.

It can also (probably) be characterized abstractly as

* The Cauchy completion of $k(t^G)$ in its valuation [[uniform space|uniformity]].

* The completion of $k(t^G)$ as a valued field, i.e. the unique (up to isomorphism) dense valued field extension without proper dense valued field extension.

## Applications

* The universal Novikov field of $\mathbb{R}$ is a natural context in which to relate [[magnitude homology]] of finite [[metric spaces]] to their [[magnitude]].  (Hahn series also suffice, but all the action actually takes place in the Novikov field.)

* In the [[Fukaya category]], the chain complexes defining Hom's between objects are defined over a Novikov ring.

## Related pages

Other rings of generalized power series include:

* [[Puiseux series]]
* [[Hahn series]]
* [[Ribenboim power series]]

Hahn series are a special kind of Ribenboim power series, but Puiseux and Novikov series are not.  However, they are all instances of the linearization of a [[finiteness space]].

## References

* [Wikipedia: Novikov ring](https://en.wikipedia.org/wiki/Novikov_ring)
* [Wikipedia: quantum cohomology](https://en.wikipedia.org/wiki/Quantum_cohomology)
* [MathOverflow: definitions of Novikov field](https://mathoverflow.net/q/13203)

[[!redirects Novikov field]]
[[!redirects Novikov fields]]
[[!redirects Novikov ring]]
[[!redirects Novikov rings]]
[[!redirects Novikov series]]
[[!redirects universal Novikov field]]
[[!redirects universal Novikov fields]]
[[!redirects universal Novikov ring]]
[[!redirects universal Novikov rings]]
[[!redirects Levi-Civita field]]


nowhere dense set

# Nowhere dense sets
* table of contents
{: toc}

## Definition

A [[subspace|subset]] $A\subseteq X$ of a [[topological space]] $X$ is called **nowhere dense** (alias **rare**) if the [[interior]] of its [[closure]] is [[empty subspace|empty]]:
$$ Int(Cl(A)) = \emptyset .$$

## Related pages

* [[dense]]

* [[dense subspace]]

[[!redirects rare]]
[[!redirects rare set]]
[[!redirects rare sets]]
[[!redirects rare subset]]
[[!redirects rare subsets]]
[[!redirects rare subspace]]
[[!redirects rare subspaces]]
[[!redirects nowhere dense]]
[[!redirects nowhere dense set]]
[[!redirects nowhere dense sets]]
[[!redirects nowhere dense subset]]
[[!redirects nowhere dense subsets]]
[[!redirects nowhere dense subspace]]
[[!redirects nowhere dense subspaces]]


Noé Bárcenas Torres

* [personal page](http://www.matmor.unam.mx/~barcenas/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=141461)

* [Google Scholar page](https://scholar.google.de/citations?user=VneBbDkAAAAJ&hl=de)

## Selected writings

On 3-[[twisted equivariant K-theory]] via the [[universal equivariant PU(H)-bundle|universal equivariant $PU(\mathcal{H})$-bundle]]:

* {#BEJU14} [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Michael Joachim]], [[Bernardo Uribe]], _Universal twist in Equivariant K-theory for proper and discrete actions_, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2014) ([arXiv:1202.1880](https://arxiv.org/abs/1202.1880), [doi:10.1112/plms/pdt061](https://doi.org/10.1112/plms/pdt061))

* [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Bernardo Uribe]], [[Mario Velasquez]], Section 3.2 of: _Segal's spectral sequence in twisted equivariant K-theory for proper and discrete actions_, Proceedings of the Edinburgh Mathematical Society **61** 1 (2018) ([arXiv:1307.1003](https://arxiv.org/abs/1307.1003), [doi:10.1017/S0013091517000281](https://doi.org/10.1017/S0013091517000281))

category: people

[[!redirects Noé Bárcenas]]
[[!redirects Noe Barcenas]]


NP

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

#Contents#
* table of contents
{:toc}

## Idea

The [[complexity class]] of "Nondeterministic Polynomial time"-[[algorithms]].

## Related concepts

* [[nondeterministic computation]]

## References

* {#Sipser2012} [[Michael Sipser]], §7.3 in: *Introduction to the Theory Of Computation*, 3rd ed: Cengage Learning (2012) [ISBN:978-1-133-18779-0,[pdf](https://www.mog.dog/files/SP2019/Sipser_Introduction.to.the.Theory.of.Computation.3E.pdf), [pdf](https://fuuu.be/polytech/INFOF408/Introduction-To-The-Theory-Of-Computation-Michael-Sipser.pdf)]

See also:

* Wikipedia, NP (complexity)

[[!redirects nondeterministic polynomial time]]


nPOV

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

[Wikipedia](http://en.wikipedia.org/wiki/Main_Page) enforces its entries to adopt an [NPOV](http://en.wikipedia.org/wiki/NPOV) -- a _neutral point of view_.  This is appropriate for an encyclopedia.

However, [[About|the nLab is not Wikipedia]], nor is it an encyclopedia, although it does aspire to provide a useful reference in many areas (among its other purposes).  In particular, the $n$Lab *has* a particular point of view, which we may call the **$n$POV**, the _[[higher algebra|higher algebraic]]_, [[homotopy theory|homotopical]], or _[[higher category theory|n-]] [[category theory|categorical]] point of view_.

To some extent the $n$POV is just the observation that [[higher algebra]], [[homotopy theory]], [[type theory]], [[category theory]] and [[higher category theory]], have a plethora of useful applications.

# $n$POV
* table of contents
{: toc}

## Idea

Around the [[HomePage|nLab]] it is believed that [[higher algebra]], [[homotopy theory]], [[type theory]], [[category theory]] and [[higher category theory]] provide a point of view on [[Mathematics]], [[Physics]] and [[Philosophy]] which is a valuable unifying point of view for the understanding of the concepts involved.

So at the $n$Lab, we don't care so much about being neutral.  Although we don't want to offend people unnecessarily, we are also not ashamed about writing from this particular point of view.  There are certainly other valid points of view on mathematics, but describing them and being neutral towards them is not the purpose of the $n$Lab.  Rather, the $n$Lab starts from the premise that higher algebra, homotopy theory, type theory, category theory and higher category theory are a true and useful point of view, and one of its aims is to expose this point of view generally and in a multitude of examples, and thereby accumulate evidence for it.

If you feel skeptical about the $n$-point of view, you may want to ignore the $n$Lab. Or you may want to take its content as a contribution to a discussion on what is behind the claim that higher algebra, homotopy theory, type theory, and/or category theory is the right language to describe the world, or at least the world of mathematical ideas.

### Category theory

As recalled in parts on the page on [[category theory]], category theorists have early on, beginning in the 1960s, proclaimed the advantages of category theory over other points of view. It has been observed that this claim, or at least the way it has been put forward, has contributed to a certain alienation of category theory in parts of the mathematical community. That may be true and is understandable. But since a claim is not false _just_ because it is put forward with possibly unpleasant boldness, all evidence for the claim deserves to be collected and exposed. We hope the $n$Lab to play a role in this effort.

In particular, there have been dramatic developments since the 1960s. Back then promoting category theory may have been as visionary as the invention of [[complex numbers]] was in the 16th century. But just as the early rejections of the complex numbers appear strangely out of place from today's perspective, where their ubiquity proves their reality to the point that it is hard to imagine how life must have been before their conception, so developments of category theory and its applications in the last years have in many areas brought it to the point that rejecting its prevalence amounts (we believe) to rejecting the obvious and ubiquitous. But also mathematics as a whole has drastically grown since then, and while category theory has become an entirely obvious ingredient in areas such as [[homotopy theory]], [[homological algebra]], [[algebraic geometry]] and even fields like [[topological quantum field theory]], its similar role to be played in many other areas has often not found wide recognition yet. But this is gradually changing.

### Higher algebra/Homotopy theory/Type theory

From the perspective of [[higher algebra]], [[homotopy theory]], and [[type theory]], the fundamental objects of mathematics are [[homotopy types]] or [[infinity-groupoid|$\infty$-groupoids]], and the $n$-truncated versions of them, the [[n-groupoid|$n$-groupoids]], due to their differing treatment of [[equality]]/[[equivalence]] of such objects. This perspective is somewhat different from the perspective of category theory, as category theory takes $n$-categories to be more fundamental than $n$-groupoids, while these branches of mathematics view $n$-categories merely as $n$-groupoids with additional higher structure on top of the [[core]] $n$-groupoid. In particular, $n$-categories are merely specific kinds of [[directed (n,r)-graph|directed $(n,r)$-graphs]].

On the other hand, from this perspective, if one were to take the directed structure to be more fundamental as one does in category theory, then the fundamental structures would end up being $n$-[[n-poset|posets]] rather than $n$-[[n-category|categories]]; in particular, taking [[preorder]] to be more fundamental than [[equivalence relations]]/[[equality]].

## The role of the $n$POV {#RoleOfnPOV}

### Category theory

Practitioners of category theory have often attempted to express the striking power of category theory (or general conceptual methods), sometimes through aphorism, sometimes through metaphor. Early on, [[Peter Freyd]] wrote

> Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial.

This can be taken to mean that one thing category theory does is help make the softer bits seem utterly natural and obvious, so as to quickly get to the heart of the matter, isolating the hard nuggets, which one may then attack with abandon. This is an invaluable service category theory performs for mathematics; therefore, category theory is plain good pragmatics.

However, it is also possible to take it a step beyond the pragmatic attitude, and see category theory (and now higher category theory) as exemplifying a style for doing even hard mathematics, as in the style for which Grothendieck is renowned. Paraphrasing from Colin McLarty's [excellent essay](http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Mathbiographies/mclarty1.pdf), let us regard the aforementioned pragmatic attitude as leading up to the _hammer-and-chisel principle_: if you think of a theorem to be proved as a nut to be opened, so as to reach "the nourishing flesh protected by the shell" (Grothendieck), then one thing to do is "put the cutting edge of the chisel against the shell and strike hard. If needed, begin again at many different points until the shell cracks -- and you are satisfied." Grothendieck points to Serre as a master of this technique. He then says:

> I can illustrate the second approach with the same image of a nut to be opened. The first analogy which came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months -- when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

> A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the resistant substance. (Translated from the French by McLarty)

This arresting metaphor of "la mer qui monte" ("[[the rising sea]]"), which over time changes the very form of the resistant substance, is very much in the style of Grothendieck himself. McLarty quotes Deligne as saying that a typical Grothendieck proof consists of a long series of trivial steps where "nothing seems to happen, and yet at the end a highly non-trivial theorem is there."

## Examples

For an (incomplete) list of examples of topics for which the $n$POV has proven to be a useful perspective, see

* [[applications of (higher) category theory]]

category: meta

[[!redirects n-category point of view]]
[[!redirects n-categorial point of view]]
[[!redirects n-categorical point of view]]
[[!redirects n-category-theoretic point of view]]
[[!redirects n-point of view]]


NQ-supermanifold

+-- {: .rightHandSide}
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###Context###
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Another term for _[[dg-manifold]]_ .

## Definition

+-- {: .un_defn}
###### Definition (NQP-supermanifold)

An **N-[supermanifold]** is a [[supermanifold]] equipped with a lift of the $\mathbb{Z}_2$-grading to a $\mathbb{N}$-grading through the standard homomorphism $even/odd : \mathbb{N} \to \mathbb{Z}_2$.

A **Q-[supermanifold]** is a supermanifold equipped with an odd-graded [[vector field]] $Q$ (i.e. an odd-graded [[derivation]] of the algebra of functions) which is **homological**, i.e. the super Lie bracket with itself vanishes: $[Q,Q] = 0$.

A **P-[supermanifold]** is a supermanifold equipped with a graded symplectic structure.

=--

## Remarks

* It is an old observation by [[Maxim Kontsevich]], amplified by [[Pavol Severa]] (ref...) that NQ-supermanifolds are precisely those supermanifolds which are equipped with an action of $End(\mathbb{R}^{0|1})$, the endomorphism monoid of the [[odd line]].

* NQ-supermanifolds are an equivalent way of thinking of [[∞-Lie algebroid]]s. See the list of references there.

* NQP-supermanifolds are hence _[[symplectic Lie n-algebroids]]_

## Related concepts

* [[dg-manifold]], [[derived smooth manifold]]

* For the analogous phenomenon relating [[super Lie algebras]] to [[dg-Lie algebras]] see [there](super+Lie+algebra#RelationToDGLieAlgebras).

## References

The "Q-manifold"-terminology is due to

* M. Alexandrov, [[Maxim Kontsevich|M. Kontsevich]], [[Albert Schwarz|A. Schwarz]], O. Zaboronsky, _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997 ([arXiv:hep-th/9502010](http://arxiv.org/abs/hep-th/9502010))

[[!redirects NQ-supermanifolds]]
[[!redirects N-supermanifold]]
[[!redirects N-supermanifolds]]
[[!redirects Q-supermanifold]]
[[!redirects Q-supermanifolds]]
[[!redirects P-supermanifold]]
[[!redirects P-supermanifolds]]


NS-brane

+-- {: .rightHandSide}
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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]] the _NS-[[branes]]_ are

* the [[string]] itself

* the [[NS5-brane]].

These are also called the **F1-brane** and the **F5-brane**, respectively, for _fundamental brane_ .

To be distinguished from the [[D-branes]] in [[type II string theory]]  and from the [[M-brane]]s in [[11-dimensional supergravity]].

## Properties

### Charges and background fields

The fundamental [[string]] carries [[electric charge]] under the [[background gauge field]] called the [[Kalb-Ramond field]]/[[B-field]].

The [[NS5-brane]] carries [[magnetic charge]] under this field.

Notice how this relates the dimensions. The [[B-field]] is a [[circle n-bundle with connection|circle 2-bundle]] with connection hence has [[higher parallel transport]] over 2-dimensional surfaces: the $(1+1)$-dimensional [[worldsheet]] of the [[string]]. Its [[electric-magnetic duality|EM dual]] in the ambient 10-dimensional [[supergravity]] [[spacetime]] is a [[circle n-bundle with connection|circle 6-bundle with connection]] with [[higher parallel transport]] over the $5+1$-dimensional [[worldvolume]] of the NS5-brane.

## Related concepts

[[!include table of branes]]

## References

* [[Clifford Johnson]], _The NS-brane_ talk notes ([web](http://arxiv.org/html/hep-th/9802001v1/page3.html))

[[!redirects NS-branes]]


NS5-brane

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[string theory]] the _NS5-brane_ is a certain extended physical objects -- a [[brane]] -- that appears in/is predicted by the theory.

There are different incarnations of this object:

For instance the [[effective QFT|effective background QFT]] of the [[type II string theory|type II string]] -- [[type II supergravity]] -- admits solutions to its generalized [[Einstein equations]] which describe higher dimensional analogs of charged [[black holes]] in ordinary gravity. Among them is a 5+1-dimensional "black brane" which is [[magnetic charge|magnetically charged]] under the [[Kalb-Ramond field]]. Since the KR field and the field of [[gravity]] constituting this solution of type II supergravity have as quanta the [[worldsheet]] excitations of the [[spinning string]] [[sigma-model]] that sit in what is called the [[Neveu-Schwarz sector]] one calls this the **NS5-brane**.

This is to distinguish it from the [[D-brane|D5-brane]] which is instead charged under the [[RR-field]] whose quanta come from the [[Ramond-Ramond sector]] of the [[superstring]].

There are other incarnations of the NS 5-brane:

by the general logic of [[Kalb-Ramond field|higher electromagnetism]] the (1+1)-dimensional string has under [[electric-magnetic duality]] a _magnetic dual_ . By dimension counting this is a 5-brane. If we think of the string this way as the structure that supports the [[sigma-model]] that defines perturbative [[string theory]],  we also call it the _F1-brane_ (the _fundamental_ 1-brane). In this sense the the corresponding magnetic dual is the _F5-brane_ -- the _fundamental_ fivebrane.

One can understand the NS5-"black brane" solution to [[type II supergravity]] as being the solitonic incarnation of the fundamental 5-brane in much the same way as an ordinary [[black hole]] in ordinary [[gravity]] is a solitonic incarnation of the [[fundamental particle]]: as the particle, the black hole it is characterized just by [[mass]], [[charge]] and [[angular momentum]].

Similarly, the "black" NS5-brane is characterizes by [[mass]], [[B-field]] charge and [[angular momentum]].

## Properties

### As a black brane

[[!include black branes in supergravity -- table]]

### Relation to little string theory

By the [[brane scan]], on the [[worldvolume]] of an NS5-brane propagates a [[superstring]]. This is called the _[[little string]]_, see there for more.

### D-branes ending on NS5-branes
 {#DBranesEndingOnNS5Branes}

The [[black brane|black]] [[D-branes]] may end on [[black brane|black]] NS5-branes ([Callan-Harvey-Strominger 91, sections IV.C and V.B](#CallanHarveyStrominger91), [Tseytlin 96](#Tseytlin96), [Argurio-Englert-Houart 97](#ArgurioEnglertHouart97), [Brodie-Hanany 97](#BrodieHanany97), [EGKRS 00](#EGKRS00)).

{#OneWayToSeeThis} One way to see this is to analyze the [[twisted de Rham cohomology|twisted]] [[Bianchi identity]] $\mathrm{d} F_{8-p} \,=\, H_3 \wedge F_{8-(p+2)}$ for the [[RR-field]] [[flux densities]] sourced by [[D-brane|D$p$-branes]], in the presence of the [[B-field]]-[[flux density]] $H_3$ which is sourced by the NS5-brane -- see at *[[Hanany-Witten effect]]* for this analysis.

Another way to see this is to assume that [[S-duality]] and [[T-duality]] work as expected, since then the B/NS5 intersections appear as "[[duality in string theory|duals]]" of the intersections of the [[fundamental string]] with D-branes which exist by definition of D-branes:

$$
  \array{
    \text{F1 on D5}
    &\overset{\text{S}}{\leftrightarrow}&
    \text{D1 on NS5}
    \\
    && \updownarrow\mathrlap{T}
    \\
    &&
    \text{D2 on NS5}
    \\
    && \updownarrow\mathrlap{T}
    \\
    &&
    \text{D3 on NS5}
    &\overset{\text{S}}{\leftrightarrow}&
    \text{D3 on D5}
    \\
    && \updownarrow\mathrlap{T}
    \\
    &&
    \vdots
  }
$$

This leads to what is called [[geometric engineering of quantum field theory]] on the [[worldvolume]] of these branes (following [Hanany-Witten 1997](#HananyWitten97), review includes [Fazzi 17](#Fazzi17)).

#### Open D$p$-brane theories on the NS5

Combining the above two items, the corresponding [[worldvolume]] theories of the NS5-branes (see also at [[little string theory]]) in the presence of light D$p$-branes ending on/inside them are also referred to as "open D$p$-brane theories" or "OD$p$ theories" ([Gopakumar-Minwalla-Seiberg-Strominger 00, section 6](#GopakumarMinwallaSeibergStrominger00), [Harmark 00, section 4.2](#Harmark00), [Lu 06, section 6](#Lu06))

#### D4-branes ending on NS5-branes
 {#D4EndingOnNS5}

See [Witten 1997](#Witten97).

\begin{imagefromfile}
    "file_name": "NS5-D4.jpg",
    "width": 300,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [EGKRS 00](#EGKRS00)"
\end{imagefromfile}

The [[duality between M-theory and type IIA string theory|M-theory lift]] is a single [[M5-brane]] branching along a [[Seiberg-Witten curve]] (see [there](Seiberg-Witten+theory#ReferencesSWCurveAsM5Worldvolume)).
In the singular limit this is the
[[intersecting brane|intersection]] locus of two [[M5-branes]] on an "[[M3-brane]]" (see there).

#### D6-branes ending on NS5-branes
  {#D6BranesEndingOnNS5Branes}

Consider a [[black brane|black]] NS5-brane with [[near horizon geometry]] $  \underset{\sim AdS_7}{\underbrace{ \mathbb{R}^{5,1} \times \mathbb{R}_{\phi} }} \times S^3$ ([EGKRS 00, p. 8](#EGKRS00)):

The [[3-sphere]] factor $S^3$ is the [[unit sphere]] around the black NS5-brane [[worldvolume]] $\mathbb{R}^{5,1}$, and $\mathbb{R}_{\phi}$ parameterizes the radial distance from it.

Placing a [[D6-brane]] at one point of the $S^3$-factor ([EGKRS 00, p. 20](#EGKRS00))

means to take its [[worldvolume]] to be the factor $\mathbb{R}^{5,1} \times \mathbb{R}_\phi$, hence extending to one side of the NS5-brane ([EGKRS 00, p. 7](#EGKRS00)):

Placing another [[D6-brane]] at the corresponding antipodal point means to have it extend also to the other side ([EGKRS 00, p. 5](#EGKRS00)):

or else to have embedded the black NS5-brane into a single [[D6-brane]].

Special properties [[D6-branes]] ending on NS5-branes were highlighted in ([Brodie-Hanany 97, section 2.4](#BrodieHanany97)): the [[worldvolume]] theory becomes [[chiral fermion|chiral]].

The [[M-theory]]-lift of this situation should be the [[4-sphere|4-spherical]] [[orbifold]] [[near horizon geometry]] of an [[M5-brane]] ([MFF 12, section 8.3](#MFF12))

$$
  \underset{\sim AdS_7}{\underbrace{\mathbb{R}^{5,1} \times \mathbb{R}_\phi}} \times S^4/G_{ADE}
$$

where $G_{ADE} \subset SU(2)$ is a [[finite group|finite]] [[subgroup]] of [[special unitary group|SU(2)]] (hence in the [[ADE classification]]), [[action|acting]] via the identification $S^4 \simeq S(\mathbb{H} \oplus \mathbb{R})$ (see at _[[4-sphere]]_ the section _[SU(2)-action](4-sphere#QuaternionAction)_).
For non-trivial $G_{ADE}$, this [[action]] has precisely two [[fixed points]] $S^0 \hookrightarrow S^4 \to S^3$. Hence $\mathbb{R}^{5,1} \times \mathbb{R}_\phi \times S^0$ must be the [[worldvolume]] of two [[KK-monopoles]] of [[11d supergravity]], which is the M-theory lift of the two [[D6-branes]]. While the M-theory lift of the NS5-brane is the [[M5-brane]] with [[worldvolume]] $\mathbb{R}^{5,1}$.
See also [Fazzi 17, p. 38](#Fazzi17):

> The NS5-D6 Hanany–Witten setup engineering six-dimensional $(1,0)$ theories is equivalent to M5-branes at an $A_k$ singularity in eleven dimensions.

Next, this construction may be repeated, having the [[D6-branes]] end on different NS5-branes, hence "suspended between NS5-branes" (graphics from [Fazzi 17, p. 33](#Fazzi17)):

#### D6 branes intersecting D8-branes over NS5-branes
 {#D6BranesIntersectingD8BranesOverNS5Branes}

...and on the other end, the D6-branes may [[intersecting branes|end on]] [[D8-branes]] over an NS5-brane ([Janssen-Meessen-Ortin 99, Section 3](#JanssenMeessenOrtin99)) yielding [[D6-D8 brane intersections]] (for more detials see at [[Hanany-Witten effect]], [here](Hanany-Witten+effect#D6StretchingBetweenNS5AndD8)):

> graphics grabbed from [Gaiotto-Tomasiello 14](#GaiottoTomasiello14)

Subject to the [[s-rule]]:

> graphics grabbed from [Fazzi 17](#Fazzi17)

> graphics grabbed from [GKSTY 02](#GKSTY02)

[[fuzzy funnel]]$\,$[[noncommutative geometry]]:

> graphics grabbed from [Fazzi 17, Fig. 3.14](#Fazzi17), taken in turn from [Gaiotto-Tomassiello 14, Figure 5](#GaiottoTomassiello14)

Generally [[Dp-D(p+2)-brane intersections]] [[geometric engineering of QFT|geometrically engineer]] [[Yang-Mills monopoles]] in the [[worldvolume]] of the higher dimensional D$(p+2)$-branes.

Specifically for $p = 6$, i.e. for [[D6-D8-brane intersections]], this fits with the [[Witten-Sakai-Sugimoto model]] [[geometric engineering of QFT|geometrically engineering]] [[quantum chromodynamics]], and then gives a [[geometric engineering of QFT|geometric engineering]] of the [[Yang-Mills monopoles]] in actual [[QCD]] ([HLPY 08, p. 16](D6-D8-brane+intersection#HLPY08)).

Here we are showing

1. the [[color charge|color]] [[D4-branes]];

1. the [[flavour (particle physics)|flavor]] [[D8-branes]];

   with

   1. the [[5d Chern-Simons theory]] on their [[worldvolume]]

   1. the corresponding [[4d WZW model]] on the boundary

   both exhibiting the [[meson fields]]

1. the [[baryon]] [[D4-branes]]

   (see at _[WSS -- Baryons](Witten-Sakai-Sugimoto+model#Baryons)_)

1. the [[Yang-Mills monopole]] [[D6-branes]]

   (see at _[[D6-D8-brane bound state]]_)

1. the [[NS5-branes]].

> graphics grabbed from [Sati-Schreiber 20]()

#### NS5 half-branes
 {#NSHalfBranes}

By the discussion [above](NS5-brane#D6BranesEndingOnNS5Branes)_, a [[black brane|black]] [[D6-brane]] may end on a [[black brane|black]] [[NS5-brane]], and in fact a priori each [[black brane|brane]] [[NS5-brane]] has to be the junction of two [[black brane|black]] [[D6-branes]].

> from [GKSTY 02](#GKSTY02)

If in addition the [[black brane|black]] [[NS5-brane]] sits at an [[O8-plane]], hence at the [[orientifold]] [[fixed point]]-locus, then in the ordinary $\mathbb{Z}/2$-[[quotient]] it appears as a "[[half-brane]]" with only one copy of [[D6-branes]] ending on it:

> from [GKSTY 02](#GKSTY02)

(In [Hanany-Zaffaroni 99](#HananyZaffaroni99) this is interpreted in terms of the [['t Hooft-Polyakov monopole]].)

{#LiftToMTheortyOfNS5Halfbrane} The lift to [[M-theory]] of this situation is an [[M5-brane]] [[brane intersection|intersecting]] an [[M9-brane]] (see at _[[MO5-plane]]_ and at _[[heterotic M-theory on ADE-orbifolds]]_):

> from [GKSTY 02](#GKSTY02)

Alternatively the [[O8-plane]] may [[brane intersection|intersect]] the [[black brane|black]] [[D6-branes]] away from the [[black brane|black]] [[NS5-brane]]:

> from [HKLY 15](#HKLY15)

In general, some of the NS5 sit away from the [[O8-plane]], while some sit on top of it:

> from [Hanany-Zaffaroni 98](#HananyZaffaroni98)

> graphics grabbed from [Apruzzi-Fazzi 17, p. 18](#ApruzziFazzi17)

The lift to [[M-theory]] under [[duality between M-theory and heterotic string theory]] is the [[half M5-brane]].

See also at _[[intersecting D-brane models]]_ the section _[Intersection of D6s with O8s](intersecting+D-brane+model#IntersectionOfD6WithO8)_.

#### Webs

corresponding [[brane webs]]:

> graphics grabbed from (Kimura 16 [pdf](http://www2.yukawa.kyoto-u.ac.jp/~qft.web/2016/slides/kimura.pdf))

### NS5/D4/D2 bound states
 {#NS5D4D2BoundStates}

[[bound states]] of NS5-branes with [[D4-branes]] and [[D2-branes]]:

[Mitra-Roy 00, section 4](#MitraRoy00), [Mukhi-Suryanarayana 00](#MukhiSuryanarayana00), [Jia-Lu-Roy 17, p. 12, Table 1](#JiaLuRoy17), also [Mitra-Roy 01](#MitraRoy01), [Alishahiha-Oz 00](#AlishahihaOz00).

> graphics grabbed from [Mukhi-Suryanarayana 00](#MukhiSuryanarayana00)

Notice that this includes configurations with the D4-branes and D2-branes contained strictly inside the NS5: [Mitra-Roy 00, section 4](#MitraRoy00)

Review includes [Camino 02, section 4.5](#Camino02)

see also [Petri 18](#Petri18)

For the lift to [[M-theory]] see at _[M2-M5 brane bound state](M2-brane#M2M5BoundStates)_

### Relation to Khovanov homology

[[Khovanov homology]] has long been expected to appear as the [[observables]] in a 4-[[dimension]]al [[TQFT]] in higher analogy of how the [[Jones polynomial]] arises as an observable in 3-dimensional [[Chern-Simons theory]]. For instance for $\Sigma : K \to K'$ a cobordism between two [[knots]] there is a natural [[morphism]]

$$
  \Phi_\Sigma : \mathcal{K}(K) \to \mathcal{K}(K')
$$

between the Khovanov homologies associated to the two knots.

In ([Witten11](#Witten11)) it is argued, following indications in ([GukovSchwarzVafa](#GukovSchwarzVafa))
that this 4d TQFT is related to the [[worldvolume]] theory of the _image_ in [[type IIA string theory|type IIA]] of [[D3-branes]] ending on NS5-branes in [[type IIB string theory|type IIB]] after one [[S-duality]] and one [[T-duality]] operation:

$$
  (D3 - NS4)
    \stackrel{S}{\mapsto}
  (D3 - D5)
    \stackrel{T}{\mapsto}
  (D4-D6)
  \,.
$$

Earlier indication for this had come from the observation that [[Chern-Simons theory]] is the [[effective QFT|effective background theory]] for the [[A-model]] 2d [[TCFT]] (see [TCFT – Worldsheet and effective background theories](http://ncatlab.org/nlab/show/TCFT#ActionFunctionals) for details).

Notice that after the above [[T-duality]] operation the $(D4-D6)$-system wraps the $S^1$ ([[circle]]) along which the T-duality takes place.

Lifting that configuration to [[11-dimensional supergravity]] gives [[M5-branes]] (the erstwhile [[D4-brane]]s) on [[Taub-NUT spacetime|Taub-NUT]] ($\times S^1$). The [[M5-branes]] wrap the circle-fiber of Taub-NUT, which shrinks to zero size at the origin (the location of the erstwhile D6, which is where the D4s "end"). The low-energy theory, on a stack of M5-branes, is the [[6d (2,0)-susy QFT]].

## Related concepts

* [[Hanany-Witten effect]]

* [[small instanton]]

* [[brane]], [[string]]

* [[fivebrane structure]], [[differential fivebrane structure]]

[[!include table of branes]]

\linebreak

## References

### As a black brane

The 5-brane in [[heterotic string theory]] was found as a [[black brane]] in

* {#Strominger91} [[Andrew Strominger]],

  _Heterotic solitons_, Nucl.Phys. B343 (1990) 167-184 Nucl.Phys. B353 (1991) 565 ([spire](http://inspirehep.net/record/27900))

  Erratum: Nucl. Phys. B353 (1991) 565-565 (doi:10.1016/0550-3213(91)90349-3, [spire:27900](http://inspirehep.net/record/27900))

* {#CallanHarveyStrominger91} [[Curtis Callan]], [[Jeffrey Harvey]], [[Andrew Strominger]], _Worldbrane actions for string solitons_, Nuclear Physics B Volume 367, Issue 1, 16 December 1991, Pages 60-82 (doi:10.1016/0550-3213(91)90041-U)

* Marco Cariglia, [[Kurt Lechner]], _NS5-branes in IIA supergravity and gravitational anomalies_ ([arXiv:hep-th/0203238](http://arxiv.org/abs/hep-th/0203238))

The [[brane intersection|intersection]] laws with [[black brane|black]] [[D-branes]] are  discussed in

* {#Tseytlin96} [[Arkady Tseytlin]], _No-force condition and BPS combinations of p-branes in 11 and 10 dimensions_, Nucl.Phys.B487:141-154,1997 ([arXiv:hep-th/9609212](https://arxiv.org/abs/hep-th/9609212))

* {#HananyWitten97} [[Amihay Hanany]], [[Edward Witten]], _Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics_, Nucl. Phys. B492:152-190, 1997 ([arXiv:hep-th/9611230](https://arxiv.org/abs/hep-th/9611230))

* {#ArgurioEnglertHouart97} R. Argurio, [[François Englert]], L. Houart, _Intersection Rules for $p$-Branes_, Phys. Lett. B398:61-68, 1997 ([arXiv:hep-th/9701042](https://arxiv.org/abs/hep-th/9701042))

* {#Witten97} [[Edward Witten]], _Solutions Of Four-Dimensional Field Theories Via M Theory_, Nucl.Phys.B500:3-42,1997 ([arXiv:hep-th/9703166](https://xxx.lanl.gov/abs/hep-th/9703166))

* {#BrodieHanany97} [[John Brodie]], [[Amihay Hanany]], _Type IIA Superstrings, Chiral Symmetry, and N=1 4D Gauge Theory Dualities_, Nucl.Phys. B506 (1997) 157-182 ([arXiv:hep-th/9704043](https://arxiv.org/abs/hep-th/9704043))

* {#EGKRS00} [[Shmuel Elitzur]], [[Amit Giveon]], [[David Kutasov]], [[Eliezer Rabinovici]], [[Gor Sarkissian]], *D-Branes in the Background of NS Fivebranes*, JHEP 0008 (2000) 046 [[arXiv:hep-th/0005052](https://arxiv.org/abs/hep-th/0005052), [doi:10.1088/1126-6708/2000/08/046](https://doi.org/10.1088/1126-6708/2000/08/046)]

Review includes

* {#Fazzi17} [[Marco Fazzi]], _Higher-dimensional field theories from type II supergravity_ ([arXiv:1712.04447](https://arxiv.org/abs/1712.04447))

The [[M-theory]]-lift of the black NS5-brane embedded into a [[D6-brane]] should be the configuration from section 8.3 of

* {#MFF12} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553))

### As a Green-Schwarz sigma-model

The [[Green-Schwarz action functionals]] for the NS5-brane:

In [[heterotic string theory]] (see also at _[[dual heterotic string theory]]_):

* [[Kurt Lechner]], [[Mario Tonin]], _Worldvolume and target space anomalies in the D=10 super--fivebrane sigma--model_ ([arXiv:hep-th/9603094](http://arxiv.org/abs/hep-th/9603094))

* J. Mourad, _Anomalies of the $SO(32)$ five-brane and their cancellation_, Nucl.Phys. B512 (1998) 199-208 ([arxiv:hep-th/9709012](https://arxiv.org/abs/hep-th/9709012))

* {#Lechner10} [[Kurt Lechner]], _Quantum properties of the heterotic five-brane_, Phys.Lett.B693:323-329, 2010 ([arxiv:1005.5719](https://arxiv.org/abs/1005.5719))

In [[type IIA string theory]]:

* [[Igor Bandos]], Alexei Nurmagambetov, [[Dmitri Sorokin]], _The type IIA NS5--Brane_ ([arXiv:hep-th/0003169](http://arxiv.org/abs/hep-th/0003169))

* Daniel Persson, _Fivebrane Instantons and Hypermultiplets_ (2010) ([pdf](http://string.lpthe.jussieu.fr/QKPHYS2010/Persson.pdf))

as a [[higher WZW model]]:

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_

### Under dualities

Discussion of the effect of [[T-duality]] on NS5-branes includes

* [[Eduardo Eyras]], [[Bert Janssen]], [[Yolanda Lozano]], _5-branes, KK-monopoles and T-duality_, Nucl. Phys. B **531** (1998) 275-301 [[arXiv:hep-th/9806169](https://arxiv.org/abs/hep-th/9806169), doi:10.1016/S0550-3213(98)00575-6]

* [[David Tong]], _NS5-Branes, T-Duality and Worldsheet Instantons_, JHEP 0207:013,2002 ([arXiv:hep-th/0204186](https://arxiv.org/abs/hep-th/0204186))

### Relation to the M5-brane

Most of the following references are more on the [[M5-brane]].

The fact that the worldvolume theory of the M5-brane should support fields that are [[self-dual higher gauge theory|self-dual]] [[connections on a 2-bundle]] ($\sim$ a [[gerbe]]) is discussed in

* {#Witten07} [[Edward Witten]], _[[Conformal field theory in four and six dimensions]]_, in [[Ulrike Tillmann]], _Topology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal_,  London Mathematical Society Lecture Note Series (2004) ([arXiv:0712.0157](http://arxiv.org/abs/0712.0157))

as well as sections 3 and 4 of

* [[Edward Witten]], _Geometric Langlands From Six Dimensions_ ([arXiv:0905.2720](http://arxiv.org/abs/0905.2720))
 {#Witten09}.

A review of some aspects is in

* [[Robbert Dijkgraaf]], _The mathematics of fivebranes_ ([pdf](http://arxiv.org/PS_cache/hep-th/pdf/9810/9810157v1.pdf))

The relation to [[Khovanov homology]] is discussed in

* {#Witten11} [[Edward Witten]], _Fivebranes and Knots_, Quantum Topology, Volume 3, Issue 1, 2012, pp. 1-137 ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216), [doi:10.4171/QT/26](https://www.ems-ph.org/journals/show_abstract.php?issn=1663-487X&vol=3&iss=1&rank=1))

* {#GukovSchwarzVafa} [[Sergei Gukov]], [[Albert Schwarz]], [[Cumrun Vafa]], _Khovanov-Rozansky Homology And Topological Strings_ , Lett. Math. Phys. 74 (2005) 53-74, ([arXiv:hep-th/0412243](http://arxiv.org/abs/hep-th/0412243))

See also

* [[Greg Moore]], _On the role of six‐dimensional $(2,0)$-theories in recent developments in
Physical Mathematics_ , talk at _Strings2011_ ([pdf slides](http://www-conference.slu.se/strings2011/presentations/3%20Wednesday/930_Moore.pdf))

The above discussion makes use of some blog comments (notably by [[Jacques Distler]]) appearing at

* [[Urs Schreiber]], _4d QFT for Khovanov Homology_ ([web](http://golem.ph.utexas.edu/category/2011/02/4d_qft_for_khovanov_homology.html))

### Intersection with O8/D8

Intersection of [[black brane|black]] NS5-branes with [[O8-planes]]/[[black brane|black]] [[D8-branes]] is discussed in

* {#HananyZaffaroni98} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Branes and Six Dimensional Supersymmetric Theories_, Nucl.Phys. B529 (1998) 180-206 ([arXiv:hep-th/9712145](https://arxiv.org/abs/hep-th/9712145))

* {#JanssenMeessenOrtin99} [[Bert Janssen]], [[Patrick Meessen]], [[Tomas Ortin]], _The D8-Brane Tied up: String and Brane Solutions in Massive Type IIA Supergravity_, Phys. Lett. B453 (1999) 229-236 ([arXiv:hep-th/9901078](https://arxiv.org/abs/hep-th/9901078))

* {#HananyZaffaroni99} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopoles in String Theory_, JHEP 9912 (1999) 014 ([arXiv:hep-th/9911113](https://arxiv.org/abs/hep-th/9911113))

* {#GKSTY02} E. Gorbatov, [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], [[Stefan Theisen]], [[Shimon Yankielowicz]], _On Heterotic Orbifolds, M Theory and Type I' Brane Engineering_, JHEP 0205:015, 2002 ([arXiv:hep-th/0108135](https://arxiv.org/abs/hep-th/0108135))

* {#GaiottoTomasiello14} [[Davide Gaiotto]], [[Alessandro Tomasiello]], _Holography for $(1,0)$ theories in six dimensions_ ([arXiv:1404.0711](https://arxiv.org/abs/1404.0711))

* {#HKLY15} Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi, _6d SCFTs, 5d Dualities and Tao Web Diagrams_, JHEP05(2019)203 ([arXiv:1509.03300](https://arxiv.org/abs/1509.03300))

* {#ApruzziFazzi17} Fabio Apruzzi, [[Marco Fazzi]], _$AdS_7/CFT_6$ with orientifolds_, J. High Energ. Phys. (2018) 2018: 124 ([arXiv:1712.03235](https://arxiv.org/abs/1712.03235))

* {#HSS18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], Example 2.2.7 of: _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, Communications in Mathematical Physics (2019) ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987), [doi:10.1007/s00220-019-03442-3](http://link.springer.com/article/10.1007/s00220-019-03442-3))

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 4 of: _[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]_ ([arXiv:1908.00042](https://arxiv.org/abs/1908.00042))

On the [[orientifold]] [[T-duality|T-dual]] of half M5-branes:

* {#FengHeKarchUranga01} Bo Feng, [[Yang-Hui He]], [[Andreas Karch]], [[Angel Uranga]], _Orientifold dual for stuck NS5 branes_, JHEP 0106:065, 2001 ([arXiv:hep-th/0103177](https://arxiv.org/abs/hep-th/0103177))

For more see at _[[M-theory on S1/G_HW times H/G_ADE]]_.

### NS5/D4/D2 bound states
 {#ReferencesNS5D4D2BoundStates}

[[bound states]] of NS5-branes with [[D4-branes]] and [[D2-branes]] (see also at _[M2-branes -- Dyonic membranes](M2-brane#ReferencesDyonic)_)

* {#MitraRoy00} Indranil Mitra, Shibaji Roy, _(NS5,Dp) and (NS5,D(p+2),Dp) bound states of type IIB and type IIA string theories_, JHEP 0102:026, 2001 ([arXiv:hep-th/0011236](https://arxiv.org/abs/hep-th/0011236))

* {#AlishahihaOz00} Mohsen Alishahiha, Yaron Oz, _Supergravity and "New" Six-Dimensional Gauge Theories_, Phys.Lett.B495:418-426, 2000 ([arXiv:hep-th/0008172](https://arxiv.org/abs/hep-th/0008172))

* {#MitraRoy01} Indranil Mitra, Shibaji Roy, _(NS5,D5,D3) bound state, OD3, OD5 limits and SL(2,Z) duality_, Phys.Rev. D65 (2002) 106001 ([arXiv:hep-th/0107127](https://arxiv.org/abs/hep-th/0107127))

* {#MukhiSuryanarayana00} [[Sunil Mukhi]], Nemani V. Suryanarayana, _Stable Non-BPS States and Their Holographic Duals_, Int. J. Mod. Phys. A16 (2001) 966-975 ([arXiv:hep-th/0011185](https://arxiv.org/abs/hep-th/0011185))

* {#Camino02} J. M. Camino, section 4.5 of _Worldvolume Dynamics of Branes_ ([arXiv:hep-th/0210249](https://arxiv.org/abs/hep-th/0210249))

* {#JiaLuRoy17} Qiang Jia, J. X. Lu, Shibaji Roy, _On 1/4 BPS ((F, D1), (NS5, D5)) bound states of type IIB string theory_, JHEP 08 (2017) 007 ([arXiv:1704.01463](https://arxiv.org/abs/1704.01463))

* Giuseppe Dibitetto, Nicolò Petrim, _6d surface defects from massive type IIA_, JHEP 01 (2018) 039 ([arXiv:1707.06154](https://arxiv.org/abs/1707.06154))

* {#Petri18} Nicolò Petri, _Surface defects in massive IIA_, talk at [Recent Trends in String Theory and Related Topics](http://physics.ipm.ac.ir/conferences/stringtheory3/) 2018 ([pdf](http://physics.ipm.ac.ir/conferences/stringtheory3/note/N.Petri.pdf))

### Open D$p$-brane theories on the NS5

* {#GopakumarMinwallaSeibergStrominger00} [[Rajesh Gopakumar]], [[Shiraz Minwalla]], [[Nathan Seiberg]], [[Andrew Strominger]], _OM Theory in Diverse Dimensions_, JHEP 0008:008, 2000 ([arXiv:hep-th/0006062](https://arxiv.org/abs/hep-th/0006062))

* {#Harmark00} Troels Harmark, _Open Branes in Space-Time Non-Commutative Little String Theory_,  Nucl.Phys. B593 (2001) 76-98 ([arXiv:hep-th/0007147](https://arxiv.org/abs/hep-th/0007147))

* {#Lu06} J. X. Lu, _$(1 + p)$-Dimensional Open $D(p - 2)$ Brane Theories_, JHEP 0108:049, 2001 ([arXiv:hep-th/0102056](https://arxiv.org/abs/hep-th/0102056))

[[!redirects NS-5-brane]]
[[!redirects F5-brane]]
[[!redirects F-5-brane]]
[[!redirects 5-brane]]
[[!redirects fivebrane]]

[[!redirects NS5-branes]]
[[!redirects NS-5-branes]]
[[!redirects F5-branes]]
[[!redirects F-5-branes]]
[[!redirects 5-branes]]
[[!redirects fivebranes]]

[[!redirects half NS5-brane]]
[[!redirects half NS5-branes]]


NSR action functional

#Contents#
* table of contents
{:toc}

## Idea

A type of [[action functional]] for [[supersymmetry|supersymmetric]] $p$-[[brane]] [[sigma-models]] with manifest [[worldvolume]] [[supersymmetry]].

In particular the _[[NSR string]]_ for $p = 1$.

## Related concepts

[[!include worldvolume-target supersymmetry of brane sigma-models]]


nuclear adjunction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

# Nuclear adjunctions
* table of contents
{: toc}

## Definition

A **nuclear adjunction** or **bimonadic adjunction** is an [[adjunction]] that is both [[monadic adjunction|monadic]] and [[comonadic adjunction|comonadic]].

One point in favour of the terminology "nuclear" over "bimonadic" is that the term **bimonadic functor** is sometimes used in the literature for a [[functor]] that is both [[monadic functor|monadic]] and [[comonadic functor|comonadic]], which is a different concept that is related to [[bimonads]].

## Properties

* Consider an adjunction $F \dashv G$, generating a [[monad]] $T$ and a [[comonad]] $D$. Under minimal assumptions on the categories, there is an adjunction between the category of $T$-algebras, and the category of $D$-coalgebras, and this adjunction is nuclear. See [Pavlovic–Hughes](#PavlovicHughes20) and below.

* Consider an adjunction $F \dashv G : B \to A$, generating a [[monad]] $T$ and a [[comonad]] $D$. Assuming that $B$ is [[Cauchy complete]], $F$ is [[comonadic]] if and only if the free–forgetful adjunction associated to the [[Eilenberg–Moore category]] of $T$ is nuclear. See [Mesablishvili](#Mesablishvili06).

## Construction of nuclear adjunction

Let $\ell \dashv r$ be an adjunction, inducing a monad $T_1$. One can form the [[Eilenberg–Moore category]] of $T$, which induces an adjunction $f_{T_1} \dashv u_{T_1}$. This in turn induces a comonad $D_2$ on the Eilenberg–Moore category. One can form the Eilenberg–Moore category of $D_2$, which induces an adjunction $u_{D_2} \dashv c_{D_2}$. This in turn induces a monad $T_3$ on the Eilenberg–Moore of $D_2$. We can then continue to iterate this construction, giving monads and comonads $T_1, D_2, T_3, D_4, T_5, \dots$.

[\begin{tikzcd}
	B && {\mathrm{Alg}(T_1)} & {\mathrm{Coalg}(D_2)} & {\mathrm{Alg}(T_3)} & \cdots \\
	& A
	\arrow[""{name=0, anchor=center, inner sep=0}, "\ell", shift left=2, from=2-2, to=1-1]
	\arrow[""{name=1, anchor=center, inner sep=0}, "r", shift left=2, from=1-1, to=2-2]
	\arrow["{!}", from=1-1, to=1-3]
	\arrow[""{name=2, anchor=center, inner sep=0}, "{f_{T_1}}", shift left=2, from=2-2, to=1-3]
	\arrow[""{name=3, anchor=center, inner sep=0}, "{u_{T_1}}", shift left=2, from=1-3, to=2-2]
	\arrow[""{name=4, anchor=center, inner sep=0}, "{u_{D_2}}"', shift right=2, from=1-4, to=1-3]
	\arrow[""{name=5, anchor=center, inner sep=0}, "{c_{D_2}}"', shift right=2, from=1-3, to=1-4]
	\arrow[""{name=6, anchor=center, inner sep=0}, "{f_{T_3}}", shift left=2, from=1-4, to=1-5]
	\arrow[""{name=7, anchor=center, inner sep=0}, "{u_{T_3}}", shift left=2, from=1-5, to=1-4]
	\arrow["\dashv"{anchor=center, rotate=45}, draw=none, from=0, to=1]
	\arrow["\dashv"{anchor=center, rotate=-49}, draw=none, from=2, to=3]
	\arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=4, to=5]
	\arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=6, to=7]
\end{tikzcd}](https://q.uiver.app/?q=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)

An obvious question is: does this process ever terminate? In unpublished work of Lack, it is [claimed](#Taylor) that this process terminates at $T_3$. In other words, $f_{T_3} \dashv u_{T_3}$ is a nuclear adjunction. Thus $T_3 \simeq T_5 \simeq \dots$ and $D_4 \simeq D_6 \simeq \dots$. Furthermore, if the domain of $\ell$ is [[Cauchy complete]], then it terminates at $D_2$.

Under the simplifying assumptions of [[Cauchy completeness]], this is proven in [Pavlovic–Hughes](#PavlovicHughes20). They observe that this construction forms a [[pseudo-idempotent 2-monad]] on a 2-category of monads on $Cat$.

Note that this construction is distinct from [[monadic decomposition]], which instead constructs new monads from comparison functors, rather than (co)free–forgetful adjunctions.

## References

* [[Michael Barr]], *Coalgebras in a category of algebras*, in: *Category Theory, Homology Theory and their Applications I*, Lecture Notes in Mathematics **86**, Springer (1969) 1-12 [[doi:10.1007/BFb0079381](https://doi.org/10.1007/BFb0079381), [pdf](https://www.math.mcgill.ca/barr/papers/coalgs.pdf), [[Barr-CoalgInAlg.pdf:file]]]

* [[Armin Frei]] and [[John L. MacDonald]], _Algebras, coalgebras and cotripleability_, Archiv der Mathematik 22 (1971): 1-6.

* {#Taylor} Paul Taylor's email to the [[categories mailing list]] "monadic completion of adjunctions" ([link](https://github.com/punkdit/categories/blob/26b751bbcbe6765fe447e805629ff6416f4c38b1/www.mta.ca/~cat-dist/catlist/1999/monadic-compl))

* Stefano Kasangian, [[Stephen Lack]], and [[Enrico Vitale]]. _Coalgebras, braidings, and distributive laws_, Theory and Applications of Categories 13.8 (2004): 129-146. ([html](https://www.emis.de/journals/TAC/volumes/13/8/13-08abs.html))

* [[Bart Jacobs]]. _Coalgebras and approximation_. Logical Foundations of Computer Science: Third International Symposium, LFCS'94 St. Petersburg, Russia, July 11–14, 1994 Proceedings. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005.

* {#Mesablishvili06} Bachuki Mesablishvili. _Monads of effective descent type and comonadicity_. Theory and Applications of Categories 16.1 (2006): 1-45. ([pdf](http://www.tac.mta.ca/tac/volumes/16/1/16-01.pdf))

* [[Matias Menni]]. _Bimonadicity and the explicit basis property_. Theory and Applications of Categories 26.22 (2012): 554-581. ([pdf](http://www.tac.mta.ca/tac/volumes/26/22/26-22.pdf))

* [[Bart Jacobs]]. _Bases as coalgebras_. Logical Methods in Computer Science 9 (2013).

* {#PavlovicHughes20} [[Dusko Pavlovic]], Dominic J. D. Hughes, _The nucleus of an adjunction and the Street monad on monads_ ([arXiv:2004.07353](https://arxiv.org/abs/2004.07353))

[[!redirects nuclear adjunctions]]
[[!redirects bimonadic adjunction]]
[[!redirects bimonadic adjunctions]]