double-negation shift

# Double-negation shift

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## Definition

In [[logic]], the principle of **double-negation shift** (**DNS**), also known as the **Kuroda Principle**, is the statement that for any [[predicate]] $P$, the following holds:

$$ (\forall x, \neg\neg P(x)) \to \neg\neg\forall x, P(x). $$

Of course, this is a tautology in [[classical logic]] where [[double negation]] $\neg\neg$ is the identity, but in [[constructive mathematics]] this is not provable (although its [[converse]] is).

It is provable even constructively, though, if the domain of [[quantifier|quantification]] of the variable $x$ is [[finite set|finite]].  In general, the principle of double-negation shift can be considered a property of this domain, which holds automatically for finite sets, and might hold for some other sets but not for others.  Sometimes by "double-negation shift" is meant specifically the version for the domain $\mathbb{N}$ of [[natural numbers]].

## Equivalence to double-negated PEM

The principle of double-negation shift (over arbitrary domains) is equivalent to the double-negation of the universally-quantified principle of [[excluded middle]]:

$$\neg\neg \forall Q:\Omega, (Q \vee \neg Q). $$

Since $\forall Q, \neg\neg(Q \vee \neg Q)$ is a constructive tautology, the above double-negated PEM follows from DNS with domain $Prop$ and $P(Q) = (Q\vee\neg Q)$.  Conversely, if double-negated PEM holds, then to prove any negated goal like $\neg\neg\forall x, P(x)$ we may assume full PEM, i.e. classical logic, in which case DNS holds.

## Validity in toposes

The interpretation of the predicate $(Q : \Omega | Q\vee\neg Q)$ in an elementary topos $\mathcal{E}$ is the subobject $[t,f] : 1+1=2\to\Omega$. Thus, $\neg\neg \forall Q:\Omega, Q \vee \neg Q $ holds in $\mathcal{E}$ precisely if there exists a subterminal $U\to 1$ such that $\neg\neg U \cong\id_1$ and
$$(*) \qquad\quad (\Omega\times U\to\Omega)\subseteq(2\to\Omega).$$
If $\mathcal{E}$ is the topos of sheaves on a locale $X$, the subterminals correspond to the opens of the locale, and $(*)$ holds for an open $U\in O(X)$ precisely if $U\subseteq V\vee\neg V$ for all $V\in O(X)$ with $V\subseteq U$. This means precisely that $U$ is boolean as an open sublocale, so double negation shift holds in ${Sh}(X)$ precisely if $X$ has a dense boolean open sublocale. If $X$ is (the set of opens in) a $T_0$-space, the booleanness condition is equivalent to discreteness, so double negation shift holds in the sheaf topos precisely if $X$ has a dense discrete open subspace. A non-trivial example satisfying $T_2$ is the 1-point compactification of an infinite set, and the real line is a counterexample (since a dense discrete open would be non-empty, and each of its element would be an isolated point, contradicting connectedness of $\mathrm{R}$).

In the case of a *presheaf topos*, say on a small category $\mathrm{C}$, truth values are precisely sieves in $\mathrm{C}$, i.e. sets of objects $U$ such that for every morphism $f: A\to B$, $B\in U$ implies $A\in U$. In this situation, $(*)$ is equivalent to $\mathrm{C}/A\simeq 1$ for all $A\in U$. From this it can be deduced that double negation shift holds in presheaves over $\mathrm{C}$ if and only if for every $A\in\mathrm{C}$ there exists an $f:B\to A$ with $\mathrm{C}/B\simeq 1$.

DNS holds in every Kripke model with finite frame; see [here](https://plato.stanford.edu/entries/logic-intuitionistic/).


double-negation topology > history

< [[double-negation topology]]


doublet-triplet splitting problem

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## Idea

One of the technical issues in the [[phenomenology|phenomenological]] viability of [[grand unified theories]].

## Related concepts

* [[proton decay]]

## References

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 2.6. (p. 48 (62)) of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

* Gero von Gersdorff, _A Clockwork Solution to the Doublet-Triplet Splitting Problem_ ([arXiv:2009.08480](https://arxiv.org/abs/2009.08480))

See also:

* Wikipedia, _[Doublet–triplet splitting problem](https://en.wikipedia.org/wiki/Doublet–triplet_splitting_problem)_

Possible resolution for GUT-models in the context of [[M-theory on G2-manifolds]] is discussed in

* {#Witten02} [[Edward Witten]], _Deconstruction, $G_2$ Holonomy, and Doublet-Triplet Splitting_, ([arXiv:hep-ph/0201018](http://arxiv.org/abs/hep-ph/0201018))

* [[Bobby Acharya]], Krzysztof Bozek, Miguel Crispim Romao, Stephen F. King, Chakrit Pongkitivanichkul, _$SO(10)$ Grand Unification in M theory on a $G_2$ manifold_ ([arXiv:1502.01727](http://arxiv.org/abs/1502.01727))


doubly closed monoidal category

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#Contents#
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## Idea
 {#Idea}

Generally, by a "doubly closed monoidal category" one may mean a [[category]] equipped with not just one but two different [[closed monoidal category]]-[[structures]] [[O'Hearn & Pym (1999), §3](#O'HearnPym99); [O'Hearn (2003), §2.2](O'Hearn03)].

A common and relevant special case is that of a [[cartesian closed category]] which is equipped with further non-cartesian [[closed monoidal category]]-[[structure]], in which case one may speak of *cartesian doubly closed monoidal categories* [[O'Hearn & Pym (1999), §3](#O'HearnPym99); [Jeltsch (2016), Def. 3](#Jeltsch16)]. Yet more specially, if the [[underlying]] category is a [[topos]] one may speak of a *[[doubly monoidal topos|(doubly) monoidal topos]]* (see there for more).

Examples of doubly closed monoidal categories commonly arise as "parameterized" generalizations of plain [[closed monoidal categories]]. For example, categories *[[VectBund]]* of [[finite number|finite]] [[rank of a vector bundle|rank]] [[vector bundles]] over base spaces taken from [[convenient categories of topological spaces]] (or just over [[Sets]]), whose [[morphisms]] admit [[base change]], are cartesian doubly closed monoidal with the further non-cartesian [[tensor product]] being the [[external tensor product]] of [[vector bundles]]. In the same vein, [[tangent (infinity,1)-toposes|tangent $\infty$-toposes]] of [[parameterized spectra|parameterized]] ([[module spectra|module]]-)[[spectra]] form cartesian doubly [[monoidal (infinity,1)-category|closed monoidal $\infty$-categories]] (and in fact [[doubly monoidal (infinity,1)-topos|doubly monoidal $\infty$-toposes]]).

By the general [[relation between category theory and type theory]], the [[internal language]] of such cartesian double closed monoidal categories is a combination of classical ([[intuitionistic type theory|intuitionistic]]) [[type theory]] for the cartesian closed [[fragment]], and some kind of [[linear type theory]] for the non-cartesian closed [[fragment]]: Where a [[context]] in [[intuitionistic type theory]] is an iterated cartesian ([[dependent product type|dependent]]) [[product type]] $(-)\times (-)$ and a context in [[linear type theory]] is an iterated [[multiplicative conjunction]] $(-) \otimes (-)$, so a context in the [[internal language]] of cartesian doubly monoidal categories is obtained as an alternation of these two construction principles, forming a [[tree]] with nodes or "bunches" labeled either "$\times$" or "$\otimes$" --- this is the topic of *[[bunched type theory]]*.

## Related concepts

* [[closed monoidal category]]

* [[doubly monoidal topos]]

* [[bunched type theory]]

## References

Discussion of doubly closed monoidal categories as [[categorical semantics]] for [[bunched logic]]/[[bunched type theory]]:

* {#O'HearnPym99} [[Peter O'Hearn]], [[David J. Pym]], §3 in: *The Logic of Bunched Implications*, The Bulletin of Symbolic Logic **5** 2 (1999) 215-244  [[pdf](http://www.lsv.ens-cachan.fr/~demri/OHearnPym99.pdf), [doi:10.2307/421090](https://doi.org/10.2307/421090)]

* {#O'Hearn03} [[Peter O'Hearn]], §2.2 in: *On bunched typing*, Journal of Functional Programming **13** 4 (2003) 747-796 [[doi:10.1017/S0956796802004495](https://doi.org/10.1017/S0956796802004495)]

* {#Jeltsch16} Wolfgang Jeltsch, *Abstract categorical semantics for resourceful functional reactive programming*, Elsevier Journal of Logical and Algebraic Methods in Programming **85** 6 (2016) 1177-1200 [[doi:10.1016/j.jlamp.2016.07.001](https://doi.org/10.1016/j.jlamp.2016.07.001)]

Discussion of cartesian doubly [[monoidal (infinity,1)-category|closed monoidal $\infty$-categories]] of [[parameterized spectra|parameterized]] ([[module spectra|module]]-)[[spectra]] as [[categorical semantics]] of bunched [[linear homotopy type theory]]:

* {#Riley22Thesis} [[Mitchell Riley]], *A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory*, PhD Thesis (2022) [[doi:10.14418/wes01.3.139](https://doi.org/10.14418/wes01.3.139), [ir:3269](https://digitalcollections.wesleyan.edu/object/ir%3A3269), [pdf](https://mvr.hosting.nyu.edu/pubs/thesis.pdf)]

[[!redirects doubly closed monoidal categories]]

[[!redirects cartesian doubly closed monoidal category]]
[[!redirects cartesian doubly closed monoidal categories]]

[[!redirects doubly closed monoidal (infinity,1)-category]]
[[!redirects doubly closed monoidal (infinity,1)-categories]]

[[!redirects doubly closed monoidal (∞,1)-category]]
[[!redirects doubly closed monoidal (∞,1)-categories]]

[[!redirects cartesian doubly closed monoidal (infinity,1)-category]]
[[!redirects cartesian doubly closed monoidal (infinity,1)-categories]]

[[!redirects cartesian doubly closed monoidal (∞,1)-category]]
[[!redirects cartesian doubly closed monoidal (∞,1)-categories]]


doubly indexed category

#Contents#
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## Idea

Informally a doubly indexed category is a unitary [[double functor|lax double functor]] $F : \mathbb{C}^\top \to \mathbf{\mathbb{C}at}$, where the latter is the [[proarrow equipment]] of categories, functors, profunctors and natural transformations.

Formally, it is a right $\mathbb{C}$-module, i.e. it's a right [[action]] of the [[double category]] $\mathbb{C}$.

With this name, they were defined by [[David Jaz Myers]] in [his book on categorical system theory](#MyersBook).

## Definition

\begin{definition}
Let $\mathbb{C} = \mathbf{C}_1 \overset{s,t}\rightrightarrows \mathbf{C}_0$ be a [[double category]], presented as a [[pseudomonad]] in $\mathbf Span(Cat)$.
A *doubly indexed category* over $\mathbb{C}$ is a right $\mathbb{C}$-[[module over a monad|module]], meaning it consists of a functor $\pi : \mathbf F \to C_0$ together with a map $* : \mathbf{C}_1 \times_{C_0} \mathbf{F} \to \mathbf{F}$ over $\mathbf{C}_0$, satisfying the laws of unitality and functoriality of a module.
\end{definition}

The reason one can see this data as given by a unitary lax double functor $F : \mathbb{C}^\top \to \mathbf{\mathbb{C}at}$ is the following.
First, the data of the functor $\pi: \mathbf F \to C_0$ in itself is equivalent to that of a unitary [[lax 2-functor]] into $\mathbf{Prof}$, i.e. a [[displayed category]] (see there for an explanation). This means objects of $\mathbb{C}$ are sent to categories (the fibers of $\pi$) while horizontal maps ([[double category|tight maps]]) in $\mathbb{C}$ act 'profunctorially' on them. This assignment is unitary and lax.

Then the action $*$ gives, for each vertical morphism (loose maps) in $\mathbb{C}$, a functor. Explicitly, if $f: x \to y$ is a vertical map in $\mathbb{C}$, and $a : F(x) = \pi^{-1}(x)$, then $F(f)(a) = f * a : F(y)$.

This part of the action is strict, or at least pseudofunctorial.

## Examples

One of the central exampls of doubly indexed category is given by the 'self double-indexing' any [[finitely complete category]] $\mathbf{C}$ induces.
Specifically, it's the action $\mathbf{C}/-$ of $\mathbf{Span(C)}$.

It is defined as follows:

1. An object $x : \mathbf{C}$ is sent to the slice category $\mathbf{C}/x$
2. A morphism $f:x \to y$ is sent to the profunctor $\mathbf{C}/f: \mathbf{C}/x^{\mathrm{op}} \times \mathbf{C}/y \to \mathbf{C}$ that sends  maps $p : a \to x$, $q:b \to y$ to the set of maps $h:a \to b$ that make the square commute:
$$
\mathbf{C}/f(p, q) := \left\{ \begin{matrix} \ a & \overset{h}\to & b\ \\ p \downarrow && \downarrow q\\ x & \overset{f}\to & y\end{matrix} \right\}
$$
The laxator is defined by sending squares on $f:x \to y$ and $g:y \to z$ agreeing on their $y$ boundary to the obvious composite square lying over $f;g : x \to z$ (thus forgetting the middle boundary).
One can see this assignment is unitary, as $\mathbf{C}/1_x$ is exactly $\mathbf{C}/x(-,=)$ (the [[hom functor|identity profunctor]] on [[slice category|$\mathbf{C}/x$]]).
3. A span $x \overset{l}\leftarrow s \overset{r}\to y$ is sent to the pull-push functor $r_* l^* : \mathbf{C}/x \to \mathbf{C}/y$ which, on a map $p : a \to x$, acts first by [[pullback]] along $l$ and then post-composition it with $r$:
$$
\mathbf{C}/(l, r)(p) = \sigma_r l^*p
$$
4. On squares, it can be defined as...

## Related concepts

* [[indexed category]]

* [[double category]]

* [[module over a monad]]

* [[categorical systems theory]], [[systems theory]]

## References

* [[David Jaz Myers]], *Double Categories of Open Dynamical Systems*, EPTCS **333** (2021) 154-167 [[arXiv:2005.05956](https://arxiv.org/abs/2005.05956), [doi:10.4204/EPTCS.333.11](https://doi.org/10.4204/EPTCS.333.11)]

* {#MyersBook} [[David Jaz Myers]], *Categorical systems theory*, book project [[github](https://github.com/DavidJaz/DynamicalSystemsBook/tree/master/book), [pdf](http://davidjaz.com/Papers/DynamicalBook.pdf)]

Exposition:

* [[David Jaz Myers]], *[Categorical systems theory](https://topos.site/blog/2021/11/categorical-systems-theory/)*, [Topos Institute Blog](https://topos.site/blog/) (Nov 2021)

* [[David Jaz Myers]], *Double Categories of Dynamical Systems* (2020) [[pdf](http://davidjaz.com/Talks/DJM_Dyn2020.pdf)]

Actions of double categories are central in some approaches to [[dependent optics]]:

* [[Matteo Capucci]], *Seeing double through dependent optics* (2022), [arXiv:2204.10708](https://arxiv.org/abs/2204.10708)


doubly infinite series

## Idea

Doubly infinite series is a series $a_n$ where $n$ ranges from $-\infty$ to $\infty$.

#### Warning

Doubly infinite series is a term which should be used with caution, as some people use the term for the double series
(series with two indices),
so that it is an infinite series in each variable, say
$\sum_{i=1}^\infty\sum_{j=1}^\infty a_{i j}$.

## Definition

Doubly infinite series

$$
\sum_{k = -\infty}^\infty a_k,
$$

is a pair of two (ordinary infinite) [[series]],
its negative part
$\sum_{k = -1}^\infty a_{-k}$
and its positive  part $\sum_{k = 0}^\infty a_k$ (one could say nonnegative part, but it is not customary).
A doubly infinite series converges
if each of its constituent parts (positive and negative), converges; the sum of the series is then, by definition, the sum of the limits of the two series.

Notion to cover later: principal value of a double series

## Related concepts

* [[finite set]], [[finite series]], [[polynomial]]

* [[natural numbers]], [[infinite series]], [[power series]]

* [[integers]], **doubly infinite series**, [[Laurent series]]

* [[rational numbers]], series indexed in rationals, [[Puiseux series]]

* [[real numbers]], [[definite integral]], [[Fourier transform]]


Doug Ravenel

* [webpage](http://www.math.rochester.edu/people/faculty/doug/)

* [[Michael Hopkins]], _The mathematical work of Douglas C. Ravenel_, Homology Homotopy Appl. Volume 10, Number 3 (2008), 1-13 ([euclid:hha/1251832464](https://projecteuclid.org/euclid.hha/1251832464))

## Selected writings

On the [[Hopf ring]] of [[MU]]:

* [[Douglas Ravenel]], [[W. Stephen Wilson]], _The Hopf ring for complex cobordism_,  Bull. Amer. Math. Soc. 80 (6) 1185 - 1189, November 1974 (doi:10.1016/0022-4049(77)90070-6, [euclid](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-80/issue-6/The-Hopf-ring-for-complex-cobordism/bams/1183536024.full?tab=ArticleLink), [pdf](https://people.math.rochester.edu/faculty/doug/mypapers/hopfring.pdf))

On the [[Adams-Novikov spectral sequence]]:

* {#Ravenel78} [[Douglas Ravenel]], _A Novice's guide to the Adams-Novikov spectral sequence_, in: [[Michael Barratt]], [[Mark Mahowald]] (eds.) _Geometric Applications of Homotopy Theory II_. Lecture Notes in Mathematics, vol 658, Springer 1978 ([doi:10.1007/BFb0068728](https://doi.org/10.1007/BFb0068728), [pdf](https://people.math.rochester.edu/faculty/doug/mypapers/Novice.pdf))

On [[chromatic homotopy theory]] and introducing [[Ravenel's spectra]] and [[Ravenel's conjectures]]:

* {#Ravenel84} [[Douglas Ravenel]], _Localization with Respect to Certain Periodic Homology Theories_, American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 ([doi:10.2307/2374308](https://doi.org/10.2307/2374308), [jstor:2374308](https://www.jstor.org/stable/2374308))

On [[stable homotopy groups of spheres]] and [[chromatic homotopy theory]]:

* {#MahowaldRavenel87} [[Mark Mahowald]], [[Doug Ravenel]], _Towards  a Global Understanding of the Homotopy Groups of Spheres_, in: [[Samuel Gitler]] (ed.): _The Lefschetz Centennial Conference: Proceedings on Algebraic Topology II_, Contemporary Mathematics volume 58, AMS 1987  ([ISBN:978-0-8218-5063-3](https://bookstore.ams.org/conm-58-2), [pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf), [[MahowaldRavenelHomotopyGroupsOfSpheres.pdf:file]])

On [[stable homotopy groups of spheres]] and [[chromatic homotopy theory]]:

* [[Doug Ravenel]], _Nilpotence and Periodicity in Stable Homotopy Theory_, Annals of Mathematics Studies **128**, Princeton University Press (1992) [[ISBN:9780691025728](https://press.princeton.edu/books/paperback/9780691025728/nilpotence-and-periodicity-in-stable-homotopy-theory-am-128-volume), [pdf](https://people.math.rochester.edu/faculty/doug/mybooks/nilpb.pdf), [webpage](https://people.math.rochester.edu/faculty/doug/nilp.html)]

  > ("the orange book")

On [[elliptic genera]]:

* {#LandweberRavenelStong93} [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], _Periodic cohomology theories defined by elliptic curves_, in [[Haynes Miller]] et al (eds.), _The Cech centennial: A conference on homotopy theory_, June 1993, AMS (1995) ([pdf](http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf))

On [[chromatic homotopy theory]], [[complex cobordism cohomology]] and [[stable homotopy groups of spheres]],:

* [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, Academic Press Orland (1986) reprinted as: AMS Chelsea Publishing, Volume 347, 2004 ([ISBN:978-0-8218-2967-7](https://bookstore.ams.org/chel-347-h), [webpage](http://www.math.rochester.edu/people/faculty/doug/mu.html), [pdf](https://web.math.rochester.edu/people/faculty/doug/mybooks/ravenel.pdf))

On the [[Morava K-theory]] of [[iterated loop spaces]] of [[spheres]] and on [[stable splitting of mapping spaces]]:

* [[Douglas Ravenel]], *What we still don't know about loop spaces of spheres*, in: [[Mark Mahowald]], [[Stewart Priddy]] (eds.), *Homotopy Theory via Algebraic Geometry and Group Representations*, Contemporary Mathematics **220**, AMS 1998  ([pdf](https://people.math.rochester.edu/faculty/doug/mypapers/loop.pdf), [[Ravenel_LoopSpacesOfSpheres.pdf:file]], [doi:10.1090/conm/220](http://dx.doi.org/10.1090/conm/220))

Solving the [[Arf-Kervaire invariant problem]] with methods of [[equivariant stable homotopy theory]]:

* {#HillHopkinsRavenel21} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], *Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem*, New Mathematical Monographs, Cambridge University Press (2021) [[doi:10.1017/9781108917278](https://doi.org/10.1017/9781108917278)]

* {#HillHopkinsRavenel09} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], *On the non-existence of elements of Kervaire invariant one*, Annals of Mathematics **184** 1 (2016)[[doi:10.4007/annals.2016.184.1.1](https://doi.org/10.4007/annals.2016.184.1.1), [arXiv:0908.3724](http://arxiv.org/abs/0908.3724), [talk slides](https://www.math.rochester.edu/people/faculty/doug/otherpapers/Skye_handout.pdf)]

* {#HillHopkinsRavenel} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire problem in algebraic topology: Sketch of the proof_, Current Developments in Mathematics, 2010: 1-44 (2011) ([[HHRKervaire.pdf:file]], [doi:10.4310/CDM.2010.v2010.n1.a1](https://dx.doi.org/10.4310/CDM.2010.v2010.n1.a1))

* {#HillHopkinsRavenelIntroduction} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire invariant problem in algebraic topology: introduction_ (2016) [[pdf](http://math.ucla.edu/~mikehill/Research/CDMHistory.pdf)]

category: people

[[!redirects Douglas Ravenel]]

[[!redirects Douglas C. Ravenel]]


Douglas Bridges

Douglas Bridges is a New Zealand based mathematician, working in the University of Canterbury.  He has worked in [[constructive mathematics]] as well as various aspects of [[computability]].

* [Home page](http://www.math.canterbury.ac.nz/~d.bridges/)

* [FAQs in constructive mathematics](http://www.math.canterbury.ac.nz/php/groups/cm/faq/)

## Selected writings

On [[constructive analysis]]:

* [[Errett Bishop]], [[Douglas Bridges]] *Constructive analysis*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

* [[Douglas Bridges]], *Constructive mathematics: a foundation for computable analysis*, Theoretical Computer Science **219** 1–2 (1999) 95-109 [doi:10.1016/S0304-3975(98)00285-0]

On [[weak countable choice]]:

* [[Fred Richman]], [[Douglas Bridges]], Peter Schuster, *A weak countable choice principle*. Proceedings of the American Mathematical Society 128(9):2749-2752, March 2000. [[doi:10.1090/S0002-9939-00-05327-2](http://dx.doi.org/10.1090/S0002-9939-00-05327-2)]

category:people


Douglas Cline

* [personal page](https://www.pas.rochester.edu/~cline/)

* [institute page](https://www.pas.rochester.edu/people/faculty/cline_douglas/index.html)

## Selected writings

Detailed expository textbook on [[classical mechanics]] with emphasis of the [[principle of extremal action]]:

* [[Douglas Cline]], *Variational Principles in Classical Mechanics*, University of Rochester (2021) [[pdf](http://classicalmechanics.lib.rochester.edu/pdf/Variational_Principles_in_Classical_Mechanics_3e_final_fbcover.pdf), online version]

category: people


Douglas Smith

* [webpage](https://www.dur.ac.uk/mathematical.sciences/people/profile/?id=1630)

## Selected writings

On [[intersecting branes]] in [[string theory]]:

* {#Smith02} [[Douglas J. Smith]], *Intersecting brane solutions in string and M-theory*, Class. Quant. Grav. **20** R233 (2003) [[arXiv:hep-th/0210157](https://arxiv.org/abs/hep-th/0210157), [doi:10.1088/0264-9381/20/9/203](https://doi.org/10.1088/0264-9381/20/9/203)]

## Related entries

* [[BPS state]]

category: people

[[!redirects Douglas J. Smith]]


Douglas Stanford

* [webpage](https://www.ias.edu/scholars/douglas-stanford)

## Selected writings

General aspects of [[nearly AdS2-CFT1]] ([[JT-gravity]]/matrix models):

* [[Juan Maldacena]], [[Douglas Stanford]], [[Zhenbin Yang]], _Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space_ ([arXiv:1606.01857](https://arxiv.org/abs/1606.01857))

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

* [[Juan Maldacena]], [[Douglas Stanford]], [[Zhenbin Yang]], *Diving into traversable wormholes*,Fortsch. Phys. **65** 5 (2017) 1700034 [[arXiv:1704.05333](https://arxiv.org/abs/1704.05333), [doi:10.1002/prop.201700034](https://doi.org/10.1002/prop.201700034)]

On [[random matrix theory]] applied to [[AdS/CFT]] and [[black holes in string theory]]:

via the [[SYK model]]:

* Jordan S. Cotler, Guy Gur-Ari, Masanori Hanada, [[Joseph Polchinski]], Phil Saad, [[Stephen Shenker]], [[Douglas Stanford]], Alexandre Streicher, Masaki Tezuka, _Black Holes and Random Matrices_,  JHEP 1705:118, 2017 ([arXiv:1611.04650](https://arxiv.org/abs/1611.04650))

via [[Jackiw-Teitelboim gravity]]:

* {#SaadShenkerStanford19} Phil Saad, [[Stephen Shenker]], [[Douglas Stanford]], _JT gravity as a matrix integral_ ([arXiv:1903.11115](https://arxiv.org/abs/1903.11115))

* {#StanfordWitten19} [[Douglas Stanford]], [[Edward Witten]], _JT Gravity and the Ensembles of Random Matrix Theory_ ([arXiv:1907.03363](https://arxiv.org/abs/1907.03363))

Claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]:

* [[Geoff Penington]], [[Stephen Shenker]], [[Douglas Stanford]], [[Zhenbin Yang]], _Replica wormholes and the black hole interior_ ([arXiv:1911.11977](https://arxiv.org/abs/1911.11977))

On the [[SYK model]] in terms of [[chord diagrams]]:

* [[Henry W. Lin]], [[Douglas Stanford]], *A symmetry algebra in double-scaled SYK* [[arXiv:2307.15725](https://arxiv.org/abs/2307.15725)]

category: people


Dov Samet

* [personal page](https://english.tau.ac.il/profile/dsamet)

* [GoogleScholar page](https://scholar.google.com/citations?user=GsWe1fgAAAAJ&hl=en)

## Selected writings

On [[geometric model of modal logic|geometric models]] of [[S5 modal logic]] (regarded as [[epistemic logic]]):

* [[Dov Samet]], *S5 knowledge without partitions*, Synthese **172** (2010) 145–155 [[doi:10.1007/s11229-009-9469-0](https://doi.org/10.1007/s11229-009-9469-0)]

category: people


Dowker space

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

#Contents#
* table of contents
{:toc}

## Idea

A _Dowker space_ is a [[normal Hausdorff topological space]] which is not [[countably paracompact topological space|countably paracompact]].

## Related concepts

* [[shrinking lemma]]

* [[K-topology]]

## References

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

[[!redirects Dowker spaces]]


Dowker's theorem

# Dowker\'s Theorem
* table of contents
{: toc}

## Note

The purpose of this entry is simply to state and prove a result of Dowker that was mentioned at [[Vietoris complex]].  Here we will look at a version of Dowker's original proof.  In later sections we will see other ways of proving it. We will also discuss some of the interpretations and applications of the result and ask about possible generalisations.

To keep the discussion fairly self contained, certain ideas will be repeated from that, and possibly other, entries.

## The two nerves of a relation: Dowker's construction

Let $X, Y$ be sets and $R$ a relation between $X$ and $Y$, so $R
\subseteq X \times Y$. We write $x R y$ for $(x, y) \in R$.

+-- {: .un_example}
###### Example
(In addition to those given in [[Vietoris complex]].)

If $K$ is a [[simplicial complex]], its structure is specified by a
collection of non-empty finite subsets of its set of vertices
namely those sets of vertices declared to be simplices. This
collection of simplices is supposed to be downward closed, i.e.,
if $\sigma$ is a simplex and $\tau \subseteq \sigma$ with $\tau \neq \emptyset$, then $\tau$
is a simplex. For our purposes here, set $X = V_K$ to be the set
of vertices of $K$ and $Y = S_K$, the set of simplices of $K$ with
$x R y$ if $x$ is a vertex of the simplex $y$.
=--

**Returning to the general situation,** we define two [[simplicial
complex|simplicial
complexes]] associated to $R$, as follows:

1. $K = K_R:$

   * the set of vertices is the set $X$;

   * a $p$-simplex of $K$ is a set $\{x_0, \cdots, x_p\}\subseteq X$ such that there is some $y \in Y$ with $x_i R y$ for $i = 0, 1, \cdots, p$.

2. $L = L_R :$

   * the set of vertices is the set $Y$;
   * a $p$-simplex of $K$ is a set $\{y_0, \cdots, y_p\}\subseteq Y$ such that there is some $x \in X$ with $x R y_j$ for $j  = 0, 1, \cdots, p$.

Clearly the two constructions are in some sense dual to each other.

+-- {: .un_remark}
###### Note

These two simplicial complexes were denoted $V(R)$ and $C(R)$ at [[Vietoris complex]].
=--

We next need some classical subdivision ideas.

### Barycentric  subdivisions

Combinatorially, if $K$ is a simplicial complex with vertex set $V_K$, then one associates to $K$ the partially ordered set of its simplices. Explicitly we write $S_K$ for the set of simplices of $K$ and $(S_K, \subseteq)$ for the partially ordered set with $\subseteq$ being the obvious inclusion. The **barycentric subdivision**, $K'$, of $K$ has $S_K$ as its set of vertices and a finite set of vertices of $K'$ (i.e. simplices of $K$) is a simplex of $K'$ if it is totally ordered by inclusion. (Thus $K'$ is the simplicial complex given by taking the nerve of the poset, $(S_K, \subseteq)$.)

+-- {: .un_remark}
###### Note

It is important to note that there is in general _no_ natural simplicial map from $K'$ to $K$. If however $V_K$ is ordered in such a way that the vertices of any simplex in $K$ are totally ordered (for instance by picking a total order on $V_K$), then one can easily specify a map
$$ \varphi : K' \rightarrow K$$
by:

* if $\sigma' = \{ x_0, \cdots, x_p\}$ is a vertex of $K'$ (so $\sigma'
\in S_K$), let $\varphi \sigma'$ be the least vertex of $\sigma'$ in
the given fixed order.

This preserves simplices, but reverses order so
if $\sigma_1' \subset \sigma_2'$ then $\varphi (\sigma_1') \geq
\varphi(\sigma_2')$.
=--

If one changes the order, then the resulting map is
_contiguous_:

## Contiguity

 Let $\varphi, \psi : K \rightarrow L$ be
two simplicial maps between [[simplicial complex|simplicial complexes]]. They are **contiguous** if for any simplex $\sigma$ of $K$, $\varphi (\sigma) \cup \psi
(\sigma)$ forms a simplex in $L$.

Contiguity gives a constructive form of homotopy applicable to
simplicial maps.

If $\psi : K \rightarrow L$ is a simplicial map, then it induces
$\psi' : K' \rightarrow L'$ after subdivision. As there is no way
of knowing / picking compatible orders on $V_K$ and $V_L$ in
advance, we get that on constructing

$$\varphi_K : K' \rightarrow K$$

and

$$\varphi_L : L' \rightarrow L$$

that $\varphi_L \psi'$ and $\psi \varphi_K$ will be contiguous to each
other but rarely equal.

**Returning to $K_R$ and $L_R$,** we order the elements of $X$ and
$Y$. Then suppose $y'$ is a vertex of $L'_R$, so $y' = \{y_0,
\cdots, y_p\}$, a simplex of $L_R$ and there is an element $x \in
X$ with $x R y_i, i = 0, 1, \cdots, p$. Set $\psi y' = x$ for one
such $x$.

If $\sigma = \{ y'_0, \cdots, y'_q\}$ is a $q$-simplex of $L'_R$,
assume $y'_0$ is its least vertex (in the inclusion ordering)

$$\varphi_L (y^\prime_0) \in y'_0 \subset y^\prime  for each  y_i \in
\sigma$$

hence $\psi y'_i R \varphi_L (y'_0)$ and the elements $\psi y'_0,
\cdots, \psi y'_q$ form a simplex in $K_R$, so $\psi : L'_R
\rightarrow K_R$ is a simplicial map. It, of course, depends on the
ordering used and on the choice of $x$,  but any other choice $\bar x$
for $\psi y'$ gives a contiguous map.

Reversing the roles of $X$ and $Y$ in the above we get a
simplicial map

$$\bar\psi : K_R' \rightarrow L_R.$$

Applying barycentric subdivisions again gives

$$\bar\psi' : K_R'' \rightarrow L_R'$$

and composing with $\psi : L_R' \rightarrow K_R$ gives a map

$$\psi \bar\psi' : K_R'' \rightarrow K_R.$$

Obviously, $K'$ is ordered by inclusion of simplices, so there is a map $\varphi_{K'}:K_R''\rightarrow K_R'$, and then we get a map

$$\varphi_K \varphi_{K'} : K_R''\rightarrow K_R.$$

+-- {: .un_proposition}
###### Proposition (Dowker, 1952, p. 88)

The two maps $\varphi_K \varphi_{K'}$ and $\psi \bar\psi'$ are contiguous.
=--

Before proving this, note that contiguity implies homotopy and that
$\varphi_K \varphi_{K'}$ is homotopic to the identity map on $K_R$ after
realisation, i.e., this proves the following modulo the proof of the proposition (see addendum below):

+-- {: .un_theorem}
###### Dowker's Theorem

$|K_R| \simeq |L_R|.$
=--

The homotopy depends on the ordering of the vertices and so is not
natural.

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

Let $\sigma''' = \{ x_0'', x_1'', \cdots, x_q''\}$ be a simplex of
$K_R''$ and as usual assume $x_0''$ is its least vertex, then for all
$i \gt 0$

$$x_0'' \subset x_i''.$$

We have that $\varphi_{K'}$ is clearly order reversing so $\varphi_{K'}
x_i'' \subseteq \varphi_{K'} x_0''$. Let $y = \bar\varphi \varphi_{K'}
x_0''$, then for each $x \in \varphi_{K'} x_0''$, $x Ry$. Since $\varphi_K
\varphi_{K'} x_i'' \in \varphi_{K'} x_i'' \subseteq \varphi_{K'} x_0''$, we
have $\varphi_K \varphi_{K'} x_i'' R y$.

For each vertex $x'$ of $x_i'', \bar \psi x' \in \bar \psi'
x_i''$, hence as $\varphi_{K'}x_0'' \in x_0'' \subset x_i'', y =
\bar \psi \varphi_{K'} xx_0'' \in \bar \psi' x_i''$ for each $x_i
''$, so for each $x_i'', \psi \bar \psi' x_i'' R y$, however we
therefore have

$$ \varphi_k \varphi_{K'} (\sigma '') \cup \psi \bar \psi (\sigma''') = \bigcup \varphi_K \varphi_{K'} (x_i'')\cup \psi \bar \psi; x_i ''$$

forms a simplex in $K_R$, i.e.
$\varphi_K \varphi_{K'}$ and $\psi \bar \psi'$ are contiguous.

=--

**Addendum.** In the current form, it is difficult to see why $\bar\psi' \psi$ should be homotopic to the identity on $|L_R|$. Notice that this is a necessary step in proving the homotopy equivalence between $|K_R|$ and $|L_R|$. The purpose of this addendum is to make a slightly different claim which yields a homotopy equivalence between $|K_R|$ and $|L_R|$.

From above, observe that we have maps $\bar\psi \varphi_{K'}:K_R'' \rightarrow L_R$ and $\varphi_L \bar\psi' : K_R'' \rightarrow L_R$. Dowker proves the following:

+-- {: .un_proposition}
###### Proposition (Dowker 1952, Lemma 6, p. 89)

The two maps $\bar\psi \varphi_{K'}$ and $\varphi_L \bar\psi'$ are contiguous.
=--

Thus upon passing to the geometric realization, we get that the realizations of the maps are homotopic, i.e. $|\bar\psi| \simeq |\bar\psi'|$. Notice that we are using the fact that $|\varphi_{K'}|$ and $|\varphi_L|$ are homotopic to the respective identity maps.

Thus we obtain $|\psi||\bar\psi'| \simeq |\psi||\bar\psi|$, and in particular, the latter composition is homotopic to the identity on $|K_R|$.

By symmetric arguments, we obtain the following results:

1. $\bar\psi \psi'$ and $\varphi_L \varphi_{L'}$ are contiguous.
2. $\varphi_K \psi'$ and $\psi \varphi_{L'}$ are contiguous.

Passing to the geometric realization, we get that $|\bar\psi| |\psi'|$ is homotopic to the identity on $|L_R|$, and also that $|\psi'|\simeq |\psi|$. Thus we see that $|\bar\psi||\psi|$ is homotopic to the identity on $|L_R|$. It follows that $|\psi|: |L_R| \rightarrow |K_R|$ is a homotopy equivalence, with $|\bar\psi|$ as its homotopy inverse.

## References

*  C. H. Dowker, _Homology Groups of Relations_, Annals of Math. 56 (1952), 84–95.

[[!redirects Dowker's theorem]]
[[!redirects Dowker's Theorem]]
[[!redirects Dowker\'s theorem]]
[[!redirects Dowker\'s Theorem]]
[[!redirects Dowker theorem]]
[[!redirects Dowker Theorem]]


down quark

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### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the [[standard model of particle physics]] the _down quark_ is the down-type [[quark]] in the first [[generation of fermions]].

## Related concepts

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

## References

See also

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

[[!redirects down quarks]]

[[!redirects down-quark]]
[[!redirects down-quarks]]


down set

In a [[partial order|poset]] or even [[preorder|proset]], the __down set__ of an element $x$ is the set
$$ x{\downarrow} = \{ y \;|\; y \leq x \} .$$

In a [[quasiorder]], the __strict down set__ of $x$ is the set
$$ x\dot{\downarrow} = \{ y \;|\; y \lt x \} .$$

If you think of a poset $P$ as a [[category]], then the down set of $x$ is the [[slice category]] $P / x$.

A down set in the [[opposite category|opposite]] $P^{op}$ of $P$ is an [[up set]] in $P$.

Note: The term 'down set' is also often used for a [[lower set]], a more general concept.  In the terminology above, the down set of $x$ is the lower set [[Moore closure|generated by]] $x$.

[[!redirects downset]]
[[!redirects down-set]]
[[!redirects down set]]
[[!redirects downsets]]
[[!redirects down-sets]]
[[!redirects down sets]]


down-up algebra

* Georgia Benkart, Tom Roby, _Down-up algebras_, [math.RT/9803159](http://arxiv.org/abs/math/9803159)
* Ellen Kirkman, Ian M. Musson, D. S. Passman, _Noetherian down-up algebras_, [pdf](http://www.math.wisc.edu/~passman/noethdu.pdf)
* Georgia Benkart, Sarah Witherspoon, _A Hopf structure for down-up algebras_, Mathematische Zeitschrift __238__:3, 523-553, [doi](http://dx.doi.org/10.1007/s002090100265)
* Ellen E. Kirkman, Ian M. Musson, _Hopf down-up algebras_, [pdf](http://www.math.wfu.edu/publications/kirkman/downupnothopf11.PDF)
* Ellen Kirkman, James Kuzmanovich, _Non-noetherian down-up algebras_, Communications in Algebra __28__:11 (2000) 5255-5268
[doi](http://dx.doi.org/10.1080/00927870008827153)
[[!redirects down-up algebras]]


doxastic logic

\tableofcontents

## Idea

*Doxastic logic* is meant to be the form of [[modal logic]] which allows to express propositions of the form: "$p$ is *believed* to be true".

## Related concepts

* [[epistemic logic]]

* [[temporal logic]]

## References

See also:

* Wikipedia, *[Doxastic logic](https://en.wikipedia.org/wiki/Doxastic_logic)*

[[!redirects doxastic modal logic]]


Dp-D(p+2)-brane bound state

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###Context###
#### String theory
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[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[bound state]] of [[D-brane|Dp-]] with [[D-brane|D(p+2)-branes]].

## Examples

* [[Dp-D(p+2)-brane bound state]]

  * [[D1-D3-brane bound state]]

  * [[D2-D4-brane bound state]]

  * [[D6-D8-brane bound state]]

## Properties

[[!include Dp-D(p+2)-brane intersections in fuzzy funnels -- section]]

### $\mathrm{D}6 \perp \mathrm{D}8$-brane intersections

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](#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 below 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]].

### The s-rule

What has come to be known as the _[[s-rule]]_ is the conjecture that the configuration of [[Dp-D(p+2)-brane bound states]] with the [[D-brane|Dp-branes]] stretching from the [[D-brane|D(p+2)-branes]] to  [[NS5-branes]], can be [[supersymmetry|supersymmetric]] only if at most one D$p$-brane ends on any one D$(p+2)$-brane.

For [[D4-D6 brane intersections]]:

> graphics grabbed from [Fazzi 17](D4-D6-brane+intersection#Fazzi17)

For [[D6-D8 brane intersections]]:

> graphics grabbed from [Fazzi 17](D6-D8-brane+intersection#Fazzi17)

> graphics grabbed from [Gaiotto-Tomasiello 14](D6-D8-brane+intersection#GaiottoTomasiello14)

\linebreak

\linebreak

### Single trace observables as $\mathfrak{su}(2)$-weight systems on chord diagrams
  {#SingleTraceObservablesAsWeightSystemsOnChordDiagrams}

We discuss how the [[single trace observables]] on the [[fuzzy 2-sphere]]-sections  of [[Dp-D(p+2) brane intersection]] [[fuzzy funnels]] are given by [[su(2)]]-[[Lie algebra weight systems]] on [[chord diagrams]] (following [Ramgoolam-Spence-Thomas 04](#RamgoolamSpenceThomas04), [McNamara-Papageorgakis 05](#McNamaraPapageorgakis05), see [McNamara 06, Section 4](#McNamara06) for review).

For more see at _[[weight systems on chord diagrams in physics]]_.

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

\linebreak

While in the commutative [[large N limit]], all powers of the [[radius]] function on the [[fuzzy 2-sphere]] are equal

$$
  \underset{N\to \infty}{\lim}
  \int_{S^2_N} R^{2 k}
  \;=\;
  4 \pi
  \,;
$$

for [[finite number|finite]] $N$ there is an ordering ambiguity: In fact, the number of functions on the [[fuzzy 2-sphere]] at [[finite number|finite]] $N$ that all go to the same function $R^{2k}$ in the [[large N limit]] grows rapidly with $k$.

At $k = 1$ there is the single radius observable (eq:RadiusOfFuzzy2Sphere)

$$
  \int_{S^2_N} R^2
  \;=\;
  \int_{S^2_N}
  \underset{i}{\sum} X_i \cdot X_i
  \;=\;
  4 \pi \tfrac{ N }{ \sqrt{N^2 -1} }
$$

At $k = 2$ there are, under the integral (eq:FuzzyS2Integration), two radius observables:

1. $ \int_{S^2_N} \underset{i,j}{\sum} X_i X_i X_j X_j$

1. $\int_{S^2_N} \underset{i,j}{\sum} X_i X_j X_j X_i$

(Here we are using that under the integral/trace, a [[cyclic permutation]] of the factors in the integrand does not change the result).

Similarly for higher $k$, where the number of possible orderings increases rapidly. The [[combinatorics]] that appears here is familiar in [[knot theory]]:

Every ordering of operators, up to cyclic permutation, in the [[single trace observable]] $Tr(R^2)^n$ is encoded in a [[chord diagram]] and the value of the corresponding [[single trace observable]] is the value of the [[su(2)]]-[[Lie algebra weight system]] on this chord diagram.

### Parallel intersection

Parallel: dissolves ([Gava-Narain-Sarmadi 97](#GavaNarainSarmadi97))

## Related concepts

* [[fuzzy funnel]]

\linebreak

[[!include brane bound states -- table]]

## References

### For parallel intersection

For parallel intersection:

* {#GavaNarainSarmadi97}  E. Gava, K.S. Narain, M.H. Sarmadi, _On the Bound States of $p$- and $(p+2)$-Branes_, Nucl. Phys. B504 (1997) 214-238 ([arxiv:hep-th/9704006](https://arxiv.org/abs/hep-th/9704006))

### For transversal intersections

#### As spikes/BIons
 {#AsSpikesBIons}

On [[Dp-D(p+2) brane intersections]] as spikes/[[BIons]]

* [[Curtis Callan]], [[Juan Maldacena]], _Brane Dynamics From the Born-Infeld Action_, Nucl. Phys. B513 (1998) 198-212 ([arXiv:hep-th/9708147](https://arxiv.org/abs/hep-th/9708147))

* [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Self-Dual String Soliton_, Nucl. Phys. B515 (1998) 203-216 ([arXiv:hep-th/9709014](https://arxiv.org/abs/hep-th/9709014))

* [[Gary Gibbons]], _Born-Infeld particles and Dirichlet p-branes_,  	Nucl. Phys. B514: 603-639, 1998 ([arXiv:hep-th/9709027](https://arxiv.org/abs/hep-th/9709027))

from the [[M5-brane]]:

* {#HoweLambertWest97} [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Self-Dual String Soliton_, Nucl. Phys. B515 (1998) 203-216 ([arXiv:hep-th/9709014](https://arxiv.org/abs/hep-th/9709014))

#### As fuzzy funnels/Yang-Mills monopoles
 {#ReferencesRelationToMonopoles}

On transversal [[Dp-D(p+2) brane intersections]] as [[Yang-Mills monopoles]] / [[fuzzy funnel]]-solutions to [[Nahm's equation]]:

For transversal [[D1-D3 brane intersections]]:

* {#Diaconescu97} Duiliu-Emanuel Diaconescu,  _D-branes, Monopoles and Nahm Equations_, Nucl. Phys. B503 (1997) 220-238 ([arxiv:hep-th/9608163](https://arxiv.org/abs/hep-th/9608163))

* [[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))

* {#ConstableMyers99} [[Neil Constable]], [[Robert Myers]], Oyvind Tafjord, _The Noncommutative Bion Core_, Phys. Rev. D61 (2000) 106009 ([arXiv:hep-th/9911136](https://arxiv.org/abs/hep-th/9911136))

* [[Robert Myers]], Section 4 of: _Nonabelian D-branes and Noncommutative Geometry_, J. Math. Phys. 42: 2781-2797, 2001 ([arXiv:hep-th/0106178](https://arxiv.org/abs/hep-th/0106178))

* [[Neil Constable]], [[Neil Lambert]], _Calibrations, Monopoles and Fuzzy Funnels_, Phys. Rev. D66 (2002) 065016 ([arXiv:hep-th/0206243](https://arxiv.org/abs/hep-th/0206243))

* Jessica K. Barrett, Peter Bowcock, _Using D-Strings to Describe Monopole Scattering_ ([arxiv:hep-th/0402163](https://arxiv.org/abs/hep-th/0402163))

* Jessica K. Barrett, Peter Bowcock, _Using D-Strings to Describe Monopole Scattering - Numerical Calculations_ ([arxiv:hep-th/0512211](https://arxiv.org/abs/hep-th/0512211))

* {#ThomasWard06} Steven Thomas, John Ward, _Electrified Fuzzy Spheres and Funnels in Curved Backgrounds_, JHEP 0611:019, 2006 ([arXiv:hep-th/0602071](https://arxiv.org/abs/hep-th/0602071))

For transversal [[D2-D4-brane bound states]] (with an eye towards [[AdS/QCD]]):

* {#GZZ09} Alexander Gorsky, Valentin Zakharov, Ariel Zhitnitsky, _On Classification of QCD defects via holography_, Phys. Rev. D79:106003, 2009 ([arxiv:0902.1842](https://arxiv.org/abs/0902.1842))

For transversal [[D3-D5 brane intersections]]:

* {#GaiottoWitten08} [[Davide Gaiotto]], [[Edward Witten]], Section 2.4 of: _Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory_, J Stat Phys (2009) 135: 789 ([arXiv:0804.2902](https://arxiv.org/abs/0804.2902))

For transversal [[D6-D8 brane intersections]] (with an eye towards [[AdS/QCD]]):

* {#HLPY08} Deog Ki Hong, Ki-Myeong Lee, Cheonsoo Park, Ho-Ung Yee, Section V of: _Holographic Monopole Catalysis of Baryon Decay_, JHEP 0808:018, 2008 ([https:arXiv:0804.1326](https://arxiv.org/abs/0804.1326))

and as transversal [[D6-D8-brane bound states]] on a [[half NS5-brane]] in [[type I' string theory]]:

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

Making explicit the completion of the $\mathfrak{su}(2)_{\mathbb{C}} \simeq \mathfrak{sl}(2,\mathbb{C})$-representation to a $\mathfrak{gl}(2,\mathbb{C})$-representation by adjoining the gauge field component $A_y$ to the scalar fields $\vec X$:

* [Gaiotto-Witten 08, Section 3.1.1](#GaiottoWitten08)

* [[Sergey Cherkis]], _Instantons on Gravitons_, around (21) in: Commun. Math. Phys. 306:449-483, 2011 ([arXiv:1007.0044](https://arxiv.org/abs/1007.0044))

#### Relation to Vassiliev braid invariants
 {#ReferencesRelationToKnotInvariants}

Relation of [[Dp-D(p+2)-brane bound states]] ([hence](#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]) to [[Vassiliev braid invariants]] via [[chord diagrams]] computing [[radii]] of [[fuzzy spheres]]:

* [[Sanyaje Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: _Resolving brane collapse with $1/N$ corrections in non-Abelian DBI_, Nucl. Phys. B703 (2004) 236-276 ([arxiv:hep-th/0405256](https://arxiv.org/abs/hep-th/0405256))

* [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))

* {#McNamara06} [[Simon McNamara]], Section 4 of: _Twistor Inspired Methods in Perturbative Field Theory and Fuzzy Funnels_, 2006 ([spire:1351861](http://inspirehep.net/record/1351861), [pdf](https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf), [[McNamara06.pdf:file]])

* [[Constantinos Papageorgakis]], p. 161-162 of: _On matrix D-brane dynamics and fuzzy spheres_, 2006 ([[Papageorgakis06.pdf:file]])

#### Lift to M2-M5-brane bound states
 {#ReferencesLiftToMTheory}

The lift of [[Dp-D(p+2)-brane bound states]] in [[string theory]] to [[M2-M5-brane bound states]]/[[E-strings]] in  [[M-theory]], under [[duality between M-theory and type IIA string theory]]+[[T-duality]], via generalization of [[Nahm's equation]] (this eventually motivated the [[BLG-model]]/[[ABJM model]]):

* [[Anirban Basu]], [[Jeffrey Harvey]], _The M2-M5 Brane System and a Generalized Nahm's Equation_, Nucl.Phys. B713 (2005) 136-150 ([arXiv:hep-th/0412310](https://arxiv.org/abs/hep-th/0412310))

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 2.2.1 of _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

### Single trace observables as weight systems on chord duagrams

Relation of [[single trace observables]] in the [[non-abelian DBI action]] on [[Dp-D(p+2)-brane bound states]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]) to [[su(2)]]-[[Lie algebra weight systems]] on [[chord diagrams]] computing [[radii]] averages of [[fuzzy spheres]]:

* {#RamgoolamSpenceThomas04} [[Sanyaje Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: _Resolving brane collapse with $1/N$ corrections in non-Abelian DBI_, Nucl. Phys. B703 (2004) 236-276 ([arxiv:hep-th/0405256](https://arxiv.org/abs/hep-th/0405256))

* {#McNamaraPapageorgakis05} [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))

* {#McNamara06} [[Simon McNamara]], Section 4 of: _Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels_, 2006 ([spire:1351861](http://inspirehep.net/record/1351861), [pdf](https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf), [[McNamara06.pdf:file]])

* [[Constantinos Papageorgakis]], p. 161-162 of: _On matrix D-brane dynamics and fuzzy spheres_, 2006 ([[Papageorgakis06.pdf:file]])

### In solid state physics

Relation of [[Yang-Mills monopoles]] as [[Dp/D(p+2)-brane intersections]] and [[Yang-Mills instantons]] as [[Dp/D(p+4)-brane intersections]] to the [[K-theory classification of topological phases of matter]] via [[AdS/CFT duality in solid state physics]]:

* [[Koji Hashimoto]], [[Taro Kimura]], _Band spectrum is D-brane_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 1 ([arXiv:1509.04676](https://arxiv.org/abs/1509.04676))

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Dp-D(p+2)-brane intersections in fuzzy funnels -- section

### Transveral Dp-D(p+2)-brane intersections in fuzzy funnels

The [[boundary condition]] in the [[nonabelian DBI model]] of [[coincident brane|coincident]] [[D-branes|Dp-branes]] describing their transversal [[intersecting branes|intersection/ending]] [[Dp-D(p+2)-brane bound states|with/on D(p+2)-branes]] is controled by [[Nahm's equation]] and thus exhibits the [[intersecting branes|brane intersection]]-locus equivalently as:

1. a [[fuzzy funnel]] [[noncommutative geometry]] interpolating between the $\mathrm{D}p$- and the $\mathrm{D}(p+2)$-brane worldvolumes;

1. [[geometric engineering of QFT|geometric engineering]] of [[Yang-Mills monopoles]] in the [[worldvolume]]-theory of the ambient $D(p+2)$-branes.

([Diaconescu 97](Dp-Dpplus2-brane+bound_state#Diaconescu97),
[Constable-Myers-Fafjord 99](Dp-Dpplus2-brane+bound+state#ConstableMyers99), [Hanany-Zaffaroni 99](Dp-Dpplus2-brane+bound+state#HananyZaffaroni99), [Gaiotto-Witten 08, Section 2.4](Dp-Dpplus2-brane+bound+state#GaiottoWitten08), [HLPY 08](Dp-Dpplus2-brane+bound+state#HLPY08), [GZZ 09](Dp-Dpplus2-brane+bound+state#GZZ09))

More explicitly, for $y \in (0,\infty ]$ the transversal distance along the stack of $N$ $\mathrm{D}p$-branes away from the $\mathrm{D}(p+2)$-brane, and for

$$
  X^i \in C^\infty\big( (0,\infty], \mathfrak{u}(N) \big)
  \phantom{AAA}
  i \in \{1,2,3\}
$$

the three scalar fields on the worldvolume, the [[boundary condition]] is:

$$
  \frac{d}{d y}
  X^3
  +
  [X^1, X^2]
  \;=\;
  0
  \,,
  \;\;\;
  \frac{d}{d y}
  X^1
  +
  [X^2, X^3]
  \;=\;
  0
  \,,
  \;\;\;
  \frac{d}{d y}
  X^2
  +
  [X^3, X^1]
  \;=\;
  0
$$

as $y \to 0$. These are _[[Nahm's equations]]_, solved by

$$
  X^i(y) = \frac{1}{y} \rho^i + \text{non-singular}
$$

where

$$
  \rho \;\colon\; \mathfrak{su}(2) \longrightarrow \mathfrak{u}(N)
$$

is a Lie algebra homomorphism from [[su(2)]] to the [[unitary Lie algebra]], and

$$
  \rho^i \coloneqq \rho(\sigma^i)
$$

is its complex-linear combination of values on the canonical [[Pauli matrix]] [[linear basis|basis]].

{#FuzzyFunnelEmerges} Equivalently. $\rho$ is an $N$-[[dimension|dimensional]] [[complex numbers|complex]] [[Lie algebra representation]] of [[su(2)]]. Any such is [[reducible representation|reducible]] as a [[direct sum]] of [[irreducible representations]] $\mathbf{N}^{(M5)}$, for which there is exactly one, up to [[isomorphism]], in each dimension $N^{(M5)} \in \mathbb{N}$:

\[
  \label{rhoAsADirectSumOfsu2Irreps}
  \rho
  \;\simeq\;
  \underset{
    i
  }{\bigoplus}
  \big(
    N_i^{(M2)}
    \cdot
    \mathbf{N}_i^{(M5)}
  \big)
  \,.
\]

(Here the notation follows the discussion at _[M2/M5-brane bound states in the BMN model](BMN+matrix+model#M2M5BraneBoundStatesInTheBMNMatrixModel)_, which is the [[duality between M-theory and type IIA string theory|M-theory lift]] of the present situation).

Now each [[irrep]] $\mathbf{N}_i^{(M5)}$ may be interpreted as a [[fuzzy 2-sphere]] of [[radius]] $\propto \sqrt{ \left( N_i^{(M5)}\right)^2 - 1 }$, hence as the section of a [[fuzzy funnel]] at given $y = \epsilon$, whence the totality of (eq:rhoAsADirectSumOfsu2Irreps) represents a system of concentric [[fuzzy 2-spheres]]/[[fuzzy funnels]].

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

\linebreak

Moreover, since the [[complexification]] of [[su(2)]] is the [[sl(2)|complex special linear Lie algebra]] $\mathfrak{sl}(2,\mathbb{C})$ ([here](su2#Complexification)) the solutions to the boundary conditions are also identified with [[finite-dimensional vector space|finite-dimensional]] $\mathfrak{sl}(2,\mathbb{C})$ [[Lie algebra representations]]:

\[
  \label{RhoAsAnsl2Rep}
  \rho \;\in\; \mathfrak{sl}(2,\mathbb{C}) Rep
  \,.
\]

\linebreak

This is what many authors state, but it is not yet the full picture:

{#ParallelCPField} Also the worldvolume [[Chan-Paton gauge field]] component $A$ along $y$ participates in the brane intersection

$$
  A \in C^\infty\big( (0,\infty], \mathfrak{u}(N) \big)
$$

its boundary condition being that

$$
  [A, X^i] \;=\; 0
  \phantom{AAAA}
  \text{for all}\; i \in \{1,2,3\}
$$

as $y \to 0$ ([Constable-Myers 99, Section 3.3](Dp-Dpplus2-brane+bound+state#ConstableMyers99), [Thomas-Ward 06, p. 16](Dp-Dpplus2-brane+bound+state#ThomasWard06), [Gaiotto-Witten 08, Section 3.1.1](Dp-Dpplus2-brane+bound+state#GaiottoWitten08))

Together with (eq:RhoAsAnsl2Rep) this means that the quadruple of fields $(X^1,X^2,X^3,A)$ constitutes a [[Lie algebra representation]] of the [[general linear Lie algebra]]

$$
  \mathfrak{gl}(2,\mathbb{C})
  \;\simeq\;
  \underset{
    \langle
      X^1, X^2, X^3
    \rangle
  }{
  \underbrace{
    \mathfrak{sl}(2,\mathbb{C})
  }
  }
  \oplus
  \underset{
    \langle A \rangle
  }{
    \underbrace{
      \mathbb{C}
    }
  }
$$

This makes little difference as far as bare [[Lie algebra representations]] are concerned, but it does make a crucial difference when these are regarded as [[metric Lie representations]] of [[metric Lie algebras]], since $\mathfrak{gl}(2,\mathbb{C})$ admits further invariant metrics...

\linebreak


Dp-D(p+4)-brane bound state

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#### String theory
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{:toc}

## Idea

[[bound states]] of [[D-branes|Dp-branes]] with [[D-brane|D(p+4)-branes]].

## Properties

### As Yang-Mills instantons in the Chan-Paton gauge field
 {#AsYangMillsInstantons}

Due to the [[higher WZW term]] $\propto \int_{D_{p+4}} C_{p+1} \wedge \langle F \wedge F \rangle$ in the [[Green-Schwarz sigma model]] for [[D-brane|D(p+4)-branes]], [[Yang-Mills instantons]] in the [[Chan-Paton gauge field]] on $D (p+4)$-branes are equivalently [[Dp-D(p+4)-brane bound states]] (see e.g. [Polchinski 96, 5.4](#Polchinski96), [Tong 05, 1.4](#Tong05)).

## Examples

[[!include Dp-D(p+4)-brane bound states -- contents]]

## Related concepts

* [[fuzzy funnel]]

\linebreak

[[!include brane bound states -- table]]

## References

### Relation to instantons

The argument that [[Yang-Mills instantons]] in the [[Chan-Paton gauge field]] on a [[D-brane|D(p+4)-brane]] are equivalent to [[Dp-D(p+4) brane bound states]] goes back to

* [[Edward Witten]], _Small Instantons in String Theory_,  	Nucl. Phys. B460:541-559, 1996 ([arXiv:hep-th/9511030](https://arxiv.org/abs/hep-th/9511030))

* [[Michael Douglas]], _Branes within Branes_, In: Baulieu L., Di Francesco P., Douglas M., Kazakov V., Picco M., Windey P. (eds.) _[Strings, Branes and Dualities](https://link.springer.com/book/10.1007/978-94-011-4730-9)_ NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 520. Springer, Dordrecht ([arxiv:hep-th/9512077](https://arxiv.org/abs/hep-th/9512077), [doi:10.1007/978-94-011-4730-9_10](https://doi.org/10.1007/978-94-011-4730-9_10))

* [[Michael Douglas]], _Gauge Fields and D-branes_, J. Geom. Phys. 28 (1998) 255-262 ([arXiv:hep-th/9604198](https://arxiv.org/abs/hep-th/9604198))

Review is in:

* {#Polchinski96} [[Joseph Polchinski]], Section 5.4 of: _TASI Lectures on D-Branes_ ([arXiv:hep-th/9611050](https://arxiv.org/abs/hep-th/9611050))

* {#Tong05} [[David Tong]], Section 1.4 of _TASI Lectures on Solitons_ ([hep-th/0509216](https://arxiv.org/abs/hep-th/0509216))

### D0-D4-brane bound states

Discussion specifically of [[D0-D4-brane bound states]]:

* [[Cumrun Vafa]], _Instantons on D-branes_, Nucl. Phys. B463 (1996) 435-442 ([arXiv:hep-th/9512078](https://arxiv.org/abs/hep-th/9512078))

with emphasis to the resulting [[configuration spaces of points]], as in

* [[Cumrun Vafa]], [[Edward Witten]], Section 4.1 of: _A Strong Coupling Test of S-Duality_, Nucl. Phys. B431:3-77, 1994 ([arXiv:hep-th/9408074](https://arxiv.org/abs/hep-th/9408074))

### D1-D5-brane bound states

Discussion specifically of [[D1-D5-brane bound states]]

* [[Neil Lambert]], _D-brane Bound States and the Generalised ADHM Construction_, Nucl. Phys. B519 (1998) 214-224 ([arXiv:hep-th/9707156](https://arxiv.org/abs/hep-th/9707156))

* {#Martinec99} [[Emil Martinec]], _Manifestations of the D1-D5 system_, 1999 ([[MartinecD1D5.pdf:file]])

* _A review of the D1/D5 system and five dimensional black hole from supergravity and brane viewpoint_ ([arXiv:hep-th/0002184](http://arxiv.org/abs/hep-th/0002184))

### D4-D8-brane bound states

Discussion specifically of [[D4-D8-brane bound states]]:

In the [[Witten-Sakai-Sugimoto model]] [[geometric engineering of QFT|geometrically engineering]] [[QCD]], where the [[D4-branes]] get interpreted as [[baryons]]:

* {#SakaiSugimoto04} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], Section 5.7 of: _Low energy hadron physics in holographic QCD_, Prog. Theor. Phys.113:843-882, 2005 ([arXiv:hep-th/0412141](https://arxiv.org/abs/hep-th/0412141))

### M5-MO9 brane bound states

Discussion specifically of [[M5-MO9 brane bound states]]:

Discussion as [[Yang-Mills instantons]] ("small instantons"):

* {#Strominger90} [[Andrew Strominger]], _Heterotic solitons_, Nucl. Phys. B343 (1990) 167-184 (doi:10.1016/0550-3213(90)90599-9) Erratum: Nucl. Phys. B353 (1991) 565-565 (doi:10.1016/0550-3213(91)90349-3) ([spire:27900](http://inspirehep.net/record/27900))

* {#Witten96} [[Edward Witten]], _Small Instantons in String Theory_,  	Nucl.Phys.B460:541-559, 1996  (doi:10.1016/0550-3213(95)00625-7)

Discussion as [[half M5-branes]]:

* {#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]]_, CMP (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))

* {#SatiSchreiber19} [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], Section 4.1 of: _[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]_ ([arXiv:1909.12277](https://arxiv.org/abs/1909.12277))

### In solid state physics

Relation of [[Yang-Mills monopoles]] as [[Dp/D(p+2)-brane intersections]] and [[Yang-Mills instantons]] as [[Dp/D(p+4)-brane intersections]] to the [[K-theory classification of topological phases of matter]] via [[AdS/CFT duality in solid state physics]]:

* [[Koji Hashimoto]], [[Taro Kimura]], _Band spectrum is D-brane_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 1 ([arXiv:1509.04676](https://arxiv.org/abs/1509.04676))

* Charlotte Kristjansen, Gordon W. Semenoff, _The D3-probe-D7 brane holographic fractional topological insulator_, JHEP10 (2016) 079 ([arXiv:1604.08548](https://arxiv.org/abs/1604.08548))

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Dp-D(p+4)-brane bound states -- contents

**[[Dp-D(p+4)-brane bound states]] and [[Yang-Mills instantons]]**

* [[D0-D4 brane bound state]]

* [[D1-D5 brane bound state]]

* [[D4-D8 brane bound state]]

* [[M5-MO9 brane bound state]]


dR-flat modality

the (delooping of) the kernel of the [[counit]] of the [[flat modality]]

see at _[[cohesive (infinity,1)-topos -- structures]]_ the section on [de Rham cohomology](cohesive+topos+--+structures#deRhamCohomology)


dR-shape modality

the (looping of) the cokernel of the [[unit]] of the [[shape modality]]

notation: $\Pi_{dR}$ or $ʃ_{dR}$

see at _[[cohesive (infinity,1)-topos -- structures]]_ the section on [de Rham cohomology](cohesive+topos+--+structures#deRhamCohomology)


dream mathematics

# Dream mathematics
* table of contents
{: toc}

## Idea

Dream mathematics is an alternative [[foundation of mathematics]] first studied (as a piece of [[metamathematics]]) by [[Robert Solovay]].  Several 'ill-behaved' counterexamples in [[analysis]] fail to exist in it.

## Definition

__Dream mathematics__ is [[mathematics]] founded on [[ZF]] (or an equivalent [[structural set theory]] such as [[SEAR]]) with [[dependent choice]] and the following axioms (any of which contradict [[full choice]]) required of every [[subset]] $A$ of the [[real line]]:

*  [[Lebesgue-measurable set|Lebesgue measurability]]: $A$ is the [[union]] of a [[Borel set]] and a [[null set]];
*  The [[perfect-set property]]: $A$ is the [[union]] of a [[perfect set]] and a [[countable set]];
*  The [[Baire property]]: $A$ is the [[union]] of an [[open set]] and a [[meager set|meagre set]].

A __dream universe__ is any [[model]] of dream mathematics.  The most well known (and the first known) is the _Solovay model_.

## Consistency

Solovay proved that dream mathematics is [[consistent theory|consistent]] if the existence of an [[inaccessible cardinal]] is consistent with [[ZFC]].  More precisely, Solovay showed how to construct a [[model]] of dream mathematics (now called the _Solovay model_) from any model of $ZFC$ with an inaccessible cardinal.

[[Saharon Shelah]] later showed that one could start with *any* model of $ZFC$ and construct a model of $ZF + DC$ in which every set of reals has the Baire property; on the other hand, [[Ernst Specker]] had already shown that an inaccessible cardinal must be consistent if the perfect set property is. Various intermediate consistency results for Lebesgue measurability are also known, but a complete characterisation is still elusive.

François G. Dorais later showed in his answer to [[Alex Simpson]]'s [MathOverflow question](https://mathoverflow.net/questions/141801/how-strong-is-all-sets-are-lebesgue-measurable-in-weaker-contexts-than-zf) that $BZ + DC + LM$ proves the consistency of $ZFC$. It remains unclear whether something similar holds in [[constructive mathematics]] ($IBZ + DC + LM$) or with weaker choice principles such as [[countable choice]] ($BZ + CC + LM$).

## Properties

Besides the axioms themselves, other nice properties hold in dream mathematics.  Examples include:

*  Every (total) [[linear function]] from a [[Fréchet space]] to any [[topological vector space]] is [[continuous map|continuous]] (so every linear mapping between [[Banach spaces]] is [[bounded map|bounded]]).

*  On a [[localizable measure|localisable measure space]], the [[dual vector space|dual]] of the [[Lebesgue space]] $L^\infty$ is $L^1$ (so these two spaces are [[reflexive Banach space|reflexive]]).

*  Any two [[complete space|complete]] [[norms]] (or even [[F-norms]]) on a given [[vector space]] over the [[real numbers]] must be [[topological equivalence|topologically equivalent]], which nicely explains why you have never seen two *specific* inequivalent complete norms on a given real vector space.

All of the above statements are theorems (in $ZF$) for vector spaces with finite [[dimension]] (or measure spaces with finite [[cardinality]]); dream mathematics makes these true in infinite dimensions as well.

## References

* Wikipedia (English); [Solovay model](https://en.wikipedia.org/wiki/Solovay_model).

* _[[HAF]]_; end of Chapter 27.

* [[Alex Simpson]], François G. Dorais, *How strong is "all sets are Lebesgue Measurable" in weaker contexts than ZF?* ([web](https://mathoverflow.net/questions/141801/how-strong-is-all-sets-are-lebesgue-measurable-in-weaker-contexts-than-zf))

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[[!redirects dream analysis]]
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[[!redirects dream universes]]

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[[!redirects Solovay universe]]


dressing action

The concept is introduced by Kyoto school (Date, Miwa, Jimbo) around 1980s

* M. A. Semenov-Tian-Shansky, _Poisson groups and dressing transformations_, Zap. Nauchn. Sem. LOMI, 150 (1986),  119–142 [mathnet.ru](http://mi.mathnet.ru/znsl4975); _Poisson Lie groups, quantum duality principle, and the quantum double_, in: Math. aspects of conformal and topological field theories and quantum groups (South Hadley, MA, 1992),  219--248, Contemp. Math. __175__, AMS 1994; _Classical $r$-matrices, Lax equations, Poisson Lie groups and dressing transformations_, in:  Field theory, quantum gravity and strings, II (Meudon/Paris, 1985/1986), 174--214, Lec. Notes in Phys. __280__, Springer 1987, [MR89g:58098](http://www.ams.org/mathscinet-getitem?mr=89g:58098)
* А.Г. Рейман, М.А. Семёнов-Тян-Шанский, _Интегрируемые системы. Теоретико-групповой подход_, monograph (2003)


Drew Heard

* [webpage](https://drew-heard.github.io/)

category: people


Drinfel'd double

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#Contents#
* table of contents
{:toc}

## Idea

The _Drinfel'd double_- or _quantum double_-construction a sends a [[Hopf algebra]] to a [[quasi-triangular Hopf algebra]] [[Drinfeld (1987)](#Drinfeld87)]; or more generally, it sends a [[quasi-Hopf algebra]] to a [[quasi-triangulated quasi-Hopf algebra]] [[Majid (1994)](#Majid)].

Geometrically, if the given [[Hopf algebra]] is the [[group algebra]] of a [[finite group]] $G$, then the quantum double is the [[groupoid convolution algebra]] of the corresponding [[inertia groupoid]] $\mathcal{L}\mathbf{B}G \simeq G \sslash_{ad} G$ (this is made almost explicit in [Dijkgraaf, Pasquier & Roche (1990), eq. 2.1.10](#DijkgraafPasquierRoche90); the [[categorification|categorified]] version of this statement is highlighted in [Hinich (2007)](#Hinich07)).

More generally, if the given [[quasi-Hopf algebra]] is the [[twisted groupoid convolution algebra]] of a [[group cohomology]] 3-[[cocycle]] $c \colon \mathbf{B}G \to \mathbf{B}^3 U(1)$, then the corresponding quantum double is the [[twisted groupoid convolution algebra]] of the [[inertia groupoid]] equipped with the [[transgression in group cohomology|transgressed]] 2-cocycle [[Willerton (2005)](#Willerton)]

## Definition

Given a [[field]] $k$, __Drinfel'd__ (or Drinfeld or quantum) __double__ of a finite dimensional Hopf $k$-algebra $H$ is the tensor product algebra $D(H) = H\otimes H^*$ where $H^* = Hom_k(H, k)$ with induced canonical Hopf algebra structure. It appears that the canonical element in $D(H)$ (the image of the identity under the isomorphism $Hom_k(H,H) \cong H\otimes H^*$) is a universal $R$-element in $D(H)$ making it into a [[quasitriangular Hopf algebra]], which is making it "almost cocommutative".

This quasitriangular structure in some infinite-dimensional versions of the construction is related to the quasitriangular structure on some [[quantum groups]].

## Properties

### Relation to Yetter-Drinfeld modules

The [[category of modules]] over a quantum double of a finite-dimensional Hopf algebra $H$ is equivalent to the category of [[Yetter-Drinfeld modules]] of $H$.

### Relation to Drinfeld center

For $H$ a [[Hopf algebra]] arising as the [[groupoid convolution algebra]] of a finite [[groupoid]], the [[category of modules]] of its Drinfeld double is equivalently the [[Drinfeld center]] of the category of modules of the original algebra.

More generally, the analog of this statement holds for [[orbifolds]] ([Hinich (2007)](#Hinich07)).

[[!include structure on algebras and their module categories - table]]

## Related concepts

* [[inertia orbifold]], [[transgression in group cohomology]]

* [[Drinfeld center]]

## References

The original articles:

* {#Drinfeld87} [[Vladimir Drinfeld]], _Quantum groups_, in: A. Gleason (ed.) *[Proceedings of the](https://archive.org/details/proceedingsofint0002inte_v5c3/mode/2up) [1986 International Congress of Mathematics](https://inspirehep.net/conferences/966215?ui-citation-summary=true)* **1** (1987) 798-820 [[pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1986.1/ICM1986.1.ocr.pdf)]

  expanded version:

  Journal of Soviet Mathematics **41** (1988) 898–915 [[doi:10.1007/BF01247086](https://doi.org/10.1007/BF01247086)]

* [[Shahn Majid]], _Doubles of quasitriangular Hopf algebras_, Comm. Algebra __19__ 11 (1991) 3061-3073 [[MR92k:16052](http://www.ams.org/mathscinet-getitem?mr=1132774) [doi:10.1080/00927879108824306](http://dx.doi.org/10.1080/00927879108824306)]

* [[Shahn Majid]], _Some remarks on the quantum double_, Czechoslovak J. Phys. __44__ 11-12 (1994) 1059-1071 [[arXiv:hep-th/9409056](https://arxiv.org/abs/hep-th/9409056), [doi:10.1007/BF01690458](https://doi.org/10.1007/BF01690458)]

Survery:

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

The generalization of the double construction to [[quasi-Hopf algebras]] motivated by

* {#DijkgraafPasquierRoche90} [[Robbert Dijkgraaf]], [[Vincent Pasquier]], [[Philippe Roche]], *QuasiHopf algebras, group cohomology and orbifold models*, Nucl. Phys. B Proc. Suppl **18** (1990) 60-72 [doi:10.1016/0920-5632(91)90123-V]

* [[Robbert Dijkgraaf]], [[Vincent Pasquier]], [[Philippe Roche]], _Quasi-quantum groups related to orbifold models_, *International Colloquium on Modern Quantum Field Theory* (Bombay, 1990), 375-383, World Sci. (1991) [[cds:206306](https://cds.cern.ch/record/206306?ln=ka), [[DPR-QuasiQuantumGroups.pdf:file]]]

which is reviewed in

*  A. Coste, J-M. Maillard, _Representation Theory of Twisted Group Double_, Annales Fond.Broglie 29 (2004) 681-694, ([arXiv:hep-th/0309257](http://arxiv.org/abs/hep-th/0309257))

was obtained in

* {#Majid} [[Shahn Majid]], _Quantum double for quasi-Hopf algebras_, Lett. Math. Phys. __45__ (1998), no. 1, 1–9, [MR2000b:16077](http://www.ams.org/mathscinet-getitem?mr=1631648), [doi](http://dx.doi.org/10.1023/A:1007450123281), [q-alg/9701002](http://arxiv.org/abs/q-alg/9701002)

* [[Shahn Majid]], _Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras_, [hep-th/9311184](http://arxiv.org/abs/hep-th/9311184)

Interpretation via [[transgression in group cohomology]]:

* {#Willerton08} [[Simon Willerton]], Section 1 of: *The twisted Drinfeld double of a finite group via gerbes and finite groupoids*, Algebr. Geom. Topol. 8 (2008) 1419-1457 [[arXiv:math/0503266](https://arxiv.org/abs/math/0503266), [doi:10.2140/agt.2008.8.1419](https://doi.org/10.2140/agt.2008.8.1419)]

The special case of the Drinfeld double of a [[finite group]] is discussed further in

* Hui-Xiang Chen, Gerhard Hiss, _Notes on the Drinfeld double of a finite-dimensional group algebra_ ([pdf](http://www.math.rwth-aachen.de/~Gerhard.Hiss/Preprints/drindouble.pdf))

A characterization of the (quasi-)Hopf algebras arising this way is in

* Sonia Natale, _On group theoretical Hopf algebras and exact factorization of finite groups_ ([arXiv:math/0208054](http://arxiv.org/abs/math/0208054))

The equivalence of category of modules over the Drinfeld double for the case of [[orbifolds]], hence representations of the [[inertia orbifold]], with the [[Drinfeld center]] of the category of representations of the original orbifold is discussed in

* {#Hinich07} [[Vladimir Hinich]], *Drinfeld double for orbifolds*, in *Quantum Groups*, Contemporary Math **433** AMS (2007) 251-265 [[math.QA/0511476](http://arxiv.org/abs/math/0511476), [doi:10.1090/conm/433](http://dx.doi.org/10.1090/conm/433)]

[[!redirects Drinfel'd doubles]]

[[!redirects Drinfeld double]]
[[!redirects Drinfeld doubles]]

[[!redirects quantum double]]
[[!redirects quantum doubles]]

[[!redirects twisted Drinfeld double]]
[[!redirects twisted Drinfeld doubles]]


Drinfel'd twist

## Definition

Given a [[bialgebra]] $H$, a __Drinfel'd twist__ is an invertible element $\chi\in H\otimes H$ satisfying

$$
(1\otimes\chi)(id\otimes\Delta)\chi = (\chi\otimes 1)(\Delta\otimes id)\chi,
$$
satisfying $(\epsilon\otimes id)\chi = (id\otimes\epsilon)\chi = 1$ (in fact it is enough to require one out of these two counitality conditions).

In [[Shahn Majid|Majid]]'s formalism of [[bialgebra cocycle]]s, this is the same as a counital 2-cocycle in $H$.

## Application to twisting

Vladimir Drinfel'd introduced $\chi$ in order to suggest a procedure of twisting [[Hopf algebra]]s. Given a bialgebra, Hopf algebra or quasitriangular Hopf algebra, one gets another such structure twisting it with a Drinfel'd twist. Let $H$ be a quasi-triangular Hopf algebra with comultiplication $\Delta$, antipode $S$, universal R-element $R$ and let $\chi$ be a Drinfel'd twist. Then the same algebra becomes another quasitriangular Hopf algebra with formulas for comultiplication $\Delta_\chi b = \chi (\Delta b)\chi^{-1}$, universal R-element $R_\chi = \chi_{21} R\chi$ and antipode $S_\chi b =
\sum \chi^{(1)} S(\chi^{(2)}) (S b)(\sum \chi^{(1)}S(\chi^{(2)}))^{-1}$. The counit is not changed. Here $\chi_{21}=\tau(\chi)$ where $\tau$ is the flip of tensor factors, and $\chi = \sum \chi^{(1)}\otimes\chi^{(2)}$ is a notation similar to Sweedler's convention.

## Generalizations

In the original Drinfel'd's work 2-cocycles for twisting quasi-Hopf algebras were also considered.

There is also a subtle generalization for bialgebroids over nonassociative base due Ping Xu.

## Literature

The original references are

* [[Vladimir Drinfeld|V. G. Drinfeld]], _Quasi-Hopf algebras and Knizhnik-Zanolodchikov equations_, Acad. Sci. Ukrainian SSR, Institute for Theoretical Physics, Preprint ITP-89-43B (Kiev 1989) [pdf](https://inis.iaea.org/collection/NCLCollectionStore/_Public/23/037/23037492.pdf)
* V. G. Drinfel'd, _Quasi-Hopf algebras_, Algebra i Analiz, 1:6 (1989) 114-148; Leningrad Math. J., 1:6 (1990) 1419-1457 ([pdf](https://www.mathnet.ru/php/getFT.phtml?jrnid=aa&paperid=53&what=fullt&option_lang=eng) in Russian)
* V. G. Drinfel'd, _Almost commutative Hopf algebras_, Algebra i Analiz, 1:2 (1989) 30-46; Leningrad Math. J., 1:2 (1990) 321-342 [mathnet.ru](https://www.mathnet.ru/eng/aa1)

Drinfeld twist and Drinfeld associator have been reproduced as special cases of higher bialgebra cocycles in works of Majid,

* [[Shahn Majid]], _Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras_, in H.-D. Doebner, V.K. Dobrev, A.G. Ushveridze, eds., Generalized symmetries in Physics. World Sci. (1994) 13-41; ([arXiv:hep.th/9311184](http://front.math.ucdavis.edu/9311.3184))
* [[Shahn Majid]], _Foundations of quantum group theory_, Cambridge UP

The bialgebroid generalization is described in

* [[Ping Xu]], _Quantum groupoids_, Commun. Math. Phys. __216__ (2001) 539--581 arXiv:[q-alg/9905192](https://arxiv.org/abs/math/9905192)

category: algebra

[[!redirects Drinfeld twist]]
[[!redirects Drinfel'd twist]]
[[!redirects Drinfelʹd twist]]
[[!redirects Дринфельд twist]]
[[!redirects Drinfeld 2-cocycle]]


Drinfel'd-Sokolov reduction

### Classical version

* [[Vladimir Drinfeld|V. G. Drinfel'd]], V.V. Sokolov, Sov. Math. Dokl. __23__ (1981) 457; J. Sov. Math. __30__ (1985) 1975.
* Alberto De Sole, Victor G. Kac, Daniele Valeri, _Classical W-algebras and Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras_, [arxiv/1207.6286](http://arxiv.org/abs/1207.6286); _Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents_, [arxiv/1306.1684](http://arxiv.org/abs/1306.1684)

### Quantum version

* [[Edward Frenkel]], [[Victor Kac]], Minoru Wakimoto, _Characters and fusion rules for W-algebras via quantized Drinfelʹd-Sokolov reduction_, Comm. Math. Phys. __147__ (1992), no. 2, 295–328.
* [[Boris Feigin]], [[Edward Frenkel]], _Quantization of the Drinfeld-Sokolov reduction_, Physics Letters B246, 1–2, 1990, 75–81, doi
* Д. Талалаев, А. Червов, _Уравнение КЗ, G-оперы, квантовая редукция Дринфельда–Соколова и квантовое тождество Гамильтона–Кэли_, Zapiski Naučnyh Seminarov POMI __360__, 2008, 246–259, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=znsl&paperid=2168&what=fullt&option_lang=rus); Engl, transl. D. Talalaev, A. Chervov, _KZ equation, G-opers, quantum Drinfeld-Sokolov reduction, and quantum Cayley-Hamilton identity_, J. Math, Sci. (New York), 2009, 158:6, 904–911, [doi](http://dx.doi.org/10.1007/s10958-009-9415-1), [hep-th/0607250](http://arxiv.org/abs/hep-th/0607250)

[[!redirects Drinfeld-Sokolov reduction]]


Drinfeld associator

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

#Contents#
* table of contents
{:toc}

## Definition

Let $\mathbf{k}$ be a [[field]] of [[characteristic]] 0 and $\lambda \in \mathbf{k}^*$. A **$\lambda$-Drinfeld associator**, or just **$\lambda$-associator**, is a [[grouplike element]] $\Phi(a,b)$ of the $\mathbf{k}$-[[associative algebra|algebra]] of [[formal power series]] in two non-commuting [[variables]] $a,b$ satisfying:

1. The pentagon equation
$$
\Phi(t_{12} , t_{23} + t_{24} )\Phi(t_{13} + t_{23} , t_{34} ) = \Phi(t_{23} , t_{34} )\Phi(t_{12} + t_{13} , t_{24} + t_{34} )\Phi(t_{12} , t_{23} )
$$
in $\widehat{U(L_4)}$
2. the hexagon equation
$$
\exp(\lambda a/2)\Phi(c,a)\exp(\lambda c/2)\Phi(b,c)\exp(\lambda b/2)\Phi(a,b)=1
$$

where $L_4$ is the fourth [[Drinfeld-Kohno Lie algebra]] and $c=-a-b$.

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

The set of "0-associators" is the what is called the _[[Grothendieck-Teichmueller group]]_. This acts freely on the set of Drinfeld associators.

=--

## Relations with braided monoidal categories

These [[equations]] are modelled on the defining [[axioms]] of [[braided monoidal categories]]. Indeed, associators provides a universal way of constructing braided monoidal categories out of some [[Lie algebra|Lie algebraic]] data.

Drinfeld associators are also used to construct [[quasi-Hopf algebra|quasi-Hopf algebras]].

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

Let $(\mathfrak{g},t)$ be a metrizable [[Lie algebra]], that is a Lie algebra $\mathfrak{g}$ together with a non-degenerate symmetric $\mathfrak{g}$-invariant 2-tensor $t$. Then if $\Phi$ is a $\lambda$-associator and $\hbar$ a formal variable, then the action of
$$
\Phi(\hbar (t \otimes 1),\hbar (1\otimes t)) \in U(\mathfrak{g})^{\otimes 3}[[\hbar]]
$$
 and $e^{\hbar \lambda t/2}\circ P$ turns the category of $U(\mathfrak{g} ) [ [ \hbar ] ] $ module into a braided monoidal category, where $P$ is the flip: $P(a\otimes b)=b\otimes a$.
=--

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

Examples of metrizable Lie algebras are provided by [[simple Lie algebras]], in which case $t$ is a scalar mutliple of the [[Killing form]]. The braided monoidal category obtained this way is equivalent to that constructed from the corresponding [[quantum group]], by the [[Drinfeld-Kohno theorem]].

=--

##Existence##
An explicit associator over $\mathbf{C}$ was constructed by [[Drinfeld]] from the [[monodromy]] of a universal version of the [[Knizhnik-Zamolodchikov equation]]. Using the non-emptiness of the set of associators, and the fact that is is a [[torsor]] under the [[action]] of the
[[Grothendieck-Teichmueller group]], he show that associators over $\mathbf{Q}$ also exists.

##Applications##
* [[Drinfeld-Kohno theorem]]
* [[Etingof-Kazhdan quantization functor]]
* [[Formality]] of the [[little 2-disk operad]]
* Construction of a functorial [[universal finite type invariant]]
* [[dilogarithm]]
* [[Hausdorff series]], [[Kashiwara-Vergne conjecture]]

## References

* [[Vladimir Drinfeld]]. On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal($\overline{\mathbf{Q}}/\mathbf{Q}$). _Leningrad Math. J._ 2:4 (1990), 829–860. ([mathnet.ru](http://mi.mathnet.ru/eng/aa199))
* [[Dror Bar-Natan]]. _On associators and the Grothendieck-Teichmuller group I._ Selecta Math. (N.S.)  4:2 (1998), 183–212. ([pdf](https://www.math.toronto.edu/drorbn/papers/GT1/GT1.pdf))
* A. Alekseev, C. Torossian, _The [[Kashiwara-Vergne conjecture]] and Drinfeld’s associators_, Ann. of Math. 175 (2012), no.2, 415–463 [pdf](http://www.math.jussieu.fr/~torossian/AT1.pdf);  _Kontsevich deformation quantization and flat connections_ [arxiv/0906.0187](https://arxiv.org/abs/0906.0187) [doi](https://doi.org/10.1007/s00220-010-1106-8)
* A. Alekseev, B. Enriques, C. Torossian, _Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations_, Publ. Math. Inst. Hautes ́Etudes Sci. 112 (2010), 143--189 [arxiv/0903.4067](https://arxiv.org/abs/0903.4067)
* [[Anton Alekseev]], Florian Naef, Xiaomeng Xu, Chenchang Zhu, _Chern–Simons, Wess–Zumino and other cocycles from Kashiwara–Vergne and associators_, Letters in Mathematical Physics __108__:3 (2018) 757–778 [doi](https://doi.org/10.1007/s11005-017-0985-4)

[[!redirects Drinfeld associators]]


Drinfeld center

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _center of a monoidal category_ or _Drinfeld center_ is the [[categorification]] of the notion of [[center]] of a [[monoid]]([[associative algebra]], [[group]], etc.) from monoids to [[monoidal categories]].

Where the [[center]] of a monoid is just a sub-monoid with the [[property]] that it commutes with everything else, under categorification this becomes a [[stuff, structure, property|structure]], since we have to specify _how_ the objects in the Drinfeld center commute ([[braided monoidal category|braid]]) with everything else.

## Definition

We first give the general-abstract definition

* [Abstract definition](#AbstractDefinition)

of Drinfeld centers. Then we spell out what this means in components in

* [Definition in components](#DefinitionInComponents).

### Abstractly
 {#AbstractDefinition}

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

For $(\mathcal{C}, \otimes)$ a [[monoidal category]], write $\mathbf{B}_\otimes \mathcal{C}$ for its [[delooping]], the pointed [[2-category]] with a single [[object]] $*$ such that $Hom_{\mathbf{B}_\otimes \mathcal{C}}(*, *) \simeq \mathcal{C}$.

The **Drinfeld center** $Z(\mathcal{C}, \otimes)$ of $(\mathcal{C}, \otimes)$ is the [[monoidal category]] of [[endofunctor|endo]]-[[pseudonatural transformations]] of the [[identity]]-[[2-functor]] on $\mathbf{B}_\otimes \mathcal{C}$:

$$
  Z(\mathcal{C}, \otimes)
  \coloneqq
  End_{\mathbf{B}_\otimes \mathcal{C}}(id_{\mathbf{B}_\otimes \mathcal{C}})
  \,.
$$

=--

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

Unwinding the definitions, we find that an [[object]] of $Z(\mathcal{C}, \otimes)$, $\Phi \colon id_{\mathbf{B}_\otimes \mathcal{C}} \to id_{\mathbf{B}_\otimes \mathcal{C}}$, has for components  pseudonaturality squares

$$
  \array{
    \ast &\stackrel{X \coloneqq \Phi(\ast)}{\to}& \ast
    \\
    {}^{\mathllap{Y}}\downarrow &\swArrow_{\Phi_Y}& \downarrow^{\mathrlap{Y}}
    \\
    \ast
    &\underset{X = \Phi(\ast)}{\to}&
    \ast
  }
$$

for each $Y \in Obj(\mathcal{C})$. As shown, these consist of a choice of an object $X \in \mathcal{C}$ together with a [[natural isomorphism]]

$$
  \Phi_{(-)} \colon X \otimes (-) \to (-) \otimes X
$$

in $\mathcal{C}$.

The [[transfor]]-property of $\Phi$ says that

$$
  \array{
    \ast &\stackrel{X \coloneqq \Phi(\ast)}{\to}& \ast
    \\
    {}^{\mathllap{Y}}\downarrow &\swArrow_{\Phi_Y}& \downarrow^{\mathrlap{Y}}
    \\
    \ast
    &\underset{X = \Phi(\ast)}{\to}&
    \ast
    \\
    {}^{\mathllap{Z}}\downarrow &\swArrow_{\Phi_Z}& \downarrow^{\mathrlap{Z}}
    \\
    \ast
    &\underset{X = \Phi(\ast)}{\to}&
    \ast
  }
  \;\;\;\;
  \simeq
  \;\;\;\;
  \;\;\;\;
  \array{
    \ast &\stackrel{X \coloneqq \Phi(\ast)}{\to}& \ast
    \\
    {}^{\mathllap{Y \otimes Z}}\downarrow &\swArrow_{\Phi_{Y \otimes Z}}& \downarrow^{\mathrlap{Y \otimes Z}}
    \\
    \ast
    &\underset{X = \Phi(\ast)}{\to}&
    \ast
  }
  \,.
$$

And so forth. Writing this out in terms of $(\mathcal{C}, \otimes)$ yields the following component characterization of Drinfeld centers, def. \ref{CompDef}.

=--

### In components
 {#DefinitionInComponents}

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

Let $(\mathcal{C}, \otimes)$ be a [[monoidal category]]. Its **Drinfeld center** is a [[monoidal category]] $Z(\mathcal{C})$ whose

* [[objects]] are pairs $(X, \Phi)$ of an object $X \in \mathcal{C}$ and a [[natural isomorphism]] (braiding morphism)

  $$
    \Phi \colon X \otimes (-) \to (-) \otimes X
  $$

  such that for all $Y \in \mathcal{C}$ we have

  $$
    \Phi_{Y \otimes Z}
    =
    (id \otimes \Phi_Z) \circ (\Phi_Y \otimes id)
  $$

* [[morphisms]] are given by

  $$
    Hom((X, \Phi), (Y,\Psi))
    =
   \left\{
     f \in Hom_{\mathcal{C}}(X,Y)
     \;|\;
     (id \otimes f) \circ \Phi_Z
     =
     \Psi_Z \circ (f \otimes id), \; \forall Z \in \mathcal{C}
   \right\}
   \,.
  $$

* the [[tensor product]] is given by

  $$
    (X, \Phi) \otimes (Y, \Psi) = (X \otimes Y, (\Phi \otimes id) \circ (id \otimes \Psi))
    \,.
  $$

=--

## Properties

### Extra structure on the Drinfeld center

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

The Drinfeld center $Z(\mathcal{C})$ is naturally a [[braided monoidal category]].

=--

\begin{proposition}
\label{DrinfeldCenterOfFusionCategory}
If $\mathcal{C}$ is a [[fusion category]]
over an [[algebraically closed field]] of [[characteristic zero]],
then the Drinfeld center $Z(\mathcal{C})$ is also naturally a fusion category.
\end{proposition}

([Etingof, Nikshych & Ostrik 2005, Thm. 2.15](#EtingofNikshychOstrik05), review in [Davydov, Mueger, Nikshych & Ostrik 2003, Sec. 2.3](#DavydovMuegerNikshychOstrik03))

See also [Drinfeld, Gelaki, Nikshych & Ostrik 2010, Cor. 3.9](#DrinfeldGelakiNikshychOstrik10),  [Mueger 2003](#Mueger03).

### Relation to Drinfeld double under Tannaka duality

Under [[Tannaka duality]], forming the Drinfeld center of a [[category of modules]] of some [[Hopf algebra]] corresponds to forming the category of modules over the corresponding [[Drinfeld double]] algebra. See there for more.

[[!include structure on algebras and their module categories - table]]

## Related concepts

* [[center of a category]], [[center of an additive category]]

* [[Drinfeld double]]

## References

Original articles:

* {#Mueger03} [[Michael Mueger]], *From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors*, J. Pure Appl. Alg. 180, 159-219 (2003) ([arXiv:math/0111205](https://arxiv.org/abs/math/0111205))

* {#EtingofNikshychOstrik05} [[Pavel Etingof]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On fusion categories*, Annals of Mathematics Second Series, Vol. 162, No. 2 (Sep., 2005), pp. 581-642 ([arXiv:math/0203060](http://arxiv.org/abs/math/0203060), [jstor:20159926](https://www.jstor.org/stable/20159926))

* {#DrinfeldGelakiNikshychOstrik10} [[Vladimir Drinfeld]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On braided fusion categories I*, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 ([doi:10.1007/s00029-010-0017-z](https://doi.org/10.1007/s00029-010-0017-z))

Review:

* {#DavydovMuegerNikshychOstrik03} [[Alexei Davydov]], [[Michael Mueger]], [[Dmitri Nikshych]], [[Victor Ostrik]], Sec. 2.3 of: *The Witt group of non-degenerate braided fusion categories*, Journal fuer reine und angewandte Mathematik, **677** (2013. 135-177), de Gruyter 2003  ([arXiv:1009.2117](https://arxiv.org/abs/1009.2117), [doi:10.1515/crelle.2012.014](https://doi.org/10.1515/crelle.2012.014))

Textbook accounts:

* [[Shahn Majid]], _Foundations of quantum group theory_, Cambridge Univ. Press

* [[Christian Kassel]], _Quantum groups_, Graduate Texts in Mathematics __155__, Springer 1995 ([doi:10.1007/978-1-4612-0783-2](https://link.springer.com/book/10.1007/978-1-4612-0783-2), [webpage](http://www-irma.u-strasbg.fr/~kassel/QGbk.html), [errata pdf](http://www-irma.u-strasbg.fr/~kassel/QGerrata030399.pdf))

A general discussion of centers of [[monoid objects]] in [[braided monoidal 2-categories]] (which reduces to the above for the 2-category [[Cat]] with its [[cartesian product]]) is in

* [[Ross Street]], _The monoidal center as a limit_, [pdf](http://maths.mq.edu.au/~street/centre.pdf)

An application to character sheaves is in

* [[George Lusztig]], _Non-unipotent character sheaves as a categorical centre_, [arxiv/1506.04598](http://arxiv.org/abs/1506.04598)

In relation to [[spectra of tensor triangulated categories]]:

* Kent B. Vashaw, _Balmer spectra and Drinfeld centers_ ([arXiv:2010.11287](https://arxiv.org/abs/2010.11287))

[[!redirects Drinfeld centers]]

[[!redirects Drinfel'd center]]
[[!redirects Drinfel'd centers]]

[[!redirects center of a monoidal category]]


Drinfeld upper half-plane

#Contents#
* table of contents
{:toc}

## Idea

The Drinfeld upper half-plane is the p-adic analogue of the complex [[upper half-plane]]. It is a [[rigid analytic space]] and may be expressed as $\mathbb{P}^{1}(\mathbb{C}_{p})\setminus \mathbb{P}^{1}(\mathbb{Q}_{p})$. There is also a function-field analogue of the construction, also referred to as the Drinfeld upper half-plane.

## Related concepts

* [[upper half-plane]]

## References

* Wikipedia, _[Drinfeld upper half plane](https://en.wikipedia.org/wiki/Drinfeld_upper_half_plane)_


Drinfeld-Kohno Lie algebra

Let $n\gt 2$. The Drinfel'd-Kohno Lie algebra is a $\mathbf{Z}$-algebra $L_n$ defined by generators $t_{ij} = t_{ji}$, $1\leq i\neq j\leq n$ subject to the relations

$$
[t_{ij}, t_{kl}] = 0, \,\,\,[t_{ij}, t_{ik}+t_{jk}] = 0, \,\,\,\,\,1\leq i\neq j\neq k\neq l\leq n
$$

It is the [[holonomy Lie algebra]] of the [[configuration space]] $X_n$ of $n$ distincts points in the complex plane. Hence, it can be used to define a flat connection on $X_n$, which is universal among [[Knizhnik-Zamolodchikov equation]]s.

Therefore, it induces a monodromy representation of $\pi_1(X_n)$ which is isomorphic to the pure [[braid group]]:
$$
PB_n \longrightarrow \exp(L_n \otimes \mathbf{C})
$$

It was shown by Kohno that the extension of this map to the $\mathbf{C}$-pro-unipotent completion of $PB_n$ is an isomorphism. Drinfeld showed using [[Drinfeld associator|associators]] that the same holds true over $\mathbf{Q}$.

In particular, $U(L_n \otimes \mathbf{Q})$ is isomorphic to the [[associated graded]] of $\mathbf{Q}[PB_n]$ with respect to the filtration induced by powers of the augmentation ideal. Since it is known that this filtration coincides with the one induced by the [[Vassiliev skein relation]], $U(L_n \otimes \mathbf{Q})$ may be identified with the algebra of horizontal [[chord diagrams]].

The universal enveloping $U(L_n)\otimes \mathbf{Q}$ is a Koszul algebra.

Cf. [[Drinfeld-Kohno theorem]]


Drinfeld-Kohno theorem

#Contents#
* table of contents
{:toc}

## Idea

The __Drinfel'd-Kohno theorem__ is about a geometric relation between classical and quantum $R$-matrices for [[quantum groups]]: the monodromies of the solutions of [[Knizhnik-Zamolodchikov equation]]s (which are equations of [[Fuchsian equation|Fuchsian type]]) for a given classical $r$-matrix give rise to the corresponding quantum $R$-matrix; more precisely it gives the same representation of the braid group.

## References

* [[Toshitake Kohno]], _Monodromy representations of braid groups and Yang-Baxter equations_, Annales de l'institut Fourier __37__, no. 4 (1987), p. 139-160, [numdam](http://www.numdam.org/item?id=AIF_1987__37_4_139_0)

* [[Toshitake Kohno]], _Conformal field theory and topology_, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs __210__. Iwanami Series in Modern Mathematics. Amer. Math. Soc. 2002. x+172 pp.

* P. Etingof, O. Shiffman, _Lectures on quantum groups_
* [[P. Etingof]], N. Geer, _Monodromy of trigonometric KZ equations_, [math.QA/0611003](http://arxiv.org/abs/math.QA/0611003)

* [[Valerio Toledano-Laredo]], _A Kohno-Drinfeld theorem for quantum Weyl groups_, [math.QA/0009181](http://arxiv.org/abs/math.QA/0009181)

* V. Chari, A. Pressley, _A guide to quantum groups_, 1994
* Uli Kraehmer, _Knizhnik-Zamolodchikov and Drinfeld-Kohno_, slides, [pdf](http://www.maths.gla.ac.uk/~ukraehmer/Warszawa0409.pdf)
* C. Kassel, _Quantum groups_, Springer 1995
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