wreath

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

# Wreaths

* table of contents
{:toc}

## Idea

A **wreath** is a generalisation of a [[distributive law]] between two [[monads]] in a [[2-category]]. While a distributive law in a [[2-category]] $K$ can be seen as an object of $Mnd(Mnd(K))$, a wreath can be seen as an object of $EM(EM(K))$, where $EM$ denotes the completion of a 2-category under [[Eilenberg–Moore objects]]. Since $EM$ is a 2-monad, the multiplication $EM \circ EM \to EM$ produces from every wreath a **composite monad**.

## Related concepts

* [[distributive law]]

## Literature

* [[Steve Lack]], [[Ross Street]], _The formal theory of monads II_, Special volume celebrating the 70th birthday of Professor Max Kelly.
J. Pure Appl. Algebra 175 (2002), no. 1-3, 243--265.

[[!redirects wreaths]]


wreath product

* [[wreath product of groups]]

* [[categorical wreath product]]

* [[weak wreath product]]

* [[wreath]] (in a 2-category)

[[!redirects wreath products]]


wreath product of groups

## Definition ##

Let $H\leq Sym(X)$ and $G\leq Sym(Y)$ be permutation groups. Then their wreath product $H \wr G$ is defined as the [[semidirect product group|semidirect product]] $H^Y \rtimes G$ where $G$ operates on $H^Y$ by permuting the components. Note that this is itself a permutation group, acting on $X\times Y$ by letting $G$ act trivially on the first and naturally on the second factor and letting $H^Y$ act on $X\times Y$ such that the $y$-th component of $H^Y$ permutes $X\times\{y\}$ naturally and fixes everything else pointwise.

## Examples ##

* Every group $G$ is a permutation group on its underlying set $|G|$ (the regular representation aka the Cayley-Yoneda embedding $G\hookrightarrow Sym(|G|)$) and often only this special case is considered so that people speak of "**the** wreath product $H \wr G$" for abstract groups $G$ and $H$ without mentioning their permutation representations.
* $Dih(4)$ the dihedral group of order $2\cdot 4=8$ is the wreath product $C_2 \wr C_2$ (in this sense of regular representations).
* The sylow-$p$-subgroups $P_n$ of the symmetric groups $Sym(n)$ can be recursively described as the wreath product $C_p \wr P_a$ where $C_p$ is the cyclic group of order $p$ and $n=ap+r$ with $0\leq r \lneq p$.
* The sylow-$\ell$-subgroups of $GL_n(q)$ for $gcd(q,\ell)=1$ can be recursively described as a wreath product of the sylow-$\ell$-subgroup of a strictly smaller $GL_{m}(q)$ and a sylow-$\ell$-subgroups of a suitable symmetric group.

## Relation to the Borel construction

The homotopy quotient described in the [[Borel construction]],

$$
  (X //G)_\bullet
    \simeq_{iso}
  X \times_G (E G)_\bullet
  \,,
$$

has the wreath product $\pi_1(X)\wr G$ for fundamental group.

(...)

## References ##

(...)


writhe

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

## Idea ##

**Writhe** is a measure of how much a knot or link _writhes_ around itself.  As a 1-dimensional line cannot actually twist, this is not a [[knot invariant]] but is an invariant of [[framed knots]] (and links).  Since a [[link diagram]] can be given a natural framing (the _blackboard framing_), it is possible to compute the writhe of a specific diagram.  One place where this is used very neatly is to convert the [[Kauffman bracket]], which is an invariant of framed links, into the [[Jones polynomial]], being an invariant of ordinary links.

## Definition ##

Recall that a [[framed link]] can be thought of as a link together with a normal direction along each component, which we call the _framing direction_.

+-- {: .num_definition #writhe}
###### Definition
The **writhe** of a framed link is the [[linking number]] of the link with its infinitesimal displacement in the framing direction.
=--

For an oriented link diagram, the writhe is defined using the orientation of the crossings.

+-- {: .num_definition #writediag}
###### Definition
The **writhe** of an oriented [[link diagram]] is defined to be the sum of the orientations of its crossings.
=--


writing in the nLab

# Contents
* table of contents
{: toc}

## Introduction

Writing in the nLab should be a pleasant and rewarding experience. We are always glad for new people to join us in this collective effort, bringing their expertise to bear on the many topics touched on at this site.  Many contributors put in a considerable amount of work to improving the nLab, and all such efforts are greatly appreciated.

However, the success of this project also depends in great degree on a certain ethos of cooperation and mutual respect that has evolved here. This page is meant to lay out some of the working principles that have helped to foster this atmosphere.

Before reading this page, you should have read or at least be aware of companion introductory nLab pages such as [[nPOV]], [[About]], and [[How to get started]] and [[HowTo]] (the last being especially useful for editing techniques). *This page* is meant to help smooth over possible misunderstandings that may arise about the nature of the project, and to help newcomers to nLab writing fit in more readily.

Perhaps the one overarching piece of advice we can impart to newcomers, which applies generally outside the nLab as well, can be condensed into a single principle (a form of the "prime directive"):

+-- {: .standout}
Let the mathematics, physics, and philosophy speak for itself; try to get your own self out of the way.
=--

The purpose of the community of nLab editors is to maximize insight, both for other members of that community and for others readers, and largely from a particular point of view called the [[nPOV]], as well as from the [[string theory]] point of view for physics related articles.  When joining that community, your purpose should be the same.  The existing community has found that this happens best in an atmosphere where we minimize distractions and help keep ourselves and our readers focused on the big picture. This advice may seem obvious, but there is a bit more to it than may appear at first glance, and so we elaborate on it below.

While scientific results are recorded or archived at the nLab, and undergo further polishing and revision there, the principal organ of communication we have is the [[nForum]]. A basic rule is that

+-- {: .standout}
With few exceptions, all edits to the nLab (either the creation of a new page or the revision of an extant one) should be announced at the nForum, in the "Latest Changes" category.
=--

The only real exceptions are very minor edits such as correction of spelling mistakes or obvious typos or indisputable grammatical errors.  However, because of this rule there can at times be a large volume of Latest Changes posts; thus a corollary is that Latest Changes posts at the forum should generally be kept very short and to the point.  They should also include a link to the nLab page in question (links at the nForum are created with the same `[[syntax]]` as on the nLab itself).

The nForum is also home to wide-ranging discussion of scientific matters related to this project, and we encourage newcomers to join these conversations as well and get to know the existing members of the community.  We ask each other questions and suggest ideas; then when those ideas yield fruit, we often record the results back in the nLab.

The genesis of every article is recorded in a revision history. Thus every contributor gets some credit (the name or alias of each contributor is shown at the bottom of each of their revisions, and the changes introduced by each revision are also faithfully recorded). But the nLab belongs to us all. Speaking optimistically, we could liken the nLab to one of the great cathedrals of Europe, where unnamed architects and artisans worked together to build a monument to a glory far surpassing that of any one individual. Feeling oneself as part of such a collective project can be a great reward.

+-- {: .num_remark}
###### Remark
There is a uniform Joker name applied to those who don't perform an edit under their own names: "[[Anonymous]]". (This was originally "Anonymous Coward", a joke.)  An edit attributed to "Anonymous" could be a case of someone who didn't submit under their real name because they forgot, or it could be someone who has good reason not to submit under their real name -- whatever. It's none of our business. We appreciate all the good edits made by the "Anonymous Cowards" out there just as we do those made by everyone else.  However, we do encourage everyone to use their own name, or at least a consistent alias, if possible.
=--

## Some obvious truisms

There are some fairly obvious corollaries to be derived from the precept "get your own self out of the way" (with the specific implication "this is really not about you").

* The nLab is not a place to promote pet theories and "revolutionary ideas".

We've had a bunch of people pass through who want to talk about e.g., their revolutionary approach to point-set topology, or how to use categorical ideas to develop a private theory of music, or deep insights into how categorical concepts shed light on the very structures of consciousness, or their ruminations on mathematical aesthetics, or their ideas on how gravitation should be understood in terms of electromagnetic effects.

In all such cases, we've had to tell them to go away and leave us alone. Not that we deem all such people to be crackpots; some may actually be fine thinkers. But we have other things we need to be doing.

* The nLab is not a place where you just plop down notes indiscriminately.

People who have seen the description at [[nLab]] that this site is a "public place where people can make notes on stuff" may get the wrong idea. For example, students have sometimes set up shop and set about writing up stuff that is currently occupying them, without regard to what is going on in the rest of the nLab. They are usually told (politely) that they are misusing the nLab, and that such articles will unfortunately have to be removed.

A more accurate description, given at the article [[About]], is that the nLab is perhaps best conceived as a group lab book, with the idea that we scribble things down that are of common interest to other researchers here. That's closer, but it's a bit simplistic. What makes the nLab somewhat different from a traditional group lab book are the numerous hyperlinks between the parts of the nLab, meaning that the scribblings should (eventually) be well integrated with what is taking place elsewhere. The integration works best when editors become somewhat aware of related articles and what has gone into them.

Of course newcomers are not expected to know everything about what related articles already exist on the nLab; the established contributors are generally quite happy to help make connections between new and old material.  In return, however, newcomers are asked to be sensitive and receptive to this integrative process. You could think of the nLab as already an evolved organism, with limbs and extensions being constantly grafted on, but in a controlled way so that the organism does not reject the graft as being of the wrong type. In Piagetian terms, new material might either be easily assimilated into the existing structure; or, some extra arrangements might need to be made to accommodate it comfortably, a process that may require patience and good will between the participants.

Again, the nForum is the place where we openly discuss such matters.

* The nLab is not a place to conduct literary experiments.

There is plenty of scope for different forms of personal expression; there is not a single uniform style throughout the nLab. That being said, idiosyncrasies in style are much the secondary consideration; getting the goods out there in useful efficient form for the reader (including one's self) is primary.

Unless you are supremely confident in your expository skills, the advice if you are a newcomer is generally to write straight mathematics (or physics, etc.), aiming for formal definitions and statements of results and the like. Not write imagined dialogues in the manner of Lakatos's Proofs and Refutations, or long essays, or stud the discussion with abundant scholarly footnotes and other marginalia. Such devices have a tendency to clutter or distract or call undue attention to themselves. Anything that distracts or deflects the reader's attention or any unnecessary digression is, in *mathematical writing*, generally bad.

Similarly, wordplay should be used very moderately, and the impulse to flaunt one's broad knowledge and culture (in the manner of David Foster Wallace, say) should almost never be indulged. A little levity can be a fine thing, but again, the overriding principle to remember is that what we want most of all is to convey scientific insight, and this is best accomplished when the English is *smooth and unobtrusive* -- doesn't call attention to itself. You may be the greatest wit since Oscar Wilde, but your privately amusing puns and learned allusions may be getting in the way here, and actually your readers might resent you for it. Just try to be sensitive to that, please. If you want a place to display your erudition, start your own blog.

To avoid any misunderstanding: we are not suggesting that one should write the mathematics (physics, philosophy) in a stuffy or formal or bureaucratic way. To the contrary, it can often be effective to adopt a light conversational tone, taking the reader into your confidence as it were; on the nLab we are often more conversational than is the traditional style in published papers. But the expression should be an unaffected one (being too chummy is as bad as being too precious). Just remember the prime directive, and you should be fine.

## Writing mathematics and physics

All nLab editors are expected to use the "iTeX" mathematics typesetting language of the instiki software, which displays in web browsers using MathML/MathJax. If you know TeX or LaTeX, this should not be very difficult to adapt to; most mathematical typesetting commands will be the same.  Some of the more important (though still fairly minor) differences are documented in the [[FAQ]].

In the first place, to get started, writing a "stub" article (even just a sentence of two, just to get an idea down) is perfectly fine. Hopefully that will grow into something more polished, maybe with the input of others, but you shouldn't worry -- just make the stub intelligible enough for someone else to pick up the thread. Some people at this stage start a list of references to come back to, which can also be a good idea. If nothing else, a link to a relevant [Wikipedia](https://en.wikipedia.org/wiki/Portal:Mathematics)/[Encyclopedia of Mathematics](https://www.encyclopediaofmath.org/index.php/Main_Page)/[MathWorld](http://mathworld.wolfram.com/)/[SEP](https://plato.stanford.edu/) page --- which should have some references of its own --- can be the germ of a reference list.

### General organization

Most nLab articles are titled by a *concept* (usually to be formally defined in the article). There are also pages titled by naming a result (e.g., Tychonoff theorem), sometimes in synopsized form (e.g., closed projection characterization of compactness). Some articles are titled by a person's name, others by a book title (sometimes as referred to colloquially, e.g. Elephant).  But nLab pages are not usually named with the sort of title you might see on a mathematics paper, such as "On a conjecture of Grothendieck".

If you look at a typical nLab entry which has grown up a little, you will frequently find it organized along the following lines. First comes an Idea or Introduction section, usually with language similar to language you might use during a hike with a colleague on mathematics, with few to no symbols and giving the general idea without getting too technical. Then you might see a Definition(s) section which gives one or several precise definitions, followed by a Properties section or Basic Results section, an Examples section, a Related Concepts section, and a References section. By no means is this (Wikipedia-like) organization rigidly adhered to, but it is certainly common. At the top (if the author(s) of the page didn't forget it), you will see a Table of Contents which serves as a broad outline of the contents (with sections, maybe some subsections, hopefully not *too* many sub-subsections -- more subs than that and it can get ridiculous).

Inside each section, the material is often arranged by making effective use of environments (definition, theorem, lemma, remark, etc.). How these are used is a matter of personal discretion. The best advice is to aim for a happy medium, between one extreme of erecting great walls of text (which can look like a sermon being preached, and where the reader can get lost), and another where the material is so finely diced into atomic environment units that the reader is in for a bumpy ride. Ideally the material is arranged somewhere in the middle, allowing for a smooth narrative flow, and where the environments are not too cluttered but act as well-placed signposts. Visually the arrangement should look appealingly smooth. Display lines (as in TeX math display mode) can often aid in that, breaking up big chunks of text and allowing for emphases where appropriate -- but too many of them and it backfires.

Choice of notation is largely up to the individual. It would be impossible to maintain a consistent choice of notation throughout the entirety of the nLab, because there are just too many authors and the nLab is too vast to keep track of it all. There is no Central Planning Committee for this type of thing. Thus, regarding notation, you should think of each article as a self-contained unit --- but do please try to maintain notational consistency within each article.

This means especially: try to be respectful of the notation others have already introduced. If they use a calligraphic font while discussing a concept, then it will help the reader if you use the same font for the same concept later in the article, even if you don't personally like calligraphic fonts that much. Above all: don't take it upon yourself to just change all the notation to suit your own liking. If you feel strongly about notation -- and many of us do -- then you can plead and argue your case at the nForum for changes you want made (within reason of course -- arguments shouldn't be overly long). Hey, you may be right, and the rest of the community will listen. But if a consensus or compromise is not reached, then please find it within yourself to adapt to what was there. (When you start an article, you can introduce your preferred notation, and then you'll be on the other side of that situation, right?)

For further words of wisdom about notation and sundry other expositional matters (many of which are being discussed without particular attribution in this article), please read [Halmos](#Halmos); it is strongly recommend as a great and helpful article written by an unquestionably excellent expositor.

### When in doubt, follow existing norms

The format you will see in a typical nLab article is somewhere between a Wikipedia article you might see on a topic in mathematics or mathematical physics, and an academic article in one of those areas. The organizational outline seen in the table of contents is very Wikipedia-like, but unlike Wikipedia, the development may be like that of an academic article with formal definition environments, theorem environments, etc., and with point of view that is not neutral with regard to where we tend to place emphasis: the [[nPOV]] and [[string theory]]. We also may conduct some original research, quite unlike Wikipedia.

But in most respects, there should not be too many surprises for people familiar with both types of articles. For example, under a References section we follow a standard type of bibliographic format (author(s), title in italics, journal with volume and number or publisher, date, page numbers), linking to online material wherever possible (always legally!!). In the text where one wants to refer (and link) to a bibliographic item, there is no official format for this, but the tendency is to use something simple like author-name-date in brackets, rather than putting a number in brackets. The page [[HowTo]] gives detailed instructions on how to do this.

Occasionally there is redundancy or repetition in the nLab; to a degree this is tolerated (a definition or proof may be recalled). However, we don't want *too* much of that. If you set out to create a new page, you may be told when you announce it at the nForum that we have that on some other page with a different title. Don't be discouraged: the material will of course not be an exact duplicate and there will surely be things which can be usefully incorporated into the extant article.

**But:** in that situation there may have to be a "redirect", where the term you used to title your article should now be a term which redirects to the extant article (or, it could go in the other direction, but usually not), and then the newly created article, after relevant material has been incorporated, is made "history". Please see [[HowTo]] for detailed instructions on how to perform this action. It's a slightly fiddly procedure though, and usually a more experienced editor will be happy to do it if needed.

### Anything I shouldn't do?

Specifically, there are some features enabled by the software, but which are somewhat deprecated or downplayed at this point in time in nLab culture.

One such feature is the "query box". This was once used so that someone reading an nLab article could insert a question or even have a running sequence of comments embedded right there within the article. But we found this didn't really work well, one reason being there it didn't notify author(s) that a question was being asked, so that you'd see the query only if you chanced to go back to the article. Some queries might sit for years before being noticed! And we're still trying to clean up query boxes.

So please don't use this the query box feature -- we ask that you ask your question or post your comment directly at the nForum. If you see an ancient query box, you or someone else should shunt it over to the nForum which serves an an archive as well.

Another thing we want to downplay (more recently) is the use of the footnote feature. It's really not a footnote at all; functionally it's more of an endnote that gets plopped slightly awkwardly at the bottom of the page in the same sized font as anything else, but without any html heading that announces: Footnotes. The bottom line is that it doesn't look very good and calls ungainly attention to itself, something that a footnote is not supposed to do.

Instead, if it's about something minor, consider using a link to another part of the nLab (or wherever) where the point is discussed. If it's less than minor, then maybe consider placing the remark in a Remark environment there in the text. In case of doubt, ask for advice at the nForum.

## Respecting the styles of other authors

This is perhaps the touchiest aspect of these recommendations, so we will say it as plainly as possible. The bottom line is this: the nLab can only function in an atmosphere of mutual consideration and respect; it will never work if there are overhanging attitudes of "my way or the highway". Somehow it *has* been at least moderately successful, we feel, despite the fact that there are already at least a few strong-willed personalities involved.

Perhaps the main reason it has worked is a prevailing ethos of cooperation, and in particular an unspoken obeisance to a "prime directive" (repeating ourselves somewhat):

+-- {: .standout}
Try to arrange nLab material so as to maximize scientific insight, letting the power of the [[nPOV]] shine through, in smooth and unobtrusive language. Try to get your own self and personal idiosyncrasies out of the way of this happening.
=--

The typical underlying tensions between

1. not wishing to step on the toes of others who have put some work into the nLab, and

1. wanting nevertheless to revise the nLab to make it better

are seemingly resolved by correctly interpreting and implementing that directive, through controlled discussion at the nForum. Of course, helping matters along is a semi-cohesive philosophy at the core of our (otherwise loosely connected) working group, of the importance of category theory and higher category theory to organize and simplify mathematics (and physics and in some respects philosophy). Those who do not share in that vision (or mission, whatever you wish to call it) might not be so easily assimilated into the nLab group -- although we are open to possibilities -- since such people may have very different ideas of what to emphasize in mathematics.

An example where a nontrivial rewriting of an article is welcome is where it makes the underlying principles clearer and the development more economical, or where it lays bare unifications with other subject areas, etc. In other words, where the *actual mathematics* is improved (from the [[nPOV]]). Sometimes entire articles may be revamped for such reasons.

Examples where rewrites are maybe not so welcome is where it comes to differences in style between otherwise reasonably competent writers. One writer may prefer a slightly more conversational tone (using 2nd person), and another a slightly more formal tone (using 3rd person). If one feels an impulse to change something in the style of another, the first question that should be asked is: is this *really* going to materially improve the reader's reception? Sometimes it might, oftentimes not really. But this is felt to be an important enough matter to make a revision, it should be discussed *beforehand* at the nForum -- don't shoot first and ask questions later. In all other cases, it's probably better to let it go. Concentrate on the mathematics.

Another small example which may be worth addressing: discrepancies between American and British (Commonwealth) spellings. One might see different spellings of the same word in the same article! Should alternate spellings (which are correct according to the author's dictionary) within the article be aligned to make them consistent? The nLab way seems to be to respect each other's differences in this (minor) respect: if one person writes 'neighborhood' and another 'neighbourhood', then that's okay: let it go.  Similarly, we don't try to enforce consistent standards of punctuation and formatting, where conventions also vary regionally: e.g. do quotations use single or double quotes, does punctuation go inside or outside the quotation marks, is an apostrophe written as `don't` or `don\'t`, etc.  This extends also to respecting each other's choice of whether to use `:` or `\colon` in mathematical notation.

Unlike the above advice about notation, such variations in spelling and punctuation may occur even within a single nLab page, and there is no need to try to make them uniform.  Think of it as a token of respect we have for one another: we may come from all corners of the earth, but we come together to build the nLab. We can tolerate a small difference in cultural background such as this.  (It was, however, agreed informally long ago to use American English spellings in page titles, although of course other spellings should be included as redirects.)

## Related pages

* [[nLab]]

* [[nForum]]

* [[nPOV]]

* [[About]]

* [[HowTo]]

* [[How to get started]]

## References

* Paul Halmos, *How To Write Mathematics*, Enseign. Math. (2) 16 (1970), 123-152. ([online pdf](http://www2.math.uu.se/~takis/ETC/Halmos_howToWriteMath.pdf))

[[!redirects writing in the nLab]]
[[!redirects writing on the nLab]]


Wronskian

Given a set of $n$ functions $f_1,\ldots,f_n$, one can define the matrix

$$
W(f_1,\ldots,f_n) = \left( \array{f_1 & f_2 & \cdots & f_n\\ f_1' & f_2' &\cdots & f_n'\\ \cdot &\cdot &\cdot &\cdots\\ f_1^{(n-1)} & f_2^{(n-1)} &\cdots &f_n^{(n-1)}}\right)
$$

The __Wronskian__ is its [[determinant]]. It is used in the study of linear independence of solution of differential equations and in mathematical physics.

* wikipedia [Wronskian](http://en.wikipedia.org/wiki/Wronskian)


Wu class

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

#Contents#
* table of contents
{:toc}

## Idea

_Wu classes_ are a type of [[universal characteristic class]] in $\mathbb{Z}_2$-[[cohomology]] that refine the [[Stiefel-Whitney classes]].

## Definition

For $X$ a [[topological space]] equipped with a class $E : X  \to B SO(n)$ (a real [[vector bundle]] of some [[rank]] $n$), write

$$
  w_k \in H^k(X, \mathbb{Z}_2)
$$

for the [[Stiefel-Whitney classes]] of $X$. Moreover, write

$$
  \cup : H^k(X, \mathbb{Z}_2) \times H^l(X, \mathbb{Z}_2)
  \to
  H^{k+l}(X, \mathbb{Z}_2)
$$

for the [[cup product]] on $\mathbb{Z}_2$-[[cohomology groups]] and write

$$
  Sq^k(-) : H^l(X, \mathbb{Z}_2) \to H^{k+l}(X, \mathbb{Z}_2)
$$

for the [[Steenrod square]] operations.

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

The **Wu class**

$$
  \nu_k
  \in
  H^k(X,\mathbb{Z}_2)
$$

is defined to be the class that "represents" $Sq^k(-)$ under the cup product, in the sense that for all $x \in H^{n-k}(X, \mathbb{Z}_2)$ where $n$ is the dimension of $X$, we have

$$
  Sq^k(x) = \nu_k \cup x
  \,.
$$

=--

(e.g. [Milnor-Stasheff 74, p. 131-133](#MilnorStasheff74))

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

In other words this says that the lifts of Wu classes to [[integral cohomology]] ([[integral Wu structures]]) are _[[characteristic element of a bilinear form|characteristic elements]]_ of the [[intersection product]] on integral cohomology, inducing [[quadratic refinements]].

=--

## Properties

### Relation to Stiefel-Whitney classes

The total [[Stiefel-Whitney class]] $w$ is the total [[Steenrod square]] of the total Wu class $\nu$.

$$
  w = Sq(\nu)
  \,.
$$

Solving this for the components of $\nu$ in terms of the components of $w$, one finds the first few Wu classes as [[polynomials]] in the Stiefel-Whitney classes as follows

* $\nu_1 = w_1$;

* $\nu_2 = w_2 + w_1^2$

* $\nu_3 = w_1 w_2$

* $\nu_4 = w_4 + w_3 w_1 + w_2^2 + w_1^4$

* $\nu_5 = w_4 w_1 + w_3 w_1^2 + w_2^2 w_1 + w_2 w_1^3$

...

### Relation to Pontryagin classes
 {#RelationToPontryaginClasses}

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

Let $X$ be an [[orientation|oriented]] [[manifold]] $T X : X \to  B SO(n)$ with [[spin structure]] $\hat T X : X \to B Spin(n)$. Then the following classes in [[integral cohomology]] of $X$, built from [[Pontryagin classes]], coincide with Wu-classes under mod-2-reduction $\mathbb{Z} \to \mathbb{Z}_2$:

* $\nu_4 = \frac{1}{2} p_1$

* $\nu_8 = \frac{1}{8}(11 p_1^2  - 20 p_2)$

* $\nu_{12} = \frac{1}{16}(37 p_1^3 - 100 p_1 p_2 + 80 p_3)$.

(all products are [[cup product|cup products]]).

=--

This is discussed in ([Hopkins-Singer, page 101](#HopkinsSinger)).

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

Suppose $X$ is 8 dimensional. Then, for $G \in H^4(X, \mathbb{Z})$ any integral 4-class, the expression

$$
  G \cup G - G \cup \frac{1}{2}p_1 \in H^4(X, \mathbb{Z})
$$

is always even (divisible by 2).

=--

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

By the basic properties of Steenrod squares, we have for the 4-class $G$ that

$$
  G \cup G = Sq^4(G)
  \,.
$$

By the definition \ref{WuClassesBySteenrodSquares} of Wu classes, the image of this integral class in $\mathbb{Z}_2$-coefficients equals the cup product with the Wu class

$$
  G \cup G - G \cup \frac{1}{2}p_1
  =
  Sq^4(G) - G \cup \nu_4
  =
  0 \; mod \; 2.
  \,,
$$

where the first step is by prop. \ref{InTermsOfPontryagin}.

=--

## Applications

### To higher dimensional Chern-Simons theory

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

The relation \ref{DivisibilityOfCupSquare} plays a central role in the definition of the [[higher dimensional Chern-Simons theory|7-dimensional Chern-Simons theory]] which is [[holographic principle|dual]] to the [[self-dual higher gauge theory]] on the [[M5-brane]]. In this context it was first pointed out in ([Witten 1996](#Witten)) and later elaborated on in ([Hopkins-Singer](#HopkinsSinger)).

Specifically, in this context $G$ is the 4-class of the [[circle n-bundle with connection|circle 3-bundle]] underlying the [[supergravity C-field]], subject to the quantization condition

$$
  G_4 = \frac{1}{2}(\frac{1}{2}p_1) + a
  \,,
$$

for some $a \in H^4(X, \mathbb{Z})$, which makes direct sense as an equation in $H^4(X, \mathbb{Z})$ if the [[spin structure]] on $X$ happens to be such $\frac{1}{2}p_1$ is further divisible by 2, and can be made sense of more generally in terms of [[twisted cohomology]] (which was suggested in ([Witten 1996](#Witten)) and made precise sense of in ([Hopkins-Singer](#HopkinsSinger)) ).

For simplicity, assume that $\frac{1}{2}p_1$ of $X$ is further divisible by 2 in the following. We then may consider direct refinements of the above ingredients to [[ordinary differential cohomology]] and so we consider [[circle n-bundle with connection|differential cocycles]] $\hat a, \hat G \in \hat H^4(X)$ with

\[
  \hat G = \frac{1}{2}(\frac{1}{2}\hat \mathbf{p}_1) + \hat a
  \in \hat H^4(X)
  \,,
  \label{DifferentialQuantizationCondition}
\]

where the differential refinement $\frac{1}{2}\hat \mathbf{p}_1$ is discussed in detail at _[[differential string structure]]_.

Now, after [[Kaluza-Klein mechanism|dimensional reduction]] on a 4-[[sphere]], the [[action functional]] of [[11-dimensional supergravity]] on the remaining 7-dimensional $X$ contains a [[higher dimensional Chern-Simons theory|higher Chern-Simons term]] which up to prefactors is of the form

$$
  \hat G \mapsto
  \exp i
  \int_X
    (
      \hat G \cup \hat G -
      (\frac{1}{4}\hat \mathbf{p}_1)^2
    )
  \,,
$$

where

* the cup product now is the differential [[Beilinson-Deligne cup product]] refinement of the integral cup product;

* the symbol $\exp(i \int_X (-))$ denotes [[fiber integration in ordinary differential cohomology]].

Using (eq:DifferentialQuantizationCondition) this is

$$
  \cdots
   =
  \exp i
  \int_X
  \left(
    \hat a \cup \hat a + \hat a \cup \frac{1}{2}\hat \mathbf{p}_1
  \right)
  \,.
$$

But by corollary \ref{DivisibilityOfCupSquare} this is further divisible by 2.
Hence the generator of the group of higher Chern-Simons action functionals is one half of this

$$
  \hat G
   \mapsto
  \exp i
  \int_X
   \frac{1}{2}
    (
      \hat G \cup \hat G -
      (\frac{1}{4}\hat \mathbf{p}_1)^2
    )
  \,.
$$

In ([Witten 1996](#Witten)) it is discussed that the space of [[states]] of this "fractional" functional over a 6-dimensional $\Sigma$ is the space of [[conformal blocks]] of the [[self-dual higher gauge theory]] on the [[M5-brane]].

=--

## Related concepts

* [[integral Wu structure]], [[twisted Wu structure]]

* [[shifted C-field flux quantization]]

* [[Pontryagin class]], [[Stiefel-Whitney class]], [[one-loop anomaly polynomial I8]]

* [[Euler class]]

## References

The original reference is

* [[Wen-Tsun Wu]], _On Pontrjagin classes: II_ Sientia Sinica 4 (1955) 455-490

See also around p. 228 of

* {#MilnorStasheff74} [[John Milnor]], [[Jim Stasheff]], _Characteristic classes_, Princeton University Press (1974)

and section 2 of

* Yanghyun Byun, _On vanishing of characteristic numbers in Poincaré complexes_, Transactions of the AMS, vol 348, number 8 (1996) ([pdf](http://www.ams.org/journals/tran/1996-348-08/S0002-9947-96-01495-X/S0002-9947-96-01495-X.pdf))

and

* [[Robert Stong]], Toshio Yoshida, _Wu classes_ Proceedings of the American Mathematical Society  Vol. 100, No. 2, (1987)  ([JSTOR](http://www.jstor.org/pss/2045970))

Details are reviewed in appendix E of

* [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_
 {#HopkinsSinger}

This is based on or motivated from observations in

* [[Edward Witten]], _Five-Brane Effective Action In M-Theory_ ([arXiv:hep-th/9610234](http://arxiv.org/abs/hep-th/9610234))
 {#Witten}

More discussion of Wu classes in this physical context is in

* [[Hisham Sati]], _Twisted topological structures related to M-branes II: Twisted $Wu$ and $Wu^c$ structures_ ([arXiv:1109.4461](http://arxiv.org/abs/1109.4461))

which also summarizes many standard properties of Wu classes.

[[!redirects Wu classes]]


WZW term of QCD chiral perturbation theory -- references

### The WZW term of QCD chiral perturbation theory
 {#WZWTermOfChiralPerturbationTheoryReferences}

The [[gauged WZW model|gauged]] [[WZW term]] of [[chiral perturbation theory]]/[[quantum hadrodynamics]] which reproduces the [[chiral anomaly]] of [[QCD]] in the [[effective field theory]] of [[mesons]] and [[Skyrmions]]:

#### General
 {#WZWTermOfChiralPerturbationTheoryReferencesGeneral}

The original articles:

* [[Julius Wess]], [[Bruno Zumino]], _Consequences of anomalous Ward identities_, Phys. Lett. B 37 (1971) 95-97 ([spire:67330](https://inspirehep.net/literature/67330), doi:10.1016/0370-2693(71)90582-X)

* {#Witten83a} [[Edward Witten]], _Global aspects of current algebra_, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 422-432 (doi:10.1016/0550-3213(83)90063-9)

See also:

* O. Kaymakcalan, S. Rajeev, J. Schechter, _Nonabelian Anomaly and Vector Meson Decays_, Phys. Rev. D 30 (1984) 594 ([spire:194756](https://inspirehep.net/literature/194756))

Corrections and streamlining of the computations:

* Chou Kuang-chao, Guo Han-ying, Wu Ke, Song Xing-kang, _On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action_, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 67-69 (doi:10.1016/0370-2693(84)90986-9))

* H. Kawai, S.-H. H. Tye,
_Chiral anomalies, effective lagrangians and differential geometry_, Physics Letters B Volume 140, Issues 5–6, 14 June 1984, Pages 403-407 (doi:10.1016/0370-2693(84)90780-9)

* J. L. Mañes, _Differential geometric construction of the gauged Wess-Zumino action_, Nuclear Physics B Volume 250, Issues 1–4, 1985, Pages 369-384 (doi:10.1016/0550-3213(85)90487-0)

* Tomáš Brauner, Helena Kolešová, _Gauged Wess-Zumino terms for a general coset space_, Nuclear Physics B Volume 945, August 2019, 114676 ([doi:10.1016/j.nuclphysb.2019.114676](https://doi.org/10.1016/j.nuclphysb.2019.114676))

See also

* Yasunori Lee, [[Kantaro Ohmori]], [[Yuji Tachikawa]], _Revisiting Wess-Zumino-Witten terms_ ([arXiv:2009.00033](https://arxiv.org/abs/2009.00033))

Interpretation as [[Skyrmion]]/[[baryon current]]:

* [[Jeffrey Goldstone]], [[Frank Wilczek]], _Fractional Quantum Numbers on Solitons_, Phys. Rev. Lett. 47, 986 (1981) ([doi:10.1103/PhysRevLett.47.986](https://doi.org/10.1103/PhysRevLett.47.986))

* {#Witten83b} [[Edward Witten]], _Current algebra, baryons, and quark confinement_, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 433-444 (doi:10.1016/0550-3213(83)90064-0)

* {#AdkinsNappi84} [[Gregory Adkins]], [[Chiara Nappi]], _Stabilization of Chiral Solitons via Vector Mesons_, Phys. Lett. 137B (1984) 251-256 ([spire:194727](http://inspirehep.net/record/194727), doi:10.1016/0370-2693(84)90239-9)

  (beware that the two copies of the text at these two sources differ!)

* {#RhoEtAl16} [[Mannque Rho]] et al., _Introduction_, In: [[Mannque Rho]] et al. (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

Concrete form for $N$-[[flavor (particle physics)|flavor]] [[quantum hadrodynamics]] in 2d:

* C. R. Lee, H. C. Yen, _A Derivation of The Wess-Zumino-Witten Action from Chiral Anomaly Using Homotopy Operators_, Chinese Journal of Physics, Vol 23 No. 1 (1985) ([spire:16389](https://inspirehep.net/literature/16389), [[LeeYenWZW85.pdf:file]])

Concrete form for 2 [[flavor physics|flavors]] in 4d:

* Masashi Wakamatsu, _On the electromagnetic hadron current derived from the gauged Wess-Zumino-Witten action_,  ([arXiv:1108.1236](https://arxiv.org/abs/1108.1236), [spire:922302](https://inspirehep.net/literature/922302))

#### Including light vector mesons
 {#WZWTermOfChiralPerturbationTheoryReferencesIncludingLightVectorMesons}

Concrete form for 2-[[flavor (particle physics)|flavor]] [[quantum hadrodynamics]] in 4d with [[light meson|light]] [[vector mesons]] included ([[omega-meson]] and [[rho-meson]]):

* {#MeissnerZahed86} [[Ulf-G. Meissner]], [[Ismail Zahed]], equation (6) in: _Skyrmions in the Presence of Vector Mesons_, Phys. Rev. Lett. 56, 1035 (1986) ([doi:10.1103/PhysRevLett.56.1035](https://doi.org/10.1103/PhysRevLett.56.1035))

* [[Ulf-G. Meissner]], [[Norbert Kaiser]], [[Wolfram Weise]], equation (2.18) in: _Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties_, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (doi:10.1016/0375-9474(87)90463-5)

* [[Ulf-G. Meissner]], equation (2.45) in: _Low-energy hadron physics from effective chiral Lagrangians with vector mesons_, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (doi:10.1016/0370-1573(88)90090-7)

* {#Kaiser00} Roland Kaiser, equation (12) in: _Anomalies and WZW-term of two-flavour QCD_, Phys. Rev. D63:076010, 2001 ([arXiv:hep-ph/0011377](https://arxiv.org/abs/hep-ph/0011377), [spire:537600](https://inspirehep.net/literature/537600))

#### Including heavy scalar mesons
 {#WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyScalarMesons}

Including [[heavy mesons|heavy]] [[scalar mesons]]:

specifically [[kaons]]:

* [[Curtis Callan]], [[Igor Klebanov]], equation (4.1) in:  _Bound-state approach to strangeness in the Skyrme model_, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (doi10.1016/0550-3213(85)90292-5)

* [[Igor Klebanov]], equation (99) of: _Strangeness in the Skyrme model_, in: D. Vauthrin, F. Lenz, J. W. Negele,  _Hadrons and Hadronic Matter_, Plenum Press 1989 ([doi:10.1007/978-1-4684-1336-6](https://link.springer.com/book/10.1007/978-1-4684-1336-6))

* N. N. Scoccola, D. P. Min, H. Nadeau, [[Mannque Rho]], equation (2.20) in: _The strangeness problem: An $SU(3)$ skyrmion with vector mesons_, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (doi:10.1016/0375-9474(89)90029-8)

specifically [[D-mesons]]:

(...)

specifically [[B-mesons]]:

* [[Mannque Rho]], D. O. Riska, N. N. Scoccola, above (2.1) in: _The energy levels of the heavy flavour baryons in the topological soliton model_, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages343–352 (1992) ([doi:10.1007/BF01283544](https://doi.org/10.1007/BF01283544))

#### Including heavy vector mesons
 {#WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyVectorMesons}

Inclusion of [[heavy mesons|heavy]] [[vector mesons]]:

specifically [[K*-mesons]]:

* S. Ozaki, H. Nagahiro, [[Atsushi Hosaka]], Equations (3) and (9) in: _Magnetic interaction induced by the anomaly in kaon-photoproductions_, Physics Letters B Volume 665, Issue 4, 24 July 2008, Pages 178-181 ([arXiv:0710.5581](https://arxiv.org/abs/0710.5581), [doi:10.1016/j.physletb.2008.06.020](https://doi.org/10.1016/j.physletb.2008.06.020))

#### Including electroweak interactions
 {#WZWTermOfChiralPerturbationTheoryReferencesIncludingElectroweakInteractions}

Including [[electroweak fields]]:

* J. Bijnens, G. Ecker, A. Picha, _The chiral anomaly in non-leptonic weak interactions_, Physics Letters B Volume 286, Issues 3–4, 30 July 1992, Pages 341-347 (doi:10.1016/0370-2693(92)91785-8)

* [[Gerhard Ecker]], [[Helmut Neufeld]], [[Antonio Pich]], _Non-leptonic kaon decays and the chiral anomaly_, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (doi:10.1016/0550-3213(94)90623-8)

Discussion for the full [[standard model of particle physics]]:

* [[Jeffrey Harvey]], Christopher T. Hill, Richard J. Hill, _Standard Model Gauging of the WZW Term: Anomalies, Global Currents and pseudo-Chern-Simons Interactions_, Phys. Rev. D77:085017, 2008 ([arXiv:0712.1230](https://arxiv.org/abs/0712.1230))


WZW-type superstring field theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### $\infty$-Wess-Zumino-Witten theory
+--{: .hide}
[[!include infinity-Wess-Zumino-Witten theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[string field theory]] for the [[superstring]] whose [[action functional]] is of the form of the [[Wess-Zumino-Witten model]], with worldsheet fields replaced by [[second quantization|second-quantum]] string fields.

## References

### Definitions

The WZW-type action functional for open superstring field theory is due to

* [[Nathan Berkovits]], _Super-Poincaré invariant superstring field theory_ , Nucl. Phys. B450 (1995) 90, ([hep-th/9503099](http://arxiv.org/abs/hep-th/9503099))

For closed superstrings, specifically [[heterotic string]]s, a WZW-type action is due to

* [[Nathan Berkovits]], Yuji Okawa, [[Barton Zwiebach]], _WZW-like Action for Heterotic String Field Theory_ ([arXiv:hep-th/0409018](http://arxiv.org/abs/hep-th/0409018))

### Computations

Derivation of [[super Yang-Mills theory]] from open WZW-type superstring field theory is in

* [[Nathan Berkovits]], [[Martin Schnabl]], _Yang-Mills Action from Open Superstring Field Theory_ ([arXiv:hep-th/0307019](http://arxiv.org/abs/hep-th/0307019))


Xavier Bekaert

* [webpage](http://dept.phys.univ-tours.fr/147-bekaert)

## Selected writings

On [[Lie algebroids]] (motivated by [[higher spin gauge theory]]):

* [[Xavier Bekaert]], *Geometric tool kit for higher spin gravity (part II): An introduction to Lie algebroids and their enveloping algebras* [[arXiv:2308.00724](https://arxiv.org/abs/2308.00724)]

## Related entries

* [[self-dual higher gauge theory]]

* [[dual graviton]]

* [[higher spin gauge theory]]

category: people


Xena project

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A project bringing [[formal proof]] and [[proof assistants]] (particularly [[Lean]]) into the practice of undergraduate [[mathematics]].

## Related projects

[[!include proof assistants and formalization projects -- list]]

## References

* [[Kevin Buzzard]], _[Xena project](https://xenaproject.wordpress.com/2018/10/07/what-is-the-xena-project/)_

* {#Buzzard19} [[Kevin Buzzard]], _Using Lean with undergraduate mathematicians_, talk at _[Lean Together 2019 ](https://lean-forward.github.io/lean-together/2019/)_  ([recording](https://av-media.vu.nl/mediasite/Play/1979034f91b640229bffabdecd4eae5c1d?playFrom=76000))

* {#Buzzard19b} [[Kevin Buzzard]], _The future of Mathematics?_, public lecture at the 80th anniversary celebration of CNRS, May 2019 ([recording](https://youtu.be/aZHbnQlFOn4))

[[!redirects Xena]]


Xenia de la Ossa

Xenia de la Ossa is professor for [[mathematics]] at Oxford, working on questions in [[mathematical physics]] and [[string theory]].

## related $n$Lab entries

* [[Calabi-Yau varieties]] over [[finite fields]]

  * [[Philip Candelas]], [[Xenia de la Ossa]], Fernando Rodriguez-Villegas, _Calabi-Yau Manifolds Over Finite Fields, I_ ([arXiv:hep-th/0012233](http://arxiv.org/abs/hep-th/0012233))

  * [[Philip Candelas]], [[Xenia de la Ossa]], Fernando Rodriguez-Villegas, _Calabi-Yau Manifolds Over Finite Fields, II_ ([arXiv:hep-th/0402133](http://arxiv.org/abs/hep-th/0402133))

* [[string phenomenology]]

category: people


Xi Dong

* [webpage](http://web.physics.ucsb.edu/~xidong/)

## Selected writings

Interpretation of [[tensor networks]] encoding [[holographic entanglement entropy]] as [[quantum error correcting codes]]:

* {#ADH14} [[Ahmed Almheiri]], [[Xi Dong]], [[Daniel Harlow]], _Bulk Locality and Quantum Error Correction in AdS/CFT_, JHEP 1504:163,2015 ([arXiv:1411.7041](https://arxiv.org/abs/1411.7041))

Introducing [[holographic Renyi entropy]]:

* {#Dong16} [[Xi Dong]], *The Gravity Dual of Renyi Entropy*, Nature Communications 7, 12472 (2016) ([arXiv:1601.06788](https://arxiv.org/abs/1601.06788), [doi:10.1038/ncomms12472]( https://doi.org/10.1038/ncomms12472))

* [[Xi Dong]], *Holographic Renyi Entropy at High Energy Density*, Phys. Rev. Lett. 122, 041602 (2019) ([arXiv:1811.04081](https://arxiv.org/abs/1811.04081))

More on [[holographic entanglement entropy]]:

* Eugenia Colafranceschi, [[Xi Dong]], [[Donald Marolf]], Zhencheng Wang, *Algebras and Hilbert spaces from gravitational path integrals: Understanding Ryu-Takayanagi/HRT as entropy without invoking holography* [[arXiv:2310.02189](https://arxiv.org/abs/2310.02189)]

category: people


Xi Yin

* [webpage](http://www.people.fas.harvard.edu/~xiyin/Site/Home.html)

## Selected writings

On [[KK-compactification|dimensional reduction]] of [[black rings]] in [[D=5 gravity]] to [[black holes]] in [[D=4 gravity]]:

* [[Davide Gaiotto]], [[Andrew Strominger]], [[Xi Yin]], _5D Black Rings and 4D Black Holes_, JHEP 0602:023, 2006 ([arXiv:hep-th/0504126](https://arxiv.org/abs/hep-th/0504126))

On the [[M5-brane elliptic genus]]:

* [[Davide Gaiotto]], [[Andrew Strominger]], [[Xi Yin]], _The M5-Brane Elliptic Genus: Modularity and BPS States_, JHEP 0708:070, 2007 ([hep-th/0607010](https://arxiv.org/abs/hep-th/0607010))

* [[Davide Gaiotto]], [[Xi Yin]], _Examples of M5-Brane Elliptic Genera_, JHEP 0711:004, 2007 ([arXiv:hep-th/0702012](https://arxiv.org/abs/hep-th/0702012))

## Related $n$Lab entries

* [[black holes in string theory]]

category: people


Xia Gu

## Selected writings

On [[braid group representations]] for [[su(2)-anyon]]-[[anyon statistics|statistics]] from the [[monodromy]] of the [[Knizhnik-Zamolodchikov connection]] of bundles of [[conformal blocks]] over [[configuration spaces of points]]:

* [[Xia Gu]], [[Babak Haghighat]], [[Yihua Liu]], *Ising- and Fibonacci-Anyons from KZ-equations*,  J. High Energ. Phys. **2022** 15 (2022) [[arXiv:2112.07195](https://arxiv.org/abs/2112.07195), doi:10.1007/JHEP09(2022)015]

On [[conformal blocks]] for [[Liouville theory]]:

* [[Xia Gu]], [[Babak Haghighat]], [[Kevin Loo]], *Irregular Fibonacci Conformal Blocks* [[arXiv:2311.13358](https://arxiv.org/abs/2311.13358)]

category: people


Xianzhe Dai

* [webpage](http://www.math.ucsb.edu/~dai/)

## related $n$Lab entries

* [[eta invariant]]

category: people


Xiao-Gang Wen

Xiao-Gang Wen is the BMO Financial Group Isaac Newton Chair in Theoretical Physics at Perimeter Institute. His expertise is in [[condensed matter]] theory of [[strongly correlated system|strongly correlated]] electronic systems and [[topological phases of matter]]. He introduced and studied the concept of *[[topological order]]* and its *[[symmetry protected topological phase|symmetry proteced]]* versions.

* [Institute page](https://physics.mit.edu/faculty/xiao-gang-wen/) at MIT

* [Wikipedia entry](https://en.wikipedia.org/wiki/Xiao-Gang_Wen)

Interview on [[topological order]]:

* [[Philip Ball]], *Making the world from topological order*, National Science Review **6** 2 (2019) 227–230 $[$[doi:10.1093/nsr/nwy116](https://doi.org/10.1093/nsr/nwy116)$]$

## Selected writings

Introducing and developing the notion of [[topological order]]:

* {#Wen89} [[Xiao-Gang Wen]], *Vacuum degeneracy of chiral spin states in compactified space*, Phys. Rev. B **40** (1989) 7387(R) $[$[doi:10.1103/PhysRevB.40.7387](https://doi.org/10.1103/PhysRevB.40.7387)$]$

* {#WenNiu90} [[Xiao-Gang Wen]],  Q. Niu, *Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces*, Phys. Rev. B **41** (1990) 9377 $[$[doi:10.1103/PhysRevB.41.9377](https://doi.org/10.1103/PhysRevB.41.9377)$]$

* {#Wen91} [[Xiao-Gang Wen]], *Non-Abelian statistics in the fractional quantum Hall states*, Phys. Rev. Lett. **66** (1991) 802 ([doi:10.1103/PhysRevLett.66.802](https://doi.org/10.1103/PhysRevLett.66.802))

* {#Wen93} [[Xiao-Gang Wen]], *Topological order and edge structure of $\nu = 1/2$ quantum Hall state*, Phys. Rev. Lett. **70** (1993) 355 $[$[doi:10.1103/PhysRevLett.70.355](https://doi.org/10.1103/PhysRevLett.70.355)$]$

* {#Wen95} [[Xiao-Gang Wen]], *Topological orders and Edge excitations in FQH states*, Advances in Physics **44** (1995) 405 $[$[arXiv:cond-mat/9506066v2](https://arxiv.org/abs/cond-mat/9506066v2), [doi:10.1080/00018739500101566](https://doi.org/10.1080/00018739500101566)$]$

Early review:

* {#Wen91Review} [[Xiao-Gang Wen]], *Topological orders and Chern-Simons theory in strongly correlated quantum liquid*,  International Journal of Modern Physics B **05** 10 (1991)  1641-1648 [[doi:10.1142/S0217979291001541](https://doi.org/10.1142/S0217979291001541)]

On [[quantum spin liquids]] as exhibiting [[topological order]]:

* [[Xiao-Gang Wen]], [[Frank Wilczek]], [[Anthony Zee]], *Chiral spin states and superconductivity*, Phys. Rev. B **39** (1989) 11413 [[doi:10.1103/PhysRevB.39.11413](https://doi.org/10.1103/PhysRevB.39.11413)]

The additional requirement that the Berry connection be non-abelian (and/or the presence of anyons):

* {#GuWen09} [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], p. 2 of:  *Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order*, Phys. Rev. B **80** 155131 (2009) $[$[arXiv:0903.1069](https://arxiv.org/abs/0903.1069), [doi:10.1103/PhysRevB.80.155131](https://doi.org/10.1103/PhysRevB.80.155131)$]$

On understanding generalized [[Laughlin wavefunctions]] as [[conformal blocks]]:

* B. Blok, [[Xiao-Gang Wen]], *Many-body systems with non-abelian statistics*, Nuclear Physics B **374** 3 (1992) 615-646 $[$doi:10.1016/0550-3213(92)90402-W$]$

* [[Xiao-Gang Wen]], Yong-Shi Wu, *Chiral operator product algebra hidden in certain fractional quantum Hall wave functions*, Nucl. Phys. B **419** (1994) 455-479 $[$doi:10.1016/0550-3213(94)90340-9$]$

Suggestion that [[topological order]] goes along with [[long-range entanglement]]:

* {#ChenGuWen2010} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], Section V of: _Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order_ Phys. Rev. B **82** 155138 (2010) $[$[arXiv:1004.3835](http://arxiv.org/abs/1004.3835)$]$

On the [[quantum adiabatic theorem]] in the context of [[topological phases of matter]]:

* [[Mathew B. Hastings]], [[Xiao-Gang Wen]], *Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance*, Phys. Rev. B **72** (2005) 045141  [[arXiv:cond-mat/0503554](https://arxiv.org/abs/cond-mat/0503554), [doi:10.1103/PhysRevB.72.045141](https://doi.org/10.1103/PhysRevB.72.045141)]

Introducing [[string-net models]] and the [[Levin-Wen model]]:

* [[Michael Levin]], [[Xiao-Gang Wen]], _String-net condensation: A physical mechanism for topological phases_, Phys.Rev. B71 (2005) 045110 ([arXiv:cond-mat/0404617](http://arxiv.org/abs/cond-mat/0404617))

Introducing the [[3d toric code]]:

* Alioscia Hamma, Paolo Zanardi, [[Xiao-Gang Wen]], *String and Membrane condensation on 3D lattices*,  Phys. Rev. B72:035307, 2005 ([arXiv:cond-mat/0411752](https://arxiv.org/abs/cond-mat/0411752), [doi:10.1103/PhysRevB.72.035307](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.72.035307))

Introducing [[topological entanglement entropy]] in view of [[string-net models]]:

* [[Michael Levin]], [[Xiao-Gang Wen]], *Detecting topological order in a ground state wave function*, Phys. Rev. Lett., 96, 110405 (2006) ([arXiv:cond-mat/0510613](https://arxiv.org/abs/cond-mat/0510613))

Introducing the notion of [[symmetry protected topological phases]]:

* {#GuWen09} [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], *Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order*, Phys. Rev. B **80** 155131 (2009) $[$[arXiv:0903.1069](https://arxiv.org/abs/0903.1069), [doi:10.1103/PhysRevB.80.155131](https://doi.org/10.1103/PhysRevB.80.155131)$]$

On [[string-net models]] via [[tensor networks]]:

* [[Zheng-Cheng Gu]], [[Michael Levin]], [[Brian Swingle]], [[Xiao-Gang Wen]], *Tensor-product representations for string-net condensed states*, Phys. Rev. B **79** (2009) 085118 $[$[doi:10.1103/PhysRevB.79.085118](https://doi.org/10.1103/PhysRevB.79.085118)$]$

On the classification of free fermion [[SPTs]]:

* {#Wen12} [[Xiao-Gang Wen]], *Symmetry-protected topological phases in noninteracting fermion systems*, Phys. Rev. B **85** (2012) 085103 $[$[doi:10.1103/PhysRevB.85.085103](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.85.085103)$]$

On [[internal symmetry|internal]]-[[symmetry protected topological phases]]:

* {#YeWen13} [[Peng Ye]], [[Xiao-Gang Wen]], *Projective construction of two-dimensional symmetry-protected topological phases with $\mathrm{U}(1)$, $SO(3)$, or $SU(2)$ symmetries*, Phys. Rev. B **87** 195128 (2013) $[$[doi:10.1103/PhysRevB.87.195128](https://doi.org/10.1103/PhysRevB.87.195128), [arXiv:1212.2121](https://arxiv.org/abs/1212.2121)$]$

Claim that [[group cohomology]] classifies some [[symmetry protected topological phases]]:

* {#CGLW11} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], _Symmetry protected topological orders and the group cohomology of their symmetry group_, Phys. Rev. B **87** (2013) 155114  $[$[arXiv:1106.4772](http://arxiv.org/abs/1106.4772)$]$

* [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], *Symmetry protected topological orders and the group cohomology of their symmetry group*,  Science **338** (2012) 1604-1606 ([arXiv:10.1103/PhysRevB.87.155114](https://doi.org/10.1103/PhysRevB.87.155114))

* {#GuWen14} [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], *Symmetry-protected topological orders for interacting fermions -- fermionic topological non-linear sigma-models and a group super-cohomology theory*, Phys. Rev. B **90** 115141 (2014) $[$[arXiv:1201.2648](http://arxiv.org/abs/1201.2648), [doi:10.1103/PhysRevB.90.115141](https://doi.org/10.1103/PhysRevB.90.115141)$]$

More on [[topological order]]/[[topological phases of matter]]:

* [[Xiao-Gang Wen]], _Topological orders and edge excitations in FQH states_, Advances in Physics __44__, 405 (1995). [cond-mat/9506066](http://arxiv.org/abs/cond-mat/9506066)

* Xiao-Gang Wen, _Vacuum degeneracy of chiral spin state in compactified spaces_, Phys. Rev. B, 40, 7387 (1989),

* Xiao-Gang Wen, _Topological orders in rigid states_, Int. J. Mod. Phys. B4, 239 (1990)

* Xiao-Gang Wen, Qian Niu,  _Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces_, Phys. Rev. B41, 9377 (1990)

* E. Keski-Vakkuri, Xiao-Gang Wen, _Ground state structure of hierarchical QH states on torus and modular transformation_, Int. J. Mod. Phys. B7, 4227 (1993) [pdf](http://dao.mit.edu/~wen/pub/kw.pdf)

* Xiao-Gang Wen, _Mean field theory of spin liquid states with finite energy gap and topological orders_, Phys. Rev. B 44 2664 (1991).
* Xiao-Gang Wen, _Non-Abelian Statistics in the FQH states_, [pdf](http://dao.mit.edu/~wen/pub/nab.pdf)
Phys. Rev. Lett. 66, 802 (1991).

* Xiao-Gang Wen,  Yong-Shi Wu, _Chiral operator product algebra hidden in certain FQH states_,  Nucl. Phys. B419, 455 (1994), [pdf](http://dao.mit.edu/~wen/pub/nabdw.pdf)

* [[Xie Chen]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], _Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order_ Phys. Rev. B 82, 155138 (2010)

* [[Juven Wang]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], _Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond_, [arxiv:1405.7689](https://arxiv.org/abs/1405.7689), [Phys. Rev. Lett. 114, 031601 (2015)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.031601)

* [[Liang Kong]], Tian Lan, [[Xiao-Gang Wen]], Zhi-Hao Zhang, Hao Zheng, _Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry_ ([arXiv:2005.14178](https://arxiv.org/abs/2005.14178))

On [[quantum information theory]] ([[entanglement entropy]]) applied to [[topological phases of matter]]/[[topological order]]:

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

## Related entries

* [[topological order]]

* [[symmetry protected trivial order]]

* [[Levin-Wen model]]

category: people


Xiao-Song Lin

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=15130)

* [[Zhenghan Wang]], *[In Memory of X.-S. Lin](http://web.math.ucsb.edu/~zhenghwa/personal.php)*

## Selected writings

On the [[loop braid group]]:

* [[Xiao-Song Lin]], *The motion group of the unlink and its representations*, in:

  *Xiao-Song Lin's Unpublished Papers* ([doi:10.1142/9789812819116_others01](https://doi.org/10.1142/9789812819116_others01)), Part B of:

  [[Kevin Lin]], [[Zhenghan Wang]], [[Weiping Zhang]] (eds.) *Topology and Physics: Proceedings of the Nankai International Conference in Memory of Xiao-Song Lin*, Tianjin, China, 27-31 July 2007. World Scientific, 2008 ([doi:10.1142/6907](https://doi.org/10.1142/6907))

category: people


Xie Chen

* [personal page](http://www.its.caltech.edu/~xcchen/)

## Selected writings

Suggestion that [[topological order]] goes along with [[long-range entanglement]]:

* {#ChenGuWen2010} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]],  _Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order_ Phys. Rev. B **82** 155138 (2010) $[$[arXiv:1004.3835](http://arxiv.org/abs/1004.3835)$]$

Claim that [[group cohomology]] classifies some [[symmetry protected topological phases]]:

* {#CGLW11} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], _Symmetry protected topological orders and the group cohomology of their symmetry group_, Phys. Rev. B **87** (2013) 155114  $[$[arXiv:1106.4772](http://arxiv.org/abs/1106.4772)$]$

* [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], *Symmetry protected topological orders and the group cohomology of their symmetry group*,  Science **338** (2012) 1604-1606 ([arXiv:10.1103/PhysRevB.87.155114](https://doi.org/10.1103/PhysRevB.87.155114))

On [[quantum information theory]] ([[entanglement entropy]]) applied to [[topological phases of matter]]/[[topological order]]:

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

category: people


Xin Wang

* [arXiv page](https://arxiv.org/search/hep-th?searchtype=author&query=Wang,+X)

## Selected writings

On [[quantum Seiberg-Witten curves]] of [[E-string]]-theories in relation to [[D6-D8-brane bound states]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], [[Xin Wang]], *E-string Quantum Curve*, Nuclear Physics B **973** (2021) 115602 [[arXiv:2103.16996](https://arxiv.org/abs/2103.16996), [doi:10.1016/j.nuclphysb.2021.115602](https://doi.org/10.1016/j.nuclphysb.2021.115602)]

category: people


Xinwen Zhu

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

* [Institute page](https://www.pma.caltech.edu/people/xinwen-zhu)

## Selected writings

On the [[moduli stack of L-parameters]]:

* {#Zhu20} [[Xinwen Zhu]], _Coherent sheaves on the stack of Langlands parameters_ ([arXiv:2008.02998](https://arxiv.org/abs/2008.02998))

Relating the [[Pontrjagin algebra]] on [[loop groups]] of [[compact Lie groups]] to their [[Langlands dual groups]]:

* [[Zhiwei Yun]], [[Xinwen Zhu]], *Integral homology of loop groups via Langlands dual groups*, Represent. Theory **15** (2011) 347-369 [[arXiv:0909.5487](https://arxiv.org/abs/0909.5487), [doi:10.1090/S1088-4165-2011-00399-X](https://doi.org/10.1090/S1088-4165-2011-00399-X)]

category: people


XTT

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

#Contents#
* table of contents
{:toc}

## Idea

XTT is an [[set-level type theory|set-level]] [[cubical type theory]] developed by [[Jonathan Sterling]], [[Carlo Angiuli]], and [[Daniel Gratzer]]. The authors have stated that the name is not short for anything, but it is a pun on "eXtensional Type Theory".

XTT features [[boundary separation]], which implies [[UIP]] as a theorem of XTT, rather than an axiom as in [[Martin-Löf type theory]]. As a result, it is an example of a [[cubical type theory]] which is not a [[homotopy type theory]].

Additionally, XTT has [[regularity]], [[canonicity]], and certain [[higher inductive types]] like [[propositional truncation]]. It is an open question if it additionally has [[decidable]] [[type checking]] and [[normal form|normalization]].

## See also

* [[set-level type theory]]

* [[cubical type theory]]

## References

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _Cubical syntax for reflection-free extensional equality_. In [[Herman Geuvers]] (ed.), _4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)_, volume 131 of _Leibniz International Proceedings in Informatics (LIPIcs)_, pages 31:1-31:25. ([arXiv:1904.08562](https://arxiv.org/abs/1904.08562), [doi:10.4230/LIPIcs.FCSD.2019.31](https://doi.org/10.4230/LIPIcs.FCSD.2019.31))

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _A Cubical Language for Bishop Sets_, Logical Methods in Computer Science, 18 (1), 2022. ([arXiv:2003.01491](https://arxiv.org/abs/2003.01491)).


xyz


XYZ particle

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

#Contents#
* table of contents
{:toc}

## Idea

_XYZ particles_ are [[charmonium]]-like [[exotic meson|exotic mesons]] which are increasingly being observed in [[experiment]] (typically in [[B-meson]]-experiments such as [[Belle collaboration|Belle]], [[LHCb experiment|LHCb]], , [[BaBar experiment|BaBar]]), but whch are hard to interpret as [[bound states]] in the traditional [[quark]] [[model (in theoretical physics)]] of [[quantum chromodynamics]].

## Related concepts

* [[tetraquark]]

* [[quantum hadrodynamics]]

## References

Review:

* Stephen Godfrey and Stephen L. Olsen, _The Exotic XYZ Charmonium-Like Mesons_, Annual Review of Nuclear and Particle Science Vol. 58:51-73 (Volume publication date 23 November 2008) ([arXiv:10.1146/annurev.nucl.58.110707.171145](https://doi.org/10.1146/annurev.nucl.58.110707.171145))

* Xiang Liu, _An overview of XYZ new particles_, Chin. Sci. Bull. 59, 3815-3830 (2014) ([arXiv:1312.7408](https://arxiv.org/abs/1312.7408))

* Stephen Lars Olsen, _The XYZ mesons: what they aren't_, EPJ Web of Conferences 202, 01003 (2019) ([doi:10.1051/epjconf/201920201003](https://doi.org/10.1051/epjconf/201920201003))

* Nora Brambilla, Simon Eidelman, Christoph Hanhart, Alexey Nefediev, Cheng-Ping Shen, Christopher E. Thomas, Antonio Vairo, Chang-Zheng Yuan, _The XYZ states: experimental and theoretical status and perspectives_, Physics Reports 2020 ([arXiv:1907.07583](https://arxiv.org/abs/1907.07583))

See also

* Wikipedia, *[XYZ particle](https://en.wikipedia.org/wiki/XYZ_particle)*

* Wikipedia, *[Exotic hadron](https://en.wikipedia.org/wiki/Exotic_hadron)*

For references specifically on [[tetraquarks]] see [there](tetraquark#References).

Discussion of exotic hadrons via [[hadron supersymmetry]]:

* [[Don Lichtenberg]],  Renato Roncaglia, Enrico Predazzi, *Predicting exotic hadron masses from supersymmetry and a quark -- diquark model*, J. Phys. G: Nucl. Part. Phys. **23** 865 ([doi:10.1088/0954-3899/23/8/001](https://iopscience.iop.org/article/10.1088/0954-3899/23/8/001))

* [[Harry J. Lipkin]], *Exotic Hadrons in the Constituent Quark Model*, Prog. Theor. Phys. Suppl. **168** (2007) 15-22 ([arXiv:hep-ph/0703190](https://arxiv.org/abs/hep-ph/0703190), [doi:10.1143/PTPS.168.15](https://doi.org/10.1143/PTPS.168.15))

[[!redirects XYZ particles]]

[[!redirects XYZ meson]]
[[!redirects XYZ mesons]]

[[!redirects exotic hadron]]
[[!redirects exotic hadrons]]


Y-system

__Y-system__ and __T-system__ are two related classes of algebraic relations associated with [[affine Lie algebra]]s and can be considered as encoding certain [[integrable system]]s. Y-system can be considered as a system of [[difference equation]]s for a finite set of commuting
variables $Y_i$, $i\in I$,

$$
Y_i(t+1)Y_i(t-1)=\prod_{j\neq i}\bigl(Y_j(t)+ 1\bigr)^{-a_{ij}}
$$

The elements of the T-system satisfy discrete [[Hirota equation]]s.

For reviews see

*  Atsuo Kuniba, Tomoki Nakanishi, Junji Suzuki, _T-systems and Y-systems in integrable systems_, J. Phys. A44 103001 (2011) [doi](http://dx.doi.org/10.1088/1751-8113/44/10/103001); _T-systems and Y-systems for quantum affinizations
of quantum Kac-Moody algebras_, SIGMA 5 (2009), 108, 23 pages

Y-system has a remarkable connection to [[cluster algebra]]s:

* [[Bernhard Keller]], _Cluster algebras, quiver representations and triangulated categories_, arXiv:0807.1960
* [[Sergey Fomin]], [[Andrei Zelevinsky]], _Y-systems and generalized associahedra_, Ann. Math. __158__ 977-1018 (2003) [doi](https://doi.org/10.4007/annals.2003.158.977)

Y-systems are relevant for integrability phenomena in superstring theory and in relation to study of spectrum of N=4 SUSY Zang-Mills theory. See survey

* Stijn J. van Tongeren, _Integrability of the $AdS_5 \times S^5$ superstring and its deformations_, J. Phys. A: Math. Theor. 47 (2014) 433001 [arxiv/1310.4854](http://arxiv.org/abs/1310.4854)

[[!redirects T-system]]
[[!redirects Y-system and T-system]]

category: physics


Yael Fregier

# Yaël Frégier

An Associate Professor  (Maître de Conférence) at the Artois University with interests in "theoretical physics, algebraic topology, machine learning and its applications".

* [webpage](https://sites.google.com/view/homepagelml/home), [publications](https://sites.google.com/view/homepagelml/publications)

## Selected publications

* Martin Callies, [[Yaël Frégier]], [[Chris Rogers]], [[Marco Zambon]], _Homotopy moment maps_, Advances in Mathematics __303__ (2016) 954--1043 ([arXiv:1304.2051](https://arxiv.org/abs/1304.2051))

category: people

[[!redirects Y. Frégier]]
[[!redirects Yaël Frégier]]


Yael Karshon

* [webpage](http://www.math.toronto.edu/karshon/)

## Selected writings

On [[geometric quantization]]:

*  [[Ana Cannas da Silva]], [[Yael Karshon]], [[Susan Tolman]], *Quantization of Presymplectic Manifolds and Circle Actions*, Transactions of the AMS **352** 2 (2000) 525-552  [[arXiv:dg-ga/9705008](http://arxiv.org/abs/dg-ga/9705008), [jstor:118052](https://www.jstor.org/stable/118052)]

On [[orbifolds]] regarded as naive local quotient [[diffeological spaces]]:

* {#IKZ10} [[Patrick Iglesias-Zemmour]], [[Yael Karshon]], Moshe Zadka, _Orbifolds as diffeologies_, Transactions of the American Mathematical Society 362 (2010), 2811-2831 ([arXiv:math/0501093](https://arxiv.org/abs/math/0501093))

## Related entries

* [[orbifold]]

* [[diffeological space]]

* [[Guillemin-Sternberg geometric quantization conjecture]]

category: people


Yakir Aharonov

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

* [Institute page](https://en-exact-sciences.tau.ac.il/profile/yakir)

* [GoogleScholar page](https://scholar.google.com/citations?user=bkVKG88AAAAJ&hl=en)

## Selected writings

Discussion of what came to be known as the [[Aharonov-Bohm effect]]:

* [[Yakir Aharonov]], [[David Bohm]], *Significance of Electromagnetic Potentials in the Quantum Theory*, Phys. Rev. **115** (1959) 485 [[doi:10.1103/PhysRev.115.485](https://doi.org/10.1103/PhysRev.115.485), [pdf](https://journals.aps.org/pr/pdf/10.1103/PhysRev.115.485)]

category: people


Yakov Eliashberg

Yakov Eliashberg (Яков Матвеевич Элиашберг) is a Russian-American mathematician who has made contributions to [[contact manifold|contact]] and [[symplectic topology]]. He is currently the Herald L. and Caroline L. Ritch Professor of Mathematics at Stanford.

* [Stanford faculty page](https://mathematics.stanford.edu/people/yakov-eliashberg)

* [Wikipedia page](https://en.wikipedia.org/wiki/Yakov_Eliashberg)

## Selected writings

On [[contact manifold|contact]] 3-manifolds:

* [[Yakov Eliashberg]], _Classification of overtwisted contact structures on 3-manifolds_, Invent. Math. 98 (1989), 623-637. ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.465.8237&rep=rep1&type=pdf))

* [[Yakov Eliashberg]], _Contact 3-manifolds twenty years since J Martinet’s work_, Ann. Inst. Fourier 42 (1992), 165-192. ([pdf](http://www.numdam.org/item/10.5802/aif.1288.pdf))

On [[quantomorphism groups]]:

* [[Yakov Eliashberg]], [[Leonid Polterovich]], _Partially ordered groups and geometry of contact transformations_. Geom. Funct. Anal. 10 (2000), no. 6, 1448-1476. ([arXiv:math/9910065](https://arxiv.org/abs/math/9910065), [doi](https://doi.org/10.1007/PL00001656))

On [[symplectic field theory]]:

* [[Yakov Eliashberg]], [[Alexander Givental]], [[Helmut Hofer]], _Introduction to symplectic field theory_, in N. Alon, J. Bourgain, A. Connes, M. Gromov, V. Milman, eds., _Visions in Mathematics: GAFA 2000 Special volume, Part II_, 2010, pp 560-673. ([arXiv:math.SG/0010059](http://arxiv.org/abs/math/0010059), [book chapter](https://link.springer.com/chapter/10.1007/978-3-0346-0425-3_4))

On Gromov's [[h-principle|$h$-principle]]:

* [[Yakov Eliashberg]], Nikolai Mishachev, _Holonomic approximation and Gromov's h-principle_, [arXiv:math/0101196](https://arxiv.org/abs/math/0101196).

* [[Yakov Eliashberg]], Nikolai Mishachev, _Introduction to the h-principle_, Graduate Studies in Mathematics 48, Amer. Math. Soc., 2002. [Google Books](http://books.google.co.uk/books?isbn=0821832271)

## Related $n$Lab pages

* [[contact manifold]]

* [[h-principle]]

* [[quantomorphism group]]

* [[symplectic field theory]]

* [[Timeline of category theory and related mathematics]]


Yakov Itin

* [institute page](http://www.math.huji.ac.il/~itin/)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=56436)

* [GoogleScholar page](https://scholar.google.com/citations?user=t6_RFl4AAAAJ&hl=en)

## Selected writings

On [[pre-metric electromagnetism]]:

* {#HehlItinObukhov16} [[Friedrich W. Hehl]], [[Yakov Itin]], [[Yuri N. Obukhov]], *On Kottler's path: origin and evolution of the premetric program in gravity and in electrodynamics*, International Journal of Modern Physics D **25** 11 (2016) 1640016  [[arXiv:1607.06159](https://arxiv.org/abs/1607.06159), [doi:10.1142/S0218271816400162](https://doi.org/10.1142/S0218271816400162)]

* [[Yakov Itin]], *Premetric representation of mechanics, electromagnetism and gravity*, International Journal of Geometric Methods in Modern Physics **15** supp01 (2018) [[doi:10.1142/S0219887818400029](https://doi.org/10.1142/S0219887818400029)]

category: people


Yakov Kremnitzer

Yakov Kremnitzer is professor for pure [[mathematics]] at Oxford.

* [webpage](https://www.maths.ox.ac.uk/contact/details/kremnitzer)

## related $n$Lab entries

* ([[non-archimedean analytic geometry|non-archimedean]]) [[analytic geometry]]

category: people

[[!redirects  Kobi Kremnitzer]]
[[!redirects Kobi Kremnizer]]
[[!redirects Yakov Kremnizer]]


Yakov Shnir

* [personal page](http://theor.jinr.ru/~shnir/)

## Selected writings

On [[magnetic monopoles]] ([[Dirac monopoles]], [[Yang-Mills monopoles]]):

* [[Yakov Shnir]], *Magnetic Monopoles*, Springer 2005 ([ISBN:978-3-540-29082-7](https://www.springer.com/gp/book/9783540252771))

category: people


Yan Liu

* [intitute page](http://sse.buaa.edu.cn/info/1009/1637.htm)

* [GoogleScholar page](https://scholar.google.com/citations?user=e0BQlXoAAAAJ&hl=en)

## Selected writings

On the [[AdS-CMT correspondence]]:

* {#ZaanenLiuSunSchalm15} [[Jan Zaanen]], [[Yan Liu]], Ya-Wen Sun, [[Koenraad Schalm]], *Holographic Duality in Condensed Matter Physics*, Cambridge University Press 2015 [[doi:10.1017/CBO9781139942492](https://doi.org/10.1017/CBO9781139942492)]

Introducing a holographic description of [[topological semimetals]] via the [[AdS-CMT correspondence]]:

* [[Karl Landsteiner]], [[Yan Liu]], *The holographic Weyl semi-metal*, Physics Letters B **753** (2016) 453-457 [[arXiv:1505.04772](https://arxiv.org/abs/1505.04772), [doi:10.1016/j.physletb.2015.12.052](https://doi.org/10.1016/j.physletb.2015.12.052)]

* [[Karl Landsteiner]], [[Yan Liu]], Ya-Wen Sun, *Quantum phase transition between a topological and a trivial semimetal from holography*, Phys. Rev. Lett. **116** 081602 (2016) [[arXiv:1511.05505](https://arxiv.org/abs/1511.05505), [doi:10.1103/PhysRevLett.116.081602](https://doi.org/10.1103/PhysRevLett.116.081602)]

* Ling-Long Gao, [[Yan Liu]], Hong-Da Lyu, *Black hole interiors in holographic topological semimetals* [[arXiv:2301.01468](https://arxiv.org/abs/2301.01468)]

category: people


Yan Soibelman

* [website](http://www.math.ksu.edu/~soibel/)

## Selected writings

On [[string theory]] via [[2-spectral triples]] reduced to [[spectral triples]]:

* {#Soibelman11} [[Yan Soibelman]], _Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry_ ([pdf](http://www.math.ksu.edu/~soibel/nc-riem-3.pdf), [pdf](https://ncatlab.org/schreiber/files/SoibelmanCFTandRicciCurvature.pdf)), in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2001)

## Related $n$Lab entries

* [[deformation theory]]

* [[spectral triple]]

* [[2-spectral triple]]

category: people


Yan Zhang

[[!redirects Yan X Zhang]]

* [institute page](http://math.mit.edu/~yanzhang/)

* [personal page](http://math.mit.edu/~yanzhang/personal.html)

## Writings

On [[adinkras]]:

* {#Zhang11} _Adinkras for Mathematicians_ ([arXiv:1111.6055](https://arxiv.org/abs/1111.6055))

* {#Zhang13} _The combinatorics of Adinkras_, PhD thesis, MIT (2013) ([pdf](http://math.mit.edu/~yanzhang/math/thesis_adinkras.pdf))

## related $n$Lab entries

* [[adinkra]]

category: people


Yang monopole

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Yang monopole_ A [[monopole]] in [[Yang-Mills theory]].
The generalization of the [[Dirac monopole]] from 3+1 dimensional [[spacetime]] to 5+1 dimensional [[spacetime]].

## Definition
 {#Definition}

### Recalling the Dirac monopole

For comparison, first note that the Dirac monopole is a [[circle group]] [[principal bundle]] with non-trivial [[first Chern class]] on a [[spacetime]] of the form

$$
  X_4 \;\coloneqq\; (\mathbb{R}^3_{space} - \{0\}) \times \mathbb{R}_{time}
$$

witnessed (via [[Chern-Weil theory]]) by the [[magnetic charge]]

$$
  Q = \int_{S^2} F_\nabla \in \mathbb{Z} \hookrightarrow \mathbb{R}
$$

which is the [[integration of differential forms|integration]] of the [[curvature]] [[differential 2-form]] $F_\nabla$ of any [[principal connection]] $\nabla$ on this bundle over any [[sphere]] wrapping the removed origin of $\mathbb{R}^3$. We may think of the Dirac monopole as being an effective [[magnetic monopole]] "[[particle]]" with [[worldline]] $\{0\} \times \mathbb{R}_{time}$. But in the above description we can just as well remove a little 3-[[ball]] $D^3_\epsilon$ from space, instead of just a point, and when viewed as such the Dirac monopole is a 2-[[brane]] with [[worldvolume]] $(\partial D_\epsilon) \times \mathbb{R}_{times}$.

### The Yang monopole

Now analogously, a _Yang monopole_ is a nontrivial [[special unitary group]]-[[principal bundle]] (for which there is no non-trivial [[first Chern class]]) with non-trivial _[[second Chern class]]_ on

$$
  X_6 \coloneqq (\mathbb{R}^5_{space} - \{0\}) \times \mathbb{R}_{time}
$$

witnessed (via [[Chern-Weil theory]]) by the [[instanton number]]

$$
  Q = \int_{S^4} \langle F_\nabla \wedge F_\nabla\rangle \in \mathbb{Z} \hookrightarrow \mathbb{R}
$$

which is the [[integration of differential forms|integration]] of the [[curvature]] [[differential 2-form]] $F_\nabla$ wedge-squared and evaluated in the canonical [[Killing form]] [[invariant polynomial]] to a [[differential 4-form]] of any [[principal connection]] $\nabla$ on this bundle over any 4-[[sphere]] wrapping the removed origin of $\mathbb{R}^5$.

As before, we may equivalently think of removing instead of just a point a small 5-[[ball]] $D_{\epsilon}^5$ and then the Yang monopole appears as a 4-[[brane]] with [[worldvolume]] $(\partial D_\epsilon^5) \times \mathbb{R}_{time}$. This is how the Yang monopole appears in [[string theory]]/[[M-theory]] ([Bergsgoeff-Gibbons-Townsend 06](BergsgoeffGibbonsTownsend06)).

In terms of [[higher gauge theory]] the Yang monopole is seen to be more directly analogous to the [[Dirac monopole]]: the 4-form $\langle F_\nabla \wedge F_\nabla\rangle$ is actually the [[curvature]] 4-form of a [[circle n-bundle with connection|circle 3-bundle with connection]] on [[spacetime]] which is induced from the given $SU(N)$-[[principal connection]], namely the [[Chern-Simons circle 3-bundle]] which is modulated by the [[universal characteristic class]] that is the differential refinement of the second Chern class:

$$
  \mathbf{c}_2 \colon \mathbf{B}SU(N)_{conn} \to \mathbf{B}^3 U(1)_{conn}
$$

(this is discussed at _[[differential string structure]]_). Hence $\mathbf{c}_2(\nabla)$ is a [[circle n-bundle with connection|3-connection]] with [[curvature]]

$$
  F_{\mathbf{c}_2(\nabla)} =  \langle F_\nabla \wedge F_\nabla \rangle
  \,.
$$

Now the analog of the [[first Chern class]] as one passes to such [[circle n-bundles]] is called the higher [[Dixmier-Douady class]], and the Yang monopole charge is just this 2-Dixmier-Douady class of the [[Chern-Simons circle 3-bundle]] induced by a $SU(N)$-[[principal connection]] with corresponding [[instanton number]]:

$$
  Q = \int_{S^4} F_{\mathbf{c}_2(\nabla)} \in \mathbb{Z} \hookrightarrow \mathbb{R}
  \,.
$$

## Examples

### In M-brane theory

The end-surface of an [[M5-brane]] ending on an [[M9-brane]] is a Yang-monopole in the M5 [[worldvolume]] ([Bergsgoeff-Gibbons-Townsend 06](#BergsgoeffGibbonsTownsend06)).

## References

### General

The Yang monopole was introduced as a generalization to 5+1 dimensional [[spacetime]] of the [[Dirac monopole]] in 3+1 dimensional spacetime in

* [[Chen Ning Yang]], _Generalization of Dirac's Monopole to $SU(2)$ Gauge Fields_,  J. Math. Phys. 19, 320 (1978).

More general discussion is in

* Tigran Tchrakian, _Dirac-Yang monopoles and their regular counterparts_ ([arXiv:hep-th/0612249](http://arxiv.org/abs/hep-th/0612249))

See also

* Frederik Nørfjand, Nikolaj Thomas Zinner, _Non-existence theorems and solutions of the Wu-Yang monopole equation_ ([arxiv:1911.08140](https://arxiv.org/abs/1911.08140))

### In string theory

Appearance of Yang monopoles in [[string theory]] goes back to

* Adil Belhaj, Pablo Diaz, Antonio Segui, _On the Superstring Realization of the Yang Monopole_ ([arXiv:hep-th/0703255](http://arxiv.org/abs/hep-th/0703255))

The appearance of the Yang monopole as the [[boundary]] of an open [[M5-brane]] ending on an [[M9-brane]] is discussed in

* [[Eric Bergshoeff]], [[Gary Gibbons]], [[Paul Townsend]], _Open M5-branes_, Phys.Rev.Lett.97:231601 2006 ([arXiv:hep-th/0607193](http://arxiv.org/abs/hep-th/0607193))
 {#BergsgoeffGibbonsTownsend06}

Based on this a Yang monopole is realized in [[type IIA string theory]] in section 2 of

* Adil Belhaj, Pablo Diaz, Antonio Segui, _The Yang Monopole in IIA Superstring: Multi-charge Disease and Enhancon Cure_ ([arXiv:1102.1538](http://arxiv.org/abs/1102.1538))

[[!redirects Yang monopoles]]


Yang Yang

* [institute page](https://grk1670.math.uni-hamburg.de/researchers/yang/)

## Selected writings

Constructing [[correlators]] in [[rational conformal field theory]] via [[string net models]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-Net Construction of RCFT Correlators*, Springer Briefs in Mathematical Physics **45**, Springer (2023) [[doi:10.1007/978-3-031-14682-4](https://doi.org/10.1007/978-3-031-14682-4), [arXiv:2112.12708](https://arxiv.org/abs/2112.12708)]

Constructing [[modular functors]] from [[pivotal category|pivotal]] [[bicategories]] using [[string net models]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-net models for pivotal bicategories* [[arXiv:2302.01468](https://arxiv.org/abs/2302.01468)]

Survey of [[algebra|algebraic]] [[structures]] in [[2d conformal field theory]] ([[vertex operator algebras]], [[conformal blocks]], [[modular tensor category|modular]] [[fusion categories]], [[modular functors]], [[FRS theorem]], [[string-net models]]):

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], *Algebraic structures in two-dimensional conformal field theory*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

category: people


Yang Zhang

* [webpage](http://www.nbi.dk/~zhang/)

## Selected writings

On [[characteristic classes]]:

* {#Zhang11} [[Yang Zhang]], _A brief introduction to characteristic classes from the differentiable viewpoint_, 2011 ([pdf](http://staff.ustc.edu.cn/~yzhphy/notes/A%20brief%20introduction%20to%20characteristic%20classes%20from%20the%20differentiable%20viewpoint.pdf), [[Zhang_CharacteristicClasses.pdf:file]])

## Related $n$Lab entries

* [[characteristic class]]

* [[splitting principle]]

category: people


Yang-Baxter equation

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

Beware that the term _Yang-Baxter equation_ can mean (or be interpreted in the context of) any of several related but different concepts:

[[!include Yang-Baxter equations -- contents]]

## References

See also

* Wikipedia, _[Yang-Baxter equation](https://en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation)_

[[!redirects Yang-Baxter equations]]


Yang-Baxter equations -- contents

**[[Yang-Baxter equations]]**

* [[classical Yang-Baxter equation]]

* [[quantum Yang-Baxter equation]]

* [[dynamical quantum Yang-Baxter equation]]

* [[associative Yang-Baxter equation]]

* [[infinitesimal braid relation]]

* [[set theoretic Yang-Baxter equation]]

* [[quasitriangular bialgebra]]


Yang-Hui He

* [webpage](https://www.city.ac.uk/people/academics/yang-hui-he)

## Selected writings

On [[toric duality]]:

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 ([arXiv:hep-th/0003085](https://arxiv.org/abs/hep-th/0003085))

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _Phase structure of D-brane gauge theories and toric duality_ , J. High Energy Phys. 08 (2001) 040 ([hep-th/0104259](https://arxiv.org/abs/hep-th/0104259))

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], [[Angel Uranga]], _Toric duality as Seiberg duality and brane diamonds, J. High Energy Phys. 12 (2001) 035 ([hep-th/0109063](https://arxiv.org/abs/hep-th/0109063))

* [[Bo Feng]], S. Franco, [[Amihay Hanany]], [[Yang-Hui He]], _Unhiggsing the del Pezzo_, J. High Energy Phys. 08 (2003) 058 ([hep-th/0209228](https://arxiv.org/abs/hep-th/0209228))

On [[discrete torsion]]:

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], Nikolaos Prezas, _Discrete Torsion, Non-Abelian Orbifolds and the Schur Multiplier_, JHEP 0101:033, 2001 ([arXiv:hep-th/0010023](https://arxiv.org/abs/hep-th/0010023))

On [[quiver gauge theories]] and [[D-branes]]:

* [[Yang-Hui He]], _Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities_ ([arXiv:hep-th/0408142](https://arxiv.org/abs/hep-th/0408142))

* {#He18} [[Yang-Hui He]], _Quiver Gauge Theories: Finitude and Trichotomoty_, Mathematics 2018, 6(12), 291 ([doi:10.3390/math6120291](https://doi.org/10.3390/math6120291))

On [[quiver gauge theory]] and [[Donaldson-Thomas theory]] in [[heterotic string theory]]:

* {#HeLee12} [[Yang-Hui He]], Seung-Joo Lee, _Quiver Structure of Heterotic Moduli_, J. High Energ. Phys. (2012) 2012: 119 ([arXiv:1208.3004](https://arxiv.org/abs/1208.3004))

On [[NS5-branes]] and [[orientifolds]] with [[RR-field tadpole cancellation]]:

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

On [[exceptional structures]] ([[exceptional Lie algebras]] and [[sporadic finite simple groups]]) via [[del Pezzo surfaces]]:

* [[Yang-Hui He]], [[John McKay]], _Sporadic and Exceptional_ [[arXiv:1505.06742](https://arxiv.org/abs/1505.06742)]

On [[heterotic string theory]] [[string phenomenology]] and the [[landscape of string theory vacua]]:

* [[Lara Anderson]], [[Yang-Hui He]], [[Andre Lukas]], _Heterotic Compactification, An Algorithmic Approach_, JHEP 0707:049, 2007 ([arXiv:hep-th/0702210](https://arxiv.org/abs/hep-th/0702210))

* {#CHE18} [[Andrei Constantin]], [[Yang-Hui He]], [[Andre Lukas]], _Counting String Theory Standard Models_, Physics Letters B Volume 792, 10 May 2019, Pages 258-262 ([arXiv:1810.00444](https://arxiv.org/abs/1810.00444))

* {#He18a} [[Yang-Hui He]], _The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning_ ([arXiv:1812.02893](https://arxiv.org/abs/1812.02893))

* {#He18b} [[Yang-Hui He]], _Deep-learning the landscape_, talk at _[String and M-Theory: The new geometry of the 21st century](https://ims.nus.edu.sg/events/2018/wstring/wk.php)_ ([pdf slides](https://ims.nus.edu.sg/events/2018/wstring/files/yang.pdf), [video recording](https://www.youtube.com/watch?v=x3ThgBgkPlE))

* [[Yang-Hui He]], _Calabi-Yau Spaces in the String Landscape_ ([arXiv:2006.16623](https://arxiv.org/abs/2006.16623))

Relating the [[abc conjecture]] to [[D=4 N=4 super Yang-Mills theory]]:

* [[Yang-Hui He]], Zhi Hu, Malte Probst, James Read, _Yang-Mills Theory and the ABC Conjecture_,  International Journal of Modern Physics A Vol. 33, No. 13, 1850053 (2018)  ([arXiv:1602.01780](https://arxiv.org/abs/1602.01780), [doi:10.1142/S0217751X18500537](https://doi.org/10.1142/S0217751X18500537))

Application of [[machine learning]] to [[mathematical structures]] in [[algebraic geometry]], [[representation theory]], [[number theory]], [[combinatorics]] and [[string theory]]:

* [[Yang-Hui He]], *Machine-Learning Mathematical Structures*, International Journal of Data Science in the Mathematical Sciences [[arXiv:2101.06317](https://arxiv.org/abs/2101.06317), [doi:10.1142/S2810939222500010](https://doi.org/10.1142/S2810939222500010)]

reviewed in:

* [[Yang-Hui He]], *Universes as Bigdata*, [talk](M-Theory+and+Mathematics#He2020) at *[M-Theory and Mathematics 2020](M-Theory+and+Mathematics#2020)*, NYU Abu Dhabi (Jan 2020) [slides: [pdf](https://ncatlab.org/nlab/files/HeSlidesAtMTheoryAndMathematics2020.pdf), video: [YT](https://www.youtube.com/watch?v=i-kuoXjcKls)]

* [[Yang-Hui He]], *Machine-Learning Mathematical Structures*, [talk](/nlab/show/M-Theory+and+Mathematics#He2023) at *[M-Theory and Mathematics 2023](M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (Jan 2023) [slides: [[He-MTheoryAndMath2023.pdf:file]], video: [YT](https://www.youtube.com/watch?v=SlF2T4fxroQ)]

Discussion via [[machine learning]] of [[connection on a bundle|connections]] on [[heterotic line bundles]] over [[Calabi-Yau manifold|Calabi-Yau 3-folds]]:

* [[Anthony Ashmore]], Rehan Deen, [[Yang-Hui He]], [[Burt Ovrut]], *Machine Learning Line Bundle Connections* ([arXiv:2110.12483](https://arxiv.org/abs/2110.12483))

## Related entries

* [[quiver gauge theory]]

* [[string phenomenology]]

* [[heterotic string]]

* [[MSSM]]

* [[landscape of string theory vacua]]

category: people


Yang-Mills equation

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

#Contents#
* table of contents
{:toc}

## Idea

The _Yang-Mills equations_ are the [[equations of motion]]/[[Euler-Lagrange equations]] of [[Yang-Mills theory]]. They generalize [[Maxwell's equations]].

## Related concepts

* [[Yang-Mills instanton]]

* [[Yang-Mills monopole]]

## References

(For full list of references see at _[[Yang-Mills theory]]_)

### General

* {#Uhlenbeck12} [[Karen Uhlenbeck]], notes by [[Laura Fredrickson]], _Equations of Gauge Theory_, lecture at Temple University, 2012 ([pdf](https://web.stanford.edu/~ljfred4/Attachments/TempleLectures.pdf), [[UhlenbeckGaugeTheory.pdf:file]])

* DispersiveWiki, _[Yang-Mills equations](http://wiki.math.toronto.edu/DispersiveWiki/index.php/Yang-Mills_equations)_

* [TP.SE](http://theoreticalphysics.stackexchange.com/), _[Which exact solutions of the classical Yang-Mills equations are known?](http://theoreticalphysics.stackexchange.com/questions/317/which-exact-solutions-of-the-classical-yang-mills-equations-are-known)_

### Solutions

#### General

Wu and Yang (1968) found a static solution to the sourceless $SU(2)$ [[Yang-Mills equations]]. Recent references include

* J. A. O. Marinho, O. Oliveira, B. V. Carlson, T. Frederico, _Revisiting the Wu-Yang Monopole: classical solutions and conformal invariance_

There is an old review,

* Alfred Actor, _Classical solutions of $SU(2)$ Yang—Mills theories_, Rev. Mod. Phys. 51, 461–525 (1979),

that provides some of the known solutions of $SU(2)$ gauge theory in [[Minkowski spacetime|Minkowski]] ([[monopoles]], plane waves, etc) and [[Euclidean space]] ([[instantons]] and their cousins). For general [[gauge groups]] one can get solutions by embedding $SU(2)$'s.

#### Instantons and monopoles

For [[Yang-Mills instantons]] the most general solution is known, first worked out by

* [[Michael Atiyah]], [[Nigel Hitchin]], [[Vladimir Drinfeld]], [[Yuri Manin]], _Construction of instantons_, Physics Letters 65 A, 3, 185--187 (1978) [pdf](http://www.new.ox.ac.uk/system/files/ADHM.pdf)

for the [[classical groups]] [[special unitary group|SU]], [[special orthogonal group|SO]] , [[symplectic group|Sp]], and then by

* C. Bernard, N. Christ, A. Guth, E. Weinberg, _Pseudoparticle Parameters for Arbitrary Gauge Groups_, Phys. Rev. __D16__, 2977 (1977)

for [[exceptional Lie groups]]. The latest twist on the [[Yang-Mills instanton]] story is the construction of solutions with non-trivial [[holonomy]]:

* Thomas C. Kraan, Pierre van Baal, _Periodic instantons with nontrivial holonomy_, Nucl.Phys. B533 (1998) 627-659, [hep-th/9805168](http://arxiv.org/abs/hep-th/9805168)

There is a nice set of lecture notes

* David Tong, _TASI Lectures on Solitons_ ([hep-th/0509216](http://arxiv.org/abs/hep-th/0509216)),

on topological solutions with different co-dimension ([[Yang-Mills instantons]], [[Yang-Mills monopoles]], vortices, domain walls). Note, however, that except for instantons these solutions typically require extra scalars and broken U(1)'s, as one  may find in [[super Yang-Mills theories]].

Some of the material used here has been taken from

* [TP.SE](http://theoreticalphysics.stackexchange.com/), _[Which exact solutions of the classical Yang-Mills equations are known?](http://theoreticalphysics.stackexchange.com/questions/317/which-exact-solutions-of-the-classical-yang-mills-equations-are-known)_

Another model featuring Yang-Mills fields has been proposed by Curci and Ferrari, see [[Curci-Ferrari model]].

[[!redirects Yang-Mills equations]]


Yang-Mills field

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Yang--Mills [[field (physics)|field]]_ is the [[gauge theory|gauge field]] of [[Yang-Mills theory]].

It is modeled by a cocycle $\hat F \in \mathbf{H}(X, \mathbf{B}U(n)_{conn})$ in differential [[nonabelian cohomology]]. Here $\mathbf{B} U(n)_{conn}$ is the [[moduli stack]] of $U(n)$-[[principal connections]], the [[stackification]] of the [[groupoid of Lie-algebra valued forms]], regarded as a [[groupoid]] [[internal category|internal to]] [[smooth spaces]].

This is usually represented by a [[vector bundle]] [[connection on a bundle|with connection]].

As a [[nonabelian cohomology|nonabelian]] [[Čech cohomology|Čech cocycle]] the Yang-Mills field on a space $X$ is represented by

* a [[cover]] $\{U_i \to X\}$

* a collection of $Lie(U(n))$-valued 1-forms  $(A_i \in \Omega^1(U_i, Lie(U(n))))$;

* a collection of $U(n)$-valued smooth functions $(g_{i j} \in C^\infty(U_{i j}, U(n)))$;

* such that on double overlaps
  $$
    A_j = Ad_{g_{i j}} \circ A_i + g_{i j} g g_{i j}^{-1}
    \,,
  $$

* and such that on triple overlaps
  $$
    g_{i j} g_{j k} = g_{i k}
    \,.
  $$

# Examples #

* For $U(n) = U(1)$ this is the [[electromagnetic field]].

* For $U(n) = SU(2) \times U(1)$ this is the "electroweak field";

* For $U(n) = SU(3) $ this is the strong nuclear force field.

[[!redirects Yang--Mills field]]
[[!redirects Yang–Mills field]]
[[!redirects Yang-Mills fields]]