FAQ

# Frequently Asked Questions
* tic
{:toc}

Also including questions that should be frequently asked but aren't.  See also [[HowTo]].

## Instiki features and "features"

### How do I make less-than and greater-than signs?

The characters < and > are interpreted as beginning or ending HTML tags.  To make a less-than or greater-than sign in iTeX math mode, use `\lt` and `\gt` respectively.  For example, `$0\lt 1$` produces $0\lt 1$.

### How do I put math inside HTML? {#markdown1}

By default, the itex syntax `$...$` is disabled inside raw HTML.  However, you can re-enable it by giving the HTML parameter `markdown="1"`.  Thus

I can has $m^a_t \hbar$!!

produces

I can has $(m^a)_t \hbar$!!

(Well, that *used* to work, but it seems to have broken along the way!)

At least for tables, you can also use the [PHP Markdown Extra](http://michelf.com/projects/php-markdown/extra/#table) syntax inside which `$...$` works without a problem.

### How do I make commutative diagrams?

See the [[HowTo]]

### My math doesn't look like math

One of the notable ways in which itex differs from latex is that in itex's math mode, a string of letters without spaces in between is interpreted as a single identifier.  This has the advantage that you can invent new identifiers without needing to use `\operatorname` or `\DeclareMathOperator`, but it means that when you _don't_ want a string of letters interpreted that way you need to put spaces in between.  For example, `$sin(x)$` produces $sin(x)$ which is probably what you want, but `$h=gf$` produces $h=gf$, whereas you probably wanted to write `$h=g f$` to get $h=g f$. On the other hand, you can (and, for the sake of the LaTeX output, probably should) use `$\sin(x)$`, etc.

### Lists are acting strangely.

There are several bugs (or "edge cases") in the handling of lists by Maruku (the formatting filter used by Instiki).  These include:

* [single-element unordered lists](http://rubyforge.org/tracker/index.php?func=detail&aid=22109&group_id=2795&atid=10735)
* [indentation by more than one space](http://rubyforge.org/tracker/index.php?func=detail&aid=8862&group_id=2795&atid=10735)

### My LaTeX produces gobbledygook in iTeX and vice versa.

iTeX is not LaTeX.  iTeX is a pure converter whereas LaTeX is a mixture of a converter and renderer (technically, latex is the rules for converting the input into pure TeX which is then rendered by tex, but since the conversion is also handled by tex, it can cheat and use information about the rendered output to reinterpret the input).  Therefore getting the two to do exactly the same is **never** going to happen.

On the other hand, the aim is to keep iTeX as close as possible to standard LaTeX whilst keeping in mind that iTeX produces MathML.  Their similarity can make the differences all the more jarring.  Here is a list of the differences that we know about together with suggested work-arounds.

* Numbers are one thing.  The syntax `$10^10$` produces $10^10$ in iTeX but $10^{1}0$ in LaTeX.  The **safe** syntax is to always use braces: `$10^{10}$` is consistent.

* Numbers include punctuation.  Periods and commas _within_ numbers are numbers.  This is so that `$1,000,000.00$` (or `$1.000.000,00$` if you are European) renders correctly as $1,000,000.00$ and not ${1},{000},{000}.{00}$.  When combined with the previous difference, this means that `$t^3.2$` renders as $t^3.2$ instead of $t^{3}.2$.  Either use braces or spaces.  (However, for some reason, `$t^3,2$` renders as $t^3,2$ like you\'d expect from TeX, so the two differences don\'t interact entirely logically.  This might have been different in earlier versions of iTeX.)

* Operator names.  Since iTeX does not have macro support, extending the list of operator names would be difficult except for the fact that neighbouring strings of letters get combined into an operator name.  Thus `$cos$` produces $cos$.  However, in LaTeX this produces $c o s$.  Where the operator name is a standard LaTeX one, such as `\cos`, the backslash can be added so `$\cos$` renders correctly in both cases.  Where the operator name is not a standard LaTeX one, say `Poly`, the **safe** syntax is to use `\operatorname`; thus `$\operatorname{Poly}$` renders correctly as $\operatorname{Poly}$ in both situations.  If you *don\'t* want an operator name, then use spaces; thus `$c o s$` produces $c o s$.

* Whitespace in embedded text.  In LaTeX `$x \text{ and } y$` renders as $x\;\text{and}\;y$ but in iTeX it renders as $x\text{ and }y$.  This is because MathML says that fore and aft whitespace on `mtext` elements should be swallowed.  The **safe** syntax is `$x\;\text{and}\;y$`.

* Two-character relations.  In LaTeX, two neighbouring relation symbols are combined into one relation symbol.  Thus `$y := f(x)$` becomes $y \mathrel{:=} f(x)$, while `$y = -f(x)$` becomes $y = -f(x)$ because a minus sign is a unary operator instead of a binary relation.  Since MathML doesn\'t know the difference between unary operators and binary relations, it is inconvenient for iTeX to do this, so `$y := f(x)$` comes out as $y := f(x)$ instead.  The __safe__ syntax is `$y \mathrel{:=} f(x)$`, which produces $y \mathrel{:=} f(x)$.  However, in many cases, there is a combined command that you can use.  In this case, `$y \coloneqq f(x)$` produces $y \coloneqq f(x)$.  This method is better when [available](http://golem.ph.utexas.edu/~distler/blog/itex2MMLcommands.html); even in LaTeX, it will adjust the vertical positioning a bit to look nice.

* Non-ascii characters.  iTeX is designed only to parse the mathematical sections of any page and so it assumes that any strange characters will be generated via iTeX commands such as `\infty` instead of being directly input as ?.  iTeX actually takes an extreme view on this and will not parse mathematics if it includes non-ascii characters.  In ordinary mathematics this should not be a problem since one should use the iTeX commands themselves to produce such characters.  Where it can be a problem is in `\text` commands embedded in mathematics because these are **still** parsed by iTeX even though it does nothing with them.  Thus `\text{?}` is not a legal construct and will simply produce $\text{?}$.  Since iTeX does not allow nesting, the construction `\text{$\infty$}` will not work.  There are two possibilities:
   1. Break the `\text` command: `\text{in~}\infty{-categories}` produces $\text{in}\;\infty{-categories}$ (note the forced space, simple whitespace there will not work).
   2. Use **numbered** entities: `\text{in ∞-categories` produces $\text{in ∞-categories}$.  Named entities will not work here as they get converted to unicode internally before being sent to the iTeX parser.

### I searched for a page that I know exists, but I couldn't find it

Instiki's search uses regular expressions.  That means you can do all sorts of fancy searches, but it also means that pages with special characters in their names are hard to search for.  For instance, if you search for `(n,r)-category` you will come up with nothing, because Instiki interprets the parentheses as a regular expression grouping construct.  To search for actual parentheses, you need to backslash them: search instead for `\(n,r\)-category`.

### Why is there no "Preview" button?

Instiki's philosophy is that the "Preview" button is the one labeled "Save".  If you don't like the result, just click "Edit" again.  Multiple edits by the same person within a 30 minute interval are collapsed into a single "edit" in the revision history.

One merit of this approach is that in the case when your preview was correct and doesn't require re-editing, it saves you an extra click.  Another is that it prevents you from ever *forgetting* to make the extra click.

### itex is a pain, why do you guys use MathML? {#WhyMathML}

Many web sites "support" LaTeX by running it through a script which converts LaTeX equations to image files for display.  While this produces acceptable results for many users, it is not true "support" for mathematics.  Images can sometimes be hard to see, and the user cannot easily resize them or change their color or font, nor can software easily read them aloud to a blind user.  In contrast, MathML is a markup language, like HTML, specifically designed to carry information not only about the display of mathematics, but its content and meaning.  A suitable client application can resize MathML along with the rest of the page, change its color or font, or even read it aloud, making MathML a much more accessible way to display mathematics.  See, for instance, [this comment](http://terrytao.wordpress.com/2009/10/29/displaying-mathematics-on-the-web/#comment-42119).

### I want to help out with the software, what can I do?

See [this discussion](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=1112).

## HTML, XML, etc.

### How do I get accented characters?

You can use HTML/XML/SGML [character entity references](https://en.wikipedia.org/wiki/List_of_XML_and_HTML_character_entity_references).  If the character you want has a mnemonic HTML name, like &eacute; for é, you can use that.  Otherwise, you can use an SGML numerical code, like &#x34AB; for 㒫, that corresponds directly to Unicode.  To look up the numbers for SGML characters, try the [Unicode Character Names Index](http://unicode.org/charts/charindex.html).  [[Toby Bartels|Toby]] has a [complete list of HTML and XML character entity references](http://tobybartels.name/characters/) (or try the Unicode Databank pages listed below) with individual pages where you can check browser compliance (originally for when you couldn\'t take even things like &harr; for granted, but even now &lrm; and &thinsp; may be lacking).  While you can use this technique for mathematical symbols too, it\'s best to do those in iTeX as explained above.

* [Unicode Data Bank](http://www.sql-und-xml.de/unicode-database/) pages:
  1.  [Math Symbols](http://www.sql-und-xml.de/unicode-database/sm.html)
  1.  [Math Operators](http://www.sql-und-xml.de/unicode-database/mathematical-operators.html)
  1.  [Math Miscellaneous A](http://www.sql-und-xml.de/unicode-database/miscellaneous-mathematical-symbols-a.html)
  1.  [Math Miscellaneous B](http://www.sql-und-xml.de/unicode-database/miscellaneous-mathematical-symbols-b.html)
  1.  [Letterlike Symbols](http://www.sql-und-xml.de/unicode-database/letterlike-symbols.html)
  1.  [Miscellaneous Symbols](http://www.sql-und-xml.de/unicode-database/miscellaneous-symbols.html)
  1.  [General Punctuation](http://www.sql-und-xml.de/unicode-database/general-punctuation.html)
  1.  [Open Punctuation](http://www.sql-und-xml.de/unicode-database/ps.html)
  1.  [Other Punctuation](http://www.sql-und-xml.de/unicode-database/po.html)
  1.  [Close Punctuation](http://www.sql-und-xml.de/unicode-database/pe.html)
  1.  [Geometric Shapes](http://www.sql-und-xml.de/unicode-database/geometric-shapes.html)
  1.  [Other Symbols](http://www.sql-und-xml.de/unicode-database/so.html)
  1.  [Dingbats](http://www.sql-und-xml.de/unicode-database/dingbats.html)

## _n_-Lab Specifics

### Why did my page get redirected?

Did you read the _Naming Conventions_ section on the [[HowTo]]?

### Where do I ask a question?
  {#AskAQuestion}

Questions, comments and other discussions are to be had on the *[nForum](https://nforum.ncatlab.org/)*.

### How do I cite a page on the nLab? {#Citing}

You should, of course, include a link or URL to the page or pages that you wish to cite.  You should also give the information about which *version* of the page you are citing, since pages change over time.  The version number can be found as follows: look at the bottom of the page where it says "Back in time ($N$ revisions)"; the current version number is $N+1$.

You can link directly to the version you want to cite by using a url such as

    http://ncatlab.org/nlab/revision/PageName/VERSIONNUMBER

We recommend that if you only include one URL, it be of the form `show/PageName` which will point to whatever version of the page is current when it is accessed.  This is because pages generally improve over time, and whoever is following your reference ought to be taken to the best, up-to-date version of the page.  Anyone who cares about finding the exact version of the page that you cited can figure out how to find it in the history.

On the other hand, if you can give two URLs (this would be cumbersome in a printed paper, but is possible with links on a web page), then it may be helpful to give the appropriate `revision` link as well as the `show` one.

*All nLab pages now include a "Cite" link at the bottom of the page which produces a BibTeX entry in accord with the suggestions above.*

### I want to help, but writing math on a wiki is scary; what can I do?

There are numerous ways you can help out and get acquainted with the community even if you don't (yet) feel comfortable writing math publically on a wiki.

1.  You can do [[lab elf]] duty, such as correcting typos and spelling errors, fixing links, creating links, creating redirects (there are many pages that still need redirects created for alternate and plural forms of the title), organizing pages, and so on.

2.  You can ask questions (see previous question), and join discussions, at the nForum, and comment at the nCafe.

3.  You can try writing small additions to existing pages, for instance a paragraph or two that helps explain something that puzzled you initially.  If you aren't entirely positive that what you wrote was correct, when you announce your edit at Latest Changes on the nForum, you can mention that you're unsure and someone will usually reassure you or give a correction.

4.  If you don\'t sign your edits, then they are attributed to the [[AnonymousCoward]].  We prefer that you sign your edits, but anonymous help is better than none at all, so that option is available to you.

### How can I get a personal section of the nLab?

Some users have personal areas of the _n_-Lab where they can have password protected pages and do work without fitting it into the rest of _n_-Lab.  If you would like to have such an area, ask the [[nlabmeta:steering committee|steering committee]].

### I got "Access denied" when editing a page.

The _n_-Lab has a spam filter that checks your IP against a blacklist.  The blacklists used are maintained by [spamcop.net](http://www.spamcop.net/) and [spamhaus.org](http://www.spamhaus.org/).  IPs are added to these lists if they are detected doing things usually associated with computers infected with viruses.  There are instructions on the webpages for finding out if your IP has been added to these lists and what to do to remove your IP from them.  Three things to point out are:

1. The _n_-Lab cannot remove your IP from the list, you have to do that yourself.
2. The _n_-Lab is not going to remove its spam protection.
3. If you got your IP dynamically (e.g. using some wireless providers) it is entirely possible that it is not your computer that was behaving badly but a previous one using the same IP.  A quick workaround is to disconnect from the network and try reconnecting (sometimes you have to wait a bit before reconnecting to get a new IP).  Of course, you should ensure that your virus software is up to date.

### I got "Internal application error" when doing something.

Sometimes something doesn't work quite right with the software and it bails out.  If you think that you were doing something that should work, please log the error message at the [n-Forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/).  The more information that you log, the easier it is for us to debug.  Useful information is: your IP, the time and date, and the URL that you were trying to access.

There is actually more information contained in the HTML source of the error message ("view source"): some errors can be down to malformed input when editing a page and that can help you fix it yourself.

category: meta


Fargues-Fontaine curve

#Contents#
* table of contents
{:toc}

## Idea

Let $C$ be an algebraically closed [[perfectoid field]] over $\mathbb{F}_{p}$. The *Fargues-Fontaine curve* $X_{C}$ is a [[complete variety|complete]] [[algebraic curve]] whose [[closed points]] parametrize the untilts of $C$. Such an untilt may be recovered as the [[residue field]] of the corresponding point.

## As an adic space

Let $E$ be a finite extension of $\mathbb{Q}_{p}$ with uniformizer $\pi$ and residue field $\kappa$. Let $S=Spa(R,R^{+})$ be a [[perfectoid space]] over $\overline{\mathbb{F}}_{q}$ with pseudouniformizer $\varpi$. We take ramified [[Witt vectors]] $W_{\mathcal{O}_{E}}(R):=W(R^{+}\otimes_{W(\kappa)}\mathcal{O}_{E})$. We define $Y_{S}$ to be the locus in $Spa(W_{\mathcal{O}_{E}}(R))$ where $\pi$ and $\varpi$ are invertible. We think of $Y_{S}$ as the "fiber product" $Spa(R^{+})\times Spa(\mathcal{O}_{E})$, even though is is not literally possible because we lack a base for the fiber product (we will define this later)!

The (relative) Fargues-Fontaine curve $X_{S}$ is then defined to be $Y_{S}/\phi_{S}^{\mathbb{Z}}$, where $\phi_{S}$ is the Frobenius on $S$.

## In the geometrization of the local Langlands correspondence

The Fargues-Fontaine curve finds application in the geometrization of the [[local Langlands correspondence]]. Note that the [[geometric Langlands correspondence]] is stated for a curve. Naively, one might think that the "curve" for the local Langlands correspondence should be $Spec(E)$, for $E$ a local field. However, the local Langlands correspondence concerns the Weil group instead of the absolute Galois group, one should instead consider $Spec(\breve{E})/\phi_{E}^{\mathbb{Z}}$, where $\breve{E}$ is the maximal unramified extension of $E$ and $\phi_{E}$ is the Frobenius.

To truly obtain a geometrization, however, one needs to "relativize" this, which is analogous to considering the stack $Bun_{G}$ in the geometric Langlands correspondence instead of just its points $Bun_{G}(\mathbb{F}_{q})$. Hence we must be able to take "base change".

If $E$ is, say $\mathbb{F}_{q}((t))$, then for a perfectoid space $S=Spa(R,R^{+})$ with pseudouniformizer $\varpi$ we may take the fiber product $S\times_{Spa(\mathbb{F}_{q})} Spa(\mathbb{F}_{q}((t)))$. However if $E$ is a finite extension of $\mathbb{Q}_{p}$, there is no analogue of the base $\mathbb{F}_{q}$ over which we take the fiber product.

However, we note that in the case where $E=\mathbb{F}_{q}((t))$, the fiber product $S\times_{Spa(\mathbb{F}_{q})} Spa(\mathbb{F}_{q}((t)))$ is the locus inside $Spa(R^{+})\times_{Spa(\mathbb{F}_{q})} Spa(\mathbb{F}_{q}[[t]])$ where both the uniformizer $\varpi$ of $R$ and the uniformizer $t$ of $\mathbb{F}_{q}$ are both invertible.

Now noting that the mixed characteristic analogue of forming power series is taking Witt vectors (compare, for instance, $\mathbb{F}_{p}[[t]]$ and $W(\mathbb{F}_{p})=\mathbb{Z}_{p}$), we think of the locus $Y_{S}$ defined above as the analogue of our fiber product, $Spa(R^{+})\times Spa(\mathcal{O}_{E})$. Again, since the local Langlands correspondence is phrased in terms of the Weil group, we have to take the quotient by the action Frobenius, and take $X_{S}=Y_{S}/\phi^{\mathbb{Z}}$, which is the Fargues-Fontaine curve.

## As a scheme

If $S=Spa(C)$ where $C$ be an algebraically closed perfectoid field over $\mathbb{F}_{p}$, we have a description of the Fargues-Fontaine curve as a scheme. Let $\mathcal{O}_{C}$ be the [[ring of integers]] of $C$, and let $W(\mathcal{O}_{C})$ be the [[Witt vectors]] of $\mathcal{O}_{C}$. Let $\phi_{C}:C\to C$ be the Frobenius morphism. We define a [[norm]] $\vert\cdot\vert_{r}$ on $W(\mathcal{O}_{C})[1/p]$ as follows:

$$\vert \sum_{n\gg -\infty} [a_{n}]p^{n}\vert_{r}=\sup_{n}\vert a_{n}\vert p^{-r n}$$

Let $B_{C}$ be the Frechet completion of $W(\mathcal{O}_{C})[1/p]$ with respect to all the norms $\vert\cdot\vert_{r}$ for every positive $r$.

Then the schematic Fargues-Fontaine curve $X_{C}^{sch}$ is defined to be
$$X_{C}^{sch}=\mathrm{Proj}(\oplus_{n\in\mathbb{N}} B_{C}^{\phi=p^{n}}).$$

## Vector Bundles on the Fargues-Fontaine Curve

[[Vector bundles]] on the Fargues-Fontaine curve admit a classification in terms of _isocrystals_ (vector spaces over the field $E$ together with a semilinear action of Frobenius). More generally for a reductive group $G$, there is a concept of $G$-isocrystals which correspond to $G$-bundles on the Fargues-Fontaine curve.

These also in turn correspond to extended pure inner forms of $G$ (which are all the inner forms if $G$ is connected). It is believed that it should not just be $G$, but all the extended pure inner forms of $G$ together, which  should appear in the geometric side of the [[local Langlands correspondence]] (for instance inner forms have the same [[Langlands dual group]]). This is another motivation for the appearance of the Fargues-Fontaine curve in the Fargues-Scholze geometrization of the local Langlands correspondence.

## Related concepts

* [[p-adic Hodge theory]]

* [[local Langlands correspondence]]

## References

* [[Jared Weinstein]], _[The Fundamental Curve of p-adic Hodge Theory, or How to Un-tilt a Tilted Field](https://galoisrepresentations.wordpress.com/2013/07/23/the-fundamental-curve-of-p-adic-hodge-theory-or-how-to-un-tilt-a-tilted-field/)_

* [[Jared Weinstein]], _[The Fundamental Curve of p-adic Hodge Theory, Part II](https://galoisrepresentations.wordpress.com/2013/08/12/the-fundamental-curve-of-p-adic-hodge-theory-part-ii/)_

* [[Dennis Gaitsgory]], [[Jacob Lurie]], _Harvard Seminar on Fargues-Fontaine Curve_ ([Informal Notes](http://www.math.harvard.edu/~lurie/FF.html))

* {#FarguesScholze21} [[Laurent Fargues]], [[Peter Scholze]], _Geometrization of the local Langlands correspondence_ ([arXiv:2102.13459](https://arxiv.org/abs/2102.13459))

* a version in global analytic geometry is mentioned at the end of [http://www-personal.umich.edu/~snkitche/Conference/notes/Kremnitzer-2.pdf](http://www-personal.umich.edu/~snkitche/Conference/notes/Kremnitzer-2.pdf) + [audio](http://leccap.engin.umich.edu/leccap/site/s79yndxt7qowoek7n3m)

* [[Federico Bambozzi]], [[Oren Ben-Bassat]], [[Kobi Kremnizer]], _Analytic geometry over $\mathbb{F}_1$ and the Fargues-Fontaine curve_, ([arXiv:1711.04885](https://arxiv.org/abs/1711.04885))


Farrell-Tate cohomology

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _Tate cohomology_ are [[cohomology groups]] $\widehat{H}(G,N)$ associated to a [[representation]] $N$ of a [[finite group]] $G$. In terms of the [[Tate spectrum]] $H N^{t G}$ of the [[Eilenberg-MacLane spectrum]] $H N$ of $N$, these may be expressed as its [[stable homotopy groups]]:

$$
  \widehat H^{-n}(G,N) \simeq \pi_n( H N^{t G})
  \,.
$$

(e.g. [Nikolaus-Scholze 17, p. 13](#NikolausScholze17))

What is called _(generalized) Farrell-Tate cohomology_ is a generalization of this construction to possibly infinite [[discrete groups]] and topological groups $G$.

(e.g. [Klein 02](#Klein02), [Nikolaus-Scholze 17, section I.4](#NikolausScholze17))

## History of the idea

Tate cohomology was introduced in [Tate52](#Tate52) by [[John Tate]] for the purposes of [[class field theory]]. When a finite group $G$ acts on an abelian group $A$, then there is a natural 'norm' map $N$ from $H_0(G, A)$ to $H^0(G,A)$, $a \mapsto \sum_g g a$.

Then the Tate cohomology groups are

* $\hat{H}^n (G, A) = H^n(G, A)$, for $n \geq 1$.
* $\hat{H}^0 (G, A) = coker N$, for $n = 0$.
* $\hat{H}^{-1} (G, A) = ker N$, for $n = 0$.
* $\hat{H}^n (G, A) = H_{-(n+1)}(G, A)$, for $n \leq -2$.

In ([Farrell78](#Farrell78)), Farrell generalized this construction to possibly infinite [[discrete groups]] of finite [[virtual cohomological dimension]].

Later this was generalized further in ([Klein 02](#Klein02)) to any topological (or discrete) group $G$ and any naive $G$-spectrum $E$.

## References

* {#NikolausScholze17} [[Thomas Nikolaus]], [[Peter Scholze]], section I-4 of _On topological cyclic homology_ ([arXiv:1707.01799](https://arxiv.org/abs/1707.01799))

* {#Tate52} [[John Tate]], _The higher dimensional cohomology groups of class field theory_, Ann. of Math. (2), 1952, 56: 294–297.

* {#Farrell78} F. Thomas Farrell,  _An extension of Tate cohomology to a class of infinite groups_, J. Pure Appl. Algebra 10 (1977/78), no. 2, 153-161.

* {#Klein02} [[John Klein]], _Axioms for generalized Farrell–Tate cohomology_, 2002, [pdf](http://www.math.wayne.edu/~klein/tate.pdf)

[[!redirects Farrell-Tate cohomologies]]

[[!redirects Tate cohomology]]
[[!redirects Tate cohomologies]]


Farzan Vafa

* [personal page](https://farzanvafa.com/)

* [institute page](https://cmsa.fas.harvard.edu/ss_people/vafa/)

## Selected writings

On [[Chern-Simons theory]] potentially applicable to a form of [[Berry curvature]] on the [[Brillouin torus]] of 3d [[topological semimetals]], with nodal lines playing the role of [[Wilson lines]]:

* [[Biao Lian]], [[Cumrun Vafa]], [[Farzan Vafa]], [[Shou-Cheng Zhang]], *Chern-Simons theory and Wilson loops in the Brillouin zone*, Phys. Rev. B **95** (2017) 094512 [[doi:10.1103/PhysRevB.95.094512](https://doi.org/10.1103/PhysRevB.95.094512)]

category: people


fat diagonal

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#### Topology
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[[!include topology - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Definition

Given a [[topological space]] $X$ and a [[natural number]] $n$, writing $X^n$ for the $n$-fold [[product topological space]] of $X$ with itself, its _fat diagonal_

$$
  \mathbf{\Delta}_X^n \hookrightarrow X^n
$$

is the [[topological subspace]] of [[n-tuples]] of points for which at least one pair of components coincide:

$$
  \mathbf{\Delta}_X^n
  \;\coloneqq\;
  \left\{
    (x_i) \in X^n
      \;\vert\;
    x_i = x_j \, \text{for some}\, i \neq j
  \right\}
  \,.
$$

## Properties

### Relation to configuration spaces
 {#RelationToConfigurationSpaces}

The [[complement]] $X^n \setminus \mathbf{\Delta}_X^n$ of the $n$-fold [[Cartesian product]] by its fat diagonal may be understood as the [[configuration space]] of $n$ _distinguishable_ points in $X$. The [[quotient space]] of that by the [[action]] of the [[symmetric group]] $S_n$ given by [[permutation]] of points in $X$ yields the [[configuration space]] of $n$ _indistinguishable_ points in $X$:

$$
  Conf_n(X)
  \;\coloneqq\;
  \left( X^n \setminus \mathbf{\Delta}_X^n\right)/S_n
  \,.
$$

Similarly, the _[[blowup]]_ of the fat diagonal in $X^n$ yields the [[Fulton-MacPherson compactification]] of [[configuration spaces of points]].

### Relation to renormalization

Closely related to the role of the fat diagonal in [[configuration spaces]] is its role in [[renormalization]] of [[perturbative quantum field theories]], which may be formulated as the choice of [[extensions of distributions]] (of [[time-ordered products]]) from the [[complement]] of a fat diagonal to the fat diagonal (the locus of coinciding "[[virtual particles]]" where [[interactions]] take place).

For more on this see at _[[geometry of physics -- perturbative quantum field theory]]_ the chapter _[Interacting quantum fields](geometry+of+physics+--+perturbative+quantum+field+theory#QuantumObservables)_

[[!redirects fat diagonals]]


fat simplex

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###Context###
#### Model category theory
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[[!include model category theory - contents]]
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#Contents#
* automatic table of contents goes here
{:toc}

## Idea

The _fat simplex functor_ is a [[cosimplicial object|cosimplicial]] [[simplicial set]]

$$
  \mathbf{\Delta} : \Delta \to sSet
$$

whose value $\mathbf{\Delta}[n]$ at $n \in \mathbb{N}$ is a [[simplicial set]] that models the $n$-[[simplex]] but is much bigger than the standard $n$-simplex $\Delta[n] = Hom_{\Delta}(-,[n])$. This is such that $\mathbf{\Delta}[-]$ is a cofibrant replacement of $*$ and of $\Delta[-] = Hom_\Delta(-,-)$ in the projective [[model structure on functors]] $\Delta \to sSet_{Quillen}$.

The fat simplex can be used to express the [[homotopy colimit]] over [[simplicial object|simplicial]] [[diagram]]s in terms of [[coend]]s of the form $\int^{[n] \in \Delta} \mathbf{\Delta}[n] \cdot F_n$. This construction is originally due to Bousfield and Kan.

## Definition

Write $\Delta$ for the [[simplex category]]. For $[n] \in \Delta$ write $\Delta/[n]$ for the corresponding [[overcategory]]. Finally write

$$
  \mathbf{\Delta}[n] :=
  N(\Delta/[n])
$$

(in [[sSet]]) for the [[nerve]] of this overcategory.

This construction is [[functor]]ial in  $[n]$:

$$
  \mathbf{\Delta}(-) =  N(\Delta/(-)) : \Delta \to sSet
  \,.
$$

## Examples

* The fat 0-simplex is $\mathbf{\Delta}[0] = N(\Delta)$, the [[nerve]] of
  the [[simplex category]] (because $[0] \in \Delta$ is the
  [[terminal object]]).

## Properties

There is a canonical morphism

$$
  \mathbf{\Delta} \to \Delta
$$

of [[cosimplicial object|cosimplicial]] [[simplicial set]], called the
[[Bousfield-Kan map]].

This exhibits $\mathbf{\Delta}$ as a cofibrant [[resolution]] of $\Delta$ and of $*$ in the projective [[model structure on functors]] on $[\Delta, sSet_{Quillen}]$.

See the discussion at [[Reedy model structure]] and at [[Bousfield-Kan map]] for details.

[[!redirects fat simplices]]


Fatimah Rita Ahmadi

* [personal page](https://www.cs.ox.ac.uk/people/fatimahrita.ahmadi/)

## Selected writings

Relating the [[ZX-calculus]] to [[braided fusion categories]] for [[anyon]] [[braiding]]:

* [[Fatimah Rita Ahmadi]], [[Aleks Kissinger]], *Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics* $[$[arXiv:2211.03855](https://arxiv.org/abs/2211.03855)$]$

category: people


Fatma Nur Ünal

* [institute page](https://www.tcm.phy.cam.ac.uk/profiles/fnu20/)

## Selected writings

On [[braid group statistics|anyonic braiding]] of nodal points in the [[Brillouin zone]] of [[semi-metals]] ("braiding in momentum space"):

* [[Robert-Jan Slager]], [[Adrien Bouhon]], [[Fatma Nur Ünal]], *Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase* [[arXiv:2208.12824](https://arxiv.org/abs/2208.12824)]

category: people


Favonia

[[!redirects Kuen-Bang Hou]]
[[!redirects Kuen-Bang Hou (Favonia)]]

Known also as Kuen-Bang Hou (Favonia).

* [page on HoTT website](http://homotopytypetheory.org/author/favonia/)

* [homepage](http://favonia.org/)

* _Higher-Dimensional Types in the
Mechanization of Homotopy Theory_, PhD thesis 2017, ([webpage](http://favonia.org/thesis.html), [pdf direct link](http://favonia.org/files/thesis.pdf))

## Selected writings

On ([[homotopy types]] of) [[covering spaces]] via [[homotopy type theory]]:

* [[Kuen-Bang Hou]], *Covering Spaces in Homotopy Type Theory*, extended abstract *Type Theory, Homotopy Theory and Univalent Foundations* (2013) $[$[doi:10.1007/978-3-319-21284-5_15](https://doi.org/10.1007/978-3-319-21284-5_15)$]$

* [[Kuen-Bang Hou]], [[Robert Harper]], *Covering Spaces in Homotopy Type Theory*, Leibniz International Proceedings in Informatics (LIPIcs) **97** (2018) $[$[doi:10.4230/LIPIcs.TYPES.2016.11](https://doi.org/10.4230/LIPIcs.TYPES.2016.11), [pdf](https://drops.dagstuhl.de/opus/volltexte/2018/9851/pdf/LIPIcs-TYPES-2016-11.pdf)$]$

Formalization of (the proof of) the [[Blakers-Massey theorem]] in [[homotopy type theory]]:

* {#FFLL16} [[Kuen-Bang Hou]] ([[Favonia]]), [[Eric Finster]], [[Dan Licata]], [[Peter LeFanu Lumsdaine]], *A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory*, LICS '16 (2016) 565–574  [[arXiv.1605.03227](https://arxiv.org/abs/1605.03227), [doi:10.1145/2933575.2934545](https://doi.org/10.1145/2933575.2934545)]

On [[cellular cohomology]] formulated in [[homotopy type theory]]:

* [[Ulrik Buchholtz]], [[Favonia|Kuen-Bang Hou (Favonia)]], _Cellular Cohomology in Homotopy Type Theory_, Logical Methods in Computer Science, Volume 16, Issue 2 (June 1, 2020) ([arXiv:1802.02191](https://arxiv.org/abs/1802.02191), [lmcs:6518](https://lmcs.episciences.org/6518))

On [[universe polymorphism]] in [[dependent type theory]] via [[monads]]:

* [[Favonia]], [[Carlo Angiuli]], [[Reed Mullanix]], *An Order-Theoretic Analysis of Universe Polymorphism*, Proceedings of the ACM on Programming Languages **7** 57 (2023) 1659–1685 [[doi:10.1145/3571250](https://doi.org/10.1145/3571250)]

## Related entries

* [[homotopy type theory]]

* [[Blakers-Massey theorem]]

category: people


Fedele Lizzi

* [webpage](http://people.na.infn.it/~lizzi/)

## Selected writings

On the [[spectral triple|spectral]] [[Riemannian geometry]] of the [[fuzzy 2-sphere]]:

* [[Francesco D'Andrea]], [[Fedele Lizzi]], [[Joseph Várilly]], _Metric Properties of the Fuzzy Sphere_, Lett. Math. Phys.103 (2013), 183-205 ([arXiv:1209.0108](https://arxiv.org/abs/1209.0108))

On the [[Connes-Lott-Chamseddine-Barrett model]]:

* Agostino Devastato, Maxim Kurkov, [[Fedele Lizzi]], _Spectral Noncommutative Geometry, Standard Model and all that_ ([arXiv:1906.09583](https://arxiv.org/abs/1906.09583))

category: people


Federica Pasquotto

* [webpage](http://www.few.vu.nl/~pasquott)

## Selected writings

On [[G-structures]]:

* {#Pasquotto} [[Federica Pasquotto]], _Linear $G$-structures by examples_ ([pdf](http://www.few.vu.nl/~pasquott/course16.pdf), [[PasquottoGStructures.pdf:file]])

category: people


Federico Ardila

__Federico Ardila__ is a mathematician at San Francisco State University with main interests in [[combinatorics]], especially [[algebraic combinatorics]].

* [web](http://math.sfsu.edu/federico)

category: people


Federico Bambozzi

* [webpage](http://www.math.unipd.it/en/department/people/user.php?usertype=7&user=523)

## related $n$Lab entries

* [[global analytic geometry]]

category: people


Federico Bonetti

* [inspire page](https://inspirehep.net/authors/1190033?ui-citation-summary=true)

## Selected writings

On [[M-theory on Spin(7)-manifolds]], [[F-theory on Spin(7)-manifolds]] and [[Witten's Dark Fantasy]]:

* [[Federico Bonetti]], [[Thomas Grimm]], [[Tom Pugh]], _Non-Supersymmetric F-Theory Compactifications on $Spin(7)$ Manifolds_, JHEP 01 (2014) 112 ([arXiv:1307.5858](https://arxiv.org/abs/1307.5858))

* [[Federico Bonetti]], [[Thomas Grimm]], [[Eran Palti]], [[Tom Pugh]], _F-Theory on $Spin(7)$ Manifolds: Weak-Coupling Limit_, JHEP 02 (2014) 076 ([arXiv:1309.2287](https://arxiv.org/abs/1309.2287))

## Selected writings

On [[M5-branes]] and their [[quantum anomaly]]-cancellation:

* {#BahBonettiMinasianNardoni18} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Class $\mathcal{S}$ Anomalies from M-theory Inflow_, Phys. Rev. D 99, 086020 (2019) ([arXiv:1812.04016](https://arxiv.org/abs/1812.04016))

* {#BBMN19} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Anomaly Inflow for M5-branes on Punctured Riemann Surfaces_,  J. High Energ. Phys. 2019, 123 (2019)
([arXiv:1904.07250](https://arxiv.org/abs/1904.07250))

* {#BBMN19b} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Anomalies of QFTs from M-theory and Holography_, J. High Energ. Phys. 2020, 125 (2020) ([arXiv:1910.04166](https://arxiv.org/abs/1910.04166))

* {#BahBonettiMinasian20} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], _Discrete and higher-form symmetries in SCFTs from wrapped M5-branes_, J. High Energ. Phys. 2021, 196 (2021) ([arXiv:2007.15003](https://arxiv.org/abs/2007.15003))

On [[5d Chern-Simons theory]] in [[5-dimensional supergravity]]:

* {#BonettiGrimmHohenegger13} [[Federico Bonetti]], [[Thomas Grimm]], [[Stefan Hohenegger]], _One-loop Chern-Simons terms in five dimensions_ ([arXiv:1302.2918](http://arxiv.org/abs/1302.2918))

On [[sphere fiber bundles]] as [[Kaluza-Klein compactifications]] in [[supergravity]] and [[string theory]]:

* [[Federico Bonetti]], [[Ruben Minasian]], Valentí Vall Camell, Peter Weck, *Consistent truncations from the geometry of sphere bundles* [[arXiv:2212.08068](https://arxiv.org/abs/2212.08068)]

category: people


Federico Camia

* [institute page](https://nyuad.nyu.edu/en/academics/divisions/science/faculty/federico-camia.html)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=64068)

## Selected writings

Relating the [[Brownian loop soup]] to [[conformal field theory]]:

* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *Scalar conformal primary fields in the Brownian loop soup* [[arXiv:2109.12116](https://arxiv.org/abs/2109.12116)]

* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *The Brownian loop soup stress-energy tensor* [[arXiv:2112.00074](https://arxiv.org/abs/2112.00074), doi:10.1007/JHEP11(2022)009]

category: people


Federico Campanini

* [institute page](https://uclouvain.be/fr/repertoires/federico.campanini)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=257634)

## Selected writings

Presenting a [[pretorsion theory]] on [[Cat]] whose torsion(-free) objects are the [[groupoids]] ([[skeletal categories]], respectively), hence whose "trivial objects" are the [[skeletal groupoids]]:

* [[Francis Borceux]], [[Federico Campanini]], [[Marino Gran]], [[Walter Tholen]], *Groupoids and skeletal categories form a pretorsion theory in $Cat$* [[arXiv:2207.08487](https://arxiv.org/abs/2207.08487)]

category: people


Federigo Enriques

[[!redirects Federigo Enriquez]]
_Federigo Enriques_ was an algebraic geometer of classical Italian school, with major contributions to the theory and classification of surfaces.

* wikipedia in [Italian](http://it.wikipedia.org/wiki/Federigo_Enriques), [English](http://en.wikipedia.org/wiki/Federigo_Enriques)

category: people


Fedor A. Smirnov

* [Math-Net.ru-page](http://www.mathnet.ru/eng/person24203)

## Selected writings

Discussion of the [hypergeometric integral solutions](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) to the [[Knizhnik-Zamolodchikov equation]] for the special case of [[level (Chern-Simons theory)|level]]$=0$ (cf. at *[logarithmic CFT -- Examples](logarithmic+CFT#Examples)*):

* [[Fedor A. Smirnov]], *Remarks on deformed and undeformed Knizhnik-Zamolodchikov equations*, $[$[arXiv:hep-th/9210051](https://arxiv.org/abs/hep-th/9210051)$]$

* [[Fedor A. Smirnov]], *Form factors, deformed Knizhnik-Zamolodchikov equations and finite-gap integration*, Communications in Mathematical Physics **155** (1993) 459–487 $[$[doi:10.1007/BF02096723](https://doi.org/10.1007/BF02096723), [arXiv:hep-th/9210052](https://arxiv.org/abs/hep-th/9210052)$]$

category: people

[[!redirects Fedor Smirnov]]


Fedor Bogomolov

A mathematician at NYU Courant and HSE University Moscow.

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

* [institute page](https://math.nyu.edu/people/profiles/BOGOMOLOV_Fedor.html)

## Selected writings

On [[elliptic fibrations]]:

* [[Fedor Bogomolov]], [[Yuri Tschinkel]], *Monodromy of elliptic surfaces* [[[MonodromyOfEllipticSurfaces.pdf:file]], [arXiv:math/0002168](https://arxiv.org/abs/math/0002168)]

On [[anabelian geometry]]:

* [[Fedor Bogomolov]], [[Yuri Tschinkel]], *Introduction to birational anabelian geometry* [[pdf](http://www.math.nyu.edu/~tschinke/papers/yuri/10msri/msri7.pdf), [arXiv:1011.0883](https://arxiv.org/abs/1011.0883)]

category: people


Fedosov's deformation quantization

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#### Physics
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[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}

## Idea

Fedosov's method ([Fedosov 94](#Fedosov94)) is a prescription for constructing [[formal deformation quantizations]] of the [[algebra of functions|algebra of]] [[smooth functions]] on any [[symplectic manifold]] $(X,\omega)$.

In particular this establishes the existence of formal deformation quantizations of all [[symplectic manifolds]]. While [Kontsevich 97](deformation+quantization#Kontsevich97) more generally proves the existence of deformation quantization of all [[Poisson manifolds]] of finite dimension, Fedosov's method, in a variant applicable to [[almost Kähler structures]] ([Karabegov-Schlichenmaier 01](#KarabegovSchlichenmaier01)) generalizes to infinite-dimensional symplectic manifolds as they appear in [[local field theory]], where it yields the quantization to [[perturbative quantum field theory]] equivalent to the method of [[causal perturbation theory]] ([Collini 16](#Collini16)). (Kontsevich's deformation quantization is however not compatible with field theory ([Hawkins-Rejzner 16, section 5.3.2](#HawkinsRejzner16)).) For more on this see at _[[locally covariant perturbative quantum field theory]]_.

Fedosov's method proceeds by the following steps:

1. For each point $x \in X$ regard the [[tangent space]] $T_x X$ as a [[symplectic vector space]] $(T_x X, \omega_x)$ and consider its [[Moyal deformation quantization]] $\mathcal{W}_x \coloneqq A_{Moy}(T_x X, \omega_x)$. In the context of [[local field theory]] these Moyal algebras are the [[Wick algebras]] of the underlying [[free field theory]].

   The union of these yelds an [[associative algebra]]-[[fiber bundle]] $\mathcal{W} \to X$. From the fiber-wise product in $A_{Moy}(T_x X, \omega_x)$ the [[space of sections]] $\Gamma_X(\mathcal{W})$ inherits itself the structure of an [[associative algebra]].

1. Find a [[flat connection]] $\nabla_{Fed}$ on $\mathcal{W}$ which respects the associative algebra structure, and such that there is a [[linear isomorphism]]

   $$
     \Gamma_X(\mathcal{W}) \supset  ker(\nabla_{Fed}) \simeq C^\infty(X) [ [ \hbar ] ]
   $$

   between the [[covariantly constant sections]] of the algebra bundle and the algebra of  smooth functions on $X$ with a [[formal power series|formal variable]] $\hbar$ adjoined.

   Such $\nabla_{Fed}$ exists, (non-uniquely), it is called a _Fedosov connection_.

1. By this linear isomorphism the algebra structure on $\Gamma_X(\mathcal{W})$ induces an [[associative algebra]] structure on $C^\infty(X)[ [ \hbar ] ]$, and this turns out to be a [[formal deformation quantization]] of $(X,\omega)$.

   In the context of [[local field theory]], this deformation quantization is equivalent to the result of quantization via [[causal perturbation theory]] ([Collini 16](#Collini16)).

## Related concepts

* [[almost Kähler geometric quantization]]

## References

The method is due to

* {#Fedosov94} [[Boris Fedosov]], _A simple geometrical construction of deformation quantization_ Journal of Differential Geometry, 40(2):213–238, 1994 ([Euclid](https://projecteuclid.org/euclid.jdg/1214455536))

with a monograph account in

* [[Boris Fedosov]], *Deformation Quantization and Index Theory*, John Wiley & Sons (1995) [ISBN:9783055017162]

Its generalization to [[almost Kähler structures]] is due to

* {#KarabegovSchlichenmaier01} [[Alexander Karabegov]], [[Martin Schlichenmaier]],  _Almost-Kähler deformation quantization_, Letters in Mathematical Physics, **57** 2 (2001) 135–148  [[doi:10.1023/A:1017993513935](https://doi.org/10.1023/A:1017993513935), [arXiv:0102169](https://arxiv.org/abs/math/0102169)]

with a survey in

* [[Martin Schlichenmaier]], _Deformation quantization for almost-Kähler manifolds_, J. of Nonlinear Math. Phys.
**11**, Supplement (2004) 49–54 [[doi:10.2991/jnmp.2004.11.s1.6](https://doi.org/10.2991/jnmp.2004.11.s1.6), [pdf](http://www.atlantis-press.com/php/download_paper.php?id=540)]

The observation that the construction of [[perturbative quantum field theory]] via [[causal perturbation theory]] is equivalent to Fedosov quantization is due to

* {#Collini16} [[Giovanni Collini]], _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

Discussion showing that this generalization to field theory is not given by Kontsevich deformation quantization:

* {#HawkinsRejzner16} [[Eli Hawkins]], [[Kasia Rejzner]], section 5.3.2 of _The Star Product in Interacting Quantum Field Theory_ ([arXiv:1612.09157](https://arxiv.org/abs/1612.09157))

[[!redirects Fedosov deformation quantization]]
[[!redirects Fedosov deformation quantizations]]

[[!redirects Fedosov quantization]]
[[!redirects Fedosov quantizations]]

[[!redirects Fedosov's formal deformation quantization]]
[[!redirects Fedosov's formal deformation quantizations]]


Fefferman-Graham ambient construction

#Contents#
* table of contents
{:toc}

## Idea

A construction in [[conformal geometry]] by which a [[conformal manifold]] is realized by embedding into a suitable ambient [[Einstein manifold]].

## Application in AdS-CFT

For application in the [[AdS-CFT correspondence]] see for instance ([deHaro-Skenderis-Solodukhin 00](#HaroSkenderisSolodukhin00), [Tetradis 11](#Tetradis11)).

## Related concepts

* [[conformal compactification]]

* [[tractor bundle]]

## References

### General

* [[Charles Fefferman]], [[Robin Graham]], _Conformal invariants_, Asterisque: 95–116.

* [[Charles Fefferman]], [[Robin Graham]],  _The Ambient Metric_, Princeton University Press., (2011)

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

### Application in AdS-CFT correspondence

References using the ambient construction in the context of the [[AdS-CFT correspondence]] include

* {#HaroSkenderisSolodukhin00} [[Sebastian de Haro]], [[Kostas Skenderis]], [[Sergey Solodukhin]], _Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence_, Commun.Math.Phys. 217 (2001) 595-622 ([arXiv:hep-th/0002230](http://arxiv.org/abs/hep-th/0002230))

* {#Tetradis11} N. Tetradis, _Holographic Horizons_ ([arXiv:1109.2335](http://arxiv.org/abs/1109.2335))

[[!redirects ambient construction]]


Fei Han

* [webpage](http://ww1.math.nus.edu.sg/onepageCV.aspx?id=mathanf)

## Selected writings

On a twisted equivariant version of the [[Bismut-Chern character]]:

* [[Fei Han]], *Supersymmetric QFT, Super Loop Spaces and Bismut-Chern Character* [[arXiv:0711.3862](https://arxiv.org/abs/0711.3862)

* [[Fei Han]], [[Varghese Mathai]], *Exotic twisted equivariant cohomology of loop spaces, twisted Bismut-Chern character and T-duality*,  Commun. Math. Phys. **337** (2015) 127-150 [[arXiv:1405.1320](https://arxiv.org/abs/1405.1320),  [doi:10.1007/s00220-014-2270-z](https://doi.org/10.1007/s00220-014-2270-z)]

On the [[Green-Schwarz mechanism]] in [[heterotic string theory]] and [[Hořava-Witten theory]]:

* [[Fei Han]], [[Kefeng Liu]], [[Weiping Zhang]], _Anomaly Cancellation and Modularity. II: $E_8 \times E_8$ case_, Sci. China Math. 60, 985–994 (2017) ([arXiv:1209.4540](https://arxiv.org/abs/1209.4540), [doi:10.1007/s11425-016-9034-1](https://doi.org/10.1007/s11425-016-9034-1))

On the [[Witten genus]] in the case of [[string^c structure]]:

* {#ChenHanZhang10} [[Qingtao Chen]], [[Fei Han]], [[Weiping Zhang]], _Generalized Witten Genus and Vanishing Theorems_, Journal of Differential Geometry 88.1 (2011): 1-39. ([arXiv:1003.2325](http://arxiv.org/abs/1003.2325))

* [[Qingtao Chen]], [[Fei Han]], _Elliptic Genera, Transgression and Loop Space Chern-Simons Forms_, Communications in Analysis and Geometry Volume 17 (2009) Number 1 ([arXiv:math/0611104](http://arxiv.org/abs/math/0611104), [doi:10.4310/CAG.2009.v17.n1.a4]( https://dx.doi.org/10.4310/CAG.2009.v17.n1.a4))

* [[Fei Han]], [[Varghese Mathai]], _Projective elliptic genera and elliptic pseudodifferential genera_, Adv. Math. 358 (2019) 106860  ([arXiv:1903.07035](https://arxiv.org/abs/1903.07035))

* Haibao Duan, [[Fei Han]], Ruizhi Huang, _$String^c$ Structures and Modular Invariants_, Trans. AMS 2020 ([arXiv:1905.02093](https://arxiv.org/abs/1905.02093))

On [[gravitino]] [[anomaly cancellation]] in [[D=11 N=1 supergravity]]:

* {#HanHuangLiuZhang20} [[Fei Han]], [[Ruizhi Huang]], [[Kefeng Liu]], [[Weiping Zhang]], *Cubic forms, anomaly cancellation and modularity*, Advances in Mathematics **394** (2022) 108023 [[arXiv:2005.02344](https://arxiv.org/abs/2005.02344), [doi:10.1016/j.aim.2021.108023](https://doi.org/10.1016/j.aim.2021.108023)]

On [[bundle gerbe modules]] with [[higher structure|higher]] [[G-structures]] (including [[String structure]], etc.) and the corresponding [[twisted Chern character]]:

* [[Fei Han]], [[Ruizhi Huang]], [[Varghese Mathai]], *Fractional structures on bundle gerbe modules and fractional classifying spaces* [[arXiv:2203.14439](https://arxiv.org/abs/2203.14439)]

category: people

[[!redirects Han Fei]]


Feit-Thompson theorem

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[[!include group theory - contents]]
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#Contents#
* table of contents
{:toc}

## Statement

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

Every [[finite group]] of [[odd number|odd]] [[order]] is a [[solvable group]].

=--

This is due to ([Feit-Thompson 62](#FeitThompson62)).

A fully formalized proof in [[Coq]] has been announced in ([INRIA 2012](#INRIA))

## References

* {#FeitThompson62} [[Walter Feit]], John Thompson,  _A solvability criterion for finite groups and some consequences_, Proc. Nat. Acad. Sci. 48 (6): 968–970, 1962 ([JSTOR]( http://www.jstor.org/stable/71265))

* {#INRIA} INRIA, _[Feit-Thompson theorem has been totally checked in Coq](http://www.msr-inria.inria.fr/events-news/feit-thompson-proved-in-coq)_


Felipe Leitner

* [webpage](http://www.igt.uni-stuttgart.de/LstDiffgeo/Leitner/)

## related $n$Lab entries

* [[conformal geometry]], [[Cartan geometry]]

category: people


Felipe Ponce

## X-Ray Transform

The X-ray transform is the operator
$$
Xf(l) := \int_l f\,d\lambda(l),
$$
where $l\subset\mathbb{R}^n$ is a line, and $\lambda$ the Lebesgue measure on the line. The manifold $M_n$ can be represented as
$$
M_n = \{(\sigma, x)\in S^{n-1}\times\mathbb{R}^n \mid \langle x,\omega\rangle = 0\},
$$
where $(\sigma,x)$ represents a line that passes through $x$ with direction $\omega$. We endow $M_n$ with a measure $\mu$ invariant under rigid motions. It is an open problem to determine the exponents $1\le p,q,r\le\infty$ for which the following inequality holds:
$$
\left\Vert Xf\right\Vert_{L^q(\omega\mapsto L^r_x(\mathbb{R}^{n-1}))} \le C\left\Vert f\right\Vert_{L^p(\mathbb{R}^n)}.
$$
For brevity we will write $L^q_\omega L^r_x := L^q(\omega\mapsto L^r_x(\mathbb{R}^{n-1}))$.

Drury proved the following Theorem.

#### Theorem

If $E\subset \mathbb{R}^n$ is a measurable set, and $1_E$ is the characteristic function of $E$, then
$$\left\Vert Xf\right\Vert_{L^{n+1,\infty}(M_n)} \le C\left\vert E\right\vert^\frac{2}{n+1},$$
where $L^{p,\infty}$ is the weak-$L^n$ space.

This inequality corresponds to the restricted weak version of the point $(p,q,r) = (\frac{n+1}{2}, n+1, n+1)$.

##### Proof

The dual operator $X^*$ is defined by duality as $\int gXf\,d\mu(l) = \int fX^*g\,dx$, where $f$ and $g$ are smooth functions. Therefore,
$$
X^*g = \int g\delta_x(l)\,d\mu,
$$
where $\delta_l(x) = \lim_{\varepsilon\to 0}\varepsilon^{-n+1}1_{\{l\mid l\cap B(x,\varepsilon)\neq\emptyset\}}$.

Set $f = 1_E$ and define the set $F := \{l\in M_n\mid Xf(l)\ge\lambda\}$, then
$$
\lambda\vert F\vert \le \int_F Xf\,d\mu = \int fX^*1_F \,dx \le \left\Vert X^*1_F\right\Vert_\infty\left\Vert f\right\Vert_1.
$$
Now choose a point $x_0$ such that $X^*1_F(x_0)\ge \frac{1}{2}\left\Vert X^*1_F\right\Vert_\infty$ ---$x$ is _the Bourgain's bush_. By translation symmetry we can assume that $x_0 = 0$. If $\delta_0$ denotes the Dirac delta at the origin, then
$$
\int 1_F Xf\,X\delta_0\,d\mu(l) \ge \lambda\int_F X\delta_0\,d\mu(l)\ge \frac{\lambda}{2}\left\Vert X^*1_F\right\Vert_\infty.
$$
On the other hand
$$
\int 1_F Xf\,X\delta_0\,d\mu(l) \le \int fX^*(1_F X\delta_0)\,dx \le  \left\Vert f\right\Vert_{L^{n,1}} \left\Vert X^*(1_F X\delta_0)\right\Vert_{L^{\frac{n}{n-1},\infty}},
$$
where $L^{p,q}(\mathbb{R}^n)$ refers to the Lorentz spaces. The function $X^*(1_F X\delta_0)$ can be explicitly computed
$$
X^*(1_F X\delta_0)(x) = \frac{1}{\vert x\vert^{n-1}}1_F(l(0,x)),
$$
where $l(0,x)$ is the line that passes through $0$ and $x$. Then, we can estimate the norm as
$$
\left\Vert X^*(1_F X\delta_0)\right\Vert_{L^{\frac{n}{n-1},\infty}} = c_n\vert X^*1_F(0)\vert^\frac{n-1}{n}\le c_n\left\Vert X^*1_F\right\Vert_\infty^\frac{n-1}{n}.
$$
Hence, we get the upper bound
$$
\int 1_F Xf\,X\delta_0\,d\mu(l) \le c_n \left\Vert f\right\Vert_{L^{n,1}}\left\Vert X^*1_F\right\Vert_\infty^\frac{n-1}{n}.
$$
Hence, we have that
$$
\left\Vert f\right\Vert_{L^{n,1}}\ge c_n\lambda \left\Vert X^*1_F\right\Vert_\infty^\frac{1}{n}.
$$
Therefore, we conclude that
$$
c_n\vert F\vert\lambda^{n+1} \le \left\Vert f\right\Vert_1\left\Vert f\right\Vert_{L^{n,1}}^n,
$$
which implies the restricted weak inequality.


Felipe Voloch

* [webpage](http://www.ma.utexas.edu/users/voloch/)

category: people


Felix Berezin

* [Wikipedia entry](http://en.wikipedia.org/wiki/Felix_Berezin)

## related $n$Lab entries

* [[Lie-Poisson structure]]

* [[Berezin integration]]

* [[Berezinian]]

* [[supermanifold]]

category: people


Felix Bloch

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

## Selected writings

Introducing [[Bloch theory]] (the [[quantum mechanics]] of [[electrons]] in [[crystals]]) into [[solid state physics]]:

* [[Felix Bloch]], *Über die Quantenmechanik der Elektronen in Kristallgittern*, Zeitschrift für Physik **52** 7-8 (1929) 555-600 $[$[doi:10.1007/BF01339455](https://www.scopus.com/record/display.uri?eid=2-s2.0-34250940025&origin=inward&featureToggles=FEATURE_NEW_DOC_DETAILS_EXPORT:1)$]$

## Related $n$Lab entries

* [[infrared divergence]], [[quantum electrodynamics]]

category: people


Felix Cherubini

[[!redirects Felix Wellen]]

(previously: Felix Wellen)

* [Website](http://felix-cherubini.de)

## Selected writings

* _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_

  PhD Thesis

  [[nLab:HoTT]]-[[nLab:Agda]] code: [DCHoTT-Agda](https://github.com/felixwellen/DCHoTT-Agda).

On [[modal homotopy type theory]] (such as concerning [[homotopy n-types]]/[[n-truncation modality]] and [[covering spaces]]):

* [[Felix Cherubini]], [[Egbert Rijke]], *Modal Descent*, Mathematical Structures in Computer Science **31** 4 (2021) 363-391  [[doi:10.1017/S0960129520000201](https://doi.org/10.1017/S0960129520000201), [arXiv:2003.09713](https://arxiv.org/abs/2003.09713)]

On [[synthetic algebraic geometry]]:

* [[Felix Cherubini]], [[Thierry Coquand]], [[Matthias Hutzler]], *A Foundation for Synthetic Algebraic Geometry* (2023) [[arXiv:2307.00073](https://arxiv.org/abs/2307.00073)]

* [[Felix Cherubini]], *A Foundation for Synthetic Algebraic Geometry*, talk at *[Homotopy Type Theory Electronic Seminar Talks](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html)* (Oct 2023) [slides:[pdf](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Cherubini-2023-10-18-HoTTEST.pdf), video:[YT](https://www.youtube.com/watch?v=lp4kcmQ0ueY)]

category: people


Felix Hausdorff

* [Wikipedia entry](http://en.wikipedia.org/wiki/Felix_Hausdorff)

## Selected writings

Early discussion of foundations of [[set theory]] and [[topology]]/[[topological spaces]]:

* [[Felix Hausdorff]], *[[Grundzüge der Mengenlehre]]*, Leipzig: Veit (1914) [:ISBN:978-0-8284-0061-9] Reprinted by Chelsea Publishing Company (1944, 1949, 1965) [[ark:/13960/t2891gn8g](https://archive.org/details/grundzgedermen00hausuoft/page/n5/mode/2up)]

## Related entries

* [[Hausdorff space]]

category: people


Felix Klein

__Felix C. Klein__ was a mathematician now famous for his study of [[group theory]] in conjunction with [[geometry]]  ([[Klein geometry]]) as laid out in his _[[Erlanger program]]_ ([Klein 72](#Klein72)).

## References

* [Wikipedia page](http://en.wikipedia.org/wiki/Felix_Klein)

* [Mathematics Genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=7401)

* John Baez, _Klein geometry XII_, [nCafe](http://golem.ph.utexas.edu/category/2011/08/klein_2geometry_xii.html)

## Writings

* {#Klein72} [[Felix Klein]], _A comparative review of recent researches in geometry_, 1872 (1892 translation: [arxiv/0807.3161](http://arxiv.org/abs/0807.3161)

  on the _[[Erlangen program]]_

* {#Klein1884} [[Felix Klein]], _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

  on the [[classification of finite rotation groups]] and specifically the [[icosahedron]]

## related $n$Lab entries

* [[stabilizer group]]

* [[classification of finite rotation groups]]

* [[Klein geometry]], [[Cartan geometry]]

[[!redirects F. Klein]]


Felix Rudolph

* [webpage](http://strings.ph.qmul.ac.uk/directory/f.j.rudolph)

## related $n$Lab entries

* [[exceptional field theory]]

category: people


Felix Wierstra

* [Institute page](https://www.uva.nl/en/profile/w/i/f.p.wierstra/f.p.wierstra.html?cb)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=252302)

## Selected writings

On [[real homotopy theory]] applied to [[Hopf invariants]] via variants of the [[Whitehead integral formula]]:

* [[Felix Wierstra]], *Hopf Invariants in Real and Rational Homotopy Theory* (2017) [[diva:146246](http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-146246), [pdf](https://www.diva-portal.org/smash/get/diva2:1136442/FULLTEXT02.pdf)]

On [[rational homotopy theory]] for possibly non-[[nilpotent spaces]] via [[deck transformation|deck]]-[[Borel-equivariant rational homotopy theory]]:

* [[Manuel Rivera]], [[Felix Wierstra]], [[Mahmoud Zeinalian]], *Rational homotopy equivalences and singular chains*, Algebr. Geom. Topol. **21** (2021) 1535-1552 [[arXiv:1906.03655](https://arxiv.org/abs/1906.03655). [doi:10.2140/agt.2021.21.1535](https://doi.org/10.2140/agt.2021.21.1535)]

category: people


Fell bundle

+-- {: .rightHandSide}
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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Operator algebra
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Fell Bundle_ is a family of [[Banach space|Banach spaces]] that varies continuously over the [[morphism]] space of a [[topological groupoid]], where nontrivial morphisms in the groupoid induce a [[bilinear map|bilinear]] [[composition]] across the [[fibers]]. They are a generalization of both [[continuous field of C*-algebras|continuous fields]] of [[C*-algebras]] over [[topological spaces]] and of $C^*$-algebraic bundles ([[fiber bundles]] of $C^*$-algebras over [[topological groups]]).

## Definition

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

A _Fell Bundle_ over a [[groupoid]] $\mathcal{G}$ is a "continuous" [[functor]] $E \colon \mathcal{G} \to\mathfrak{Corr}$, where $\mathfrak{Corr}$ is the $2$-[[2-category|category]] of $C^*$-[[C-star-algebra|algebras]] and $C^*$- [[C-star-correspondence|correspondences]].

## Banach algebra bundles

Mulvey considers the special case where the groupoid is a topological space and proves that such Banach space bundles are equivalent to sheaves of Banach spaces, and that both are equivalent to Banach spaces internal to the sheaf topos.

=--

## References

* Alex Kumjian, _Fell bundles over groupoids_, Proceedings of the AMS, volume 126 (1998) ([JSTOR](http://www.jstor.org/stable/118621))

* Alcides Buss, Chenchang Zhu, Ralf Meyer, _A higher category approach to twisted actions on $C^*$-algebras_, [arxiv/0908.0455](http://arxiv.org/abs/0908.0455)

There is a variant notion of Fell bundles over [[inverse semigroup]]s. Those are related to Fell bundles over the corresponding étale groupoids:

* Alcides Buss, Ruy Exel, _Fell bundles over inverse semigroups and twisted étale groupoids_, J. Oper. Theory __67__, No. 1, 153-205 (2012) [MR2821242](http://www.ams.org/mathscinet-getitem?mr=2821242) [Zbl 1249.46053](http://zbmath.org/?q=an:1249.46053) [arxiv/0903.3388](http://arxiv.org/abs/0903.3388)[journal](http://www.mathjournals.org/jot/2012-067-001/2012-067-001-007.html); _Twisted actions and regular Fell bundles over inverse semigroups_, [arxiv/1003.0613](http://arxiv.org/abs/1003.0613)

* Christopher J. Mulvey, _Banach sheaves_, Journal of Pure and Applied Algebra 17 (1980) 69-83 doi

[[!redirects Fell bundles]]
[[!redirects fell bundle]]
[[!redirects fell bundles]]


Fell's theorem

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###Context###
#### Measure and probability theory
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=--
#### AQFT
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[[!include AQFT and operator algebra contents]]
=--
#### Physics
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[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##
Fell's theorem is about a property of vector [[state on a star-algebra|states]] of a [[C-star algebra]], it says that if the kernels of two representations of the algebra coincide, then the vector [[states]] are mutually weak-* dense. This has a profound consequence for the [[AQFT]] interpretation: A state represents the physical state of a physical system. Since one can always only perform a finite number of measurements, with a finite precision, it is only possible to determine a weak-* neigborhood of a given state. This means that it is not possible - not even in principle - to distinguish representations with coinciding kernels by measurements.

For this reason representations with coinciding kernels are sometimes called **physically equivalent** in the [[AQFT]] literature.

## Properties ##
Let A be a unital $C^*-$algebra and $\pi_1, \pi_2$ be two representations of A on a [[Hilbert space]] H.

+-- {: .un_theorem}
###### Theorem
**equivalence theorem**
Every vector state of $\pi_1$ is the weak-* limit of vector states of $\pi_2$ iff the kernel of $\pi_1$ contains the kernel of $\pi_2$.
=--

## Related concepts

Other theorems about the foundations and [[interpretation of quantum mechanics]] include:

* [[order-theoretic structure in quantum mechanics]]

  * [[Kochen-Specker theorem]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Gleason's theorem]]

* [[Wigner theorem]]

* [[Bell's theorem]]

* [[Bub-Clifton theorem]]

* [[Kadison-Singer problem]]

## References ##

* J.G.M. Fell: _The Dual Spaces of $C^*$-algebras_ [online available here] (http://www.ams.org/journals/tran/1960-094-03/S0002-9947-1960-0146681-0/S0002-9947-1960-0146681-0.pdf)


femtometer

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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _femtometer_ is a [[physical unit]] of [[length]].

## Definition

A **femtometer** (abbreviated $fm$) is one $10^{15}$th of a [[meter]]:

$$
  fm \coloneqq 10^{-15} m
  \,.
$$

The [[radius]] of the [[proton]] is about $0.8 \, fm$.

## Related concepts

* [[meter]]

* [[micrometer]]

* [[nanometer]]

* [[Planck length]]

## References

See also

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

[[!redirects femtometers]]

[[!redirects femtometre]]
[[!redirects femtometres]]

[[!redirects fm]]


Fenchel-Nielsen coordinates

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

The _Fenchel-Nielson coordinates_ are certain [[coordinates]] on [[Teichmüller space]].

They parameterize Teichmüller space by cutting surfaces into pieces with
[[geodesic]] boundaries and [[Euler characteristic]] $\xi = -1$. These building blocks  (of hyperbolic 2d geometry) are precisely

* the 3-holed sphere;

* the 2-holed cusp;

* the 1-holed 2-cusp;

* the 3-cusp

Each surface of [[genus]] $g$ with $n$ marked points will have

* $2g - 2 + n$ generalized pants;

* $3 g - 3 + n$ closed curves.

The boundary lengths $\ell_i \in \mathbb{R}_+$ and twists
$t_i \in \mathbb{R}$  of these pieces
for

$$
  1 \leq i \leq 3g-3+n
$$

constitute the [[Fenchel-Nielsen coordinates]] on
[[Teichmüller space]] $\Tau$.

Also use $\theta_i := t_i/\ell_i \in \mathbb{R}/\mathbb{Z}$

This constitutes is a real analytic [[atlas]] of Teichmüller space. On $M$ this  reduces to coordinates $t_i \in \mathbb{R}/{\ell_i \mathbb{Z}}$,
and these constitute a real analytic atlas of moduli space.

## References

* Kathy Paur, _The Fenchel-Nielson coordinates of Teichmüller spaces_ ([pdf](http://www-math.mit.edu/phase2/UJM/vol1/PAUR-F.PDF))
* Werner Fenchel, [[Jakob Nielsen]], reprinted in _Discontinuous groups of isometries in the hyperbolic plane_, edited by Asmus L. Schmidt; De Gruyter Studies in Math. __29__, 2003.

[[!redirects Fenchel-Nielsen coordinates]]
[[!redirects Fenchel-Nielsen coordinate]]


Ferdinand Frobenius

* [Wikipedia entry](http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius)

## Related entries

* [[Frobenius algebra]], [[Frobenius manifold]]

* [[Frobenius theorem]]

* [[Frobenius homomorphism]]

* [[Frobenius reciprocity]]

category: people

[[!redirects Ferdinand Georg Frobenius]]
[[!redirects Frobenius]]


Ferdinand Veldkamp

* [Mathematics Genealogy Page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=51034)

## selected writings

* {#SpringerVeltkamp00} [[Tonny Springer]], [[Ferdinand Veldkamp]], _Octonions, Jordan Algebras, and Exceptional Groups_, Springer Monographs in Mathematics, 2000

## related $n$Lab entries

* [[octonions]]

* [[composition algebra]]

* [[G2]]

* [[Jordan algebra]]

* [[Albert algebra]]

* [[triality]]

* [[exceptional Lie group]]

category: people


Fermat curve

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

#Contents#
* table of contents
{:toc}

## Idea

The _Fermat curve_ is a projective [[algebraic curve]] over a [[ground field]] $k$ given by the Fermat equation

$$
  x^n + y^n = z^n
  \,,
$$

where $n$ is a positive [[integer]] greater or equal $2$.

_Fermat's last theorem_ is the statement, that for $n\gt 2$ this equation has no solutions in rationals (or, equivalently, has no solutions in integers), other than when at least one of $x,y$ or $z$ vanishes. In other words, the Fermat curves with $n\gt 2$ have no nontrivial $\mathbb{Q}$-[[rational points]]. For $n=2$, the Fermat curve is just the [[projective line]] $\mathbb{P}^1$, hence is much less interesting.

Famously a [[proof]] of this theorem had been announced by [[Pierre de Fermat]], but no proof was recorded before [[Andrew Wiles]] with aid of [[Richard Taylor]] proved the [[modularity theorem]] for semistable [[elliptic curves]], which by [[Ribet's theorem]] implies Fermat's last theorem (see [[Wiles' proof of Fermat's last theorem]]).

Therefore, while Fermat's theorem is generally accepted as proven now, the methods used in the proof go far beyond the mathematics known at [[Pierre de Fermat]]'s time. Moreover [[Ribet's theorem]] and the [[modularity theorem]] are major conceptual statements in [[arithmetic geometry]], quite different in character from the content of Fermat's last theorem in itself:

> I confess that Fermat's Last Theorem, as an isolated proposition, has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of. ([[Carl Friedrich Gauss]], quoted in J. R. Newman (ed.) _The World of Mathematics_ (1956))

It is open whether there exists a more direct proof, in particular a proof using only some system of arithmetic. Harvey Friedman has gone so far as to conjecture (his 'Grand Conjecture') that [[Elementary Function Arithmetic]] should be enough to prove FLT, among many other 'finitary' major theorems in mathematics).

## References

### General

* wikipedia: [Fermat curve](http://en.wikipedia.org/wiki/Fermat_curve), [Fermat's last theorem](http://en.wikipedia.org/wiki/Fermat%27s_last_theorem)

* [[eom]]: [Fermat's last theorem](http://www.encyclopediaofmath.org/index.php?title=p/f110070)

* {#McLarty10} [[Colin McLarty]], _What does it take to prove Fermat's last theorem? Grothendieck and the logic of number theory_, Bull. Symbolic Logic __16__ (2010), no. 3, 359–377 [jstor](http://www.jstor.org/stable/20749620)

### Applications

Discussion of [[anomaly cancellation]] on the [[standard model of particle physics]]:

* Nakarin Lohitsiri, [[David Tong]], _Hypercharge Quantisation and Fermat's Last Theorem_ ([arXiv:1907.00514](https://arxiv.org/abs/1907.00514))

[[!redirects Fermat's last theorem]]
[[!redirects Fermat equation]]


Fermat number

A **Fermat number** is a number of the form
$$
F_k = 2^{2^k}+1
$$
for integers $k \geq 0$. A Fermat number that is [[prime number|prime]] is called a **Fermat prime**.

[[Pierre de Fermat]] conjectured (1640) that all $F_k$ are prime, but $F_5=4294967297$ is composite (641 is a factor, discovered by [[Euler]] in 1732), and in fact (as of 2014) only $F_k$ for $k=0,1,2,3,4$ are known to be prime. Basic questions remain open, such as:

* Are there infinitely many Fermat primes?
* Are there infinitely many composite Fermat numbers?

The Fermat numbers $F_k$ for $k=5,\ldots,32$ are known to be composite, though in some cases no explicit factors are known ($k=20,24$), or incomplete factorisations ($12 \leq k \leq 32$). Isolated other cases are known to be composite, for instance $k=3329780$.

There are heuristic (nonrigorous) arguments that there should be only finitely many Fermat primes, _and_ that Fermat primes have positive [[density of a subset|density]] in the primes; it is impossible that both of these hold, since finite sets have zero density in a countable set like that of the primes. In any case, it is conjectured by various people that: there are only finitely many Fermat primes; there are only five Fermat primes; and that there are finitely many, but more than five (this last position is argued by [[Emil Artin]]).

## Related entries

* [[prime number]]
* [[constructible polygon]]
* [[Mersenne prime]]

## References

* [Wikipedia](http://en.wikipedia.org/wiki/Fermat_number)
* [[OEIS]] [A000215](http://oeis.org/A000215)
* Křížek, Michal; Luca, Florian; Somer, Lawrence _17 lectures on Fermat numbers. From number theory to geometry. With a foreword by Alena Šolcová._ CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 9. Springer-Verlag, New York, 2001. xxiv+257 pp. ISBN: 0-387-95332-9 doi:[10.1007/978-0-387-21850-2](http://dx.doi.org/10.1007/978-0-387-21850-2)

The following page lists heuristic arguments as to the finiteness of the Fermat primes, and also that there are only five such:

* The prime puzzles and conjectures connection, [Conjecture 4. Fermat primes are finite](http://www.primepuzzles.net/conjectures/conj_004.htm)

New factors of composite Fermat numbers are announced here:

* FermatSearch.org [News](http://www.fermatsearch.org/news.html)

[[!redirects Fermat prime]]


Fermat quotient

+-- {: .rightHandSide}
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###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $p$ a [[prime number]], the _Fermat quotient_ of any [[integer]] $a$ is the quotient

$$
  \partial_p a \coloneqq
  \frac{a - a^p}{p}
$$

of the difference between the $p$th power of $a$ and $a$ itself by $p$. By [[Fermat's little theorem]] this is indeed an [[integer]], i.e. conversely one has for all $a\in \mathbb{Z}$ that

$$
  a^p = a + p (\partial_p a)
$$

for a uniquely defined integer $\partial_p a$.

(The $p$-power operation $(-)^p$ here is the one that restricts to the [[Frobenius homomorphism]] after [[p-localization]] and the one which is a shadow of the [[power operations]] in [[E-infinity arithmetic geometry]] ([Lurie, remark 2.2.7](#Lurie)).)

As the notation is meant to suggest, the Fermat quotient as a map $\mathbb{Z} \to \mathbb{Z}$ from the (underlying set of the) [[ring]] of [[integers]] to itself is [[analogy|analogous]] to a [[derivation]]. It is not quite an ordinary derivation, but satisfies conditions of what has been called a __$p$-[[p-derivation|derivation]].

In view of this, in the context of [[arithmetic differential equations]] the Fermat quotient is interpreted as an analog in [[arithmetic geometry]] of actual [[derivations]] in [[algebraic geometry]]/[[complex analytic geometry]]. For more on this see also at _[[Borger's absolute geometry]]_ the section _[Motivation](Borger's%20absolute%20geometry#Motivation)_.

## Related concepts

* [[p-derivation]]

* [[arithmetic differential equation]]

* [[Borger's absolute geometry]]

* [[arithmetic differential geometry]]

## References

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

* {#Lurie} [[Jacob Lurie]], _[[Rational and p-adic Homotopy Theory]]_

[[!redirects Fermat quotients]]


Fermat theory

+-- {: .rightHandSide}
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### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A 'Fermat theory' is a [[Lawvere theory]] that extends the usual theory of [[commutative rings]] by permitting [[differentiation]].

The term _Fermat theory_ seems to have been introduced in ([Kock 09](#Kock09)) based on ([Dubuc-Kock 84](#DubucKock84)).  But as the name suggests, it has its roots in an old observation of [[Fermat]].  Namely: if $f \;\colon\; \mathbb{R} \longrightarrow \mathbb{R}$ is a [[polynomial]] function, then

$$f (x+y) = f(x) + y \tilde{f}(x,y) $$

for a unique polynomial function $\tilde{f} \colon \mathbb{R}^2 \to \mathbb{R}$.  Clearly

$$ \tilde{f}(x,y) = \frac{f(x+y) - f(x)}{y} $$

for $y \ne 0$, but the interesting thing is that

$$\tilde{f}(x,0) = f'(x)$$

So, the function $\tilde{f}$ knows about the [[derivative]] of $f$!  (This can be done for polynomials over any [[commutative ring]], although Fermat wasn\'t working in that generality.)

Later [[Jacques Hadamard]] generalized this observation from a polynomial function $f$ to a [[continuously differentiable function]] $f$, where now $\tilde{f}$ is unique if required to be [[continuous map|continuous]].  This is the statement of the _[[Hadamard lemma]]_. (For a merely [[differentiable function]] $f$, require $\tilde{f}$ to be continuous in $y$ alone.)  The function $\tilde{f}$ is thus called a [[Hadamard quotient]].  If $\tilde{f}$ is to be the same class of function as $f$, then we need [[smooth functions]], and that will be our motivating context from now on.

If we take $\tilde{f}(x,0) = f'(x)$ as a _definition_ of the derivative, we can derive many of the basic rules for derivatives from the formula

$$f(x+y) = f(x) + y \tilde{f}(x,y) $$

using just algebra --- no [[limit of a sequence|limits]]!  As an exercise, the reader should check these rules:

$$(f + g)' = f' + g'$$

$$(f g)' = f' g + f g' $$

$$(f \circ g)' = (f' \circ g) g' $$

These ideas continue to work if $f$ is a smooth function from $\mathbb{R}^n$ to $\mathbb{R}$; focussing on one variable and treating the others as [[parameter]]s, we have [[partial derivative|partial differentiation]].

## Definition

The above observations suggest defining the following kind of [[Lawvere theory]].  A **Fermat theory** is an extension of the [[algebraic theory]] of [[commutative rings]], such that for any $(n+1)$-ary operation $f$ there is a unique $(n+2)$-ary operation $\tilde{f}$ such that

$$ f(x + y, \vec{z}) =
f(x, \vec{z}) + y \tilde{f}(x,y,\vec{z})$$

where $\vec{z}$ is a list of $n$ variables which act as parameters.  (Here we are abusing language by writing the operations $f$ and $\tilde{f}$ as if they were functions, to avoid unintuitive commutative diagrams.)

## Examples

### $C^\infty$-rings

There is a [[Lawvere theory]] called **the theory of $C^\infty$-rings**, whose $n$-ary operations are the [[smooth maps]] $f: \mathbb{R}^n \to \mathbb{R}$,

$$
  T(n) \coloneqq C^\infty(\mathbb{R}^n, \mathbb{R})
  \,,
$$

with composition of operations defined in the obvious way.  An algebra of this Lawvere theory is called a **[[generalized smooth algebra|C^∞-ring]]**.

The theory of $C^\infty$-rings is a Fermat theory.  For any [[smooth manifold]] $M$, the algebra of smooth real-valued functions $C^\infty(M)$ is a $C^\infty$ ring.  More generally, if $M$ is any [[diffeological space]], [[Chen space]] or [[Frolicher space]], we can define $C^\infty(M)$, and this will be a $C^\infty$-ring.

In formulas, and even more generally: for any generalized [[space]] given by a [[presheaf]] $X$ on [[CartSp]], the corresponding $C^\infty$-ring is the copresheaf

$$
  C^\infty(X) : \mathbb{R}^n \mapsto [CartSp^{op},Set](X,Y(\mathbb{R}^n))
$$

that sends each object $\mathbb{R}^n \in CartSp$ to the [[hom-set]] in the [[functor category]] $[CartSp^{op},Set]$ from $X$ to the presheaf [[representable functor|represented]] by $\mathbb{R}^n$ under the [[Yoneda embedding]]. By the canonical right [[exact functor|exactness]] of the [[hom-functor]], this preserves [[limit]]s and hence in particular [[product]]s in [[CartSp]].

## Partial derivatives {#PartialDerivative}

Let $T$ be a Fermat theory and let $f$ be an $(n+1)$-ary operation, then we may define an operation $\partial_1 f$

by

$$\partial_1(x, \vec{z})  = \tilde{f}(x,0,\vec{z})$$

This acts like the [[partial derivative]] of $f$ with respect to its first argument. With a bit of more work we get a list of $n$-ary operations $\partial_i f$.   So, if $T(n)$ denotes the set of $n$-ary operations in the algebraic theory $T$, we get maps

$$\partial_i : T(n) \to T(n) $$

for $1 \le i \le n$.

Now $T(n)$ is automatically an algebra of $T$ (this is true for any Lawvere theory: it is the free algebra on $n$ generators), whence $T(n)$ is a commutative ring.  One can check that each map

$$\partial_i : T(n) \to T(n) $$

is a [[derivation]] of this ring --- this is really just the [[chain rule]].

## Modules and derivations {#ModulesDerivations}

Let $T$ be a Fermat theory, and let $A$ be a $T$-algebra. A [[module]] $N$ over $A$ is simply a module for the underlying [[ring]] of $A$.

But the notion of [[derivation]] $\delta : A \to N$ of such modules depends on the $T$-structure:

To motivate the concept, let first $A$ be an ordinary [[ring]] and $N$ an ordinary [[module]]. Then the three axioms of an ordinary derivation $\delta : A \to N$

1. $\delta(a + b  ) = \delta(a) + \delta(b)$

2. $\delta(\lambda a) = \lambda \delta(a)$

3. $\delta(a \cdot b) = a \delta(a) + b \delta(b)$

are equivalent to the condition that for any polynomial $p \in \mathbb{R}[x_1, \cdots, x_n]$ and ring elements $a_i$ we have

$$
  \delta\left(
    p(a_1, \cdots, a_n)
  \right)
  =
  \sum_{i= 1 }^{n}
  \frac{\partial p}{\partial x_i}
  \left(
    a_1, \cdots, a_n
  \right)
  \delta(a_i)
  \,.
$$

(It is immediate that the first three axioms imply this one. To see the converse, apply the latter to  the polynomials $p_1(x,y) = x + y$, $p_2(x) = \lambda a$ and $p_3(x,y) = x y$.)

The definition of derivations for general $T$-algebras now follows the last expression, using the notion of [partial derivatives](#PartialDerivative) from above:

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

For $T$ a Fermat theory, $A$ a $T$-algebra and $N$ an $A$-module, a **derivation**  $\delta : A \to N$ is a map such that for each $f \in T(n)$ and elements $(a_i \in A)$ we have

$$
  \delta\left(
    f(a_1, \cdots, a_n)
  \right)
  =
  \sum_{i= 1 }^{n}
  \frac{\partial f}{\partial x_i}
  \left(
    a_1, \cdots, a_n
  \right)
  \delta(a_i)
  \,.
$$

=--

Notice that in particular such a derivation of a $T$-algebra $A$ is a derivation of the underlying ring. (This follows again by using the above three polynomials and remembering that by definition $T(n)$ at least contains all polynomials.)

### Kähler differentials {#KählerDiffs}

The sets $T(n)$ for $n \in \mathbb{N}$ canonically have the structures of [[module]]s over $T(n)$.

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

The map

$$
  d :=
\langle \partial_1, \dots, \partial_n \rangle : T(n) \to \prod_{i = 1}^{n}  T(n) $$

obtained from the partial derivatives is the universal
$T$-derivation of $T(n,1)$.

=--

This means that if $N$ is a module of $T(n)$ and $\delta : T(n) \to N$ is a derivation in the above sense, then $\delta$ factors uniquely through the map $\langle \partial_1, \dots, \partial_n \rangle$.

The point of this theorem is that it gives us a version of [[Kähler differentials]] for $T(n)$.

We may think of an element  $(f_i) \in \prod_{i = 1}^{n}  T(n)$ as the [[Kähler differential]] 1-form $f_1 d x^1 + f_2 d x^2 + \cdots + f_n d x^n$ and of the derivation $d := \langle \partial_1, \dots, \partial_n \rangle$ as the operation

$$
  d  : f \mapsto \sum_i \frac{\partial f_i}{ \partial x^i} d x^i
  \,.
$$

Indeed, when the Fermat theory is that of [[generalized smooth algebra|C-infinity rings]], then this notion of Kähler differentials does coincide with the ordinary notion of smooth [[differential form|1-form]]. The same is not true, in general, if one instead forms ring-theoretic Kähler differentials.

## Related concepts

* [[model structure on differential graded-commutative superalgebras]]

## References

The original reference is

* {#DubucKock84} [[Eduardo Dubuc]], [[Anders Kock]], _On 1-form classifiers_, Comm. Alg. __12__  (1984),  no. 11-12, 1471--1531 ([pdf](http://home.imf.au.dk/kock/DK84.pdf))

Parts of the above material are a summary of the following talk:

* {#Kock09} [[Anders Kock]], _Kähler differentials for Fermat theories_, talk at  _Fields Workshop on Smooth Structures in Logic, Category Theory and Physics_, May 1, 2009, University of Ottawa.  ([abstract](http://aix1.uottawa.ca/~scpsg/Fields09/smooth.abstracts.html))

For more, see:

* [[Alex Hoffnung]], _Smooth Structure in Ottawa II_  ([blog entry](http://golem.ph.utexas.edu/category/2009/05/smooth_structures_in_ottawa_ii.html#c030745))

and the comments on this blog entry.

Refinement to [[supergeometry]] and extension to a notion of _super Fermat theory_ is discussed in

* [[David Carchedi]], [[Dmitry Roytenberg]], _On theories of superalgebras of differentiable functions_, ([arxiv:1211.6134](http://arxiv.org/abs/1211.6134))

Something similar appears in def. 1.1, 1.2 of

* [[David Yetter]], _Models for synthetic supergeometry_, Cahiers, 29, 2 (1988) ([NUMDAM](http://www.numdam.org/item?id=CTGDC_1988__29_2_87_0))

For more on this see at _[[synthetic differential supergeometry]]_.

[[!redirects Fermat theory]]
[[!redirects Fermat theories]]
[[!redirects fermat theory]]


Fermat's little theorem

# Contents #
* table of contents
{: toc}

## Statement

For $x$ an [[integer]] and $p$ a [[prime number]], if $x \nequiv 0 \pmod{p}$, then $x^{p-1} \equiv 1 \pmod{p}$.

This useful result is in a sense trivial, since the [[ring]] $\mathbb{Z}/(p)$ is a finite [[field]], with [[group of units]] $G$ of [[order of a group|order]] $p-1$; it is just a matter of recalling $x^{ord(G)} = 1$ for all $x$ in a [[finite group]] $G$. For the same reason, $a^{q-1} = 1$ for any nonzero $a$ in a finite field with $q$ elements.

## Refinements

A stronger result is that the group of units of a finite field with $q$ elements is _[[cyclic group|cyclic]]_ of order $q-1$, or indeed that every finite subgroup of the group of units of any field is cyclic. A proof may be found [here](/nlab/show/root#rootsunity).

If $A$ is a commutative $\mathbb{F}_p$-[[associative unital algebra|algebra]], then the map $a \mapsto a^p$ is an algebra endomorphism $\sigma: A \to A$ (the preservation of addition follows easily from the [[binomial theorem]] and the fact that $\binom{p}{k} \equiv 0 \pmod{p}$ when $0 \lt k \lt p$; apparently this is also known as the [[freshman's dream]]). It follows that $\sigma: \mathbb{F}_q \to \mathbb{F}_q$ is a field _automorphism_ on a field with $q = p^k$ elements, since after all

$$\sigma^k(a) = a^{p^k} = a^q = a^{q-1}a = a$$

for any $a \in \mathbb{F}_q$. See also [[Frobenius automorphism]].

## Pseudoprimes and Carmichael numbers

Fermat's little theorem says that in order for a positive integer $p$ to be prime, it is necessary that $a^p \equiv a \pmod{p}$ for any integer $a$ (one may as well assume $0 \leq a \lt p$). This gives a way of showing that an integer $n$ is _not_ prime (by finding an $a$ less than $n$ such that $a^{n-1} \nequiv 1 \pmod{n}$) that, especially for large $n$, is more efficient than  actually factoring $n$.

One type of (probabilistic) primality test for $p$ is to take a base, for example $a = 2$, and check whether $a^{p-1} = 1 \pmod{p}$. Chances are good that $p$ is in fact prime if this is satisfied, although there certainly exist composite numbers which pass this test (called pseudoprimes base $a$). The smallest pseudoprime base $2$ is $341 = 11 \cdot 31$. One effective primality test for "small" $p$ (e.g., less than $10^{15}$) is to use such a primality test coupled with a table of pseudoprimes.

There are numbers such as $n = 561 = 3 \cdot 11 \cdot 17$ which are pseudoprimes in any base; these are called _Carmichael numbers_. A positive integer $n$ is Carmichael iff it is square-free and for each prime divisor $p$ of $n$, we have that $p-1$ is a divisor of $n-1$. They are comparatively rare, but it is known there are infinitely many (for sufficiently large $n$, there are at least $n^{2/7}$ Carmichael numbers between $1$ and $n$).

## AKS primality test

A generalization of Fermat's little theorem can be used to give a deterministic test for primality that can be carried out on any integer $n$ in "polynomial time" (bounded by a polynomial applied to the number of digits of $n$); it was first published only as recently as 2002.

+-- {: .num_lemma}
###### Lemma
An integer $n \geq 2$ is prime if and only if

$$(x-a)^n \equiv x^n - a \pmod{n}$$

for every or even any $a$ coprime to $n$ (so that, for example, the case $a=1$ would be a sufficient criterion for primality).
=--

+-- {: .proof}
###### Proof
By the [argument at freshman's dream](nlab/show/freshman's+dream#primality), primality of $n$ is equivalent to $(x-a)^n \equiv x^n - a^n \pmod{n}$ (for every or any $a$), and then primality of $n$ implies the further reduction $a^n \equiv a \pmod{n}$ by Fermat's little theorem.
=--

Given an integer $r$, and an integer $a$ relatively prime to $r$, let $ord_r(a)$ denote the order of $a \pmod{r}$ as a unit in $\mathbb{Z}/(r)$. We let $log(n)$ denote the base $2$ [[logarithm]] of $n$, and $\phi$ denotes the [[Euler totient function]] (so $\phi(r)$ is the cardinality of the [[group of units]] of $\mathbb{Z}/(r)$).

+-- {: .num_theorem}
###### Theorem
**(Agrawal, Kayal, Saxena)**
For given $n$, let $r$ be the least positive integer such that $ord_r(n) \gt (log(n))^2$. Then $n$ is prime if and only if either

* $n \leq r$ and whenever $1 \leq a \leq r$, either $1 = \gcd(a, n)$ or $n = \gcd(a, n)$, or

* $r \lt n$ and whenever $a$ is an integer such that $1 \leq a \leq \sqrt{\phi(r)} \cdot log(n)$, we have
$$(x - a)^n \equiv x^n - a \pmod{x^r - 1, n}.$$
=--

From this result, one can extract an algorithm that decides primality of $n$ in time bounded by a polynomial in $log(n)$, invoking help from the following result.

+-- {: .num_lemma}
###### Lemma
For each $n$ there exists $r \leq \max \{3, (log(n))^5\}$ such that $ord_r(n) \gt (log(n))^2$.
=--

These results form the basis for the first algorithm for deciding primality that is general (applies to any integer $n$), runs in polynomial time, is deterministic (other known efficient tests were randomized and only guaranteed high probability of primality), and unconditional (does not depend on conjectured number-theoretic results such as forms of the [[Riemann hypothesis]]).

## Related concepts

+ [[Fermat quotient]]

## References

Named after _[[Pierre de Fermat]]_.

*  Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, _PRIMES is in P_,  Annals of Mathematics 160 (2) (2004), 781–793. doi:10.4007/annals.2004.160.781. JSTOR 3597229. ([pdf](http://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf))


Fermi theory of beta decay

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Fermi theory of weak $\beta$-decay_ is an [[effective quantum field theory]] approximation -- valid to an [[energy]] of about 10 [[TeV]] -- to that part of the [[standard model of particle physics]] which is relevant for the decay of [[hadron]]s by the [[weak interaction]].

## References

For instance section 2.3 of

* A. Pich, _Effective Field Theory_ ([arXiv:hep-ph/9806303](http://arxiv.org/abs/hep-ph/9806303))

[[!redirects Fermi theory of weak interactions]]


fermion

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

#Contents#
* table of contents
{:toc}

## Idea

A __fermion__ is a [[particle]]/[[quantum field]] that obeys [[Fermi-Dirac statistics]] ([[Pauli exclusion principle]]).  By the [[spin-statistics theorem]], this is the same thing as a particle whose [[spin]] is half an [[integer]] but not itself an integer.

In the [[standard model of particle physics]] and the [[standard model of cosmology]] fermions are the [[matter]] constituents of the [[observable universe]], as opposed to [[bosons]], which are the [[quanta]] of [[force]] [[gauge fields]].

The formalization of fermions is in [[supergeometry]]: fermion fields are [[sections]] of [[spinor bundles]] with odd-degree fibers in [[supergeometry]]. This is exhibited by what is called _[[Fermi-Dirac statistics]]_ and the [[Pauli exclusion principle]].

## Examples

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

## Related entries

* [[Pauli exclusion principle]], [[degeneracy pressure]]

* [[chiral fermion]]

* [[particle]], [[matter]]

* [[boson]], **fermion**, [[anyon]]

* [[parafermion]]

* [[fermionic modality]]

* [[Pauli exclusion principle]]

* [[fermion sea]]

* [[Landau's Fermi liquid model]]

* [[spin-statistics theorem]]

* [[spinors in Yang-Mills theory]]

* [[generation of fermions]]

* [[fermionic path integral]], [[Pfaffian line bundle]]

* [[quantum anomaly|fermion anomaly]]

## References

Fermions are named after [[Enrico Fermi]].

For accounts in [[mathematical physics]] see for instance

* [[Daniel Freed]], _Lecture 1 of [[Five lectures on supersymmetry]]_

[[!redirects fermions]]

[[!redirects fermion field]]
[[!redirects fermion fields]]

[[!redirects fermionic field]]
[[!redirects fermionic fields]]

[[!redirects Fermion]]
[[!redirects Fermions]]

[[!redirects Fermion field]]
[[!redirects Fermion fields]]

[[!redirects Fermionic field]]
[[!redirects Fermionic fields]]

[[!redirects fermionic particle]]
[[!redirects fermionic particles]]


fermionic path integral

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[path integral]] over the [[fermion]]ic variables of the standard kinetic [[action functional]] for [[fermion]]s (see for instance [[spinors in Yang-Mills theory]]) has a well-defined meaning as a [[section]] of the [[Pfaffian line bundle]] of the corresponding [[Dirac operator]].

## Setup

For definiteness, we consider a [[sigma model]] [[quantum field theory]] on a worldvolume $\Sigma$ and [[pseudo-Riemannian manifold|pseudo-Riemannian]] target [[spacetime]] $X$ with fields

* [[boson]]s: [[smooth function]]s $\phi : \Sigma \to X$

* [[fermion]]s ([[gravitino]]): $\psi \in \Gamma(S(\Sigma) \otimes \phi^* T X)$, [[section]]s of the [[tensor product]] of a [[spinor bundle]] on $\Sigma$ and the [[pullback]] of the [[cotangent bundle]] of $X$ along the given bosonic field $\phi$.

The [[action functional]]

$$
  S : (\phi, \psi) \mapsto S^{bos}(\phi) + S^{ferm}_{\phi}(\psi)
$$

is the sum of the

* bosonic action

  $$
    S^{bos})(\phi)
     =
    \int_\Sigma \langle d \phi \wedge \star d \phi\rangle
  $$

* fermionic action

  $$
    S^{ferm}_\phi(\psi)
     = \int_\Sigma \langle \psi, D_\phi \psi\rangle
  $$

  where $D_\phi$ is a [[Dirac operator]] on $S \otimes \phi^* T M$ (the Dirac operator on $S$ twisted by the pullback of the [[Levi-Civita connection]] on $T^* X$ ).

One imagines than that the hypothetical [[path integral]] symboilically written as

$$
  \int [d \phi] [d \psi] \exp(S(\psi)(\phi,\psi))
$$

can be computed in two steps

$$
  \cdots = \int [d \phi] \exp(S^{bos}(\phi))
   \left(
     \int [d \psi]
     \exp(S^{ferm}_\phi(\psi))
   \right)
$$

by first computing the integral over the fermions

$$
  pfaff(\phi)
   :=
  \int [d \psi]
     \exp(S^{ferm}_\phi(\psi))
$$

and then inserting this into the remaining bosonic integral. Now, as opposed to the bosonic integral, this fermionic integral can be given well-defined sense by interpreting it as an infinite-dimensional [[Berezinian integral]].

However, while this makes the expression well defined, the result is not quite a function of $\phi$, but is instead a [[section]] $pfaff$ of a [[Pfaffian line bundle]]

$$
  \array{
    && Pfaff
     \\
     {}^{pfaff := Z_{eff}^{ferm}}\nearrow & \downarrow
    \\
   C^{\infty}(\Sigma, X)
 &= &  C^{\infty}(\Sigma, X)
  }
$$

over the space of bosonic field configurations.

If $Pfaff$ is not isomorphic to the trivial line bundle, we say the system has a fermionic [[quantum anomaly]]. If instead $Pfaff$ is trivializable, any choice of trivialization

$$
  t : Paff \stackrel{\simeq}{\to} C^\infty(\Sigma, X) \times \mathbb{C}
$$

makes the fermionic path integral into a genuine function

$$
  Z_{eff}^{ferm}
   : =
 (t \circ pfaff) :
  C^\infty(\Sigma, X) \to \mathbb{C}
  \,.
$$

Any such choice of $t$ is called a choice of [[quantum integrand]].

With this one can then try to enter the remaining bosonic path integral

$$
  \int [d \phi] \exp(S^{bos}(\phi)) Z_{eff}^{ferm}(\phi)
$$

## Pfaffian bundles

> We are implicitly assuming that $dim \Sigma = 2$ or maybe $8 n + 2$ in the following. Needs to be generalized.

For $n \in \mathbb{N}$, there the [[square root]] of the determinant of a skew symmetric $(n\times n)$-[[matrix]] $D$ -- the [[Pfaffian]] of the matrix -- can be understood as the [[Berezinian integral]]

$$
  pfaff(D)
    =
   \int [d \vec \theta]
   \exp( \langle \theta , D \theta\rangle)
  \in
  det \mathbb{R}^n
$$

over the [[Grassmann algebra]] elements $\theta_i$. Written this way this is an element of the [[determinant line]] of $\mathbb{R}^n$: its identification with a number depends on the choice of basis for $\mathbb{R}^n$. For this case this is unproblematic, since there is a canonical choice of basis for the single vector space $\mathbb{R}^n$, but when $D$ instead depends on a parameter $\phi$, then in general its Pfaffian can at best be a section of a [[determinant line bundle]].

We now generalize this to the case that $D$ is not a finite-dimensional matrix, but a [[Dirac operator]] acting on spaces of sections of a [[spinor bundle]]. We discuss that we can reduce this "infinite-dimensional matrix" in a sense _locally_ to a finite dimensional one in a consistent way, such that the above ordinary construction of Pfaffians applies.

In the above setup, write

$$
  \mathcal{H}_\phi^{\pm} := \Gamma(S^{\pm} \otimes \phi^* T^* X)
$$

for the space of spinor sections for given $\phi : \Sigma \to X$. Then the choral [[Dirac operator]]s a maps

$$
  D_\phi^{\pm} : \mathcal{H}_\phi^{\pm} \to \mathcal{H}^{\mp}_\phi
  \,.
$$

We also have a "quaternionic structure"

$$
  J_\phi^{\pm} : \mathcal{H}_\phi^{\pm} \to \mathcal{H}^{\mp}_\phi
$$

Define then an [[open cover]] of the space $C^\infty(\Sigma,X)$
of the space of bosonic fields with [[open set]]s  $U_\mu$ for $(0 \leq \mu)$ given by

$$
  U_\mu := \{ \phi \in C^\infty(\Sigma,X)  | \mu \nin Spec D_\phi^2\}
  \,,
$$

hence the collection of bosonic field configurations such that $\mu$ is not in the [[operator spectrum]] of the squared [[Dirac operator]].

Over these open subsets we have the _finite_ [[rank]] [[vector bundle]]s

$$
  \mathcal{H}_\phi^{\mu \pm}
  :=
   \oplus_{0 \leq \epsilon \leq \mu}
  Eig(D_\phi^2, \epsilon)
$$

of [[eigenspace]]s of $D_\phi^2$ for [[eigenvalue]]s bounded by $\mu$.

The Dirac operator that we are interested in is

$$
  D_\phi^\mu
  :=
  J_\phi^- \circ D_\phi^+ : \mathcal{H}_\phi^{\mu,+} \to \mathcal{H}_\phi^{\mu,+}
 \,.
$$

This defines now a finite-dimensional [[matrix]]

$$
  \langle -, D_\phi^\mu -\rangle
$$

whose [[Berezinian integral]] is the [[Pfaffian]]

$$
  \int [d \psi] \exp(\langle \psi , D^\mu_\phi \psi \rangle )
  =
  pfaff(D^\mu_\phi) \in det \mathcal{H}^{\mu \pi}_\phi
  \,.
$$

One shows that these constructions for each $\mu$ glue together to define

* a smooth [[line bundle]] $Pfaff \to C^\infty(\Sigma, X)$

* with a smooth [[section]] $pfaff(D)$.

Moreover, there is canonically a [[Hermitian form|Hermitian metric]]  and a  canonical unitary [[connection on a bundle]] (the [[Freed-Bismut connection]]) on this bundle.

## Examples

For the [[sigma model]] describing the [[brane|heterotic superstring]] propagating on a [[pseudo-Riemannian manifold]] $X$, the trivialization of the Pfaffian line bundle, hence the cancellation of its fermionic [[quantum anomaly]] is related to the existence of a (twisted) [[differential string structure]] on $X$. See there for more details.

## Related concepts

* [[Nicolai map]]

* [[path integral]]

* [[spinors in Yang-Mills theory]]

[[!redirects fermionic path integrals]]


Fernando Lucatelli Nunes

* [GoogleScholar page](https://scholar.google.com/citations?user=QoW_v8QAAAAJ&hl=en)

* [institute page](https://www.uu.nl/staff/FLucatelliNunes)

## Related entries

* [[absolutely dense functor]]

* [[inserter]]

* [[non-canonical isomorphism]]

* [[adjoint triangle theorem]]

* [[automatic differentiation]]

* [[descent]]

* [[lax-idempotent 2-adjunction]]

* [[Kan extension]]

category: people

[[!redirects Fernando Nunes]]