André Joyal

André Joyal is a Canadian mathematician, a professor at Université du Québec à Montréal.

He got his PhD in [1971](https://web.archive.org/web/20060325080542/http://132.208.138.87/_joyal/) from Université de Montréal.

His wide mathematical work, is mainly in [[category theory]], [[topos theory]] and abstract [[homotopy theory]]. His works include a wide generalization of [[Galois theory]] with [[Myles Tierney]], the combinatorial ideas of "Joyal's [[species]]", discovery of the category structure on the collection of Conway combinatorial [[game theory|games]], the discovery of Kripke-Joyal semantics, a series of works (mainly with [[Ross Street]]) about (braided, tortile etc.) [[monoidal category|monoidal categories]] prompted partly by methods and motivation in theoretical physics, much of his work for about last 30 years centered on developing the theory of [[quasicategory|quasicategories]], after the first ideas of Boardman and Vogt. In the 1980s Joyal invented a Quillen [[model category]] [[model structure for quasi-categories|structure on the category of simplicial sets]] (and [[model structure on simplicial presheaves|categories of simplicial presheaves]]). Joyal and J. Kock more recently proved [[Simpson's conjecture]] (on higher categories via weak units) in categorical dimension 3.

Joyal  promoted [[quasi-categories]], greatly extending their theory, as a basis for [[(∞,1)-category theory]].

Joyal has contributed to the $n$Lab as 'joyal'; he once began a project at [[joyalscatlab:HomePage|joyalscatlab]].

[webpage](http://professeurs.uqam.ca/professeur?c=joyal.andre), [wikipedia](http://en.wikipedia.org/wiki/Andr%C3%A9_Joyal)

[Old webpage](https://web.archive.org/web/19990221035246/http://www.math.uqam.ca/_joyal/).

[CV](https://web.archive.org/web/20030707225620/http://132.208.138.87/_joyal/joyal.pdf) from 1995.

## Selected writings

On [[forcing]] via [[classifying toposes]] and the [[classifying topos of a localic groupoid]]:

* {#JoyalTierney84} [[André Joyal]], [[Myles Tierney]], *An extension of the Galois theory of Grothendieck*,  Mem. Amer. Math. Soc. **309** (1984) [[ISBN:978-1-4704-0722-3](https://bookstore.ams.org/memo-51-309)]

  (historical note: according to MR756176 (86d:18002) by [[Peter Johnstone]], the main results of this monograph were obtained by the authors around 1978-1979, typed version circulated from 1982, and the results influenced the field much before the actual publication)

On [[algebraic set theory]]:

* A. Joyal, [[Ieke Moerdijk|I. Moerdijk]], _Algebraic set theory_. London Mathematical Society Lecture Note Series __220__. Cambridge Univ. Press (1995) viii+123 pp. ISBN: 0-521-55830-1

On [[transition systems]], [[bisimulations]] and [[open morphisms]]:

* {#JoyalNielsenWisnkel94} [[André Joyal]], [[Mogens Nielsen]], [[Glynn Winskel]]: *Bisimulation from open maps*, BRICS report series RS-94-7 (1994) [[doi:10.7146/brics.v1i7.21663](https://doi.org/10.7146/brics.v1i7.21663), [pdf](http://www.brics.dk/RS/94/7/BRICS-RS-94-7.pdf)], Information and Computation **127** 2 (1996) 164-185 [[doi:10.1006/inco.1996.0057](https://doi.org/10.1006/inco.1996.0057)]

Introducing the notion of [[traced monoidal categories]]:

* {#JoyalStreetVerity96} [[André Joyal]], [[Ross Street]], [[Dominic Verity]], _Traced monoidal categories_, Math. Proc. Camb. Phil. Soc. **119** (1996) 447-468 [[pdf](http://sci-prew.inf.ua/v119/3/S0305004100074338.pdf), [doi:10.1017/S0305004100074338](https://doi.org/10.1017/S0305004100074338)]

On the [[Dwyer-Kan loop groupoid]]-construction ([[path groupoid|path]]-[[simplicial groupoids]]):

* [[André Joyal]], [[Myles Tierney]], *On the theory of path groupoids*, Journal of Pure and Applied Algebra **149** 1 (2000) 69-100 [doi:10.1016/S0022-4049(98)00164-9]

On [[quasi-categories]]:

* [[André Joyal]], *Quasi-categories and Kan complexes*, J. Pure Appl. Algebra **175** (2002) 207-222 (special volume celebrating the 70th birthday of Prof. Max Kelly) [doi:10.1016/S0022-4049(02)00135-4]

On the [[Quillen equivalence]] between the [[model categories]] for [[quasi-categories]] and [[complete Segal spaces]]:

* {#JoyalTierney07} [[Andre Joyal]], [[Myles Tierney]], _Quasi-categories vs. Segal spaces_, in *Categories in Algebra, Geometry and Mathematical Physics*, Contemporary Mathematics **431** (2007) [[arXiv:math/0607820](http://arxiv.org/abs/math/0607820), [doi:10.1090/conm/431](https://doi.org/10.1090/conm/431)]

On [[quasi-categories]]:

* {#Joyal08} [[André Joyal]], *[[The Theory of Quasi-Categories and its Applications]]*, lectures at _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, CRM 2008 ([pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [[JoyalTheoryOfQuasiCategories.pdf:file]])

On [[simplicial homotopy theory]], the [[classical model structure on simplicial sets]] and the [[classical model structure on topological spaces]]:

* {#JoyalTierney08} [[André Joyal]], [[Myles Tierney]] _Notes on simplicial homotopy theory_, Lecture at _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, CRM 2008  ([[JoyalTierneyNotesOnSimplicialHomotopyTheory.pdf:file]])

* {#JoyalTierney09} [[André Joyal]], [[Myles Tierney]] _An introduction to simplicial homotopy theory_, 2009  ([web](http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01), [[JoyalTierneySimplicialHomotopyTheory.pdf:file]])

On [[homotopy theory]] via [[category theory]]:

* {#JoyalCategorialHoTT} [[André Joyal]], *[[Categorical Homotopy Type Theory.pdf:file]]*, MIT Topology Seminar (March 17, 2014)

Proving the [[Blakers-Massey theorem]] in any [[(infinity,1)-topos|$(\infty,1)$-topos]] and with the [[(n-connected, n-truncated) factorization system]] allowed to be replaced by more general [[modal homotopy type theory|modalities]]:

* {#AnelBiedermanFinsterJoyal17a} [[Mathieu Anel]], [[Georg Biedermann]], [[Eric Finster]], [[André Joyal]], _A Generalized Blakers-Massey Theorem_, Journal of Topology **13** 4 (2020) 1521-1553 $[$[arXiv:1703.09050](https://arxiv.org/abs/1703.09050), [doi:10.1112/topo.12163](https://doi.org/10.1112/topo.12163)$]$

On [[(infinity,1)-topos|$(\infty,1)$-toposes]] understood as *[[logoi]]*:

* {#AnelJoyal19} [[Mathieu Anel]], [[André Joyal]], *Topo-logie*, in *[[New Spaces for Mathematics and Physics]]*, Cambridge University Press (2021) 155-257 [[doi:10.1017/9781108854429.007](https://doi.org/10.1017/9781108854429.007), [pdf](http://mathieu.anel.free.fr/mat/doc/Anel-Joyal-Topo-logie.pdf)]

See also:

* A. Joyal, M. Tierney, _Quasi-categories vs Segal spaces_, Categories in algebra, geometry and mathematical physics,  277--326, Contemp. Math. __431__, Amer. Math. Soc., Providence, RI, 2007. [math.AT/0607820](http://arxiv.org/abs/math.AT/0607820).

* A. Joyal, M. Tierney, _On the theory of path groupoids_,  J. Pure Appl. Algebra __149__  (2000),  no. 1, 69--100, [doi](http://dx.doi.org/10.1016/S0022-4049%2898%2900164-9)

* A. Joyal, R. Street, _Pullbacks equivalent to pseudopullbacks_, Cahiers topologie et géométrie différentielle catégoriques 34 (1993) 153-156; [numdam](http://www.numdam.org/item?id=CTGDC_1993__34_2_153_0) MR94a:18004.

* A. Joyal, M. Tierney, _Strong stacks and classifying spaces_,  Category theory (Como, 1990),  213--236, Lecture Notes in Math. __1488__, Springer 1991.

* A. Joyal, [[Ross Street|R. Street]], _An introduction to Tannaka duality and quantum groups_,  Category theory (Como, 1990),  413--492, Lecture Notes in Math. __1488__, Springer 1991 ([pdf](http://www.math.mq.edu.au/~street/CT90Como.pdf)).

* A. Joyal, R. Street, _The geometry of tensor calculus I_,  Adv. Math.  __88__(1991),  no. 1, 55--112, [doi](http://dx.doi.org/10.1016/0001-8708%2891%2990003-P); _Tortile Yang-Baxter operators in tensor categories_,  J. Pure Appl. Algebra  __71__  (1991),  no. 1, 43--51, [doi](http://dx.doi.org/10.1016/0022-4049%2891%2990039-5); _Braided tensor categories_, Adv. Math. __102__ (1993),  no. 1, 20--78, [doi](http://dx.doi.org/10.1006/aima.1993.1055).

* A. Joyal, R. Street, _Braided monoidal categories_, Macquarie Math Reports 860081 (1986) [pdf](http://www.maths.mq.edu.au/~street/JS1.pdf); Macquarie Math Reports 850067 (1985) [pdf](http://www.math.mq.edu.au/~street/BMC850067.pdf).

* A Joyal, _Notes on quasicategories_, ([draft](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf))

* A. Joyal, Letter to [[Alexander Grothendieck]], 11.4.1984, [[lettrejoyal.pdf:file]].

* A. Joyal, _Disks, duality and $\Theta$-categories_, preprint (1997)  (contains an original definition of a weak $n$-category: for a short account see Leinster's [book](http://arxiv.org/abs/math.CT/0305049), 10.2).

* A. Joyal, _Remarques sur la théorie des jeux à deux personnes_, Gazette des
Sciences Mathematiques du Québec 1(4):46–52, 1977; Robin Houston's rough translation [can be found here](https://bosker.wordpress.com/2009/11/16/the-category-of-conway-games/)
* André Joyal, _Free lattices, communication and money games_, in: Logic and
scientific methods. Volume one of the proceedings of the tenth international
congress of logic, methodology and philosophy of science, Florence, Italy, Synth.
Libr. 259, pages 29–68. Dordrecht: Kluwer Academic Publishers, 1997.

category: people

[[!redirects Andre Joyal]]
[[!redirects A. Joyal]]
[[!redirects Joyal]]


André Juan Ferreira-Martins

* [GoogleScholar page](https://scholar.google.co.uk/citations?user=tyTlYvcAAAAJ&hl=de)

## Selected writings

On the [[Randall-Sundrum model]] and on the [[AdS/CFT correspondence]]:

* [[André Juan Ferreira-Martins]], *Gravity and its wonders: braneworlds and holography* ([arXiv:2105.10062](https://arxiv.org/abs/2105.10062))

category: people

[[!redirects Andre Juan Ferreira-Martins]]

[[!redirects Andre Ferreira-Martins]]

[[!redirects A. J. Ferreira-Martins]]


André Lichnerowicz

* [wikipedia](http://en.wikipedia.org/wiki/Andr%C3%A9_Lichnerowicz)

## Selected writings

Introducing the notion of [[Poisson manifolds]]:

* [[André Lichnerowicz]], *Les variétés de Poisson et leurs algèbres de Lie associées*, Journal of Differential Geometry **12** 2 (1977) 253–300 [[doi:10.4310/jdg/1214433987](http://dx.doi.org/10.4310/jdg/1214433987)]

Introducing [[formal deformation quantization]]:

* {#BayenFlatoFronsdalLichnerowiczSternheimer78} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], [[André Lichnerowicz]], [[Daniel Sternheimer]], _Deformation theory and quantization. I. Deformations of symplectic structures._, Annals of Physics **111** 1 (1978) 61-110 [doi:10.1016/0003-4916(78)90224-5]

* {#BayenFlatoFronsdalLichnerowiczSternheimer78b} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], [[André Lichnerowicz]], [[Daniel Sternheimer]], _Deformation theory and quantization. II. Physical applications_, Annals of Physics **111** 1 (1978) 111-151 [doi:10.1016/0003-4916(78)90225-7]

category: people

[[!redirects Andre Lichnerowicz]]


André Miemiec

* [webpage](http://andre-miemiec.de/)

## Selected writings

On [[D=11 supergravity]] (supposedly the low-energy limit of [[M-theory]]):

* [[André Miemiec]], [[Igor Schnakenburg]], _Basics of M-Theory_, Fortsch. Phys. 54 (2006) 5-72 ([arXiv:hep-th/0509137](http://arxiv.org/abs/hep-th/0509137), [doi:10.1002/prop.200510256]( https://doi.org/10.1002/prop.200510256))

category: people

[[!redirects Andre Miemiec]]


André Neveu

* [Wikipedia entry](https://en.wikipedia.org/wiki/Andr%C3%A9_Neveu)

## Selected writings

On the [[NSR superstring]]:

* [[André Neveu]], [[John Schwarz]], _Factorizable dual model of pions_, Nucl. Phys. B31, 86 (1971) (doi:10.1016/0550-3213(71)90448-2)

Early discussion of [[quantization]] of the [[Polyakov action]] for the [[bosonic string]] with attention to the [[Liouville field]]:

* [[Jean-Loup Gervais]], [[André Neveu]], *The dual string spectrum in Polyakov's quantization (I)*, Nuclear Physics B **199** 1 (1982) 59-76 [doi:10.1016/0550-3213(82)90566-1]

* [[Jean-Loup Gervais]], [[André Neveu]], *Dual string spectrum in Polyakov's quantization (II). Mode separation*, Nuclear Physics B **209** 1 (1982) 125-145 [doi:10.1016/0550-3213(82)90105-5]

## Related entries

* [[spinning string]]

* [[supersymmetry]]

category: people

[[!redirects Andre Neveu]]


André Néron

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

## related $n$Lab entries

* [[elliptic fibration]]

category: people


André Petermann

* [Wikipedia entry](https://en.wikipedia.org/wiki/Andr%C3%A9_Petermann)

## Related $n$Lab entries

* [[Stückelberg-Petermann renormalization group]]

* [[renormalization]]

category: people

[[!redirects Andre Petermann]]


André Weil

* [Wikipedia entry](http://en.wikipedia.org/wiki/Andr%C3%A9_Weil)

## Selected writings

Introducing the [[Chern-Weil homomorphism]]:

* [[André Weil]], _Géométrie différentielle des espaces fibres_, unpublished, item [1949e] in: _André Weil Oeuvres Scientifiques / Collected Papers_, vol. 1 (1926-1951), 422-436, Springer 2009 ([ISBN:978-3-662-45256-1](https://www.springer.com/gp/book/9783662452561))

## Related entries

* [[Weil conjecture]]

* [[Weil cohomology theory]]

category: people

[[!redirects Andre Weil]]
[[!redirects Weil]]


André-Quillen homology

#Contents#
* table of contents
{:toc}

## Idea

André-Quillen cohomology is the [[cochain cohomology]] of the [[derived functor|derived]] [[Kähler differentials]], hence of the [[cotangent complex]] (for the moment see there for more).

## Related concepts

* [[topological André-Quillen homology]]

## References

The orginal work comes from two sources, the work of [[Michel André]] and that of [[Dan Quillen]]:

* [[Michel André]], _Méthode simpliciale en algèbre homologique et algèbre commutative_, Springer Lecture Notes in Mathematics, Vol 32, 1967.

* Michel André, _Homologie des algèbres commutatives_ Grundlehren der mathematischen Wissenschaften, Band 206. Springer. 1974

and then

* [[D. G. Quillen]], 1970, _On the (co-)homology of commutative rings_, in _Proc. Symp. on Categorical Algebra_, 65–87, American Math. Soc.

[[!redirects André-Quillen cohomology]]
[[!redirects Andre-Quillen homology]]
[[!redirects Andre-Quillen cohomology]]


Andrés Villaveces

[[Andrés Villaveces]] is a mathematician at Universidad Nacional de Colombia sede Bogotá ([web](https://bogota.unal.edu.co)) specialized in [[model theory]]. He is also interested in philosophy and history of mathematics.

* personal [homepage](https://avillavecesn.net)

### Selected writing and expositions

* John T. Baldwin, Andrés Villaveces, _[[Boris Zilber|Zilber]]’s notion of logically perfect structure: Universal Covers_, [arXiv:2302.04650](https://arxiv.org/abs/2302.04650)
* Alexander Berenstein, Andrés Villaveces, _Hilbert spaces with random predicates_,  [pdf](http://matematicas.uniandes.edu.co/~aberenst/amalg10.pdf)
* Fernando Zalamea (editor), _Rondas en Sais. Ensayos sobre matemáticas y cultura contemporánea._ (Essays on mathematics and contemporary culture, by Moreno, Javier; de Lorenzo, Javier; Villaveces, Andrés; Pérez, Jesús Hernando; Restrepo, Gabriel; Cruz Morales, John Alexánder; Vargas, Francisco; Oostra, Arnold; Ferreirós, José; Zalamea, Fernando; Martín, Alejandro) Universidad Nacional de Colombia, Facultad de Ciencias Humanas 2012 [pdf](https://repositorio.unal.edu.co/bitstream/handle/unal/78631/9789587618693.pdf)

#### Presentations

* _Two logics, and their connections with large cardinals_, online talk at NY Logics Seminar, 2021, part1.[yt](https://www.youtube.com/watch?v=X2sWl6HGlpc), part2.[yt](https://www.youtube.com/watch?v=TVQoCKwSx7o)
* _Simplicity via complexity via simplicity? Sandboxes for simplicity_, Simplicity conference (multidisciplinary), [yt](https://www.youtube.com/watch?v=p42oxR1kwV8)
* _Galoisian model theory: the role(s) of Grothendieck (à son insu !)_, Los Angeles, May 22, 2022 [yt](https://www.youtube.com/watch?v=4NCk8ItLMR0) [slides](https://avillavecesn.net/2022/05/21/galoisian-model-theory-the-roles-of-grothendieck-a-son-insu-los-angeles-5-22)

category: people

[[!redirects Andres Villaveces]]


Andy Tonks

* [website](https://www2.le.ac.uk/departments/mathematics/extranet/staff-material/staff-profiles/andy-tonks)

* [thesis (Bangor, 1993)](https://groupoids.org.uk/pdffiles/tonksthesis.pdf)

## Selected writings

On [[cohomology]] of [[monoids in monoidal categories]] (specifically in [[distributive monoidal categories]]):

* [[Hans-Joachim Baues]], [[Mamuka Jibladze]], [[Andy Tonks]], *Cohomology of monoids in monoidal categories*, in: *Operads: Proceedings of Renaissance Conferences*, Contemporary Mathematics **202**, AMS (1997) 137-166 [[doi:10.1090/conm/202](https://doi.org/10.1090/conm/202), preprint:[pdf](https://archive.mpim-bonn.mpg.de/id/eprint/1484/1/preprint_1995_121.pdf)]

On the [[Adams-Hilton model]] for the [[Pontrjagin ring]]-structure on the [[singular chain complex]] of a [[based loop space]]:

* [[Kathryn Hess]], [[Paul-Eugène Parent]], [[Jonathan Scott]], [[Andrew Tonks]], *A canonical enriched Adams-Hilton model for simplicial sets*, Advances in Mathematics **207** 2 (2006) 847-875 [[doi:10.1016/j.aim.2006.01.013](https://doi.org/10.1016/j.aim.2006.01.013), [arXiv:math/0408216](https://arxiv.org/abs/math/0408216)]

See also:

* Andrew P. Tonks, _On the Eilenberg-Zilber Theorem for crossed complexes_, J. Pure Appl. Algebra, 179~(1-2) (2003) 199--220,

* [[Imma Gálvez-Carrillo]], [[Joachim Kock]], Andrew Tonks. _Homotopy linear algebra_. Proc. Roy. Soc. Edinburgh Sect. A, 148(2):293--325, 2018.

* [[Imma Gálvez-Carrillo]], [[Joachim Kock]], Andrew Tonks. _Groupoids and Faà di Bruno formulae for Green functions in bialgebras of trees_, Adv. Math., 254:79--117, 2014.

* [[Imma Gálvez-Carrillo]], [[Andy Tonks]], [[Bruno Valette]], _Homotopy Batalin-Vilkovisky algebras_, J. Noncomm. Geom. __6__:3, 539--602 [arXiv:0907.2246](https://arxiv.org/abs/0907.2246) [doi](https://doi.org/10.4171/JNCG/99)

* Imma Gálvez, [[Vassily Gorbounov]], Andrew Tonks, _Homotopy Gerstenhaber structures and vertex algebras_, [math/0611231.QA](https://arxiv.org/abs/math/0611231)

* [[Imma Gálvez-Carrillo]], [[Joachim Kock]], Andrew Tonks, _Decomposition spaces, incidence algebras and Möbius inversion_, [arXiv:1404.3202](https://arxiv.org/abs/1404.3202)

## Related entries

* [[Eilenberg-Zilber theorem]]

* [[homotopy BV-algebra]]

category: people

[[!redirects Andrew Tonks]]
[[!redirects A. P. Tonks]]


Angel Toledo

PhD student, interested in (NC)DAG.

[website](http://math.unice.fr/~toledo)


Angelo Vistoli

* [website](http://homepage.sns.it/vistoli)

## Selected writings

On basics of [[algebraic geometry]] following [[Grothendieck]]'s [[FGA]]:

* [[Barbara Fantechi]], [[Lothar Göttsche]], [[Luc Illusie]], [[Steven L. Kleiman]], [[Nitin Nitsure]], [[Angelo Vistoli]]:

  **[[FGA explained]]** (ICTP, Trieste 2003--2005)

  Mathematical Surveys and Monographs __123__

  Amer. Math. Soc. 2005. x+339 pp.

  * [ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s)

    [MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)

    [lecture notes](http://indico.ictp.it/event/a0255/other-view?view=ictptimetable)

## Related $n$Lab entries

* [[Grothendieck fibration]]
* [[Grothendieck topology]]
* [[stack]]
* [[descent]]
* [[FGA explained]]
* [[descent along a torsor]]

category: people
[[!redirects A. Vistoli]]


angle

+-- {: .rightHandSide}
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### Context
#### Trigonometry
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[[!include trigonometry -- contents]]
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## Related concepts

* [[distance]]

* [[Euler angle]]

* [[trigonometry]]

  [[trigonometric function]]

## References

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

[[!redirects angles]]

[[!redirects arc length]]
[[!redirects arc lengths]]

[[!redirects arclength]]
[[!redirects arclengths]]


angular momentum

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

## Idea

In [[classical mechanics]], the analog of [[momentum]] for rotational dynamics is called _angular momentum_, and is defined as twice the product of the [[mass]] and the [[areal velocity]].

In [[quantum mechanics]], the angular momentum [[quantum observables]] constitute a [[representation]] of the ([[special orthogonal group|special]]) [[orthogonal group]] $SO(n)$ of $n$-dimensional [[Euclidean space]], in applications typically considered for $n = 3$ or $n = 2$.

Therefore the theory of quantum angular momentum is that of the [[irreducible representation]] of the [[rotation group]].

## Related concepts

* [[angular velocity]], [[moment of inertia]]

* [[areal velocity]]

* [[Clebsch-Gordan coefficient]]

* [[spin]]

* [[helicity]]

* [[Pauli-Lubanski vector]]

* [[spin-orbit coupling]]

* [[moment map]]

## References

### Classical angular momentum

### Representation theory of the special orthogonal group

* Wheeler, _Irreducible representation of the rotation group_ ([pdf](http://www.physics.usu.edu/Wheeler/GaugeTheory/RotationGroupIrreps.pdf))

[[!redirects angular momenta]]


angular velocity

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

## Idea

The analog of [[velocity]] for rotational movement.

For rotation in a [[plane]] inside a [[Cartesian space]] $\mathbb{R}^n$ the angular velocity is a [[bivector]] in $\wedge^2 \mathbb{R}^n$ of the form

$$
  \dot \omega \,\, e_1 \wedge e_2
  \,,
$$

where $e_1$ and $e_2$ are unit vector spanning the plane of rotation, and where $\dot \omega$ is the magnitude of the angular velocity.

Of $n = 3$ (and only then) can we identify [[bivector]]s with plain [[vector]]s (by the dual operation induced by the [[Hodge star operator]]). Often in the literature only this "angular velocity vector" in 3 dimensions is considered.

## Related concepts

* [[rigid body dynamics]]

  * [[angular momentum]]

  * [[moment of inertia]]

* [[galaxy rotation curve]]

## References

Standard discussion of angular velocity in $d \leq 3$ is for instance in

* wikipedia: [angular velocity](http://en.wikipedia.org/wiki/Angular_velocity)

The more general discussion in terms of [[bivector]]s is found for instance in [[Geometric Algebra]]-style documents, such as

* Chris Doran, Anthony Lasenby, _Geometric Algebra for Physicists_ Cambridge University Press

  _Physical applications of geometric algebra_ ([pdf](https://dspace.ist.utl.pt/bitstream/2295/52599/1/Lectures_on_Geometric_Algebra.pdf#page=56))


Angélica M. Osorno

* [personal page](https://people.reed.edu/~aosorno/)

## Selected wrirings

On [[iterated loop spaces]] and [[coherence theorems]] for [[monoidal 2-categories]] ([[braided monoidal 2-category|braided]], [[sylleptic monoidal 2-category|sylleptic]], [[symmetric monoidal 2-category|symmetric]]), and, somewhat implicitly: [[sylleptic 3-groups]]:

* [[Nick Gurski]], [[Angélica M. Osorno]], *Infinite loop spaces, and coherence for symmetric monoidal bicategories*, Adv. Math. 246 (2013) 1-32 ([arXiv:1210.1174](https://arxiv.org/abs/1210.1174))

category: people

[[!redirects Angelica M. Osorno]]

[[!redirects Angélica Osorno]]
[[!redirects Angelica Osorno]]


Angélica Osorno

* [webpage](http://people.reed.edu/~aosorno/)

category: people


Anibal M. Medina-Mardones

[Website](https://www.medina-mardones.com/)

Obtained his PhD degree in 2015 at Stony Brook, advised by [[Dennis Sullivan]].

## Selected writings

* [[Anibal M. Medina-Mardones]], _New formulas for cup-$i$ products and fast computation of Steenrod squares_, [arXiv](arXiv:2105.08025).

* [[Ralph M. Kaufmann]], [[Anibal M. Medina-Mardones]], _Cochain level May-Steenrod operations_, [arXiv](https://arxiv.org/abs/2010.02571).

* [[Anibal M. Medina-Mardones]], _An axiomatic characterization of Steenrod's cup-i products_, [arXiv](https://arxiv.org/abs/1810.06505).

[[!redirects Anibal Medina-Mardones]]

category: people


Aniceto Murillo

* [webpage](http://agt.cie.uma.es/~aniceto/)

## Selected writings

On [[rational models of mapping spaces]]:

* [[Urtzi Buijs]], [[Aniceto Murillo]], _Basic constructions in rational homotopy theory of function spaces_,  Annales de l'Institut Fourier, Volume 56 (2006) no. 3, p. 815-838 ([doi:10.5802/aif.2201](https://doi.org/10.5802/aif.2201))

* [[Urtzi Buijs]], [[Aniceto Murillo]], _The rational homotopy Lie algebra of function spaces_, Comment. Math. Helv. 83 (2008), 723–739 ([pdf](https://pdfs.semanticscholar.org/d404/657ccc24b0c06434086485d3e528e0316e26.pdf))

* {#BuijsFelixMurillo12} [[Urtzi Buijs]], [[Yves Félix]], [[Aniceto Murillo]], _$L_\infty$-rational homotopy of mapping spaces_,  published as _$L_\infty$-models of based mapping spaces_,  J. Math. Soc. Japan Volume 63, Number 2 (2011), 503-524 ([arXiv:1209.4756](https://arxiv.org/abs/1209.4756), [doi:10.2969/jmsj/06320503](https://doi.org/10.2969/jmsj/06320503))

On [[Whitehead L-infinity algebras|Whitehead $L_\infty$-algebras]] in [[rational homotopy theory]]:

* [[Francisco Belchí]], [[Urtzi Buijs]], [[José M. Moreno-Fernández]], [[Aniceto Murillo]], _Higher order Whitehead products and $L_\infty$ structures on the homology of a DGL_, Linear Algebra and its Applications, **520** (2017) 16-31 [[arXiv:1604.01478](https://arxiv.org/abs/1604.01478), [doi:10.1016/j.laa.2017.01.008](https://doi.org/10.1016/j.laa.2017.01.008)]

On [[rational homotopy theory]] with general [[fundamental groups]]:

* [[Urtzi Buijs]], [[Yves Félix]], [[Aniceto Murillo]], [[Daniel Tanré]], _Homotopy theory of complete Lie algebras and Lie models of simplicial sets_, Journal of Topology (2018) 799-825 ([arXiv:1601.05331](https://arxiv.org/abs/1601.05331), [doi:10.1112/topo.12073](https://doi.org/10.1112/topo.12073))

## Related $n$Lab entries

* [[rational homotopy theory]]

* [[L-infinity algebra]]

category: people


Anirban Basu

* [webpage](http://www.hbni.ac.in/faculty/HRI/hrial_phys_basu_anirban.htm)

## Selected writings

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

* {#BasuHarvey05} [[Anirban Basu]], [[Jeffrey Harvey]], _The M2-M5 Brane System and a Generalized Nahm's Equation_, Nucl.Phys. B713 (2005) 136-150 ([arXiv:hep-th/0412310](https://arxiv.org/abs/hep-th/0412310))

## Related $n$Lab entries

* [[fuzzy 3-sphere]]

category: people


Anisomorphism

I collect high quality mathematics resources and try to bridge basics to advanced topics seen here on nlab. Current book lists are at [the portal](https://theportal.wiki/wiki/Read) and [sheafification's fast track](http://sheafification.com/the-fast-track/)

category: people


ANITA experiment

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Experiments
+-- {: .hide}
[[!include experiments -- contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[experiment]] studying [[neutrino]] [[physics]]:

The _Antarctic Impulsive Transient Antenna_ (ANITA) experiment has been designed to study ultra-high-energy (UHE) cosmic neutrinos by detecting the radio pulses emitted by their interactions with the Antarctic ice sheet.

In 2018 the experiment saw some events that were interpreted as signaling New Physics beyond the [[standard model of particle physics]] ([Fox et. al.  18](#FoxEtAl18)).

Possible [[model (in theoretical physics)]] that have been suggested might explain this include [[sterile neutrinos]] ([Huang 18](#Huang18)) or [[leptoquarks]] ([Chauhan-Mohanty 18](#ChauhanMohanty18))

## Related concepts

* [[Michelson-Morley experiment]]

* [[LHC]]

  [[LHCb experiment]], [[Belle experiment]]

* [[RHIC]]

* [[Super-Kamiokande]]

* [[LIGO]]

* [[EDGES]]

## References

* University of Hawaii, _[ANITA Webpage](https://www.phys.hawaii.edu/~anita/)_

* {#ANITA20} [[ANITA Collaboration]], _Ultra-high Energy Air Showers Observed by ANITA-IV_ ([arXiv:2008.05690](https://arxiv.org/abs/2008.05690))

Review:

* {#Wissel20} Stephanie Wissel, _Prospects in UHE Neutrino Astronomy_, talk at [NEUTRINO 2020](https://conferences.fnal.gov/nu2020/), June 2020 ([pdf](https://indico.fnal.gov/event/43209/contributions/187883/attachments/130573/159213/Wissel_UHEnus_Neutrino2020.pdf), [[WisselNeutrino2020.pdf:file]])

See also

* Wikipedia, _[Antarctic Impulse Transient Antenna](https://en.wikipedia.org/wiki/Antarctic_Impulse_Transient_Antenna)_,

On the potentially anomalous events recorded:

* {#FoxEtAl18} Derek B. Fox, Steinn Sigurdsson, Sarah Shandera, Peter Mészáros, Kohta Murase, Miguel Mostafá, Stephane Coutu, _The ANITA Anomalous Events as Signatures of a Beyond Standard Model Particle, and Supporting Observations from IceCube_ ([arXiv:1809.09615](https://arxiv.org/abs/1809.09615))

Speculation that [[sterile neutrinos]] could explain the ANITA anomaly:

* {#Huang18} Guo-yuan Huang, _Sterile neutrinos as a possible explanation for the upward air shower events at ANITA_, Phys. Rev. D 98, 043019 (2018) ([arXivL1804.05362](https://arxiv.org/abs/1804.05362))

Speculation that [[leptoquarks]] could (not only explain the [[flavour anomalies]] but also) explain the ANITA anomaly:

* {#ChauhanMohanty18} Bhavesh Chauhan, Subhendra Mohanty, _A common leptoquark solution of flavor and ANITA anomalies_ ([arXiv:1812.00919](https://arxiv.org/abs/1812.00919))

* P. S. Bhupal Dev, Rukmani Mohanta, Sudhanwa Patra, Suchismita Sahoo, _Unified explanation of flavor anomalies, radiative neutrino mass and ANITA anomalous events in a vector leptoquark model_ ([arXiv:2004.09464](https://arxiv.org/abs/2004.09464))

Critical discussion:

* {#Anchordoqui19} L. A. Anchordoqui et al. _The pros and cons of beyond standard model interpretations of ANITA events_, PoS ICRC2019 (2020) 884, 884 ([arXiv:1907.06308](https://arxiv.org/abs/1907.06308), [spire:1744116](https://inspirehep.net/literature/1744116))

[[!redirects Antarctic Impulsive Transient Antenna]]

[[!redirects ANITA Collaboration]]


Anna Beliakova

* [institute page](https://www.math.uzh.ch/index.php?id=people&key1=578)

## Selected writings

On [[handlebodies]]

* [[Anna Beliakova]], Marco De Renzi, *Refined Bobtcheva-Messia Invariants of 4-Dimensional 2-Handlebodies* [[arXiv:2205.11385](https://arxiv.org/abs/2205.11385)]

Exposition in:

* [[Anna Beliakova]], *On algebraisation of low-dimensional Topology*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#BeliakovaMar2023), video:[YT](https://www.youtube.com/watch?v=7I5526YkI44)]

category: people


Anna Ceresole

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

## Selected writings

On ([[super Yang-Mills theory|super]]-)[[Yang-Mills theory]], [[AdS/CFT correspondence]] and [[GUTs]] and [[string theory]]:

* [[Anna Ceresole]] C. Kounnas [[Dieter Lüst]], [[Stefan Theisen]], _Quantum Aspects of Gauge Theories, Supersymmetry and Unification_ ([doi:10.1007/BFb0104238](https://link.springer.com/book/10.1007/BFb0104238))

On the work of [[Tullio Regge]] (such as [[Regge calculus]], [[geometric supergravity]]):

* [[Leonardo Castellani]],  [[Anna Ceresole]], [[Riccardo D'Auria]], [[Pietro Fré]] (eds.): _Tullio Regge: An Eclectic Genius_, World Scientific 2019 ([doi:10.1142/11643](https://doi.org/10.1142/11643))

category: people


Anna Labella

* [GoogleScholar page](https://scholar.google.com/citations?user=Zjh1hLEAAAAJ)

## Selected writings

On [[distributive monoidal categories]]:

* [[Anna Labella]], *Categories with sums and right distributive tensor product*, Journal of Pure and Applied Algebra **178** 3 (2003) 273-296 [doi:10.1016/S0022-4049(02)00169-X]

category: people


Anna Marie Bohmann

* [website](https://math.vanderbilt.edu/bohmanar/index.html)

## Selected writings

On [[commutative ring]]-structure ([[E-infinity ring|$E_\infty$-ring]]-structure) in [[rational equivariant stable homotopy theory]] in general and in [[rational equivariant K-theory]] in particular:

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Naive-commutative structure on rational equivariant K-theory for abelian groups* ([arXiv:2002.01556](https://arxiv.org/abs/2002.01556))

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Genuine-commutative ring structure on rational equivariant K-theory for finite abelian groups* ([arXiv:2104.01079](https://arxiv.org/abs/2104.01079))

## Related $n$Lab entries

* [[equivariant stable homotopy theory]]

category: people


Anne Broadbent

* [personal page](https://mysite.science.uottawa.ca/abroadbe/)

* [GoogleScholar page](https://scholar.google.com/citations?user=oDLJjc0AAAAJ&hl=en)

## Idea

On [[software verification|certification]] of [[quantum computations]]:

* [[Anne Broadbent]], Arthur Mehta, Yuming Zhao, *Quantum delegation with an off-the-shelf device* [[arXiv:2304.03448](https://arxiv.org/abs/2304.03448)]

category: people


Anne Marie Svane

* [webpage](https://vbn.aau.dk/en/persons/142961)

## Selected writings

On the [[homotopy groups]] of the [[embedded cobordism category]]:

* [[Marcel Bökstedt]], [[Anne Marie Svane]], _A geometric interpretation of the homotopy groups of the cobordism category_, Algebr. Geom. Topol. 14 (2014) 1649-1676 ([arXiv:1208.3370](https://arxiv.org/abs/1208.3370))

* [[Marcel Bökstedt]], [[Johan Dupont]], [[Anne Marie Svane]], _Cobordism obstructions to independent vector fields_, Q. J. Math. 66 (2015), no. 1, 13-61 ([arXiv:1208.3542](https://arxiv.org/abs/1208.3542))

On the [[EHP spectral sequence]]:

* [[Marcel Bökstedt]], [[Anne Marie Svane]], _A generalization of the stable EHP spectral sequence_ ([arXiv:1208.3938](http://arxiv.org/abs/1208.3938))

category: people

[[!redirects Anne Svane]]


Anne Sjerp Troelstra

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

## Selected writings

On [[intuitionistic mathematics]], [[constructive mathematics]] and the [[BHK interpretation]]:

* {#Troelstra69} [[Anne Sjerp Troelstra]], *Principles of Intuitionism*, Lecture Notes in Mathematics **95** Springer Heidelberg (1969)  [[doi:10.1007/BFb0080643](https://link.springer.com/book/10.1007/BFb0080643)]

* {#Troelstra77} [[Anne Sjerp Troelstra]], *Aspects of Constructive Mathematics*, Studies in Logic and the Foundations of Mathematics **90** 973-1052 (1977) [doi:10.1016/S0049-237X(08)71127-3]

* [[Anne Sjerp Troelstra]], [[Dirk van Dalen]], *Constructivism in Mathematics -- An introduction*, Vol 1, Studies in Logic and the Foundations of Mathematics **121**, North-Holland (1988) [[ISBN:9780444702661](https://www.elsevier.com/books/constructivism-in-mathematics-vol-1/troelstra/978-0-444-70266-1)]

On [[intuitionistic logic]]:

* {#Troelstra90} [[Anne Sjerp Troelstra]], *On the Early History of Intuitionistic Logic*, in *Mathematical Logic*, Springer, (1990) [[doi:10.1007/978-1-4613-0609-2_1](https://doi.org/10.1007/978-1-4613-0609-2_1)]

On [[constructivism]]:

* {#Troelstra91} [[Anne Sjerp Troelstra]], *History of Constructivism in the Twentieth Century* (1991) [[pdf](https://www.illc.uva.nl/Research/Publications/Reports/ML-1991-05.text.pdf), [[Troelstra-HistoryOfConstructivism.pdf:file]]]

On [[linear logic]]:

* [[Anne Sjerp Troelstra]], *Lectures on Linear Logic*, CSLI Lectures notes **29** (1992) [[ISBN:0937073776](https://web.stanford.edu/group/cslipublications/cslipublications/site/0937073776.shtml)]

On [[proof theory]] with emphasis on [[natural deduction]] and [[sequent calculi]]:

* [[Anne Sjerp Troelstra]], [[Helmut Schwichtenberg]], _Basic Proof Theory_ , Cambridge University Press (2000, 2012) [[doi:10.1017/CBO9781139168717](https://doi.org/10.1017/CBO9781139168717)]

## Related entries

* [[constructivism]]

* [[exponential modality]]

category: people

[[!redirects Troelstra]]
[[!redirects Anne Troelstra]]


Annette Huber

* [Web](http://home.mathematik.uni-freiburg.de/arithgeom/huber.html)

* [[motives]]


annulus

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

## Idea

An (open) **annulus** is a [[topological space]] that is [[homeomorphic]] to the [[disk]] with an interior point removed: $D^2 \setminus \{0\}$.

An often used  model for the corresponding closed annulus is the subspace $\{(x,y)\mid 1\leq x^2 + y^2 \leq 4\} \subset \mathbb{R}^2$, of the plane consisting of the point lying between a unit circle and a circle of radius 2, both centred on the origin.

## Related concepts

* [[trinion]]

* [[Riemann surface]]


anodyne morphism

+-- {: .rightHandSide}
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###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The original definition by [Gabriel–Zisman](#GabrielZisman67) (Definition IV.2.1.4) defined __anodyne extensions__ as the [[weak saturation]] of [[simplicial horn]] inclusions.

More generally, the same definition can be used to talk about the [[weak saturation]] of any set $S$ of morphisms
in any category.  One also talks about _anodyne maps_ or _anodyne morphisms_.

If the [[small object argument]] is applicable,
anodyne maps are precisely maps with a [[left lifting property]] with respect to all [[fibrations]], where the latter is defined as morphisms with a [[right lifting property]] with respect to $S$.

In particular, if $S$ is a set of generating [[acyclic cofibrations]] in a [[model category]] with applicable [[small object argument]], then anodyne maps are precisely [[acyclic cofibrations]].

The standard example, often taken to be the default, is that of morphisms in the category [[sSet]] of [[simplicial sets]] which have the [[left lifting property]] against all [[Kan fibrations]]. In this case, anodyne morphisms ([Gabriel-Zisman 67, chapter IV.2](#GabrielZisman67)) are equivalent to [[acyclic cofibrations]] in the [[classical model structure on simplicial sets]].

So in the standard example of left lifting against Kan fibrations, one typically speaks of _anodyne extensions_ if one produces morphisms by these operations from the set of [[horn]] inclusions. (see for instance ([Jardine](#Jardine))).

## Definition

### Relative to Kan fibrations of simplicial sets

A [[morphism]] $f : A \to B$ of [[simplicial sets]] is called **anodyne** if it has the left [[lifting property]] with respect to all [[Kan fibrations]].

So $f$ is anodyne if for every [[Kan fibration]] $X \to Y$ and every commuting diagram

$$
  \array{
    A &\to& X
    \\
    \downarrow^f && \downarrow
    \\
    B &\to& Y
  }
$$

there exists a lift

$$
  \array{
    A &\to& X
    \\
    \downarrow^f &\nearrow& \downarrow
    \\
    B &\to& Y
  }
  \,.
$$

See for instance ([Jardine](#Jardine)) for details.

### Relative to left/right inner Kan fibrations of simplicial sets

Similarly a morphism is called

* **left anodyne** if it has the left [[lifting property]] with respect to all [[left Kan fibration]]s

* **right anodyne** if it has the left [[lifting property]] with respect to all [[right Kan fibration]]s

* **inner anodyne** if it has the left [[lifting property]] with respect to all [[inner Kan fibration]]s

See ([Lurie](#Lurie)) (following Joyal).

### Relative to inner Kan fibrations of dendroidal sets

In the category of [[dendroidal sets]] there is a notion of horn inclusions that generazies that of simplicial sets. The corresponding saturated class of morphisms is called that of dendroidal **inner anodyne morphisms**.

See ([Cisinski-Moerdijk 09](#CisinskiMoerdijk09)).

## Properties

### Pushout-products with inclusions

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

The [[pushout product]] $f \Box g$ of two [[monomorphisms]] $f,g$ in [[sSet]] is again a monomorphism, which is anodyne (a weak homotopy equivalence) if $f$ or $g$ is so.

=--

This is due to ([Gabriel-Zisman 67, IV.2, prop. 2.2](#GabrielZisman67)). The argument is somewhat more streamlined form is also in [Joyal-Tierney 05, theorem 3.2.2](#JoyalTierney05)

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

Prop. \ref{PushoutProductOfSimplicialCofibrationWithAcyclicCOfibIsAcyclic} is the key lemma which implies (is effectively equivalent to) the statement that the [[classical model structure on simplicial sets]] is an [[enirched model category]] over itself.

=--

## Related concepts

* [[factorization system]], [[small object argument]]

* [[Kan fibration]], **anodyne morphism**

* [[right/left Kan fibration]], [[right/left anodyne map]]

* [[inner fibration]]

* [[Cartesian fibration]]

## References

The original concept of anodyne extensions as morphisms in the saturation class of the [[simplicial set|simplicial]] [[horn]] inclusions originates in

* {#GabrielZisman67} [[Pierre Gabriel]], [[Michel Zisman]], chapter IV.2 of _[[Calculus of fractions and homotopy theory]]_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf))

Review includes

* {#Jardine} [[John Frederick Jardine]], _Homotopy theory, lecture 5_ ([pdf](http://www.math.uwo.ca/~jardine/papers/HomTh/lecture005.pdf))

* {#JoyalTierney05} [[André Joyal]], [[Myles Tierney]], _An introduction to simplicial homotopy theory_, 2005  ([chapter I](http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01), more notes [pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern47.pdf))

Left/right and inner anodyne morphisms of simplicial sets are discussed in section 2 of

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_

Inner anodyne morphisms of [[dendroidal sets]] are discussed in

* {#CisinskiMoerdijk09} [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal sets as models for homotopy operads_ ([arXiv:0902.1954](http://arxiv.org/abs/0902.1954))

[[!redirects anodyne morphisms]]
[[!redirects anodyne map]]
[[!redirects anodyne maps]]

[[!redirects anodyne extension]]
[[!redirects anodyne extensions]]

[[!redirects inner anodyne morphism]]
[[!redirects inner anodyne morphisms]]
[[!redirects inner anodyne map]]
[[!redirects inner anodyne maps]]


AnodyneHoward

Nobody, with no scholarship let alone authority of his own, but the relative aptitude (or willingness) for mending the odd broken link.

category: people


anomalous diffusion

The usual diffusion comes from [[Brownian motion]] -- the random walk with equal steps. Anomalous diffusion is about more general case which comes from more complicated [[random process]]es, including nonlocal in time and those having jumps.

* J-P Bouchaud, A Georges, _Anomalous diffusion in disordered media: statistical mechanisms_, Phys. Rep. 195 (1990) pdf
* wikipedia [anomalous diffusion](https://en.wikipedia.org/wiki/Anomalous_diffusion)
* Ralf Metzler, Joseph Klafter, _The random walk's guide to anomalous diffusion: a fractional dynamics approach_, Physics Reports __339__:1 (2000) 1–77 doi
* Michael F. Shlesinger, Joseph Klafter, Gert Zumofen, _Above, below and beyond Brownian motion_, Amer Jour Phys 67(12) (1999) 1253 [doi](http://dx.doi.org/10.1119/1.19112)
* J. Klafter, M. F. Shlesinger, G. Zumofen, _Beyond Brownian motion_, Physics Today 49(2) (1996) 33 [doi](http://dx.doi.org/10.1063/1.881487)
* Ralf Metzler, Eli Barkai, Joseph Klafter, _Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach_, Physical Review Letters 82:18, 3563-3567 (1999)
* R. Metzler, E. Barkai, J. Klafter, _Deriving fractional Fokker-Planck equations from a generalised master equation_, Europhys. Lett. 46 431 (1991) [doi](http://dx.doi.org/10.1209/epl/i1999-00279-7)

* V. Zaburdaev, S. Denisov, J. Klafter, _Lévy walks_, [arxiv/1410.5100v1](https://arxiv.org/abs/1410.5100v1)

* Eric Lutz, _Fractional Langevin equation_, Phys. Rev. E 64, 051106 [doi](http://dx.doi.org/10.1103/PhysRevE.64.051106) [cond-math/0103128v1](http://arxiv.org/abs/cond-mat/0103128v1)

> We investigate fractional Brownian motion with a microscopic random-matrix model and introduce a fractional Langevin equation. We use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath. We further compare fractional Brownian motion with the fractal time process. The respective mean-square displacements of these two forms of anomalous diffusion exhibit the same power-law behavior. Here we show that their lowest moments are actually all identical, except the second moment of the velocity. This provides a simple criterion that enable us to distinguish these two non-Markovian processes.

> A well–known example of a process which is non–local in space is Lévy stable motion, for which the mean–square displacement is actually infinite due to the occurrence of very long jumps. In this Letter we focus on processes which are nonlocal in time and whence show memory effects. Specifically, we shall discuss and compare fractional Brownian motion (fBm) and the fractal time process (ftp). These two forms of anomalous diffusion are fundamentally different...


anomalous magnetic moment

+-- {: .rightHandSide}
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### Context
#### Algebraic Qunantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]], the [[magnetic moment]] of [[particles]] as predicted by [[classical field theory]] may receive corrections due to [[quantum physics|quantum effects]]. Such corrections are also called _[[quantum anomalies]]_, and hence one speaks of the _anomalous magnetic moments_, traditionally denoted by "$g-2$".

The archetypical example is the anomalous magnetic moment of the [[electron]] in [[quantum electrodynamics]], which is famous as the [[pQFT]]-prediction that matches [[experiment]] to an accuracy of about $10^{-12}$ (e.g.[Scharf 95, (3.10.20)](#Scharf95)).

Similarly there is the anomalous magenetic moment $g_\mu - 2$ of the [[muon]] and the  other [[leptons]].

## Anomalies
 {#Anomalies}

In fact, the anomalous magnetic moment of the [[muon]] $g_\mu - 2$ has become notorious for apparently showing a noticeable _discrepancy_ between [[theory (physics)|theoretic prediction]] from the [[standard model of particle physics]] and its value as determined in [[experiment]]. The discrepancy is now found to have [[statistical significance]] around 3.5[[standard deviation|σ]] ([DHMZ 17](#DHMZ17)) or 4[[standard deviation|σ]] ([Jegerlehner 18a](#Jegerlehner18a)).

In April 2021, after re-doing the Brookhaven experiment, Fermilab confirms these findings and states a statistical significance of the deviation of 4.2 sigma ([Abi et al. 21](#AbiEtAl21)).

Recent measurements may even show a possible deviation around 2.5 [[standard deviation|σ]] for the electron's anomalous magnetic moment (see [Falkowski 18](#Falkowski18)). Details depend on understanding of [[non-perturbative effects]] ([Jegerlehner 18b, section 2](#Jegerlehner18b)).

In particular, there seems to be inconsistencies in the theoretical understanding of the relevant [[lattice QCD]]-computations:

\begin{imagefromfile}
    "file_name": "LehnerMeyermug2Lattice.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -50,
        "right": 10,
        "bottom": -40,
        "left": 20
    }
\end{imagefromfile}

> graphics from [Lehner-Meyer 20, Fig 14](#LehnerMeyer20)

If these experimental "anomalies" (in the sense of [[phenomenology]]) in the anomalous magnetic moment $g_\mu - 2$ (and possibly even in $g_e -2$) are real (the established rule of thumb is that deviations are established once their [[statistical significance]] reaches 5[[standard deviation|σ]], see [here](statistical+significance#ParticlePhysics)), they point to "new physics" beyond the [[standard model of particle physics]].

In fact [Lyons 13b](#Lyons13b) argued that the detection-threshold of the [[statistical significance]] of anomalies here should be just 4[[standard deviation|σ]], which would mean that they should already count as being detected:

\begin{center}

\end{center}

> table taken from [Lyons 13b, p. 4](#Lyons13b)

Together with the [[flavour anomalies]], these anomalies relate to the [[flavour problem]] in the [[standard model of particle physics]]:

> graphics from [Crivellin-Hoferichter 20](flavour+anomaly#CrivellinHoferichter20)

> (here "$R$" refers to [[flavour anomalies]] in various channels, "$a$" refers to anomalies in the the [[anomalous magnetic moments]] of the [[electron]] and the [[muon]], "LFUV" is shoft for "Lepton Flavor Universality Violation", and the numbers are the [[statistical significances]] of the effects seen)

Possible explanations for the anomalies in the anomalous magnetic moments is the existence of [[leptoquarks]] ([Bauer-Neubert 15](#BauerNeubert15), [CCDM 16](#CCDM16), [Falkowski 17](#Falkowski17), [Müller 18](#Mueller18)), which at the same time are a candidate for explaining the [[flavour anomalies]] (see also [Chiang-Okada 17](#ChiangOkada17)).

## Contributions

### QED contributions

(...)  for the [[electron]] see e.g. [Scharf 95, section 3.10](#Scharf95) (...)

### Quantum gravity contributions
 {#QuantumGravityCorrection}

The further corrections of [[loop order|1-loop]] [[perturbative quantum gravity]] to the anomalous magnetic moment of the [[electron]] and the [[muon]] have been computed in ([Berends-Gastman 75](#BerendsGastman75)) and found to be finite without need for [[renormalization]]. These [[Feynman diagrams]] contribute:

### Axion contributions

Possible contributions to and xconstraints on $g_{lep}-2$ from hypothetical [[axions]] are discussed in [ACGM 08](#ACGM08), [MMPP 16](#MMPP16), [BNT 17](#BNT17)...

## Related concepts

* [[electric dipole moment]]

## References

### General

Basic discussion:

* {#Steinmann02} [[Othmar Steinmann]], _What is the Magnetic Moment of the Electron?_, Commun.Math.Phys. 237 (2003) 181-201 ([arXiv:hep-ph/0211187](https://arxiv.org/abs/hep-ph/0211187))

* Kirill Melnikov, [[Arkady Vainshtein]], _Theory of the Muon Anomalous Magnetic Moment_, Springer Tracts in Modern Physics 216, 2006

* [[Friedrich Jegerlehner]], _The Anomalous Magnetic Moment of the Muon_, Springer Tracts in Modern Physics 226, Springer-Verlag Berlin Heidelberg, 2008

* Song Li, Yang Xiao, Jin Min Yang, *A pedagogical review on muon $g-2$, Modern Physics 4 (2021) 40-47 ([arXiv:2110.04673](https://arxiv.org/abs/2110.04673))

Discussion of detection-threshold for the [[statistical significance]] of anomalies:

* {#Lyons13b} [[Louis Lyons]], _Discovering the Significance of 5 sigma_ ([arXiv:1310.1284](https://arxiv.org/abs/1310.1284))

See also

* Wikipedia, _[Anomalous magnetic dipole moment](https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment)_

Prediction to [[loop order]] 5 in [[QED]]:

* Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita, Makiko Nio, *Tenth-Order QED Contribution to the Electron $g-2$ and an Improved Value of the Fine Structure Constant*, Phys. Rev. Lett. 109, 111807 ([arXiv:1205.5368](https://arxiv.org/abs/1205.5368), [doi:10.1103/PhysRevLett.109.111807](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.111807))

Comprehensive discussion for the [[muon]]:

* T. Aoyama et al., _The anomalous magnetic moment of the muon in the Standard Model_ ([arXiv:2006.04822](https://arxiv.org/abs/2006.04822))

  > Discussion up to 12th [[loop order]] in section 6.3.

* Alex Keshavarzi, Kim Siang Khaw, Tamaki Yoshioka, *Muon $g-2$: current status* ([arXiv:2106.06723](https://arxiv.org/abs/2106.06723))

### Experiment and deviation

Discussion of precision experiment and possible deviation from theory:

* {#DHMZ17} Michel Davier, Andreas Hoecker, Bogdan Malaescu, Zhiqing Zhang, _Reevaluation of the hadronic vacuum polarisation contributions to the Standard Model predictions of the muon g-2 and alpha(mZ) using newest hadronic cross-section data_, Eur. Phys. J. C (2017) 77: 827 ([arXiv:1706.09436](https://arxiv.org/abs/1706.09436))

* J. L. Holzbauer on behalf of the Muon g-2 collaboration, _The Muon g-2 Experiment Overview and Status_, Proceedings for The 19th International Workshop on Neutrinos from Accelerators (NUFACT 2017) ([arXiv:1712.05980](https://arxiv.org/abs/1712.05980))

* {#Jegerlehner18a} [[Fred Jegerlehner]], _The Muon g-2 in Progress_, Acta Physica Polonica 2018 ([doi:10.5506/APhysPolB.49.1157](https://arxiv.org/ct?url=https%3A%2F%2Fdx.doi.org%2F10.5506%252FAPhysPolB.49.1157&v=01073571), [arXiv:1804.07409](https://arxiv.org/abs/1804.07409))

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

* {#Falkowski18} [[Adam Falkowski]], _[Both $g-2$ anomalies](http://resonaances.blogspot.com/2018/06/alpha-and-g-minus-two.html)_, June 2018

* {#AbiEtAl21} B. Abi et al. (Muon g−2 Collaboration), _Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm_, Phys. Rev. Lett. 126, 141801 2021 ([doi:10.1103/PhysRevLett.126.141801](https://doi.org/10.1103/PhysRevLett.126.141801))

  $\,$

  Exposition:

  Priscilla Cushman, _Muon’s Escalating Challenge to the Standard Model_,  Physics 14, 54, April 2021 ([web](https://physics.aps.org/articles/v14/54))

* Maarten Golterman, *Theory review for hadronic corrections to $g-2$* [[arXiv:2208.05560](https://arxiv.org/abs/2208.05560)]

* Ashutosh Kotwal, Joaquim Matias, Andrea Mauri, Tom Tong, Lukas Varnhorst, *Round table on Standard Model Anomalies* [[arXiv:2211.13030](https://arxiv.org/abs/2211.13030)]

### Relation to flavour anomalies

Possible explanation of the anomaly in the anomalous magnetic moments in terms of [[leptoquarks]]:

* {#BauerNeubert15} Martin Bauer, Matthias Neubert, _One Leptoquark to Rule Them All: A Minimal Explanation for $R_{D^{(\ast)}}$, $R_K$ and $(g-2)_\mu$_, Phys. Rev. Lett. 116, 141802 (2016) ([arXiv:1511.01900](https://arxiv.org/abs/1511.01900))

* {#CCDM16} Estefania Coluccio Leskow, Andreas Crivellin, Giancarlo D'Ambrosio, Dario Müller, _$(g-2)_\mu$, Lepton Flavour Violation and Z Decays with Leptoquarks: Correlations and Future Prospects_, Phys. Rev. D 95, 055018 (2017) ([arXiv:1612.06858](https://arxiv.org/abs/1612.06858))

* {#BiswasShaw19} Anirban Biswas, Avirup Shaw, _Reconciling dark matter, $R_{K^{(\ast)}}$ anomalies and $(g-2)_\mu$ in an $L_\mu-L_\tau$ scenario_ ([arXiv:1903.08745](https://arxiv.org/abs/1903.08745))

* {#Falkowski17} [[Adam Falkowski]], _[Leptoquarks strike back](http://resonaances.blogspot.com/2015/11/leptoquarks-strike-back.html)_, November 2017

* {#ChiangOkada17} Cheng-Wei Chiang, Hiroshi Okada, _A simple model for explaining muon-related anomalies and dark matter_ ([arXiv:1711.07365](https://arxiv.org/abs/1711.07365))

* {#Mueller18} Dario Müller, _Leptoquarks in Flavour Physics_, EPJ Web of Conferences 179, 01015 (2018) ([arXiv:1801.03380](https://arxiv.org/abs/1801.03380))

* Junichiro Kawamura, Stuart Raby, Andreas Trautner, _Complete Vector-like Fourth Family and new $U(1)'$ for Muon Anomalies_ ([arXiv:1906.11297](https://arxiv.org/abs/1906.11297))

Further possible joint explanation of the [anomalies](anomalous+magnetic+moment#Anomalies) observed in the [[muon]] [[anomalous magnetic moment]] and the [[flavour anomalies]]:

* Geneviève Bélanger, Cédric Delaunay, Susanne Westhoff, _A Dark Matter Relic From Muon Anomalies_,  Phys. Rev. D 92, 055021 (2015) ([arXiv:1507.06660](https://arxiv.org/abs/1507.06660))

* {#ChiangOkada17} Cheng-Wei Chiang, Hiroshi Okada, _A simple model for explaining muon-related anomalies and dark matter_ ([arXiv:1711.07365](https://arxiv.org/abs/1711.07365))

* Junichiro Kawamura, Stuart Raby, Andreas Trautner, _Complete Vector-like Fourth Family and new $U(1)'$ for Muon Anomalies_ ([arXiv:1906.11297](https://arxiv.org/abs/1906.11297))

* Lorenzo Calibbi, M.L. López-Ibáñez, Aurora Melis, Oscar Vives, _Muon and electron $g-2$ and lepton masses in flavor models_ ([arXiv:2003.06633](https://arxiv.org/abs/2003.06633))

* A. S. de Jesus, S. Kovalenko, F. S. Queiroz, K. Sinha, C. Siqueira, _Vector-Like Leptons and Inert Scalar Triplet: Lepton Flavor Violation, $g-2$ and Collider Searches_ ([arXiv:2004.01200](https://arxiv.org/abs/2004.01200))

* Shaikh Saad, _Combined explanations of $(g-2)_\mu$, $R_{D^\ast}$, $R_{K^\ast}$ anomalies in a two-loop radiative neutrino mass model_ ([arXiv:2005.04352](https://arxiv.org/abs/2005.04352))

* Da Huang, António P. Morais, Rui Santos, _Anomalies in $B$ Decays and Muon $g-2$ from Dark Loops_ ([arXiv:2007.05082](https://arxiv.org/abs/2007.05082))

* K.S. Babu, P.S. Bhupal Dev, Sudip Jana, Anil Thapa, _Unified Framework for $B$-Anomalies, Muon $g-2$, and Neutrino Masses_ ([arXiv:2009.01771](https://arxiv.org/abs/2009.01771))

* Sang Quang Dinh, Hieu Minh Tran, _Muon $g-2$ and semileptonic $B$ decays in BDW model with gauge kinetic mixing_ ([arXiv:2011.07182](https://arxiv.org/abs/2011.07182))

* Mingxuan Du, Jinhan Liang, Zuowei Liu, Van Que Tran, *A vector leptoquark interpretation of the muon $g-2$ and $B$ anomalies* ([arXiv:2104.05685](https://arxiv.org/abs/2104.05685))

A [[leptoquark]] model meant to address all of the [[flavour anomalies]], the [[(g-2)-anomaly]] and the [[Cabibbo anomaly]] at once:

* {#MarzoccaTrifinopoulos21} [[David Marzocca]], Sokratis Trifinopoulos, *A Minimal Explanation of Flavour Anomalies: B-Meson Decays, Muon Magnetic Moment, and the Cabbibo Angle* ([arXiv:2104.05730](https://arxiv.org/abs/2104.05730))

Realization in [[F-theory]] of [[GUT]]-models with [[Z'-bosons]] and/or [leptoquarks]] addressing the [[flavour anomalies]] and the [(g-2) anomalies](anomalous+magnetic+moment#Anomalies):

* Miguel Crispim Romao, Stephen F. King, George K. Leontaris, _Non-universal $Z'$ from Fluxed GUTs_, Physics Letters B Volume 782, 10 July 2018, Pages 353-361 ([arXiv:1710.02349](https://arxiv.org/abs/1710.02349))

* A. Karozas, G. K. Leontaris, I. Tavellaris, N. D. Vlachos, _On the LHC signatures of $SU(5) \times U(1)'$ F-theory motivated models_ ([arXiv:2007.05936](https://arxiv.org/abs/2007.05936))

### QED contributions

The computation of the anomalous magnetic dipole moment of the [[electron]] in [[QED]] is spelled out (via [[causal perturbation theory]]) in

* {#Scharf95} [[Günter Scharf]], section 3.10, culminating in (3.10.20), of _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

### QCD contribution

#### General

Discussion of [[QCD]] contributions via [[lattice QCD]]:

* {#LehnerMeyer20} Christoph Lehner, Aaron S. Meyer, _Consistency of hadronic vacuum polarization between lattice QCD and the R-ratio_ ([arXiv:2003.04177](https://arxiv.org/abs/2003.04177))

#### Via holographic QCD

Application of [[holographic QCD]] to [[anomalous magnetic moment]] of the [[muon]]:

* Luigi Cappiello, _What does Holographic QCD predict for anomalous $(g-2)_\mu$?_, 2015 ([pdf](https://agenda.infn.it/getFile.py/access?contribId=19&sessionId=5&resId=0&materialId=paper&confId=9430))

* [[Josef Leutgeb]], [[Anton Rebhan]], _Axial vector transition form factors in holographic QCD and their contribution to the anomalous magnetic moment of the muon_ ([arXiv:1912.01596](https://arxiv.org/abs/1912.01596))

* [[Josef Leutgeb]], [[Anton Rebhan]], _Axial vector transition form factors in holographic QCD and their contribution to the muon $g-2$_ ([arXiv:2012.06514](https://arxiv.org/abs/2012.06514))

* [[Josef Leutgeb]], [[Anton Rebhan]], *Hadronic light-by-light contribution to the muon $g-2$ from holographic QCD with massive pions* ([arXiv:2108.12345](https://arxiv.org/abs/2108.12345))

* [[Josef Leutgeb]], Jonas Mager, [[Anton Rebhan]], *Holographic QCD and the muon anomalous magnetic moment* ([arXiv:2110.07458](https://arxiv.org/abs/2110.07458))

### Gravity contributions

Corrections at [[loop order|1-loop]] from [[quantum gravity]] are discussed in

* {#BerendsGastman75} F. A. Berends, R. Gastmans, _Quantum gravity and the electron and muon anomalous magnetic moment_, Phys. Lett. B55 Issue 3 Feb 1975 311-312 (doi:10.1016/0370-2693(75)90608-5)

This discussion is adapted to [[supergravity]] in

* F. del Aguila, A. Culatti, R. Munoz-Tapia, M. Perez-Victoria, _Supergravity corrections to $(g-2)_l$ in differential renormalization_, Nuclear Physics  B  504  (1997)  532-550 ([arXiv:hep-ph/9702342](https://arxiv.org/abs/hep-ph/9702342))

### Axion contributions
 {#ReferencesAxionContributions}

Contribution of hypothetical [[axions]] to the [[anomalous magnetic moment]]
of the [[electron]] and [[muon]] in [[QED]]:

* Yannis Semertzidis, _Magnetic and Electric Dipole Moments in Storage Rings_, chapter 6 of Markus Kuster, Georg Raffelt, Berta Beltrán (eds.), _Axions: Theory, cosmology, and Experimental Searches_, Lect.
Notes Phys. 741 (Springer, Berlin Heidelberg 2008) (doi:10.1007/978-3-540-73518-2_2)

* {#ACGM08} Roberta Armillis, Claudio Coriano, Marco Guzzi, Simone Morelli, _Axions and Anomaly-Mediated Interactions: The Green-Schwarz and Wess-Zumino Vertices at Higher Orders and g-2 of the muon_, JHEP 0810:034,2008 ([arXiv:0808.1882](https://arxiv.org/abs/0808.1882))

* {#MMPP16} W.J. Marciano, A. Masiero, P. Paradisi, M. Passera, _Contributions of axion-like particles to lepton dipole moments_, Phys. Rev. D 94, 115033 (2016) ([arXiv:1607.01022](https://arxiv.org/abs/1607.01022))

* {#BNT17} Martin Bauer, Matthias Neubert, Andrea Thamm, _Collider Probes of Axion-Like Particles_, J. High Energ. Phys. (2017) 2017: 44. ([arXiv:1708.00443](https://arxiv.org/abs/1708.00443), doi:10.1007/JHEP12(2017)044)

The basic relevant [[Feynman diagrams]] are worked out here:

* [pdf](http://www-personal.umich.edu/~jbourj/peskin/6-3.pdf)

[[!redirects anomalous magnetic moments]]

[[!redirects anomalous magnetic dipole moment]]
[[!redirects anomalous magnetic dipole moments]]

[[!redirects electron anomalous magnetic moment]]
[[!redirects muon anomalous magnetic moment]]

[[!redirects anomalous magnetic moment of the electron]]
[[!redirects anomalous magnetic dipole moment of the electron]]

[[!redirects g-2]]


anomaly cancellation

+-- {: .rightHandSide}
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#### Algebraic Qunantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

Given an [[action functional]] with a [[quantum anomaly]], an _anomaly cancellation_ is a modification of this such that the anomaly disappears. Usually this is specifically understood to be the modification given by adding another action functional which is also anomalous, such that the total anomaly vanishes, hence that both contributions cancel each other.

For more see at _[[quantum anomaly]]_.

## Related concepts

* [[tadpole cancellation]]

## References

### General

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Chapter 13 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

Discussions of [[spin structures]] as [[anomaly cancellation]] for the [[spinning particle]] (see also the references [here](supersymmetric quantum mechanics#ReferencesRelationToMorseTheory) at *[[supersymmetric quantum mechanics]]*):

* {#Witten85} [[Edward Witten]], p. 65-68 in: *Global anomalies in string theory*, in: W. Bardeen and A. White (eds.) *[Symposium on Anomalies, Geometry, Topology](https://inspirehep.net/conferences/965785?ui-citation-summary=true)*, World Scientific (1985) 61-99 [[[WittenGlobalAnomaliesInStringTheory.pdf:file]], [spire:214913](https://inspirehep.net/literature/214913)]

* [[Luis Alvarez-Gaumé]], p. 165 of: *Supersymmetry and the Atiyah-Singer index theorem*,  Comm. Math. Phys. **90** 2  (1983) 161-173 [[doi:10.1007/BF01205500](https://doi.org/10.1007/BF01205500), [euclid](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-90/issue-2/Supersymmetry-and-the-Atiyah-Singer-index-theorem/cmp/1103940278.full)]

* [[Daniel Friedan]], P. Windey, *Supersymmetric derivation of the Atiyah-Singer index and the chiral anomaly*,  Nucl. Phys. B **235** (1984) 395 [doi:10.1016/0550-3213(84)90506-6]

### In the standard model of particle physics

Discussion of [[anomaly cancellation]] in 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))

  > (relating to [[Fermat's last theorem]])

### In string theory

Review in [[string theory]]:

* [[John H. Schwarz]], *Anomaly Cancellation: A Retrospective From a Modern Perspective* (2001) [[doi:10.1142/9789812778185_0014](https://doi.org/10.1142/9789812778185_0014), [pdf](https://cds.cern.ch/record/508470/files/0107059.pdf)]

[[!redirects anomaly cancellations]]


Anonymous Hero > history

< [[Anonymous Hero]]
[[!redirects Anonymous Hero -- history]]


Anonymous User

Many edits to the [[HomePage|nLab]] are anonymous and/or signed by pseudonyms.

#Content#
* table of contents
{: toc}

## Generalities
 {#Generalities}

In choosing a name with which to sign their edits on the $n$Lab, users are free to pick almost any string of characters and without providing any justification. In this sense all user accounts are *de facto* anonymous: Even if signed with what may look like the name of an existing person there are no checks in place to associate any edit with any existing person.

(This policy is clearly both convenient for users as well as somewhat risky for the $n$Lab community: We rely on users not to abuse this liberty.)

Indeed, in the practice of editing the $n$Lab, the point of signing edits with a string of characters (a "name") is not to identify a legal person behind that name (though that is, of course, for many users a desired side effect) but rather:  To allow relating edits that come from the same source, in order to facilitate communicating about edits (such as on the [nForum](https://nforum.ncatlab.org)).

Therefore we kindly ask all users to *choose a unique pseudonym* for signing their edits and stick to using it consistently, as far as reasonable. That pseudonym might be the name which you read in your passport (we wouldn't know, either way) or it might be any other string of characters which you fancy (we are relying on users' decency in choosing reasonable handles), where the main point in the actual practice of communicating about $n$Lab edits is just that: it be unique.

In other words, we don't care who you are elsewhere, but we like to know you as the $n$Lab contributor that you are, hence, if you will, as the scholarly persona which is embodied by the collection of your nLab edits.

Finally, people who deeply care about their edits being unrelatable to their whereabouts should know that every editor\'s IP address is (usually) listed in the entry's history (found by clicking _history_ at the bottom of any page).

## In practice

In short, please choose a unique identifiable pseudonym with which to sign your edits.

## In the past

From its inception up to some years ago, the nLab software would offer the string "AnonymousCoward" as the default for filling in the "Submit as..."-field. Eventually we found this was too condescending   (though in hindsight there may have been some value to it) and then changed the default to "Anonymous".

This worked fine until yet a few years later there occurred a sudden flood of edits of apparently (but who knows) multiple users who all signed with the exact same string "Anonymous" and most of whose edits were of decidedly low quality and kept needing intervention from regulars. But the real problem was these users would not react to commentary which their edits received on the [nForum](https://nforum.ncatlab.org).
Therefore, the only way we saw left to handle this problem was to try and block the string "Anonymous". Remarkably  enough, that was sufficient to stop the rogue editor(s).

This is why currently the nLab software will not allow you to sign with that exact string of characters. Notice that any variant of "Anonymous" still works.

So if you do insist on using something like "Anonymous" as your signature, you may regard it as a minimalistic [CAPTCHA](https://en.wikipedia.org/wiki/CAPTCHA) that you have to be able to find and choose a variant of that character string.
But in the interest of the nLab project, it would be helpful if you solved a slighly more ambitious CAPTCHA and chose a pseudonym which is not just a small variant of "Anonymous". Thanks!

(In fact, the same comments now apply also to the string "Guest", which is the default name on the nForum.)

category: people

[[!redirects Anonymous]]
[[!redirects AnonymousCoward]]
[[!redirects Anonymous Coward]]
[[!redirects Anonymous coward]]
[[!redirects AnotherAnonymousCoward]]
[[!redirects InterestedAnonymousCoward]]
[[!redirects AnonymousHero]]
[[!redirects Anonymous Hero]]
[[!redirects AnonymousHeroicSpeller]]
[[!redirects NeedNotBeConcrete]]
[[!redirects sandboxfan]]
[[!redirects typoHunter]]
[[!redirects Spelling freak]]


Another page > history

< [[Another page]]
[[!redirects Another page -- history]]


Ansten Klev

* [webpage](https://sites.google.com/site/anstenklev/)

## Selected publications

* [[Ansten Klev]], *A Comparison of Type Theory with Set Theory*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[doi:10.1007/978-3-030-15655-8_10](https://doi.org/10.1007/978-3-030-15655-8_10), [pdf](https://drive.google.com/file/d/1FIfAvSkt9uJh6R2uS7gz-fWh5hLKVx5w/view)]

category: people


antecedent

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

#Contents#
* table of contents
{:toc}

## Idea

A [[judgement]] on the left of the "$\vdash$"-symbol of a [[sequent]]/[[hypothetical judgement]].

A _[[hypothesis]]_ or _[[context]]_.

## Related concepts

**antecedents** $\vdash$ [[succedents]]

**antecedents** $\vdash$ [[consequent]]

[[!redirects antecedents]]


Anthony Ashmore

* [personal webpage](https://anthonyashmore.com/)

* [Google Scholar page](https://scholar.google.com/citations?user=XwF3jY4AAAAJ&hl=en)

## Selected writings

On [[heterotic line bundles]] in the [[hidden sector]] of [[heterotic M-theory]]:

* [[Anthony Ashmore]], [[Sebastian Dumitru]], [[Burt Ovrut]], Section 4.2 of: _Line Bundle Hidden Sectors for Strongly Coupled Heterotic Standard Models_ ([arXiv:2003.05455](https://arxiv.org/abs/2003.05455))

* [[Anthony Ashmore]], [[Sebastian Dumitru]], [[Burt Ovrut]], _Explicit Soft Supersymmetry Breaking in the Heterotic M-Theory B−L MSSM_ ([arXiv:2012.11029](https://arxiv.org/abs/2012.11029))

* [[Anthony Ashmore]], [[Sebastian Dumitru]], [[Burt Ovrut]], *Hidden Sectors from Multiple Line Bundles for the B−L MSSM* ([arXiv:2106.09087](https://arxiv.org/abs/2106.09087))

* [[Sebastian Dumitru]], [[Burt A. Ovrut]], *Heterotic M-Theory Hidden Sectors with an Anomalous $U(1)$ Gauge Symmetry* ([arXiv:2109.13781](https://arxiv.org/abs/2109.13781))

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

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

category: people


Anthony Bahri

Anthony Bahri is an algebraic topologist working on toric topology and geometry.

* [website](http://users.rider.edu/~bahri/)

### See also

* [[Handbook of Homotopy Theory]]

category: people


Anthony Bak

* [personal page](https://www.math.uni-bielefeld.de/~bak/)

## Selected writings

On [[Hermitian K-theory]]:

* [[Anthony Bak]], *Grothendieck Groups of Modules and Forms Over Commutative Orders*, American Journal of Mathematics, **99** 1 (1977) 107-120 [[jstor:2374010](https://www.jstor.org/stable/2374010), [doi:10.2307/2374010](https://doi.org/10.2307/2374010)]

category: people


Anthony Blanc

Anthony Blanc is a former student of [[Bertrand Toën]].

* [web](http://www.ihes.fr/~ablanc/)
* [math genealogy](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=177054)

* [[dg-categories]]
* [[noncommutative motives]]
* [[topological K-theory]], [[semi-topological K-theory]]


Anthony Bordg

* [webpage](https://sites.google.com/site/anthonybordg/home)

## Selected publications

On the [[UniMath]] library:

* [[Anthony Bordg]], *Univalent Foundations and the UniMath Library*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[doi:10.1007/978-3-030-15655-8_8](https://doi.org/10.1007/978-3-030-15655-8_8), [arXiv:1710.02723](https://arxiv.org/abs/1710.02723)]

category: people


Anthony Durity

## where i'm at

[PhD, Digital Arts and Humanities, University College Cork](https://www.ucc.ie/en/dah/current/phddigitalartsandhumanities/)

## how to get in touch with me

Drop me a line at <anthony@durity.com>


Anthony Elmendorf

* [website](https://academics.pnw.edu/engineering-sciences/member/anthony-elmendorf/)

## Selected writings

Introducing [[Elmendorf's theorem]]:

* [[Anthony Elmendorf]], _Systems of fixed point sets_, Trans. Amer. Math. Soc., 277(1):275–284, 1983 ([jstor:1999356](https://www.jstor.org/stable/1999356))

On [[higher algebra]] ([[brave new algebra]]) in [[stable homotopy theory]], i.e. on [[ring spectra]], [[module spectra]] etc.:

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]],
_[[Rings, modules and algebras in stable homotopy theory]]_,
Mathematical Surveys and Monographs Volume 47, AMS 1997 ([ISBN:978-0-8218-4303-1](https://bookstore.ams.org/surv-47-s), [pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))

## Related entries

* [[Elmendorf's theorem]]

* [[cycle category]]

category: people


Anthony Houppe

* [InSpire page](https://inspirehep.net/authors/1804720)

## Selected writings

On [[type II supergravity]]-solutions corresponding to [[D1-D5-P bound states]] in non-supersymmetric generality via [[AdS3-CFT2 duality]] ("[[microstrata]]"):

* [[Bogdan Ganchev]], [[Anthony Houppe]], [[Nicholas P. Warner]], *Q-Balls Meet Fuzzballs: Non-BPS Microstate Geometries*, J. High Energ. Phys. **2021** 28 (2021)
[[arXiv:2107.09677](https://arxiv.org/abs/2107.09677)]

* [[Bogdan Ganchev]], [[Stefano Giusto]], [[Anthony Houppe]], [[Rodolfo Russo]], *$AdS_3$ holography for non-BPS geometries*, European Physical Journal C **82** 217 (2022)
 [[https://arxiv.org/abs/2112.03287](https://arxiv.org/abs/2112.03287), [doi:10.1140/epjc/s10052-022-10133-2](https://doi.org/10.1140/epjc/s10052-022-10133-2)]

* [[Bogdan Ganchev]], [[Stefano Giusto]], [[Anthony Houppe]], [[Rodolfo Russo]], [[Nicholas P. Warner]], *Microstrata* [[arXiv:2307.13021](https://arxiv.org/abs/2307.13021)]

On [[M2-M5-brane bound states]]:

* [[Iosif Bena]], [[Anthony Houppe]], [[Dimitrios Toulikas]], [[Nicholas P. Warner]], *Maze Topiary in Supergravity* [[arXiv:2312.02286](https://arxiv.org/abs/2312.02286)]

category: people


Anthony Knapp

* [personal page](https://www.math.stonybrook.edu/~aknapp/)

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

## Selected writings

On basic [[group theory]] ([[groups]], [[group actions]], [[linear representations]]) and basic [[algebra]] ([[rings]], [[modules]], [[fields]], [[Galois theory]]):

* [[Anthony Knapp]], *Basic Algebra*, Springer (2006) [[doi:10.1007/978-0-8176-4529-8](https://doi.org/10.1007/978-0-8176-4529-8), [pdf](https://www.math.mcgill.ca/darmon/courses/19-20/algebra2/knapp.pdf)]

category: people


Anthony Licata

* [webpage](http://math.stanford.edu/~amlicata/Anthony_Licata_homepage/Home.html)

category: people


Anthony Morse

Anthony Morse and John L. Kelly created the Morse-Kelly set theory.

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

category: people


Anthony P. Morse

> Not to be confused with [[Marston Morse]].

Anthony Perry Morse was a mathematician at the University of California, Berkeley, primarily working in [[geometric measure theory]].

He got his PhD degree in 1937 from Brown University, advised by Clarence Raymond Adams.

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

## Selected writings

On the [[Morse–Kelley set theory]]:

* [[Anthony P. Morse]], _A theory of sets_, Pure and Applied Mathematics XVIII, Academic Press (1965), xxxi+130 pp.   Second Edition, Pure and Applied Mathematics 108, Academic Press (1986), xxxii+179 pp. ISBN: 0-12-507952-4

On the [[Morse–Sard lemma]]:

* [[Anthony P. Morse]], _The Behavior of a Function on Its Critical Set_, Annals of Mathematics 40:1 (1939), 62–70.  [doi](http://dx.doi.org/10.2307/1968544).

[[!redirects Anthony Morse]]
[[!redirects A. P. Morse]]
[[!redirects A. Morse]]

category: people


Anthony Scholl

* [webpage](https://www.dpmms.cam.ac.uk/~ajs1005/)

## related $n$Lab entries

* [[special values of L-functions]]

* [[Beilinson conjectures]]

category: people


Anthony Sudbery

* [webpage](https://www-users.york.ac.uk/~as2/welcome.htm)

## Selected writings

On [[supersymmetry and division algebras]]:

* {#Sudbery84} [[Anthony Sudbery]], _Division algebras, (pseudo)orthogonal groups and spinors_, Jour. Phys. A17 (1984),
939–955 ([doi:10.1088/0305-4470/17/5/018](
https://iopscience.iop.org/article/10.1088/0305-4470/17/5/018))

* {#ChungSudbery87} K.-W. Chung, [[Anthony Sudbery]], _Octonions and the Lorentz and conformal groups of ten-dimensional space-time_, Phys. Lett. B 198 (1987), 161–164 (doi:10.1016/0370-2693(87)91489-4)

* {#ManogueSudbery89} [[Corinne Manogue]], [[Anthony Sudbery]], _General solutions of covariant superstring equations of motion_, Phys. Rev. D 12 (1989), 4073–4077 ([doi:10.1103/PhysRevD.40.4073](https://doi.org/10.1103/PhysRevD.40.4073))

category: people


Anthony Voutas

* [personal page](https://voutasaur.us)

## Selected writings

On [[monads]] and their [[Eilenberg-Moore categories]] in [[universal algebra]]:

* [[Anthony Voutas]], *The basic theory of monads and their connection to universal algebra*, (2012) [[pdf](https://voutasaur.us/monad-algebra.pdf), [[Voutas-Monads.pdf:file]]]

category: people


Anthony Zee

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

## Selected writings

On [[photon]]/[[pion]] [[interaction]]:

* Ruvi Aviv, [[Anthony Zee]], _Low-Energy Theorem for $\gamma \to 3 \pi$_
Phys. Rev. D 5, 2372 (1972) ([doi:10.1103/PhysRevD.5.2372](https://doi.org/10.1103/PhysRevD.5.2372))

The model of [[anyon statistics]] as an [[Aharonov-Bohm effect]] of a "[[fictitious gauge field]]" sourced by and coupled to each anyon:

* {#ArovasScriefferWilczekZee85} [[Daniel P. Arovas]], [[John Robert Schrieffer]], [[Frank Wilczek]], [[Anthony Zee]], *Statistical mechanics of anyons*, Nuclear Physics B **251** (1985) 117-126 (reprinted in [Wilczek 1990, p. 173-182](Frank+Wilczek#Wilczek90)) $[$doi:10.1016/0550-3213(85)90252-4$]$

Discussion of [[Spin(16)]]-[[grand unified theories]] (traditionally called "SO(16)" GUT):

* {#WilczekZee82} [[Frank Wilczek]], [[Anthony Zee]], _Families from spinors_, Phys. Rev. D 25, 553 (1982) ([doi:10.1103/PhysRevD.25.55310.1103/PhysRevD.25.553](https://doi.org/10.1103/PhysRevD.25.553))

* Goran Senjanović, [[Frank Wilczek]], [[Anthony Zee]], _Reflections on mirror fermions_,  Physics Letters B Volume 141, Issues 5–6, 5 July 1984, Pages 389-394 Physics Letters B (doi:10.1016/0370-2693(84)90269-7)

On [[Hopf-Wess-Zumino terms]]:

* {#WilczekZee84} [[Frank Wilczek]], [[Anthony Zee]], _Linking Numbers, Spin, and Statistics of Solitons_, Phys. Rev. Lett. 51, 2250, 1983 ([doi:10.1103/PhysRevLett.51.2250](https://doi.org/10.1103/PhysRevLett.51.2250))

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

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

category: people


anti D-brane

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* table of contents
{:toc}

## Idea

An _anti-D-brane_ is the higher dimensional analog for [[D-branes]] of what [[antiparticles]] are for [[fundamental particles]].

In [[perturbative string theory]] the strings stretching between a D-brane and an anti-D-brane have a [[tachyon]] mode. The analog of [[Sen's conjecture]] for this case is the statement that the tachyon potential energy is precisely the energy density of the brane and that the [[condensation]] of the tachyon mode witnesses the annihiliation of the brane/anti-brane pair. ([Sen 98](#Sen98))

## Relation to K-theory

In general there are $n$ D-branes and $n'$ anti D-branes coinciding, carrying
[[nLab:Chan-Paton gauge fields]] $V_{brane}$ (of [[rank]] $n$) and $V_{\text{anti-brane}}$, respectively,
yielding a pair of [[vector bundles]]

$$
  (V_{\text{brane}}, V_{\text{anti-brane}})
  \,.
$$

Such pairs are also called [[nLab:virtual vector bundles]].

Now branes annihilate with anti-branes if they have exact opposite [[nLab:D-brane charge]], which here
means that they carry the same [[nLab:Chan-Paton gauge field|Chan-Paton]] vector bundle. In other words, pairs as above of the special form
$(W,W)$ are equivalent to pairs of the form $(0,0)$.

$$
  (W,W) \sim 0
  \,.
$$

More generally, since there is arbitrary brane/anti-brane pair creation/annihilation, the actual net Chan-Paton charge
of coincident branes and anti-branes is the [[nLab:equivalence class]] of $(V_{\text{brane}}, V_{\text{anti-brane}})$
under the [[nLab:equivalence relation]] which is generated by the relation

$$
  (V_{\text{brane}} \oplus W, V_{\text{anti-brane}} \oplus W)
    \;\sim\;
  (V_{brane}, V_{anti-brane})
$$

for all [[nLab:complex vector bundles]] $W$ ([Witten 98, Section 3](#Witten98)).

For a fixed brane [[nLab:worldvolume]] $X$, the additive [[abelian group]] of such equivalence classes of [[nLab:virtual vector bundles]] is called the [[nLab:topological K-theory]] of $X$, denoted $K(X)$.

This is one of the arguments which suggest that the true home of the gauge field on multiple D-branes is in [[nLab:generalized cohomology theory]] called [[nLab:topological K-theory]].  It follows that also the [[RR-fields]] are in K-theory ([Moore-Witten 00](#MooreWitten99)).

## References

### Anti D-branes

The version of [[Sen's conjecture]] for brane/anti-brane annihilation is due to

* {#Sen98} [[Ashoke Sen]], _Tachyon Condensation on the Brane Antibrane System_, JHEP 9808:012,1998 ([arXiv:hep-th/9805170](https://arxiv.org/abs/hep-th/9805170))

Textbook accounts on anti-D-branes include

* [[Koji Hashimoto]], section 5.3.1 of _D-brane_, Springer 2012

The relation between brane/anti-brane annihilation and the [[topological K-theory]] nature of [[D-brane charge]] is due to

* {#Witten98} [[Edward Witten]], section 3 of _D-Branes And K-Theory_, JHEP 9812:019,1998 ([arXiv:hep-th/9810188](http://arxiv.org/abs/hep-th/9810188))

and the argument that this implies that also the [[RR-fields]] are in K-theory is due to

* {#MooreWitten99} [[Gregory Moore]], [[Edward Witten]], p. 6 of _Self-Duality, Ramond-Ramond Fields, and K-Theory_, JHEP 0005:032 (2000) ([arXiv:hep-th/9912279](https://arxiv.org/abs/hep-th/9912279))

Review of this is in

* [[Edward Witten]], _Overview Of K-Theory Applied To Strings_,  	Int.J.Mod.Phys.A16:693-706,2001 ([arXiv:hep-th/0007175](https://arxiv.org/abs/hep-th/0007175))

### Anti M-branes

Similarly for lifts to [[M-branes]]:

anti-[[M2-branes]]:

* [[Mohammad Garousi]], _A proposal for M2-brane-anti-M2-brane action_, Phys. Lett.B686:59-63, 2010 ([arXiv:0809.0381](https://arxiv.org/abs/0809.0381))

...

anti-[[M5-branes]]:

* [[Seiji Terashima]], footnote 2 on section 4 of _On M5-branes in N=6 Membrane Action_, JHEP0808:080,2008 ([arXiv:0807.0197](https://arxiv.org/abs/0807.0197))

[[!redirects anti D-branes]]

[[!redirects anti-D-brane]]
[[!redirects anti-D-branes]]

[[!redirects anti M-brane]]
[[!redirects anti M-branes]]

[[!redirects anti p-brane]]
[[!redirects anti p-branes]]

[[!redirects anti-brane]]
[[!redirects anti-branes]]

[[!redirects anti brane]]
[[!redirects anti branes]]


anti de Sitter group

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=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

The _anti de Sitter group_ is the [[isometry group]] $O(d-1,2)$ of $d$-dimensional [[anti de Sitter spacetime]]. ([[orthogonal group]] for [[signature]] $(d-1,2)$)

This is [[analogy|analogous]] to how the [[Poincare group]] is the isometry group of [[Minkowski spacetime]].

The connected component $SO(d-1,2)$ of the anti de Sitter group is [[isomorphism|isomorphic]] to the connected component of the [[conformal group]] of $\mathbb{R}^{d-2,1}$. This is the basis of the [[AdS-CFT correspondence]].

## Properties

### Exceptional isomorphisms

* $SO(6,2) \simeq SO(4,\mathbb{H})$ (where $\mathbb{H}$ is the [[quaternions]])

## Related concepts

* [[anti de Sitter spacetime]]

* [[singleton representation]]

[[!include table of orthogonal groups and related]]

[[!include local and global geometry - table]]

## References
 {#References}

The anti de Sitter [[Lie algebra]] is discussed for instance in

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], volume 1, chapter I.3.8 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

The [[representation theory]] and its [[Inönü-Wigner contraction]] to that of the [[Poincaré group]] is discussed in

* [[Jouko Mickelsson]], J. Niederle, _Contractions of Representations of de Sitter Groups_, Comm. Math. Phys. Volume 27, Number 3 (1972), 167-180. ([Euclid](http://projecteuclid.org/euclid.cmp/1103858248))

* Mauricio Ayala, Richard Haase, _Group contractions and its consequences upon representations of different spatial symmetry groups_ ([arXiv:hep-th/0206037](https://arxiv.org/abs/hep-th/0206037))

* Francisco J. Herranz, Mariano Santander, section 4 of _(Anti)de Sitter/Poincare symmetries and representations from Poincare/Galilei through a classical deformation approach_, J.Phys.A41:015204,2008 ([arXiv:math-ph/0612059](https://arxiv.org/abs/math-ph/0612059))

[[!redirects AdS group]]

[[!redirects anti de Sitter Lie algebra]]
[[!redirects AdS Lie algebra]]


anti de Sitter space > history

see [[anti de Sitter spacetime]]


anti de Sitter spacetime

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#### Gravity
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=--
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#Contents#
* table of contents
{: toc}

## Definition

Up to [[isometry]], the **anti de Sitter spacetime** of [[dimension]] $d$, $AdS_d$, is the [[pseudo-Riemannian manifold]] whose underlying [[manifold]] is the [[submanifold]] of the [[Minkowski spacetime]] $\mathbb{R}^{d-1,2}$ that solves the equation

$$
  \sum_{i = 1}^{d-1} (x_i)^2 - (x_d)^ 2 - (x_0)^2 = -R^2
$$

for some $R \neq 0$ (the "radius" of the spacetime) and equipped with the metric induced from the ambient metric, where $\{x^0, x^1, x^2, \cdots, x^d\}$ denote the canonical [[coordinates]]. $AdS_d$ is [[homeomorphic]] to $\mathbb{R}^{d-1} \times S^1$, and its isometry group is $O(d-1, 2)$.

More generally, one may define the anti de Sitter space of signature $(p,q)$ as isometrically embedded in the space $\mathbb{R}^{p,q+1}$ with coordinates $(x_1, ..., x_p, t_1, \ldots, t_{q+1})$ as the sphere $\sum_{i=1}^p x_i^2 - \sum_{j=1}^{q+1} t_j^2 = -R^2$.

> graphics grabbed from [Yan 19](holographic+entanglement+entropy#Yan19)

## Properties

### Coordinate charts

(...)

in _horospheric coordinates_ the AdS [[metric tensor]] is

$$
  g_{AdS}
  \;=\;
  \frac{1}{z^2}
  \left(
    g_{(\mathbb{R}^{p,1})} + (d z)^2
  \right)
$$

In terms of

$$
  y \coloneqq 1/z
$$

this becomes

$$
  g_{AdS}
  \;=\;
  y^2 \, g_{(\mathbb{R}^{p,1})} + \frac{1}{y^2}(d y)^2
$$

and with

$$
  y \coloneqq \tfrac{1}{n} r^n
$$

for $n \neq 0$

we get

$$
  g_{AdS}
  \;=\;
  \tfrac{1}{n^2}r^{2n} \, g_{(\mathbb{R}^{p,1})} + \frac{1}{r^2}(d r)^2
$$

### Conformal boundary

(...)

### Holography

Asymptotically anti-de Sitter spaces play a central role in the realization of the [[holographic principle]] by [[AdS/CFT correspondence]].

### In $p$-adic geometry

A [[p-adic geometry|2-adic]] [[arithmetic geometry]]-version of [[AdS spacetime]] is identified with the [[Bruhat-Tits tree]] for the [[projective general linear group]] $PGL(2,\mathbb{Q}_p)$:

> graphics from [Casselman 14](Bruhat-Tits+tree#Casselman14)

In the [[p-adic AdS/CFT correspondence]] this may be regarded (at some finite depth truncation) as a [[tensor network state]]:

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

and as such validates the [[Ryu-Takayanagi formula]] for [[holographic entanglement entropy]].

## Related concepts

* [[thermal AdS3]]

* [[anti de Sitter group]]

* [[de Sitter spacetime]]

* [[super anti de Sitter spacetime]]

* [[near-horizon geometry]]

* [[singleton representation]]

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

## References

### General

Reviews:

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], volume 1, chapter I.3.8 of: _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[Ingemar Bengtsson]], _Anti-de Sitter space_, lecture notes 1998 ([[Bengtsson98.pdf:file]])

* [[Matthias Blau]], chapter 38 of: _Lecture notes on general relativity_ ([web](http://www.blau.itp.unibe.ch/GRLecturenotes.html))

* [[Gary Gibbons]], _Anti-de-Sitter spacetime and its uses_ ([arXiv:1110.1206](http://arxiv.org/abs/1110.1206))

* {#Natsuume15} [[Makoto Natsuume]], section 6 of: _AdS/CFT Duality User Guide_, Lecture Notes in Physics 903, Springer 2015 ([arXiv:1409.3575](https://arxiv.org/abs/1409.3575))

* Leszek M. Sokolowski, _The bizarre anti-de Sitter spacetime_, International Journal of Geometric Methods in Modern Physics 13 no.9 (2016) 1630016 ([arXiv:1611.01118](https://arxiv.org/abs/1611.01118))

See also:

* Wikipedia, _[anti de Sitter space](http://en.wikipedia.org/wiki/Anti_de_Sitter_space)_

Further discussion:

* Abdelghani Zeghib, _On closed anti de Sitter spacetimes_, Math. Ann. 310, 695–716 (1998) ([pdf](http://www.umpa.ens-lyon.fr/~zeghib/Anti.de.Sitter.pdf))

* C. Frances, _The conformal boundary of anti-de Sitter space-times_, in  _AdS/CFT correspondence: Einstein metrics and their conformal boundaries_ , 205--216, IRMA Lect. Math. Theor. Phys., 8, Eur. Math. Soc., Zürich, 2005 ([pdf](http://mahery.math.u-psud.fr/~frances/ads-cft2.pdf))

* Jiri Podolsky, Ondrej Hruska, _Yet another family of diagonal metrics for de Sitter and anti-de Sitter spacetimes_, Phys. Rev. D 95, 124052 (2017) ([arXiv:1703.01367](https://arxiv.org/abs/1703.01367))

Discussion of thermal [[Wick rotation]] on global [[anti-de Sitter spacetime]]  (which is already periodic in _real_ time) to [[Euclidean field theory]] with periodic _imaginary_ time is in

* {#AllenFolacciGibbons87} B. Allen, A. Folacci, [[Gary Gibbons]], _Anti-de Sitter space at finite temperature_, Physics Letters B Volume 189, Issue 3, 7 May 1987, Pages 304-310 (doi:10.1016/0370-2693(87)91437-7)

Discussion of [[black holes in anti de Sitter spacetime]]:

* Hawking, Stephen W., and Don N. Page. "Thermodynamics of black holes in anti-de Sitter space." Communications in Mathematical Physics 87.4 (1983): 577-588.

* {#Socolovsky17} M. Socolovsky, _Schwarzschild Black Hole in Anti-De Sitter Space_ ([arXiv:1711.02744](https://arxiv.org/abs/1711.02744))

* Peng Zhao, _Black Holes in Anti-de Sitter Spacetime_ ([pdf](https://pdfs.semanticscholar.org/9939/335e0d282e70201d6087988bf4f5aedfc8f6.pdf))

* Jakob Gath, _The role of black holes in the AdS/CFT correspondence_ ([pdf](https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/dissertations/2009/Jakob-Gath-Dissertation-2009.pdf))

Relation to [[Teichmüller theory]]:

* Francesco Bonsante, Andrea Seppi, _Anti-de Sitter geometry and Teichmüller theory_ ([arXiv:2004.14414](https://arxiv.org/abs/2004.14414))

### Phenomenology

* {#SenAdilSen21} Anjan A. Sen, Shahnawaz A. Adil, Somasri Sen, *Do cosmological observations allow a negative $\Lambda$?* ([arXiv:2112.10641](https://arxiv.org/abs/2112.10641))

### As string vacua

On (in-)stability of non-[[supersymmetry|supersymmetric]] [[AdS spacetime|AdS]] [[string theory vacua|vacua in string theory]]:

* [[Iosif Bena]], [[Krzysztof Pilch]], [[Nicholas Warner]], _Brane-Jet Instabilities_,  J. High Energ. Phys. 2020, 91 (2020) ([arXiv:2003.02851](https://arxiv.org/abs/2003.02851))

[[!include pp-waves as Penrose limits of AdS spacetimes -- references]]

[[!redirects anti de Sitter space]]
[[!redirects anti de Sitter spaces]]
[[!redirects anti-de Sitter space]]
[[!redirects anti-de Sitter spaces]]
[[!redirects anti de Sitter spacetime]]
[[!redirects anti de Sitter spacetimes]]
[[!redirects anti-de Sitter spacetime]]
[[!redirects anti-de Sitter spacetimes]]

[[!redirects anti de-Sitter spacetime]]

[[!redirects AdS-spacetime]]
[[!redirects AdS-spacetimes]]

[[!redirects horospheric coordinates]]
[[!redirects horospheric coordinate chart]]
[[!redirects horospheric coordinate charts]]

[[!redirects AdS spacetime]]
[[!redirects AdS spacetimes]]

[[!redirects AdS]]


anti-commutator

#Contents#
* table of contents
{:toc}

## Definition

### In rings and algebras

For $A$ a [[ring]] or [[associative algebra]], the **anti-commutator** of two elements $x,y \in A$ is the element

$$
  \{x,y\} \coloneqq x y + y x
  \,.
$$

## Related concepts

* [[commutator]]

[[!redirects anti-commutators]]

[[!redirects anticommutator]]
[[!redirects anticommutators]]


anti-cyclotomic field

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=--
=--
=--

## Idea

A [[field extension]] with [[Galois group]] a [[dihedral group]].

## Related concepts

* [[field extension]], [[Galois group]]

* [[cyclotomic extension]]

* [[dihedral group]]

## Referenes

* [[Groupprops]], _[Galois extensions for dihedral groups:D8](http://groupprops.subwiki.org/wiki/Galois_extensions_for_dihedral_group:D8)_

* Hirotada Naito, _Dihedral extensions of degree 8 over the rational p-adic fields_, Proc. Japan Acad. Ser. A Math. Sci. Volume 71, Number 1 (1995), 17-18. ([Euclid](http://projecteuclid.org/euclid.pja/1195510841))

* David Brink, _Prime decomposition in the anti-cyclotomic extension_, Mathematics of Computation, vol 76, number 260 (2007) ([pdf](http://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-01964-3/S0025-5718-07-01964-3.pdf))

[[!redirects anti-cyclotomic fields]]

[[!redirects anti-cyclotomic extension]]
[[!redirects anti-cyclotomic extensions]]

[[!redirects anticyclotomic extension]]
[[!redirects anticyclotomic extensions]]

[[!redirects anti-cyclotomic field extension]]
[[!redirects anti-cyclotomic field extensions]]