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André Henriques | https://ncatlab.org/nlab/source/Andr%C3%A9+Henriques | * [personal website](http://andreghenriques.com/)
* [university website](https://www.maths.ox.ac.uk/people/andre.henriques)
## Selected writings:
On [[tmf]]:
* [[Christopher Douglas]], [[John Francis]], [[André Henriques]], [[Michael Hill]], _Topological Modular Forms_, Mathematical Surveys and Monographs Volume 201, AMS 2014 ([ISBN:978-1-4704-1884-7](https://bookstore.ams.org/surv-201))
On [[2d CFT]] as an [[functorial quantum field theory]] on a [[(infinity,n)-category of cobordisms|2-category of complex cobordisms]]:
* [[André Henriques]], *The complex cobordism 2-category*, 2021 ([video](http://andreghenriques.com/ComplexCob2CatandCentralExt.mp4))
## Related $n$Lab-entries
* [[Lie integration]]
* [[string Lie 2-algebra]], [[string Lie 2-group]]
* [[conformal net]]
* [[Chern-Simons theory]]
* [[orbispace]]
category: people
[[!redirects Andre Henriques]]
|
André Hirschowitz | https://ncatlab.org/nlab/source/Andr%C3%A9+Hirschowitz |
* [webpage](http://math.unice.fr/~ah/)
category: people
[[!redirects Andre Hirschowitz]] |
André Joyal | https://ncatlab.org/nlab/source/Andr%C3%A9+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 [<a href="https://doi.org/10.1016/S0022-4049(98)00164-9">doi:10.1016/S0022-4049(98)00164-9</a>]
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) [<a href="https://doi.org/10.1016/S0022-4049(02)00135-4">doi:10.1016/S0022-4049(02)00135-4</a>]
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 | https://ncatlab.org/nlab/source/Andr%C3%A9+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 | https://ncatlab.org/nlab/source/Andr%C3%A9+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 [<a href="https://doi.org/10.1016/0003-4916(78)90224-5">doi:10.1016/0003-4916(78)90224-5</a>]
* {#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 [<a href="https://doi.org/10.1016/0003-4916(78)90225-7">doi:10.1016/0003-4916(78)90225-7</a>]
category: people
[[!redirects Andre Lichnerowicz]]
|
André Miemiec | https://ncatlab.org/nlab/source/Andr%C3%A9+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 | https://ncatlab.org/nlab/source/Andr%C3%A9+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) (<a href="https://doi.org/10.1016/0550-3213(71)90448-2">doi:10.1016/0550-3213(71)90448-2</a>)
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 [<a href="https://doi.org/10.1016/0550-3213(82)90566-1">doi:10.1016/0550-3213(82)90566-1</a>]
* [[Jean-Loup Gervais]], [[André Neveu]], *Dual string spectrum in Polyakov's quantization (II). Mode separation*, Nuclear Physics B **209** 1 (1982) 125-145 [<a href="https://doi.org/10.1016/0550-3213(82)90105-5">doi:10.1016/0550-3213(82)90105-5</a>]
## Related entries
* [[spinning string]]
* [[supersymmetry]]
category: people
[[!redirects Andre Neveu]] |
André Néron | https://ncatlab.org/nlab/source/Andr%C3%A9+N%C3%A9ron |
* [Wikipedia entry](https://en.wikipedia.org/wiki/André_Néron)
## related $n$Lab entries
* [[elliptic fibration]]
category: people
|
André Petermann | https://ncatlab.org/nlab/source/Andr%C3%A9+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 | https://ncatlab.org/nlab/source/Andr%C3%A9+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 | https://ncatlab.org/nlab/source/Andr%C3%A9-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 | https://ncatlab.org/nlab/source/Andr%C3%A9s+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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/Angel+Toledo | PhD student, interested in (NC)DAG.
[website](http://math.unice.fr/~toledo) |
Angelo Vistoli | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/angle |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Trigonometry
+-- {: .hide}
[[!include trigonometry -- contents]]
=--
=--
=--
## 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 | https://ncatlab.org/nlab/source/angular+momentum |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#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 | https://ncatlab.org/nlab/source/angular+velocity |
+-- {: .rightHandSide}
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###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#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 | https://ncatlab.org/nlab/source/Ang%C3%A9lica+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 | https://ncatlab.org/nlab/source/Ang%C3%A9lica+Osorno |
* [webpage](http://people.reed.edu/~aosorno/)
category: people
|
Anibal M. Medina-Mardones | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 [<a href="https://doi.org/10.1016/S0022-4049(02)00169-X">doi:10.1016/S0022-4049(02)00169-X</a>]
category: people |
Anna Marie Bohmann | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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) [<a href="https://doi.org/10.1016/S0049-237X(08)71127-3">doi:10.1016/S0049-237X(08)71127-3</a>]
* [[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 | https://ncatlab.org/nlab/source/Annette+Huber | * [Web](http://home.mathematik.uni-freiburg.de/arithgeom/huber.html)
* [[motives]] |
annulus | https://ncatlab.org/nlab/source/annulus |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###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 | https://ncatlab.org/nlab/source/anodyne+morphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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) <a href="http://pmc.polytechnique.fr/pagesperso/dg/cours/biblio/PhysRep%20195,%20127%20(1990)%20Bouchaud,%20Georges%20%5BAnomalous%20diffusion%20in%20disordered%20media%20-%20Statistical%20mechanisms,%20models%20and%20physical%20applications%5D.pdf">pdf</a>
* 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 <a href="http://dx.doi.org/10.1016/S0370-1573(00)00070-3">doi</a>
* 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 | https://ncatlab.org/nlab/source/anomalous+magnetic+moment |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### 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}
<img src="https://ncatlab.org/nlab/files/LyonsGMinus2DetectionThreshold.jpg" width="600">
\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]]:
<center>
<img src="https://ncatlab.org/nlab/files/CrivellinRandAHints.jpg" width="300">
</center>
> 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:
<img src="https://ncatlab.org/nlab/files/GravityVertexCorrectionsForQED.png" width="750">
### 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 (<a href="https://doi.org/10.1016/0370-2693(75)90608-5">doi:10.1016/0370-2693(75)90608-5</a>)
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) (<a href="https://doi.org/10.1007/978-3-540-73518-2_2">doi:10.1007/978-3-540-73518-2_2</a>)
* {#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), <a href="https://doi.org/10.1007/JHEP12(2017)044">doi:10.1007/JHEP12(2017)044</a>)
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 | https://ncatlab.org/nlab/source/anomaly+cancellation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Qunantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#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), <a href="http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf">pdf</a>)
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 [<a href="https://doi.org/10.1016/0550-3213(84)90506-6">doi:10.1016/0550-3213(84)90506-6</a>]
### 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 | https://ncatlab.org/nlab/source/Anonymous+Hero+%3E+history | < [[Anonymous Hero]]
[[!redirects Anonymous Hero -- history]] |
Anonymous User | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/Another+page+%3E+history | < [[Another page]]
[[!redirects Another page -- history]] |
Ansten Klev | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/Anthony+Licata |
* [webpage](http://math.stanford.edu/~amlicata/Anthony_Licata_homepage/Home.html)
category: people |
Anthony Morse | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 (<a href="https://doi.org/10.1016/0370-2693(87)91489-4">doi:10.1016/0370-2693(87)91489-4</a>)
* {#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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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)) $[$<a href="https://doi.org/10.1016/0550-3213(85)90252-4">doi:10.1016/0550-3213(85)90252-4</a>$]$
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 (<a href="https://doi.org/10.1016/0370-2693(84)90269-7">doi:10.1016/0370-2693(84)90269-7</a>)
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 | https://ncatlab.org/nlab/source/anti+D-brane |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* 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 | https://ncatlab.org/nlab/source/anti+de+Sitter+group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#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 | https://ncatlab.org/nlab/source/anti+de+Sitter+space+%3E+history |
see [[anti de Sitter spacetime]] |
anti de Sitter spacetime | https://ncatlab.org/nlab/source/anti+de+Sitter+spacetime |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Riemannian geometry
+-- {: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Gravity
+-- {: .hide}
[[!include gravity contents]]
=--
=--
=--
#Contents#
* table of contents
{: toc}
<div style="float:right;margin:0 10px 10px 0;">
<img src="https://ncatlab.org/nlab/files/AdSSlicingByHyperbolic.jpg" width="400">
</div>
## 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]] $PGL(2,\mathbb{Q}_p)$:
<center>
<img src="https://ncatlab.org/nlab/files/BruhatTitsTreeOfSL2.jpg" width="600">
</center>
> 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]]:
<center>
<img src="https://ncatlab.org/nlab/files/BruhatTitsTreeTensorNetworkStateFromMetricLie.jpg" width="300">
</center>
> 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 (<a href="https://doi.org/10.1016/0370-2693(87)91437-7">doi:10.1016/0370-2693(87)91437-7</a>)
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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/anti-cyclotomic+field |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
## 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]] |
anti-dual linear space | https://ncatlab.org/nlab/source/anti-dual+linear+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include higher linear algebra - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
If the [[ground ring]] ([[ground field]]) is equipped with a [[star-ring|star structure]] (an [[anti-involution]], such as [[complex conjugation]] in the complex numbers), then the operation of forming [[dual linear spaces]] may be "twisted" by this involution to yield a notion of "anti-dual" spaces. These relate to [[Hermitian forms]] as ordinary dual spaces relate to ordinary [[inner products]].
## Definition
Let $\mathbb{K}$ be a [[star-ring]], hence a [[unital ring]] (often a [[field]], whence our notation, and here often the [[complex numbers]]) equipped with an [[anti-involution]] (for instance [[complex conjugation]]):
$$
\overline{(-)} \,\colon\, \mathbb{K} \to \mathbb{K}
$$
$$
\overline{x \cdot y}
\,=\,
\overline{y} \cdot \overline{x}
\,.
$$
Consider a $\mathbb{K}$-[[module]] $\mathscr{V} \,\in\, \mathbb{K}Mod$ (say *right* modules for possibly non-commutative $\mathbb{K}$, but being just $\mathbb{K}$-[[vector spaces]] if $\mathbb{K}$ is a [[field]]).
\begin{definition}
The *anti-dual* space $\overline{\mathscr{V}^\ast}$ of $\mathscr{V}$ is the space of [[anti-linear maps]] to the [[ground ring]] ([[ground field]]):
$$
\overline{\mathscr{V}^\ast}
\;\coloneqq\;
\Big\{
\phi \,\colon\,
\mathscr{V} \to \mathbb{K}
\,\Big\vert\,
\underset{v, v' \in \mathscr{V}}{\forall}
\,
\phi(v + v') = \phi(v) + \phi(v')
,\;
\underset{
{ v \in \mathscr{V} }
\atop
{ \lambda \in \mathbb{K} }
}{\forall}
\,
\phi(v \cdot \lambda) \,=\, \overline{\lambda} \cdot v
\Big\}
$$
equipped itself with the structure of a (right) module by
$$
\phi \in \overline{\mathscr{V}^\ast}
,\,
\,\lambda \in \mathbb{K}
\;\;\;
\vdash
\;\;\;
(\phi \cdot \lambda)(c)
\;=\;
\phi(v) \cdot \lambda
\,.
$$
\end{definition}
(eg. [Karoubi & Villamayor 1973, Ex. 1](KaroubiVillamayor73); [Karoubi 2010, §1](#Karoubi10))
As the notation already indicates, an equivalent definition is:
\begin{definition}
The anti-dual $\overline{\mathscr{V}^\ast}$ has as [[underlying]] [[abelian group]] the ordinary [[linear dual ]] $\mathscr{V}^\ast$ but equipped with its anti-linear structure, namely with the (right) module action twisted by the involution as:
$$
\phi \,\in\, Hom_{\mathbb{K}}(\mathscr{V}, \mathbb{K})
\;\;\;
\vdash
\;\;\;
\underset{v \in \mathscr{V}}{\forall}
\,
(\phi \cdot \lambda)(v)
\;=\;
\overline{\lambda} \cdot \phi(v)
\,.
$$
\end{definition}
(eg. [Mishchenko 1976 §1.1](#Mishchenko76)).
An [[isomorphism]] between the two definitions
$$
\array{
\mathscr{V}^\ast
&\longrightarrow&
\mathscr{V}^\ast
\\
\phi &\mapsto& \overline{\phi}
}
$$
is established by postcomposing linear forms with the star-involution
$$
\overline{\phi}
\,\colon\,
v \,\mapsto\, \overline{\phi(v)}
$$
which respects the above (right) $\mathbb{K}$-actions due to
$$
\array{
\big(
v \,\mapsto\,
\phi(v)\cdot \lambda
\big)
&\mapsto&
\big(
v
\,\mapsto\,
\overline{\phi(v) \cdot \lambda}
\big)
\\
& = &
\big(
v
\,\mapsto\,
\overline{\lambda} \cdot \overline{\phi(v)}
\big)
\mathrlap{\,.}
}
$$
## Related concepts
* [[hyperbolic functor]]
## References
Discussion in the context of [[Hermitian K-theory]]:
* {#KaroubiVillamayor73} [[Max Karoubi]], [[Orlando Villamayor]], Ex. 1 in: *K-théorie algébrique et K-théorie topologique II*, Math. Scand. **32** (1973) 57-86 [[jstor:24490565](https://www.jstor.org/stable/24490565)]
* {#Mishchenko76} [[Alexandr S. Mishchenko]], §1.1 (pp. 76) in: *Hermitian K-Theory. The Theory of characteristic classes and methods of functional analysis*, Uspeki Mat. Nauk **31** 2 (1976) 69-134, Russ. Math. Surv. **31** 71 (1976) [[doi:10.1070/RM1976v031n02ABEH001478](https://iopscience.iop.org/article/10.1070/RM1976v031n02ABEH001478), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/mishch5.pdf)]
* {#Karoubi10} [[Max Karoubi]], §1 in: *Le théorème de périodicité en K-théorie hermitienne*, Quanta of Maths **1**, AMS and Clay Math Institute Publications (2010) [[arXiv:0810.4707](https://arxiv.org/abs/0810.4707)]
See also:
* Wikipedia, *[Anti-dual space](https://en.wikipedia.org/wiki/Antilinear_map#Anti-dual_space)*
[[!redirects anti-dual linear spaces]]
[[!redirects anti-dual vector space]]
[[!redirects anti-dual vector spaces]]
|
anti-ghost field | https://ncatlab.org/nlab/source/anti-ghost+field |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[BV-BRST formalism]], for [[gauge fixing]] [[Yang-Mills theory]] (to [[Lorenz gauge]] or similar) a [[contractible chain complex]] of [[auxiliary field|auxiliary]] [[field bundles]] is introduced for two [[Lie algebra]]-valued [[field (physics)|fields]], one in degree zero, called the _[[Nakanishi-Lautrup field]]_, usually denoted "$B$" and one in degree -1, called the _antighost field_, usually denoted $\overline{C}$.
Beware that there are also the [[antifields]] of the [[ghost fields]], which technically are hence "anti-ghostfields" as opposed to the Nakanishi-Lautrup "antighost-fields".
Whoever is responsible for this bad terminology should be blamed.
## Related concepts
* [[field (physics)]]
* [[ghost]],
* [[antifield]], **antighost**
## References
Review for the case of [[electromagnetism]] and with [[path integral]] terminology is in
* {#Henneaux90} [[Marc Henneaux]], section 9.1 of _Lectures on the Antifield-BRST formalism for gauge theories_, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 ([pdf](http://www.math.uni-hamburg.de/home/schweigert/ws07/henneaux2.pdf))
while discussion for general [[Yang-Mills theory]] in the context of [[causal perturbation theory]]/[[perturbative algebraic quantum field theory]] is in
* {#Rejzner16} [[Katarzyna Rejzner]], section 7.2 of _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))
[[!redirects antighost]]
[[!redirects antighosts]]
[[!redirects anti-ghost fields]]
[[!redirects antighost field]]
[[!redirects antighost fields]]
|
anti-ideal predicate | https://ncatlab.org/nlab/source/anti-ideal+predicate |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
\tableofcontents
## Idea
In [[material set theory]], an [[anti-ideal]] of a [[commutative ring]] $R$ with an [[apartness relation]] $\#$ is an $\#$-[[open subset]] $I \subseteq R$ which satisfies
* $\neg(0 \in I)$
* for all $x \in R$ and $y \in R$, $x + y \in I$ implies that $x \in I$ or $y \in I$
* for all $x \in R$ and $y \in R$, $x \in I$ and $y \in I$ implies that $x \cdot y \in I$
In [[dependently sorted set theory]], where membership $x \in S$ is not a [[relation]], the above statement that $x \in I$ for every element $x \in R$ in material set theory is equivalently a predicate in the logic $x \in R \vdash P_I(x)$. The given [[anti-ideal]] $I$ is then defined by [[restricted separation]] as the set $I \coloneqq \{x \in R \vert P_I(x)\}$, which in structural set theory automatically comes with an [[injection]]
$$i:\{x \in R \vert P_I(x)\} \hookrightarrow R$$
such that
$$\exists y \in \{x \in R \vert P_I(x)\}.x = i(y) \iff P(x)$$
Hence, the notion of [[anti-ideal]] predicate, a formulation of the notion of anti-ideal as a predicate rather than a subset.
## Definition
Given a commutative ring $R$ with an [[apartness relation]] $\#$, an **anti-ideal predicate** on $R$ is an $\#$-[[apartness-open predicate|open predicate]] $x \in R \vdash P_I(x)$ which satisfies
* $\neg P_I(0)$
* for all $x \in R$ and $y \in R$, $P_I(x + y)$ implies that $P_I(x)$ or $P_I(y)$
* for all $x \in R$ and $y \in R$, $P_I(x)$ and $P_I(y)$ implies that $P_I(x \cdot y)$
The **[[anti-ideal]]** $I$ is then defined by [[restricted separation]] as
$I \coloneqq \{x \in R \vert P_I(x)\}$
## Examples
The various definitions of anti-ideals translate over from material set theory to anti-ideal predicates in dependently sorted structural set theory by replacing $x \in I$ with $P_I(x)$ throughout the definition:
* A **[[proper anti-ideal]] predicate** on a commutative ring $R$ with apartness relation $\#$ is an anti-ideal predicate $P_I$ where $P_I(1)$ is true.
* A **[[anti-prime anti-ideal]] predicate** on a commutative ring $R$ with apartness relation $\#$ is a proper anti-ideal predicate $P_I$ where for all $x \in R$ and $y \in R$, $P_I(x)$ and $P_I(y)$ implies that $P_I(x \cdot y)$.
$$\forall x \in R.\forall y \in R.P_I(x) \wedge P_I(y) \implies P_I(x \cdot y)$$
* A **[[principal anti-ideal]] predicate** on a commutative ring $R$ with apartness relation $\#$ anti-generated by an element $a \in R$ is an ideal predicate $P_I$ where for all $x \in R$ and for all $y \in R$, $P_I(x)$ implies that $x \# a \cdot y$.
$$\forall x \in R.\forall y \in R.P_I(x) \implies x \# a \cdot y$$
## See also
* [[anti-ideal]]
* [[predicate]]
* [[ideal predicate]]
* [[restricted separation]]
* [[antisubalgebra]]
[[!redirects anti-ideal predicate]]
[[!redirects anti-ideal predicates]] |
anti-reduced type | https://ncatlab.org/nlab/source/anti-reduced+type | [[!redirects anti-reduced object]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesion
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _anti-reduced object_ or simple _infinitesimal type_ is one whose [[reduced object|reduction]] is the point, hence one consisting entirely of "[[infinitesimal object|infinitesimal]] extension", i.e. an [[infinitesimally thickened point]].
## Definition
In the context of [[differential cohesion]], an anti-reduced obect is an [[comodal type]] $X$ for the [[infinitesimal shape modality]] $\Im$
$$
\Im(X) \simeq \ast
\,.
$$
## Examples
### Formal moduli problems
In [[homotopy type theory]]/[[higher topos theory]] anti-reduced types are essentially what is also called "[[formal moduli problems]]" (these are typically required to satisfy one more condition besides being anti-reduced, namely being [[cohesive (∞,1)-presheaf on E-∞ rings|infinitesimally cohesive]] in the sense of Lurie).
## Related concepts
[[!include cohesion - table]]
[[!redirects anti-reduced obects]]
[[!redirects infinitesimal type]]
[[!redirects infinitesimal types]]
|
antibracket | https://ncatlab.org/nlab/source/antibracket |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[BV-BRST formalism]] the _antibracket_ is a canonical product operation on an [[associative algebra]] generated by [[field (physics)|fields]] and [[antifields]]. If antifields are regarded as [[vector fields]] on the space of fields, then the antibracket is just the (graded) _[[Schouten bracket]]_.
There are different incarnations of the antibracket associated with different incarnations of the algebra of fields/antifields:
Applied to [[horizontal differential forms]] on the [[jet bundle]] of the [[field bundle]] this refines to the _[[local antibracket]]_.
By [[transgression of variational differential forms]] this yields the ("global") antibracket on [[polynomial observables]] of a [[Lagrangian field theory]].
For details see at _[[A first idea of quantum field theory]]_ the chapter _[Reduced phase space](A+first+idea+of+quantum+field+theory#ReducedPhaseSpace)_ for the antibracket before [[quantization]], and the chapter _[Free quantum fields](A+first+idea+of+quantum+field+theory#FreeQuantumFields)_ for the ([[time-ordered product|time-ordered]]) antibracket after quantization.
The global [[antibracket]] is closely related to the _[[BV-operator]]_. See there fore more.
## Related concepts
* [[BV-operator]]
## References
Review includes
* {#Henneaux90} [[Marc Henneaux]], section 7.3 of _Lectures on the Antifield-BRST formalism for gauge theories_, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 ([pdf](http://www.math.uni-hamburg.de/home/schweigert/ws07/henneaux2.pdf))
* {#HenneauxTeitelboim92} [[Marc Henneaux]], [[Claudio Teitelboim]], section 15.5.2 of _[[Quantization of Gauge Systems]]_, Princeton University Press 1992.
* {#GomisParisSamuel94} [[Joaquim Gomis]], Jordi Paris, Stuart Samuel, section 4.2 of _Antibracket, Antifields and Gauge-Theory Quantization_, Phys. Rept. 259 (1995) 1-145 ([arXiv:hep-th/9412228](https://arxiv.org/abs/hep-th/9412228))
* {#CattaneoRossi01} [[Alberto Cattaneo]], Carlo Rossi, section 4.2 of _Higher-dimensional BF theories in the Batalin-Vilkovisky formalism: The BV action and generalized Wilson loops_, Commun.Math.Phys. 221 (2001) 591-657 ([arXiv:0010172](https://arxiv.org/abs/math/0010172))
[[!redirects antibrackets]] |
antichain | https://ncatlab.org/nlab/source/antichain | #Contents#
* table of contents
{:toc}
In a [[preorder]] or [[poset]] $P$, an **antichain** is a subset $S \subseteq P$ such that no two distinct elements of $S$ are comparable.
Assuming $P$ has a bottom element $0$, a *strong antichain* is a subset $S \subseteq P$ such that for distinct $a, b \in S$, the only lower bound of $\{a, b\}$ is $0$. This definition may be extended to posets $P$ *without* a bottom element, by declaring $A \subseteq P$ to be a strong antichain if $A$ is a strong antichain in $P^+$, the poset formed by freely adjoining a bottom element to $P$.
In the context of [[set theory]], for example in discussions of [[forcing]] and [[countable chain conditions]], "strong antichain" is often abbreviated to just "antichain".
[[!redirects antichains]] |
anticommutative graded codifferential category | https://ncatlab.org/nlab/source/anticommutative+graded+codifferential+category | [[!redirects Anticommutative graded codifferential category]]
## Idea
This is a notion in development which is meant to link [[differential categories]] with [[homological algebra]].
For example, the category of modules over a [[commutative ring]], equipped with the [[exterior powers]] is an anticommutative graded codifferential category.
It is hoped for example that the concept of [[differential algebra]] can be studied in this context.
## Related concepts
* [[differential category]]
* [[Koszul complex]] |
antifield | https://ncatlab.org/nlab/source/antifield |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Content#
* table of contents
{:toc}
## Idea
In a [[BV-complex]] the dual of an element under the [[antibracket]] is called its _antifield_.
(There is _no_ relation to [[antiparticle]].)
## Example
In the standard example of the [[BV-complex]] of [[multivector fields]] over a [[smooth manifold]] with [[volume form]], the antifields of the functions on the manifolds are the [[vector fields]].
## Related concepts
[[BV-BRST formalism]]
* [[antibracket]]
* [[field (physics)]]
* [[ghost]],
* **antifield**, [[antighost]]
[[!redirects antifields]]
|
antihomomorphism | https://ncatlab.org/nlab/source/antihomomorphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
# Anti-Homomorphisms
* table of contents
{: toc}
## Idea
While a [[homomorphism]] of [[magmas]] (including [[groups]], [[rings]], etc) must preserve multiplication, an _antihomomorphism_ must instead _reverse_ multiplication.
## Definitions
Let $A$ and $B$ be [[magmas]], or more generally [[magma objects]] [[internalisation|in]] any [[symmetric monoidal category]] $C$. (Examples include [[groups]], which are magmas with extra properties; [[rings]], which are magma objects in [[Ab]] with extra proprties; etc.)
An __antihomomorphism__ from $A$ to $B$ is a [[homomorphism]] $f\colon A \to B^\op$ where $B^\op$ is the [[opposite magma]] of $B$, or equivalently, it is a [[function]] (or $C$-morphism) $f\colon A \to B$ such that:
* for every two ([[generalised element|generalised]]) elements $x, y$ of $A$, $f(x y) = f(y) f(x)$.
Note that for magma objects in $C$, the left-hand side of this equation is a generalised element of $B$ whose source is ${|x|} \otimes {|y|}$ (where ${|x|}$ and ${|y|}$ are the sources of the generalised elements $x$ and $y$ and $\otimes$ is the [[tensor product]] in $C$), while the right-hand side is a generalised element of $B$ whose source is ${|y|} \otimes {|x|}$. Therefore, this definition only makes unambiguous sense because $C$ is symmetric monoidal, using the unique [[natural isomorphism]] ${|x|} \otimes {|y|} \cong {|y|} \otimes {|x|}$.
An __antiautomorphism__ is an antihomomorphism whose underlying $C$-morphism is an [[automorphism]].
## Examples
* The [[inverse function]] of a [[group]] is a [[monoid]] anti-homomorphism, and in fact an anti-automorphism, hence an [[anti-involution]], since $(x y) (x y)^{-1} = 1 = x x^{-1} = x y y^{-1} x^{-1}$, which means that $(x y)^{-1} = y^{-1} x^{-1}$.
* The [[antipode]] in a [[Hopf algebra]] is an anti-homomorphism (by [this Prop.](Hopf+algebra#AntipodeIsAnAntihomomorphism)). The same is separately required for antipodes on associative [[bialgebroid]]s.
* In a [[star-algebra]] the star-operation is an anti-homomorphism, in fact an anti-automorphism, hence an anti-[[involution]].
* Combining these two examples, in an [[involutive Hopf algebra]] the [[antipode]] is an anti-automorphism.
## Related concepts
* [[opposite magma]]
[[!redirects antihomomorphism]]
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|
antilinear map | https://ncatlab.org/nlab/source/antilinear+map |
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### Context
#### Linear algebra
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#### Complex geometry
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[[!include complex geometry - contents]]
=--
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#Contents#
* table of contents
{:toc}
## Idea
An *antilinear map* or *conjugate linear map* is much like a [[linear map]], but instead of commuting with "[[scalar]] [[multiplication]]" it "anti-commutes" with it, in that multiplication by a scalar $c$ is mapped to multiplication by the scalar's [[star-conjugation|conjugate]] $\overline{c}$.
In order to make sense of this notion, the [[ground ring]] of the [[modules]] (or [[ground field]] of the [[vector spaces]]) that serve as the map's [[source|domain]] and [[codomain]] must have the additional structure of an [[involution]], to serve as the [[star-conjugation|conjugation]] map $c \mapsto \overline{c}$.
An antilinear map has a central role in the concept of [[star-algebra]]. Conversely, an antilinear map can be seen as built on a star-algebra, in that the [[involution]] makes the [[ground ring]] into a [[star-algebra]] over itself.
## Definition
Given a [[commutative ring]] (often a [[field]], or possibly just a [[rig]]) $K$, equipped with an [[involution]] $x \mapsto \overline{x}$, meaning an [[endomorphism]] with $\overline{\overline{x}} = x$ for all $x \in K$.
Then for $K$-[[modules]] (or $K$-[[linear spaces]]) $V, W$, a __$K$-antilinear map__ is a function $T \colon V \to W$ such that for all $x, y \in V$ and $r \in K$,
$$
T(r \cdot x + y)
\;=\;
\overline{r} \cdot T(x) + T(y)
\,.
$$
This differs from the definition of a [[linear map]] in the appearance of $\overline{(-)}$ on the right-hand side.
## Examples
### Simple general examples
Every $K$-linear map is also a $K$-antilinear map, for $K$ regarded as equipped with the [[identity function|identity]] [[involution]].
Any involution $\overline{(-)} \colon K \to K$ is itself an antilinear map.
### Complex vector spaces
A motivating class of examples is when $K = \mathbb{C}$ is the [[complex numbers]], and $\overline{(-)}$ is [[complex conjugation]].
In particular, the [[Hermitian adjoint]] is an antilinear map from a space of $\mathbb{C}$-linear operators to itself.
### Further examples
A [[star-algebra|$*$-algebra]] requires by definition its [[anti-involution]] to be antilinear.
## Related concepts
* [[real structure]]
* [[anti-dual linear space]]
* [[antiunitary operator]]
* [[star-algebra]]
## References
See also
* Wikipedia, *[Antilinear map](https://en.wikipedia.org/wiki/Antilinear_map)*
and see also the references at *[[Wigner's theorem]]*.
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|
antiparticle | https://ncatlab.org/nlab/source/antiparticle |
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### Context
#### Fields and quanta
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[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
If [[particles]] species are identified [[simple objects]] of a [[DHR category]], then the elements of the corresponding [[dual object]] are called **antiparticles**.
Similarly, in the [[topological quantum field theory | (2+1)-TQFT]] setting isomorphism classes of quasiparticles will correspond to simple objects of a [[modular tensor category]]. The anti-quasiparticle of a quasiparticle is its dual, under the [[rigid category | rigidity]] structure of your category.
## Examples
In the [[standard model of particle physics]], every [[fundamental particle | particle]] has an antiparticle. For example, the antiparticle of the [[electron]] is the [[positron]].
In the realm of (2+1)-TQFTs, the most simple example is the [[quantum double anyon model]] applied to an abelian group. For every finite abelian group $G$ such a model can be created, and quasiparticles can be explicitly described. They correspond to pairs $(g,\chi)$ where $g\in G$ is a group element and $\chi\in \widehat{G}$ is a [[character]]. The antiparticle is $(g^{-1},\chi^{-1})$, where both the inverses are taken with respect to the group operation.
Just like inverses play a key role in the theory of finite groups, antiparticles play a key role in the theory of (2+1)-TQFTs and modular tensor categories.
## Related concepts
* [[dilepton]]
* [[CPT theorem]]
* [[anti-D-brane]]
* [[topological quantum field theory]]
* [[rigid category]]
## References
A discussion of traditional and of formalized discussions of antimatter, with an eye towards [[AQFT]], is in
* David Baker, [[Hans Halvorson]], _Antimatter_ ([pdf](http://philsci-archive.pitt.edu/4467/1/Antimatter.pdf))
In
* [[Hans Halvorson]], _Algebraic quantum field theory_ ([pdf](http://www.princeton.edu/~hhalvors/aqft.pdf))
the topic appears around remark 8.79.
See also
* R. Ascoli, G. Teppati und S. Termini, _Some remarks about particle-antiparticle superselection rules_, Lettere Al Nuovo Cimento (1969 - 1970)
Volume 1, Number 4 (1969), 223-227, DOI: 10.
[[!redirects antiparticles]]
[[!redirects anti-particle]]
[[!redirects anti-particles]]
[[!redirects antimatter]]
|
antipode | https://ncatlab.org/nlab/source/antipode |
* **In [[topology]]:**
Regarding the [[n-sphere]] as the [[unit sphere]] inside [[Cartesian space]], $S^n \,\simeq\, S(\mathbb{R}^{n+1})$, the *antipode* of any point $p \,\in\, S^n \hookrightarrow \mathbb{R}^{n=1}$ is the point $-p \,\in\, S^n \hookrightarrow \mathbb{R}^{n+1}$ obtained by sending the [[coordinates]] of $p$ to their negatives.
The [[quotient space]] of the [[n-sphere]] by the [[cyclic group of order two|$\mathbb{Z}/2$]]-[[group action|action]] given by switching antipodes is the [[real projective space]] $\mathbb{R}P^{n-1}$.
See also *[[equator]]*, *[[hemisphere]]*, *[[meridian]]*.
* **In [[algebra]]:**
The *antipode* of a *[[Hopf algebra]]* is the [[formal duality|formal dual]] of the operation of passing to [[inverse elements]] in a [[group]].
If the Hopf algebra is the [[Pontrjagin ring]]-structure on the [[homology of loop spaces|homology of loop spaces]], then this antipode-operation corresponds to reversal of orientation of loops.
## Literature
See also
* Wikipedia, *[Antipodes](https://en.wikipedia.org/wiki/Antipodes)*
category: disambiguation
[[!redirects antipodes]]
|
antisubalgebra | https://ncatlab.org/nlab/source/antisubalgebra |
# Antisubalgebras
* table of contents
{: toc}
## Idea
In [[constructive mathematics]], we often do [[algebra]] by equipping an [[gebra|algebra]] with a [[tight apartness]] (and requiring the algebraic operations to be [[strongly extensional function|strongly extensional]]). In this context, it is convenient to replace [[subalgebras]] with _anti_-subalgebras, which [[classical mathematics|classically]] are simply the [[complements]] of subalgebras.
## Definitions
Let us work in the context of [[universal algebra]], so an __algebra__ is a [[set]] $X$ equipped with a family of [[functions]] $f_i\colon X^{n_i} \to X$ (where each [[arity]] $n_i$ is a [[cardinal number]]) that satisfy certain equational identities (which are irrelevant here). As usual, a __subalgebra__ of $X$ is a [[subset]] $S$ such that $f_i(p_1,\ldots,p_{n_i}) \in S$ whenever each $p_k \in S$. There is no need, in general, to require that any arity $n_i$ be finite or that there be finitely many $f_i$; however, for a few results, we will need a special case of these that we will call having __well-behaved constants__:
* each arity is either $0$ or at least $1$ (so each operation either is a constant or has at least one operand), and
* the number of constants is [[Kuratowski-finite]] (so there is an exhaustive list $c_1, c_2, c_3, \ldots c_n$ of constants for some [[natural number]] $n$, where it remains possible that $c_i = c_j$ might be an equational law).
The first item is true of all derived operations in the theory as long as it is true of the fundamental operations in the signature; but in this last item, we\'re counting all derived constants, not just the fundamental ones. For example, the theory of (unital) [[rings]] does *not* have well-behaved constants, because there are infinitely many constants (one for each [[integer]]).
Now we require $S$ to have a [[tight apartness]] $\ne$, which induces a tight apartness on each $X^{n_i}$ (via [[existential quantification]]), and we require the operations $f_i$ to be [[strongly extensional function|strongly extensional]]. An algebra $X$ with these properties is called an **inequality algebra**. (For much of the theory we don't need the apartness to be tight, but for some purposes it is necessary.)
A [[subset]] $A$ of $X$ is __[[open subset|open]]__ (or $\ne$-open) if, whenever $p \in A$, $q \in A$ or $p \ne q$. An __antisubalgebra__ of $X$ is an open subset $A$ such that $p_j \in A$ for some $j$ whenever $f_i(p_1,\ldots,p_{n_i}) \in A$ for any $i$. By taking the [[contrapositive]], we see that the [[complement]] of $A$ is a subalgebra $S$, but we cannot (in general) start with a subalgebra $S$ and get an antisubalgebra $A$. (Impredicatively, we can take the antisubalgebra generated, as described below, by the $\ne$-complement of $S$, that is the set of those elements of $X$ that $\ne$ every element of $S$, but its complement will generally only be a superset of $S$.)
## Examples
Unless otherwise noted, all of the constructions in these examples should be [[predicative mathematics|predicative]].
The [[empty subset]] of any algebra is an antisubalgebra, the __empty antisubalgebra__ or __improper antisubalgebra__, whose complement is the [[improper subset|improper]] subalgebra (which is all of $X$). An antisubalgebra is __[[proper subset|proper]]__ if it is [[inhabited subset|inhabited]]; the ability to have a positive definition of when an antisubalgebra is proper is a significant motivation for the concept.
If $A$ is an antisubalgebra and $c$ is a constant (given by an operation $X^0 \to X$ or a composite of same with other operations), then $p \ne c$ whenever $p \in A$. If the theory has well-behaved constants, then we can define the __trivial antisubalgebra__ to be the subset of those elements $p$ such that $p \ne c$ for each constant $c$ (the $\ne$-complement of the [[trivial subalgebra]]). In general, we may also take the __trivial antisubalgebra__ to be the [[union]] of all antisubalgebras (but this is not predicative).
Instead of [[subgroups]], use antisubgroups. In this case the definition can be simplified a bit: a subset $A$ of an inequality group $X$ is an __antisubgroup__ if $p \ne 1 $ whenever $p \in A$, $p \in A$ or $q \in A$ whenever $p q \in A$, and $p \in A$ whenever $p^{-1} \in A$. We need not assume that $A$ is open; this can be proved from strong extensionality of the group operations on $X$ and the stronger form of the nullary anticlosure condition ("$p \ne 1 $ whenever $p \in A$" is a strengthening of the condition $\neg (1\in A)$ that would be the literal nullary case of the general definition.) An antisubgroup $A$ is __[[normal subgroup|normal]]__ if $p q \in A$ whenever $q p \in A$. The __[[trivial subgroup|trivial]] antisubgroup__ is the $\ne$-complement of $\{1\}$.
Instead of [[ideals]] (of [[rings]]), use antiideals. (Technically, these are antisubalgebras of the ring as a module over itself.) Again we can omit $\ne$-openness by strengthening the nullary condition. In detail, a subset $A$ of $X$ is a __two-sided antiideal__ (or simply an __antiideal__ in the commutative case) if $p \ne 0 $ whenever $p \in A$, $p \in A$ or $q \in A$ whenever $p + q \in A$, and $p \in A$ and $q \in A$ whenever $p q \in A$. $A$ is a __left antiideal__ if instead the last condition requires only that $p \in A$, and $A$ is a __right antiideal__ if instead the last condition requires only that $q \in A$. It follows that an antiideal $A$ is proper iff $1 \in A$. $A$ is __[[prime ideal|prime]]__ (or _antiprime_) if it is proper and $p r q \in A$ for some $r$ whenever $p \in A$ and $q \in A$; in the commutative case, we can say that $p q \in A$ whenever $p \in A$ and $q \in a$. $A$ is __[[maximal ideal|minimal]]__ (or _antimaximal_) if it is proper and, for each $p \in A$, for some $q$, for each $r \in A$, $p q + r \ne 1$ and $q p + r \ne 1$ (which is constructively stronger than being prime and [[minimal element|minimal]] among proper ideals); of course, we only need one of these two inequalities in the commutative case. The __[[trivial ideal|trivial]] antiideal__ is the $\ne$-complement of $\{0\}$.
Note that a [[union]] of antisubalgebras is again an antisubalgebra. Given any subset $B$ of $X$, the antisubalgebra __generated__ by $B$ is the union of all antisubalgebras contained in $B$. (This construction is not predicative, although it may still be true predicatively that the generated subalgebra exists in some situations.)
In some cases, we may prefer to anti-generate: given any subset $B$ of $X$, the antisubalgebra __antigenerated__ by $B$ is the union of all antisubalgebras whose elements are all distinct from ($\ne$) each element of $B$, in other words the antisubalgebra generated by the $\ne$-complement of $B$. For example, the trivial antisubgroup of a group is antigenerated by $\{1\}$, and the trivial antiideal of a ring is antigenerated by $\{0\}$. More generally than these examples, we may talk of the __[[cyclic subgroup|cyclic antisubgroup]]__ or __[[principal ideal|principal antiideal]]__ antigenerated by a given element of the group or ring.
## Quotient algebras
To form a [[quotient group]] or a [[quotient ring]], it\'s enough to have a [[normal subgroup]] or a [[two-sided ideal]]. However, if we want the quotient algebra to inherit an apartness from the original algebra, then we need antisubgroups and antiideals.
In general, instead of [[congruence relations]], use anticongruence relations. An __anticongruence relation__ $K$ on $X$ is an [[apartness relation]] on $X$ that is also an antisubalgebra of $X \times X$. Given this, let $R$ be the [[negation]] of $K$; then $R$ is a congruence relation, giving a quotient algebra $X/R$. Furthermore, $K$ becomes a [[tight relation|tight]] apartness on $X/R$, relative to which the algebra operations on $X/R$ are strongly extensional. We denote the resulting algebra-with-apartness by $X/K$. (This notation should cause no confusion; if an apartness relation on a set $X$ is also an equivalence relation, then $X$ must be the [[empty set]], which has a unique apartness and at most one algebra structure, and the only [[quotient set]] of the empty set is itself.) The quotient map $X \twoheadrightarrow X/K$ is also strongly extensional.
Conversely, any strongly extensional map $f\colon X \to Y$ between algebras with apartness gives rise to an anticongruence $\aker f$ on $X$ (the __antikernel__ of $f$), where $(p, q) \in \aker f$ iff $f(p) \ne f(q)$. The complement of the antikernel is (because the apartness of $Y$ is tight) the [[kernel]] in the usual sense of universal algebra. Thus, the quotient algebra $X/(\aker f)$ is [[naturally isomorphic]] to a [[subalgebra]] $im f$ of $Y$; the maps $X \twoheadrightarrow X/(\aker f) \cong \im f \hookrightarrow Y$ are strongly extensional. Similarly, a sequence $X \overset{f}\to Y \overset{g} \to Z$ is [[exact sequence|exact]] iff $\im f$ is the complement of $\aker g$.
(We would like to say that there is an antisubalgebra $\aim f$ of $Y$ whose complement is $\im f$; then we could, for example, define a stronger notion of exactness requiring that $\aker g$ equal the antiimage of $f$. In principle, $\aim f$ should be the $\ne$-complement of $\im f$. If $X$ is Kuratowski-finite, then this works, but in general, we can prove neither that this is open nor that its complement is all of $\im f$.)
Given a group-with-apartness and a normal antisubgroup $A$, we define an anticongruence $K$, where $(p, q) \in K$ iff $p q^{-1} \in A$. Similarly, given a ring-with-apartness and a two-sided antiideal $A$, we define an anticongruence $K$, where $(p, q) \in K$ iff $p - q \in A$. This allows us to form quotient groups or quotient rings by modding out by normal antisubgroups or two-sided antiideals. Conversely, we can interpret the antikernel as a normal antisubgroup or two-sided antiideal: $p \in \aker f$ iff $f(p) \ne 1$, $p \in \aker f$ iff $f(p) \ne 0$, etc. In general, this works for any [[Omega-group]] structure.
## Localic point of view
As noted at [[apartness relation]], an apartness relation on a set $X$ is equivalent to a (strongly) closed [[equivalence relation]] on the corresponding [[discrete locale]], and the $\ne$-open subsets are those whose complementary closed sublocales are stable under this equivalence relation, and the $\ne$-topology itself is the corresponding quotient locale. From this point of view, an algebra structure is strongly extensional if it respects the equivalence relation, hence passes to the quotient; and an antisubalgebra is an $\ne$-open set whose complementary closed sublocale is additionally a localic subalgebra, since the operation $\mathsf{C}$ from open sublocales to closed ones takes arbitrary (not only finite) unions to intersections.
In other words, antisubalgebras of an inequality algebra are equivalent to closed subalgebras of a localic algebra, in the case when the latter is the quotient of a discrete algebra by a closed localic congruence.
## References
According to [Troelstra and van Dalen](#TvD):
> The study of algebraic structures in an intuitionistic setting was undertaken by Heyting ([1941](#Heyting1941))... in full generality, equipped with an apartness relation. The notion of an antisubstructure, implicit in Heyting's treatment of ideals in polynomial rings, was formulated explicitly by D.S. Scott ([1979](#Scott1979)) (N.B. the first draft of this paper contains a good deal more than the published version). Ruitenburg ([1982](#Ruitenberg1982Thesis), [1982A](#Ruitenberg1982)) deals with intuitionistic algebra in the spirit of Heyting and Scott.
* [[Arend Heyting]], _Untersuchungen über intuitionistische Algebra_, 1941
{#Heyting1941}
* [[Dana Scott]], _Identity and existence in intuitionistic logic_, 1979
{#Scott1979}
* [[Wim Ruitenberg]], _Intuitionistic Algebra_, Ph.D. Thesis, Rijksuniversiteit Utrecht, 1982
{#Ruitenberg1982Thesis}
* [[Wim Ruitenberg]], _Primality and invertibility of polynomials_, 1982
{#Ruitenberg1982}
* Surprisingly, antisubalgebras make hardly any appearence in [[Ray Mines]], [[Fred Richman]], [[Wim Ruitenburg]]. _A Course in Constructive Algebra_. Springer, 1987.
* More can be found in [[Anne Troelstra]] and [[Dirk van Dalen]], _Constructivism in Mathematics_ (volume 2).
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antisymmetric relation | https://ncatlab.org/nlab/source/antisymmetric+relation |
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#### Relations
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[[!include relations - contents]]
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A (binary) [[relation]] $\sim$ on a set $A$ is __antisymmetric__ if any two elements that are related in both orders are [[equality|equal]]:
$$\forall (x, y: A),\; x \sim y \;\wedge\; y \sim x \;\Rightarrow\; x = y$$
In the language of the $2$-poset-with-duals [[Rel]] of sets and relations, a relation $R: A \to A$ is __antisymmetric__ if its intersection with its reverse is contained in the identity relation on $A$:
$$R \cap R^{op} \subseteq \id_A$$
If an antisymmetric relation is also [[reflexive relation|reflexive]] (as most are in practice), then this containment becomes an equality.
## See also ##
* [[internal antisymmetric relation]]
[[!redirects antisymmetry]]
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[[!redirects antisymmetric relations]] |
antiunitary operator | https://ncatlab.org/nlab/source/antiunitary+operator |
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#### Linear algebra
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#### Operator algebra
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[[!include AQFT and operator algebra contents]]
=--
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=--
#Contents#
* table of contents
{:toc}
## Definition
An *anti-unitary operator* on a [[Hilbert space]] is an [[anti-linear map]] which preserves the [[inner product]]/[[norm]] up to [[complex conjugation]].
## Related concepts
* [[antilinear map]], [[anti-dual linear space]]
* [[time-reversal symmetry]]
* [[Wigner's theorem]]
## References
See also:
* Wikipedia, *[Antiunitary operator](https://en.wikipedia.org/wiki/Antiunitary_operator)*
and see the references at *[[Wigner's theorem]]*.
[[!redirects antiunitary operators]]
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|
Antoine Allioux | https://ncatlab.org/nlab/source/Antoine+Allioux |
## Selected writings
On combining [[homotopy type theory]] with [[opetopic type theory]]:
* {#AlliouxFinsterSozeau21} [[Antoine Allioux]], [[Eric Finster]], [[Matthieu Sozeau]], _Types are internal infinity-groupoids_, 2021 ([hal:03133144](https://hal.inria.fr/hal-03133144), [pdf](https://hal.inria.fr/hal-03133144/document))
category: people |
Antoine Van Proeyen | https://ncatlab.org/nlab/source/Antoine+Van+Proeyen |
* [webpage](http://itf.fys.kuleuven.be/~toine/home.htm)
## Selected writings:
Formulation of ([[Lagrangian densities]] for) [[type II supergravity]] with "democratic"/"[[pregeometric RR-field|pregeometric]]" [[RR-fields]] subject to [[self-dual higher gauge theory|self-duality]]:
* [[Eric Bergshoeff]], [[Renata Kallosh]], [[Tomas Ortin]], [[Diederik Roest]], [[Antoine Van Proeyen]], *New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls*, Class. Quant. Grav. **18** (2001) 3359-3382 [[arXiv:hep-th/0103233](https://arxiv.org/abs/hep-th/0103233), [doi:10.1088/0264-9381/18/17/303](https://doi.org/10.1088/0264-9381/18/17/303)]
On [[supergravity]]:
* [[Antoine Van Proeyen]], [[Daniel Freedman]], _Supergravity_, Cambridge University Press (2012) [[doi:10.1017/CBO9781139026833]( https://doi.org/10.1017/CBO9781139026833)]
On [[D=4 supergravity]], [[D=5 supergravity]], [[D=6 supergravity]]:
* Edoardo Lauria, [[Antoine Van Proeyen]], _$\mathcal{N}=2$ Supergravity in $D=4,5,6$ Dimensions_ ([arXiv:2004.11433](https://arxiv.org/abs/2004.11433))
## Rekated entries
* [[supersymmetry]]
category: people
|
Anton Alekseev | https://ncatlab.org/nlab/source/Anton+Alekseev | __Anton Alekseev__ is a mathematician at the [Université de Genève](http://www.unige.ch), working mainly on the interface of mathematical physics, differential geometry and algebra.
* [web](http://www.unige.ch/math/people/alekseev.html)
* [arxiv preprints](http://arxiv.org/find/math/1/au:+Alekseev_A/0/1/0/all/0/1)
category: people |
Anton Deitmar | https://ncatlab.org/nlab/source/Anton+Deitmar |
* [webpage](http://www.math.uni-tuebingen.de/user/deitmar/)
## related $n$Lab entries
* [[field with one element]]
* [[algebraic K-theory]], [[K-theory of a permutative category]]
category: people |
Anton Geraschenko | https://ncatlab.org/nlab/source/Anton+Geraschenko | [[!redirects Anton Greschenko]]
[[!redirects Anton Greschenko]]
* [webpage](http://stacky.net/wiki/index.php?title=Main_Page)
category: people
|
Anton Gerasimov | https://ncatlab.org/nlab/source/Anton+Gerasimov |
* [webpage](http://www.maths.tcd.ie/people/AntonGerasimov.php)
category: people
|
Anton Hilado | https://ncatlab.org/nlab/source/Anton+Hilado |
* [personal page](https://sites.google.com/view/antonhilado)
* [blog](https://ahilado.wordpress.com/author/ahilado/)
* [institute page](https://www.uvm.edu/cems/mathstat/profiles/anton-hilado)
## Selected writings
On [[Mochizuki's corollary 3.12]] and [[Szpiro's conjecture]]:
* [[Taylor Dupuy]], [[Anton Hilado]], _Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12_ [[arxiv:2004.13108](https://arxiv.org/abs/2004.13108)]
* [[Taylor Dupuy]], [[Anton Hilado]], _The Statement of Mochizuki's Corollary 3.12, Initial Theta Data, and the First Two Indeterminacies_ [[arxiv:2004.13228](https://arxiv.org/abs/2004.13228)]
## Related pages
* [[inter-universal Teichmüller theory]]
* [[initial Θ-data]]
category: people
|
Anton Kapustin | https://ncatlab.org/nlab/source/Anton+Kapustin | __Anton Kapustin__ is a Russian mathematical physicist, now a Professor at Caltech University in USA. His works include the study of noncommutative analogues of ADHM construction, [[homological mirror symmetry]], derived categories of [[coherent sheaves]] on algebraic varieties, Landau-Ginzburg models, TQFT-s and Langlands duality.
* [homepage](http://www.theory.caltech.edu/~kapustin) at Caltech
* [wikipedia](http://en.wikipedia.org/wiki/Anton_Kapustin)
## Selected writings
A popular exposition of formal deformation quantization with emphasis of the [[string theory|stringy]] [[Poisson sigma-model]]-construction:
* [[Anton Kapustin]], _Quantum geometry: The reunion of math and physics_ [[[Kapustin-QuantumGeometry.pdf:file]]]
On [[topological quantum field theory]]:
* [[Sergei Gukov]], [[Anton Kapustin]], _[[Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories]]_ ([arXiv:1307.4793](http://arxiv.org/abs/1307.4793))
On [[Rozansky-Witten theory]] as a [[boundary field theory]]:
* [[Anton Kapustin]], [[Lev Rozansky]], [[Natalia Saulina]],
_Three-dimensional topological field theory and symplectic algebraic geometry I_ ([arXiv:0810.5415](https://arxiv.org/abs/0810.5415))
On the relation between [[Rozansky-Witten theory]] and [[Chern-Simons theory]]:
* [[Anton Kapustin]], [[Natalia Saulina]], _Chern-Simons-Rozansky-Witten topological field theory_, Nucl. Phys. B823 (2009) 403-427 ([arXiv:0904.1447](https://arxiv.org/abs/0904.1447), [spire:817599/](http://inspirehep.net/record/817599/))
On [[generalized global symmetries]]:
* {#GKSW14} [[Davide Gaiotto]], [[Anton Kapustin]], [[Nathan Seiberg]], [[Brian Willett]], *Generalized Global Symmetries* J. High Energ. Phys. **2015** 172 (2015) [[arXiv:1412.5148](https://arxiv.org/abs/1412.5148), <a href="https://doi.org/10.1007/JHEP02(2015)172">doi:10.1007/JHEP02(2015)172</a>]
On the [[theta angle]] in [[QCD]]:
* [[Davide Gaiotto]], [[Anton Kapustin]], [[Zohar Komargodski]], [[Nathan Seiberg]], _Theta, Time Reversal, and Temperature_, JHEP05(2017)091 ([arXiv:1703.00501](https://arxiv.org/abs/1703.00501))
category: people |
Anton Petrunin | https://ncatlab.org/nlab/source/Anton+Petrunin |
* [webpage](https://www.math.psu.edu/petrunin/)
## related $n$Lab entries
* [[Euclidean geometry]]
* [[synthetic geometry]]
category: people |
Anton Rebhan | https://ncatlab.org/nlab/source/Anton+Rebhan |
* [personal page](http://www.itp.tuwien.ac.at/Homepage_Anton_Rebhan)
* [Wikipedia entry](https://de.wikipedia.org/wiki/Anton_Rebhan) (Deutsch)
## Selected writings
On the [[quark-gluon plasma]]:
* {#BlaizotIancuRebhan03} Jean-Paul Blaizot, Edmond Iancu, [[Anton Rebhan]], _Thermodynamics of the high temperature quark gluon plasma_, Quark–Gluon Plasma 3, pp. 60-122 (2004) ([arXiv:hep-ph/0303185](https://arxiv.org/abs/hep-ph/0303185), [spire:615570](http://inspirehep.net/record/615570))
On the [[Witten-Sakai-Sugimoto model]] for [[holographic QCD]]:
* {#Rebhan14} [[Anton Rebhan]], _The Witten-Sakai-Sugimoto model: A brief review and some recent results_, 3rd International Conference on New Frontiers in Physics, Kolymbari, Crete, 2014 ([arXiv:1410.8858](https://arxiv.org/abs/1410.8858))
On the [[anomalous magnetic moment]] of the [[muon]] via [[holographic QCD]]:
* [[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))
category: people |
Anton Setzer | https://ncatlab.org/nlab/source/Anton+Setzer |
* [webpage](http://www.cs.swan.ac.uk/~csetzer/)
## Selected writings
Introducing the notion of [[inductive-recursive types]]:
* [[Peter Dybjer]], [[Anton Setzer]], *Indexed induction-recursion*, in *Proof Theory in Computer Science* PTCS 2001. Lecture Notes in Computer Science**2183** Springer (2001) [[doi:10.1007/3-540-45504-3_7](https://doi.org/10.1007/3-540-45504-3_7), [pdf](http://www.cse.chalmers.se/~peterd/papers/InductionRecursionInitialAlgebras.pdf)]
## Related entries
* [[inductive-inductive type]]
category: people |
Anton Suschkewitsch | https://ncatlab.org/nlab/source/Anton+Suschkewitsch | Anton Suschkewitsch (Антон Казимирович Сушкевич) was a Russian mathematician working on [[semigroup]] theory.
[Math-Net.Ru page](http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=42597).
[Wikipedia page](https://en.wikipedia.org/wiki/Anton_Sushkevich).
## Selected writings
* [[Anton Suschkewitsch]], _On a generalization of the associative law_. Transactions of the American Mathematical Society 31:1 (1929), 204–204. [doi](http://dx.doi.org/10.1090/s0002-9947-1929-1501476-0).
* [[А. К. Сушкевич]], _Теория обобщенных групп_ (Theory of generalized groups), Государственное научно-техническое издательство Украины, 1937.
[[!redirects Антон Казимирович Сушкевич]]
[[!redirects А. К. Сушкевич]]
[[!redirects А. Сушкевич]]
[[!redirects Anton Sushkevich]]
[[!redirects A. Sushkevich]]
[[!redirects A. K. Sushkevich]]
|
Anton Zabrodin | https://ncatlab.org/nlab/source/Anton+Zabrodin |
* [webpage](https://www.hse.ru/en/org/persons/35436251)
## Selected writings
Suggestion that the [[disk]] [[worldsheet]] of the [[open string|open]] [[p-adic string]] is to be identified with the [[Bruhat-Tits tree]] $T_p$:
* [[Anton Zabrodin]], _Non-Archimedean strings and Bruhat-Tits trees_, Comm. Math. Phys.
Volume 123, Number 3 (1989), 463-483 ([euclid.cmp/1104178891](https://projecteuclid.org/euclid.cmp/1104178891))
category: people |
Anton Zeilinger | https://ncatlab.org/nlab/source/Anton+Zeilinger |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Anton_Zeilinger)
* [research group](https://www.iqoqi-vienna.at/people/staff/anton-zeilinger)
## Selected writings
On [[experiment|experimental]] [[quantum physics]] and its [[epistemology|epistemological]] [[interpretation of quantum mechanics]]:
* [[Daniel Greenberger]], [[Wolfgang L. Reiter]], [[Anton Zeilinger]], *Epistemological and Experimental Perspectives on Quantum Physics*, Vienna Circle Institute Yearbook (VCIY) **7** (1999) [[doi:10.1007/978-94-017-1454-9](https://doi.org/10.1007/978-94-017-1454-9)]
On [[quantum teleportation]] realized in [[experiment]]:
* [[Dirk Bouwmeester]], Jian-Wei Pan, Klaus Mattle, Manfred Eibl, [[Harald Weinfurter]], [[Anton Zeilinger]], *Experimental quantum teleportation*, Nature **390** 575–579 (1997) [[doi:10.1038/37539](https://doi.org/10.1038/37539)]
On the [[quantum physics]] of [[quantum information]]:
* [[Dirk Bouwmeester]], [[Artur Ekert]], [[Anton Zeilinger]] (eds.), *The Physics of Quantum Information -- Quantum Cryptography, Quantum Teleportation, Quantum Computation*, Springer (2020) [[doi:10.1007/978-3-662-04209-0](https://doi.org/10.1007/978-3-662-04209-0)]
## Related entries
* [[quantum teleportation]]
category: people
|
Anton Zeitlin | https://ncatlab.org/nlab/source/Anton+Zeitlin |
* [webpage](http://users.math.yale.edu/~az84/)
## Selected writings
On [[Maurer-Cartan equations]] as [[equations of motion]] of [[Yang-Mills theory]] and [[gravity]] (by truncation of [[string field theory]]):
* [[Anton M. Zeitlin]], *Formal Maurer-Cartan Structures: from CFT to Classical Field Equations*, JHEP 0712:098, 2007 ([arXiv:0708.0955](https://arxiv.org/abs/0708.0955))
On [[super Riemann surfaces]] and [[fat graphs]]:
* [[Albert S. Schwarz]], [[Anton M. Zeitlin]], *Super Riemann surfaces and fatgraphs* [[arXiv:2307.02706](https://arxiv.org/abs/2307.02706)]
## Related $n$Lab pages
* [[beta-gamma system]]
* [[vertex operator algebra]]
* [[super Riemann surface]]
category: people
[[!redirects Anton M. Zeitlin]]
|
Antonella Grassi | https://ncatlab.org/nlab/source/Antonella+Grassi |
* [webpage](https://www.math.upenn.edu/~grassi/)
## related $n$Lab entries
* [[M-theory on G2-manifolds]], [[ADE singularity]]
category: people |
Antoni Kosinski | https://ncatlab.org/nlab/source/Antoni+Kosinski |
* [Institute page](https://www.math.rutgers.edu/component/comprofiler/userprofile/kosinski?Itemid=774)
* [Mathematics Genealogy Page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=5936)
## Selected writings
On [[differential geometry]] ([[differentiable manifolds]], [[smooth manifolds]]) with applications to [[cobordism theory]] ([[Cohomotopy]], [[Pontryagin-Thom theorem]]):
* {#Kosinski93} [[Antoni Kosinski]], _Differential manifolds_, Academic Press 1993 ([pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf), [ISBN:978-0-12-421850-5](https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/138/suppl/C))
## Related $n$Lab entries
* [[differentiable manifold]], [[smooth manifold]]
* [[cobordism ring]],
* [[cohomotopy]]
* [[Pontrjagin-Thom construction]]
category: people
|
Antonin Delpeuch | https://ncatlab.org/nlab/source/Antonin+Delpeuch |
* [personal page](https://antonin.delpeuch.eu/index.html)
* [institute page](https://www.cs.ox.ac.uk/people/antonin.delpeuch/)
## Selected writings
On the [[free construction]] of [[autonomous categories]] (aka [[rigid monoidal categories]]) from plain [[monoidal categories]]:
* {#Delpeuch2020} [[Antonin Delpeuch]], *Autonomization of monoidal categories*, EPTCS 323 (2020) 24-43 [[arXiv:1411.3827](https://arxiv.org/abs/1411.3827), [doi:10.4204/EPTCS.323.3](https://dx.doi.org/10.4204/EPTCS.323.3)]
category: people
|
Antonio Alarcón | https://ncatlab.org/nlab/source/Antonio+Alarc%C3%B3n |
* [webpage](http://www.ugr.es/~alarcon/invest.html)
## Selected writings
On the [[Oka principle]] applied to discussion of [[minimal surfaces]]:
* [[Antonio Alarcon]], [[Franc Forstnerič]], *New complex analytic methods in the theory of minimal surfaces: a survey*, Journal of the Australian Mathematical Society, 106(3), 287-341 ([arXiv:1711.08024](https://arxiv.org/abs/1711.08024), [doi:10.1017/S1446788718000125](https://doi.org/10.1017/S1446788718000125))
category: people
[[!redirects Antonio Alarcon]]
|