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Wolfgang Soergel | https://ncatlab.org/nlab/source/Wolfgang+Soergel | [[!redirects Wolgang Soergel]]
* [webpage](http://home.mathematik.uni-freiburg.de/soergel/)
category: people |
Wolfgang Steinicke | https://ncatlab.org/nlab/source/Wolfgang+Steinicke |
* [webpage](http://www.klima-luft.de/steinicke/)
## related $n$Lab entries
* [[gravitational waves]]
category: people |
Wolfgang Ziller | https://ncatlab.org/nlab/source/Wolfgang+Ziller |
* [webpage](https://www.math.upenn.edu/~wziller/)
## Selected writings
On [[Hopf fibrations]] and notably the [[quaternionic Hopf fibration]]:
* {#GluckWarnerZiller86} [[Herman Gluck]], [[Frank Warner]], [[Wolfgang Ziller]], _The geometry of the Hopf fibrations_, L'Enseignement Mathématique, **32** (1986), 173-198 [[ResearchGate](https://www.researchgate.net/publication/266548925_The_geometry_of_the_Hopf_fibrations), [[GluckWarnerZiller-HopfFibrations.pdf:file]]]
On [[equivariant differential topology]]:
* {#Ziller13} [[Wolfgang Ziller]], _Group actions_, 2013 ([pdf](https://www.math.upenn.edu/~wziller/math661/LectureNotesLee.pdf), [[ZillerGroupActions.pdf:file]])
category: people |
Wolfhart Zimmermann | https://ncatlab.org/nlab/source/Wolfhart+Zimmermann |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Wolfhart_Zimmermann)
## Selected writings
Introducing the [[LSZ reduction formula]]:
* {#LehmannSymanzikZimmermann55} [[Harry Lehmann]], [[Kurt Symanzik]], [[Wolfhart Zimmermann]], _Zur Formulierung quantisierter Feldtheorien_, Nuovo Cimento 1(1), 205 (1955) ([doi:10.1007/BF02731765](https://doi.org/10.1007/BF02731765))
## Related $n$Lab entries
* [[retarded product]]
* [[LSZ reduction formula]]
category: people |
Wolfram Schwabhäuser | https://ncatlab.org/nlab/source/Wolfram+Schwabh%C3%A4user |
* [webpage](https://de.wikipedia.org/wiki/Wolfram_Schwabhäuser)
## related $n$Lab entries
* [[Euclidean geometry]], [[synthetic geometry]]
category: people |
Wolfram Weise | https://ncatlab.org/nlab/source/Wolfram+Weise |
* [webpage](https://www.professoren.tum.de/en/weise-wolfram)
## Selected writings
On the equivalence between [[hidden local symmetry]]- and [[massive Yang-Mills theory]]-description of [[Skyrmion]] [[quantum hadrodynamics]]:
* [[Atsushi Hosaka]], H. Toki, [[Wolfram Weise]], _Skyrme Solitons With Vector Mesons: Equivalence of the Massive Yang-Mills and Hidden Local Symmetry Scheme, 1988, Z. Phys. A332 (1989) 97-102 ([spire:24079](http://inspirehep.net/record/24079))
On [[nucleons]] as [[Skyrmions]] in [[quantum hadrodynamics]] with [[pions]], [[omega-mesons]] and [[rho-mesons]] ($\pi$-$\rho$-$\omega$-model):
* [[Ulf-G. Meissner]], [[Norbert Kaiser]], [[Wolfram Weise]], _Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties_, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (<a href="https://doi.org/10.1016/0375-9474(87)90463-5">doi:10.1016/0375-9474(87)90463-5</a>)
* [[Ulf-G. Meissner]], [[Norbert Kaiser]], Andreas Wirzba, [[Wolfram Weise]], _Skyrmions with $\rho$ and $\omega$ Mesons as Dynamical Gauge Bosons_, Phys. Rev. Lett. 57, 1676 (1986) ([doi:10.1103/PhysRevLett.57.1676](https://doi.org/10.1103/PhysRevLett.57.1676))
On [[vector meson dominance]]:
* {#PillerWeise90} G. Piller, [[Wolfram Weise]], _Vector meson dominance: Selected topics_ 1990 ([spire310958](https://inspirehep.net/literature/310958), [[PillerWeiseVMD.pdf:file]])
category: people |
wonderful compactification | https://ncatlab.org/nlab/source/wonderful+compactification |
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###Context###
#### Geometry
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[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A certain kind of [[compactification]] of a [[variety]] used for [[resolution of singularities]].
## Related concepts
* [[Fulton-MacPherson compactification]]
## References
See also
* Wikipedia _[Wonderful compactification](https://en.wikipedia.org/wiki/Wonderful_compactification)_
In generalization of [[Fulton-MacPherson compactifications]] of [[configuration spaces of points]]:
* {#deConciniProcesi95} [[Corrado de Concini]], [[Claudio Procesi]], _Wonderful models of subspace arrangements_, Selecta Mathematica, New Series Vol. 1, No. 3, 1995 ([pdf](https://link.springer.com/content/pdf/10.1007/BF01589496.pdf))
see also
* {#Feichtner05} Eva Maria Feichner, _De Concini–Procesi Wonderful Arrangement Models: A Discrete Geometer’s Point of View_, Math. Sci. Res. Inst. Publ 52, 2005 ([pdf](http://library.msri.org/books/Book52/files/17feich.pdf))
In relation to [[Feynman amplitudes on compactified configuration spaces of points]]:
* [[Marko Berghoff]], _Wonderful compactifications in quantum field theory_, Communications in Number Theory and Physics Volume 9 (2015) Number 3 ([arXiv:1411.5583](https://arxiv.org/abs/1411.5583))
[[!redirects wonderful compactifications]] |
word | https://ncatlab.org/nlab/source/word | \tableofcontents
\section{Introduction}
A _word_ in the [[elements]] of a [[set]] is, roughly speaking, a concatenation of elements of that set. To make this precise, one typically uses the machinery of [[free object|free algebraic structures]]. One may allow in the concatenation certain canonical elements of the free algebraic gadget constructed from the elements of the set one started with, such as inverses.
By extension, one may also refer to elements of any algebraic structure, at least when described in terms of generators and relations (i.e. explicitly as a quotient of a free algebraic structure), as words.
\section{Prototypical example}
The prototypical algebraic structure with which to make sense of the notion of a word is that of a [[monoid]]. If 'word' is used in a context where the intended algebraic structure is not made clear, use of [[free monoids]] is likely intended.
\begin{defn} Let $X$ be a set. A _word_ in the elements of $X$ is an element of the [[free monoid]] on $X$. \end{defn}
\begin{rmk} A free monoid has in particular an identity element, which is the _empty word_. \end{rmk}
\begin{rmk} We do not assume commutativity. \end{rmk}
\begin{example} Let $X = \{a, b\}$ be a set. Examples of words in $X$ are the empty word, $a$, $b$, $a b$, $a^{5}$, $a b^{3}$, $a b a b a b$, $b^{3}a^{2}b^{5}$ and so on. \end{example}
\section{In groups}
Another common case is that in which the algebraic structure is that of groups.
\begin{defn} Let $X$ be a set. A _word_ in the free group on the elements of $X$ is an element of the [[free group]] on $X$. \end{defn}
\begin{rmk} As for monoids, a free group has in particular an identity element, which is the _empty word_. \end{rmk}
\begin{rmk} As for monoids, we do not assume commutativity. \end{rmk}
\begin{example} Let $X = \{a, b\}$ be a set. Examples of words in the free group on $X$ are the empty word, $a$, $b$, $a b$, $a^{5}$, $a^{-5}$, $a b^{3}$, $a b a b a b$, $a^{-1}b^{-1}a^{-1}b^{-1}a^{-1}b^{-1}$, $a b^{-1}$, $b^{3}a^{2}b^{5}$, $a^{-3}b^{2}a^{7}b^{-2}$, and so on. \end{example}
## Related concepts
* [[free monoid]]
* [[string (computer science)]]
* [[syntax]]
* [[language]]
[[!redirects words]]
|
world sheets for world sheets | https://ncatlab.org/nlab/source/world+sheets+for+world+sheets |
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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A mechanism in the context of [[string theory]]:
before [[gauge fixing]], the [[worldsheet]] [[QFT]] describing the first-quantized [[string]] (see also at _[What is a particle?](particle#WhatIsAParticle)_ for conceptual background is a [[2d quantum gravity]] coupled to [[scalar fields]] (the embedding fields that say how the string sits in [[target space|target]] [[spacetime]]). Being [[quantum gravity]], even if in the comparatively extremely simple case of only two dimensions, one may ask if this arises as the [[effective field theory]]/[[second quantization]] of _another_ kind of string, one whose target spacetime is the [[worldsheet]] of the original string.
This idea is known as "world sheets for world sheets", going back to ([Green87](#Green87)).
This may be compared to how the [[worldvolume]] theory of the [[topological membrane]] which is a [[Chern-Simons theory]], arises as the second quantization of [[topological strings]] ([[A-model]]) for which that 3d worldvolume of the membrane is their [[target space]] ([Witten 92](#Witten92)), see at _[TCFT – Worldsheet and effective background theories](TCFT#ActionFunctionals)_ for more on this.
## References
* {#Green87} [[Michael Green]], _World Sheets for World Sheets_, Nucl.Phys. B293 (1987) 593-611
**Abstract** In this paper I suggest that the two-dimensional world sheet underlying the dynamics of relativistic strings may itself emerge from an approximation to a two-dimensional string theory. Coordinates that represent space-time and internal symmetries should emerge as wave functions of the massless states of that theory. This is motivated by studying various string theories with two-dimensional target spaces and internal symmetry which are related to generalized nonlinear sigma models.
See also
* [[Paul Ginsparg]], [[Gregory Moore]], _Lectures on 2D gravity and 2D string theory (TASI 1992)_ ([arXiv:hep-th/9304011](http://arxiv.org/abs/hep--th/9304011))
* [[Edward Witten]], _Chern-Simons Gauge Theory As A String Theory_, Prog.Math. 133 (1995) 637-678 ([arXiv:hep-th/9207094](http://arxiv.org/abs/hep-th/9207094))
{#Witten92}
* Eliezer Rabinovici, _World sheets for world sheets revisited_ ([talk](http://pirsa.org/09070008/), [slides](http://pirsa.org/pdf/files/91367ab3-1691-4b7d-a5d8-10bcae327dc0.pdf))
|
worldline formalism | https://ncatlab.org/nlab/source/worldline+formalism |
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###Context###
#### Algebraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Traditionally, the [[Feynman perturbation series]]/[[scattering amplitudes]] in [[perturbative quantum field theory]] are defined -- given a [[free field theory|free field]] [[vacuum]] and a [[local observable|local]] [[interaction]] [[action functional]] -- by applying the _[[Feynman rules]]_ ([this prop.](A+first+idea+of+quantum+field+theory#FeynmanPerturbationSeriesAwayFromCoincidingPoints)) to the monomial terms in the [[interaction]] [[Lagrangian density]] and deriving from that *[[Feynman rules]]* for how to weigh each [[Feynman diagram]] by a [[probability amplitude]] -- its _[[Feynman amplitude]]_ -- subject to [[renormalization]] choices.
In contrast, in what is called the _worldline formalism_ of [[perturbative quantum field theory]] this assignment is obtained instead more conceptually as the [[correlators]]/[[n-point functions]] of a 1-dimensional QFT that lives _on_ the Feynman graphs, namely the [[worldline]] theory (usually a [[sigma-model]] into the given [[target|target]] [[spacetime]]), which may be thought of as being the ([[relativistic particle|relativistic]]) [[quantum mechanics]] of the [[particles]] that are the quanta of the [[field (physics)|fields]] in the field theory.
One may think of this as realizing the intuition that the [[edges]] in a [[Feynman diagram]] correspond to worldlines of [[virtual particles]].
Mathematically, the key step here is a [[Mellin transform]] -- introducing a "[[Schwinger parameter]]" -- which realizes the [[Feynman propagator]] $\Delta_F(x,y)$ as a [[path integral]] for a [[relativistic particle]] travelling from $y$ to $x$.
This worldline formalism is equivalent to the traditional formulation. It has the conceptual advantage that it expresses the [[Feynman perturbation series]] of [[perturbative quantum field theory]] manifestly as a [[second quantization]] of its particle content given explicitly as the [[superposition]] of all 1-particle processes, and the calculational advantage of automatically summing over subsets of Feynman diagrams related by exchange of external legs, thus maintaining [[permutation]] [[symmetry]] and explicit [[gauge invariance]] of [[on-shell]] [[scattering amplitudes]].
The worldline formulation of QFT has an evident generalization to higher dimensional [[worldvolumes]]: in direct analogy one can consider summing the [[correlators]]/[[n-point functions]] over [[worldvolume]] [[theory (physics)|theories]] of "higher dimensional particles" ("[[branes]]") over all possible worldvolume geometries. Indeed, for 2-[[dimensional]] branes this is precisely the way in which [[perturbative string theory]] is defined: the [[string scattering amplitudes]] are given by the analogous "worldsheet formalism" known as the [[string perturbation series]] as the sum over all surfaces of the [[correlators]]/[[n-point functions]] of of a 2d [[SCFT]] of central charge 15.
[[!include second quantization -- table]]
Indeed, after decades of [[Feynman rules]], the worldline formalism for QFT was found only _via_ [[string theory]] in ([Bern-Kosower 91](#BernKosower91)), by looking at the point particle limit of [[string scattering amplitudes]].
<img src="https://ncatlab.org/nlab/files/StringFeynmanDiagrams.png" width="300">
> graphics grabbed from [Jurke 10](https://benjaminjurke.com/content/articles/2010/string-theory/)
<img src="https://ncatlab.org/nlab/files/PointParticleLimitOfStringDiagram.png" width="300">
> graphics grabbed from [Schubert 96](#Schubert96)
Then ([Strassler 92](#Strassler92), [Strassler 93](#Strassler93)) observed that generally the worldline formlism is obtained from the correlators of the 1d QFT of [[relativistic particles]] on their [[worldline]].
<img src="http://ncatlab.org/nlab/files/worldlineformalismoverview.jpg">
> graphics grabbed from ([Schmidt-Schubert 94](#SchmidtSchubert94))
The first calculuation along these lines was actually done earlier in ([Metsaev-Tseytlin 88](#MetsaevTseytlin88)) where the [[1-loop]] [[beta function]] for pure [[Yang-Mills theory]] was obtained as the point-particle limit of the [[partition function]] of a [[bosonic string|bosonic]] [[open string]] in a Yang-Mills [[background field]]. This provided a theoretical explanation for the observation, made earlier in ([Nepomechie 83](#Nepomechie83)) that when computed via [[dimensional regularization]] then this [[beta function]] coefficient of [[Yang-Mills theory]] vanishes in [[spacetime]] [[dimension]] 26. This of course is the critical dimension of the [[bosonic string]].
## Related entries
* [[virtual particle]]
* [string theory FAQ -- What is the relationship between quantum field theory and string theory?](string+theory+FAQ#RelationshipBetweenQuantumFieldTheoryAndStringTheory)
## References
Precursor observations include
* [[Richard Feynman]], Appendix of _Mathematical formulation of the quantum theory of electromagnetic interaction_, Physics Review Volume 80 (1950) 440.
* {#Nepomechie83} R.I. Nepomechie, _Remarks on quantized Yang-Mills theory in 26 dimensions_, Physics Letters B Volume 128, Issues 3–4, 25 August 1983, Pages 177-178 Phys. Lett. B128 (1983) 177 (<a href="https://doi.org/10.1016/0370-2693(83)90385-4">doi:10.1016/0370-2693(83)90385-4</a>)
* {#MetsaevTseytlin88} [[Ruslan Metsaev]], [[Arkady Tseytlin]], _On loop corrections to string theory effective actions_, Nuclear Physics B Volume 298, Issue 1, 29 February 1988, Pages 109-132 (<a href="https://doi.org/10.1016/0550-3213(88)90306-9">doi:10.1016/0550-3213(88)90306-9</a>)
The worldline formalism as such was first derived from the point-particle limit of [[string scattering amplitudes]] in
* {#BernKosower91} [[Zvi Bern]], D. Kosower, _Efficient calculation of one-loop QCD amplitudes_ Phys. Rev. D 66 (1991),([journal](http://prl.aps.org/abstract/PRL/v66/i13/p1669_1))
* {#BernKosower92} [[Zvi Bern]], D. Kosower, _The computation of loop amplitudes in gauge theories_, Nucl. Phys. B379 (1992) ([journal](http://ccdb4fs.kek.jp/cgi-bin/img_index?9108076))
Then it was related to actual worldline quantum field theory in
* {#Strassler92} [[Matthew Strassler]], _Field Theory Without Feynman Diagrams: One-Loop Effective Actions_, Nucl. Phys. B385:145-184,1992 ([arXiv:hep-ph/9205205](http://arxiv.org/abs/hep-ph/9205205))
* {#Strassler93} [[Matthew Strassler]], _The Bern-Kosower Rules and Their Relation to Quantum Field Theory_, PhD thesis, Stanford 1993 ([spires](http://inspirehep.net/record/364079?ln=de))
Review:
* {#SchmidtSchubert94} M. G. Schmidt, [[Christian Schubert]], _The Worldline Path Integral Approach to Feynman Graphs_ ([arXiv:hep-ph/9412358](http://arxiv.org/abs/hep-ph/9412358))
* {#Schubert96} [[Christian Schubert]], _An Introduction to the Worldline Technique for Quantum Field Theory Calculations_, Acta Phys. Polon.B27:3965-4001, 1996 ([arXiv:hep-th/9610108](https://arxiv.org/abs/hep-th/9610108))
* [[Christian Schubert]], _Perturbative Quantum Field Theory in the String-Inspired Formalism_, Phys.Rept. 355 (2001) 73-234 ([arXiv:hep-th/0101036](https://arxiv.org/abs/hep-th/0101036))
* Olindo Corradini, [[Christian Schubert]], James P. Edwards, Naser Ahmadiniaz, *Spinning Particles in Quantum Mechanics and Quantum Field Theory* ([arXiv:1512.08694](https://arxiv.org/abs/1512.08694))
> (with emphasis on [[spinning particles]])
* James P. Edwards, C. Moctezuma Mata, U. Müller, [[Christian Schubert]], *New techniques for worldline integration*, SIGMA 17 (2021), 065, 19 pages ([arXiv:2106.12071](https://arxiv.org/abs/2106.12071))
* James P. Edwards, [[Christian Schubert]],
*Quantum mechanical path integrals in the first quantised approach to quantum field theory*, ([arXiv:1912.10004](https://arxiv.org/abs/1912.10004))
Exposition with an eye towards [[quantum gravity]] is in
* {#Witten11} [[Edward Witten]], from 30:40 on in _Quantum Gravity_, Solomon Lefschetz Memorial Lecture Series, November 2011 ([video](https://www.youtube.com/watch?v=uRdCJMYc2Ds#t=15))
* {#Witten15} [[Edward Witten]], _What every physicist should know about string theory_, talk at [Strings2015](https://strings2015.icts.res.in) ([pdf](https://strings2015.icts.res.in/talkDocuments/26-06-2015-Edward-Witten.pdf), [[WittenWorldlineFormalism.pdf:file]], [video recording](https://www.youtube.com/watch?v=H0jLD0PphTY))
See also:
* [[Thanu Padmanabhan]], *World-line Path integral for the Propagator expressed as an ordinary integral: Concept and Applications*, Found Phys., 51 35 (2021) ([arXiv:2104.07041](https://arxiv.org/abs/2104.07041), [doi:10.1007/s10701-021-00447-8]( https://doi.org/10.1007/s10701-021-00447-8))
Discussion of the [[Schwinger effect]] via [[worldline formalism]]:
* [[Ian Affleck]], [[Orlando Alvarez]], [[Nicholas S. Manton]], _Pair Production At Strong Coupling In Weak External Fields_, Nuclear Physics B Volume 197 (1982) 509 (<a href="https://doi.org/10.1016/0550-3213(82)90455-2">doi:10.1016/0550-3213(82)90455-2</a>)
* [[Gerald Dunne]], [[Christian Schubert]], _Worldline Instantons and Pair Production in Inhomogeneous Fields_, Phys. Rev. D72 (2005) 105004 ([arXiv:hep-th/0507174](https://arxiv.org/abs/hep-th/0507174))
* [[Christian Schubert]], *The worldline formalism in strong-field QED* [[arXiv:2304.07404](https://arxiv.org/abs/2304.07404)]
Discussion for [[QCD]] with emphasis of [[2d QCD]] and [[AdS/QCD]]:
* Adi Armoni, Oded Mintakevich, _Comments on Mesonic Correlators in the Worldline Formalism_, Nuclear Physics B, Volume 852, Issue 1, 1 November 2011, Pages 61-70 ([arxiv:1102.5318](https://arxiv.org/abs/1102.5318))
Further development:
* M. G. Schmidt, [[Christian Schubert]], _Worldline Green Functions for Multiloop Diagrams_, Phys.Lett. B331 (1994) 69-76 ([arXiv:hep-th/9403158](https://arxiv.org/abs/hep-th/9403158))
* [[Eric D'Hoker]], Darius G. Gagne, _Worldline Path Integrals for Fermions with General Couplings_, Nucl.Phys. B467 (1996) 297-312 ([arXiv:hep-th/9512080](https://arxiv.org/abs/hep-th/9512080))
* C. Alexandrou, R. Rosenfelder, A. W. Schreiber, _Worldline path integral for the massive Dirac propagator: A four-dimensional approach_, Phys. Rev. A59 (1999) 1762-1776 ([arXiv:hep-th/9809101](http://arxiv.org/abs/hep-th/9809101))
* Fiorenzo Bastianelli, Olindo Corradini, Andrea Zirotti, _BRST treatment of zero modes for the worldline formalism in curved space_, JHEP 0401 (2004) 023 ([arXiv:hep-th/0312064](https://arxiv.org/abs/hep-th/0312064))
* Peng Dai, [[Warren Siegel]], _Worldline Green Functions for Arbitrary Feynman Diagrams_, Nucl.Phys.B770:107-122,2007 ([arXiv:hep-th/0608062](https://arxiv.org/abs/hep-th/0608062))
* S. A. Franchino-Viñas, S. Mignemi, _Worldline Formalism in Snyder Spaces_ ([arXiv:1806.11467](https://arxiv.org/abs/1806.11467))
* Olindo Corradini, Maurizio Muratori, _String-inspired Methods and the Worldline Formalism in Curved Space_ ([arXiv:1808.05401](https://arxiv.org/abs/1808.05401))
* James P. Edwards, Urs Gerber, [[Christian Schubert]], Maria Anabel Trejo, Thomai Tsiftsi, Axel Weber, _Applications of the worldline Monte Carlo formalism in quantum mechanics_, ([arXiv:1903.00536](https://arxiv.org/abs/1903.00536))
* Olindo Corradini, Maurizio Muratori, _A Monte Carlo Approach to the Worldline Formalism in Curved Space_ ([arXiv:2006.02911](https://arxiv.org/abs/2006.02911))
* Naser Ahmadiniaz, Victor Miguel Banda Guzman, Fiorenzo Bastianelli, Olindo Corradini, James P. Edwards, [[Christian Schubert]], _Worldline master formulas for the dressed electron propagator, part 1: Off-shell amplitudes_ ([arXiv:2004.01391](https://arxiv.org/abs/2004.01391))
* Naser Ahmadiniaz, Victor Miguel Banda Guzman, Fiorenzo Bastianelli, Olindo Corradini, James P. Edwards, [[Christian Schubert]], _Worldline master formulas for the dressed electron propagator, part 2: On-shell amplitudes_ ([arXiv:2107.00199](https://arxiv.org/abs/2107.00199))
* [[Christian Schubert]] et al., *Summing Feynman diagrams in the worldline formalism* [[arXiv:2208.06585](https://arxiv.org/abs/2208.06585) ]
As a means of constructing [[UV-completions]]:
* Steven Abel, Nicola Andrea Dondi, _UV Completion on the Worldline_ ([arXiv:1905.04258](https://arxiv.org/abs/1905.04258))
For [[black hole]] [[scattering]]:
* {#MogullPlefkaSteinhoff20} Gustav Mogull, [[Jan Plefka]], Jan Steinhoff, _Classical black hole scattering from a worldline quantum field theory_ ([arXiv:2010.02865](https://arxiv.org/abs/2010.02865))
and for photon-graviton scattering:
* Fiorenzo Bastianelli, Francesco Comberiati, Leonardo de la Cruz, *Light bending from eikonal in worldline quantum field theory* ([arXiv:2112.05013](https://arxiv.org/abs/2112.05013))
A list of more literature is at
* The Tangent Bundle, _[QFT Worldline formalism](http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/worldline_formalism)_
[[!redirects worldline formalisms]]
[[!redirects worldline theory]]
[[!redirects worldline theories]]
[[!redirects worldline method]]
[[!redirects worldline methods]] |
worldsheet instanton | https://ncatlab.org/nlab/source/worldsheet+instanton |
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###Context###
#### String theory
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[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[perturbative string theory]], _worldsheet instantons_ are contributions to the [[string scattering amplitudes]] of [[sigma-model]] (i.e. "geometric") [[perturbative string theory vacua]] which correspond to [[worldsheet]] configurations that may not be connected continuously to configurations constant on a point, hence which [[wrapped brane|wrap]] some 2-[[cycle]] in the [[target spacetime]].
Worldsheet instantons contribute [[non-perturbative effects]] to the [[string scattering amplitudes]] with respect to the [[string length scale]] $\ell_s$, see [there](non-perturbative+effect#WorldsheetAndBraneInstantons).
In contrast, [[non-perturbative effects|non-perturbative contributions]] to [[string scattering amplitudes]] $g_s$ in the [[string coupling constant]] come from [[D-brane instantons]].
These two kinds of stringy non-perturbative effects unify under the [[duality between M-theory and type IIA string theory]] to [[membrane instantons]], depending on whether the [[M2-brane]] configuration [[wrapped brane|wraps]] the M-theory circle fiber $S^1_{10}$ or not, see [there](non-perturbative+effect#WorldsheetAndBraneInstantons).
## Yukawa couplings in intersecting D-brane models
In [[intersecting D-brane models]] [[Yukawa couplings]] are encoded by [[worldsheet instantons]] of open strings stretching between the [[brane intersection|intersecting]] [[D-branes]] (see [Marchesano 03, Section 7.5](#Marchesano03)). Mathematically this is encoded by [[derived hom-spaces]] in a [[Fukaya category]] (see [Marchesano 03, Section 7.5](#Marchesano03)).
\begin{center}
<img src="https://ncatlab.org/nlab/files/YukawaFukaya.jpg" width="600">
\end{center}
> table grabbed from [Marchesano 03](#Marchesano03)
## References
In [[intersecting D-brane models]]:
* {#Marchesano03} [[Fernando Marchesano]], section 7.1.1 of _Intersecting D-brane Models_ ([arXiv:hep-th/0307252](https://arxiv.org/abs/hep-th/0307252))
[[!redirects worldsheet instantons]] |
worldsheet parity operator | https://ncatlab.org/nlab/source/worldsheet+parity+operator |
In [[orientifold]] [[string theory]] [[vacua]] the [[real space]]-structure of [[target space]] is accompanied by a corresponding [[involution]] of the [[worldsheet]]. This is known as the _worldsheet parity operator_.
[[!redirects worldsheet parity]]
|
worldvolume | https://ncatlab.org/nlab/source/worldvolume |
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###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In a [[sigma-model]] [[quantum field theory]] a [[field history]] is a [[morphism]] $\phi \colon \Sigma \to X$ for $\Sigma$ an $n$-[[dimension|dimensional]] [[manifold]] or similar. One is to think of this as being the _trajectory: of an $(n-1)$-[[brane]] propagating in the [[target space]] $X$.
For the case $n = 1$ (for instance the [[relativistic particle]], a 0-[[brane]]) the term **worldline** for $\Sigma$ has a long tradition. Accordingly one calls $\Sigma$ the **worldvolume** of the given $(n-1)$-[[brane]] when $n \gt 1$. For the case $n=2$ (the case of relevance in [[string theory]]) one also says **worldsheet**.
Hence generally for any [[field theory]] defined on a [[worldvolume]] or [[spacetime]] $\Sigma$, and with type of fields determined by a [[field bundle]] $E \overset{fb}{\to} \Sigma$, one may think of a [[section]] of the field bundle as a _field trajectory_.
The [[space]] of all these is the _[[space of trajectories]]_ (space of histories).
## Related concepts
* [[worldline formalism]] (for [[scattering amplitudes]] in [[QFT]])
* [[sigma-model]]
* **worldvolume**
* worldsheet
* [[Riemann surface]], [[elliptic curve]], [[algebraic curve]]
* [[worldsheet parity operator]]
* [[target space]]
* [[background gauge field]]
* [[genus]]
* [[worldsheet instanton]]
* [[world sheets for world sheets]]
[[!redirects worldline]]
[[!redirects worldsheet]]
[[!redirects worldsheets]]
[[!redirects worldvolumes]]
[[!redirects worldlines]]
[[!redirects world line]]
[[!redirects world volume]]
[[!redirects world sheet]]
[[!redirects world-sheet]]
[[!redirects world-sheets]]
|
worldvolume-target supersymmetry of brane sigma-models | https://ncatlab.org/nlab/source/worldvolume-target+supersymmetry+of+brane+sigma-models |
**manifest [[supersymmetry]] for [[brane]] [[sigma-models]]**:
| manifest [[worldvolume]] supersymmetry | manifest target+worldvolume supersymmetry | manifest [[target space]] supersymmetry |
|---|---|---|
| [[NSR action functional]] | [[superembedding approach]] | [[Green-Schwarz action functional]] |
<center>
<img src="https://ncatlab.org/nlab/files/pBraneEmbedding.jpg" width="800">
</center>
> graphics grabbed from [[schreiber:Super-exceptional embedding construction of the M5-brane|FSS19c]] |
wormhole | https://ncatlab.org/nlab/source/wormhole |
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###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Related pages
* [[black hole]]
* [[ER = EPR]]
## References
The original article:
* {#EinsteinRosen35} [[Albert Einstein]], [[Nathan Rosen]], *The Particle Problem in the General Theory of Relativity*, Phys. Rev. **48** 73 (1935) ([doi:10.1103/PhysRev.48.73](https://doi.org/10.1103/PhysRev.48.73))
See also:
* Wikipedia, _[Wormhole](http://en.wikipedia.org/wiki/Wormhole)_
Argument that the image of *wormhole traversal* under [[AdS-CFT duality]] is *[[quantum teleportation]]*:
* [[Ping Gao]], [[Daniel Louis Jafferis]], [[Aron C. Wall]], *Traversable Wormholes via a Double Trace Deformation*, Journal of High Energy Physics **2017** 151 (2017) [[arXiv:1608.05687](https://arxiv.org/abs/1608.05687), <a href="https://doi.org/10.1007/JHEP12(2017)151">doi:10.1007/JHEP12(2017)151</a>]
* [[Juan Maldacena]], [[Douglas Stanford]], [[Zhenbin Yang]], *Diving into traversable wormholes*,Fortsch. Phys. **65** 5 (2017) 1700034 [[arXiv:1704.05333](https://arxiv.org/abs/1704.05333), [doi:10.1002/prop.201700034](https://doi.org/10.1002/prop.201700034)]
[[!redirects wormholes]]
[[!redirects Einstein-Rosen bridge]]
[[!redirects Einstein-Rosen bridges]] |
Wout Merbis | https://ncatlab.org/nlab/source/Wout+Merbis |
## Selected writings
On [[3d gravity]] via [[Chern-Simons theory]]:
* [[Wout Merbis]], _Chern-Simons-like Theories of Gravity_ ([arXiv:1411.6888](https://arxiv.org/abs/1411.6888))
* [[Marc Henneaux]], [[Wout Merbis]], Arash Ranjbar, _Asymptotic dynamics of $AdS_3$ gravity with two asymptotic regions_ ([arXiv:1912.09465](https://arxiv.org/abs/1912.09465))
Discussion of [[holographic entanglement entropy]] in asymptotically flat [[3d quantum gravity]] via [[Wilson line observables]] in [[Chern-Simons theory]]:
* [[Wout Merbis]], [[Max Riegler]], _Geometric actions and flat space holography_ ([arXiv:1912.08207](https://arxiv.org/abs/1912.08207))
category: people |
Wouter Swierstra | https://ncatlab.org/nlab/source/Wouter+Swierstra |
* [personal page](https://webspace.science.uu.nl/~swier004/)
* [institute page](https://www.uu.nl/staff/WSSwierstra)
## Selected writings
Introducing [[observational type theory]]:
* {#AltenkirchMcBrideSwierstra07} [[Thorsten Altenkirch]], [[Conor McBride]], [[Wouter Swierstra]], *Observational Equality, Now!*, PLPV '07: Proceedings of the 2007 workshop on Programming languages meets program verification (2007) 57-68 [ISBN:978-1-59593-677-6, [doi:10.1145/1292597.1292608](http://doi.org/10.1145/1292597.1292608), [pdf](https://www.cs.nott.ac.uk/~psztxa/publ/obseqnow.pdf)]
category: people
|
wrapped brane | https://ncatlab.org/nlab/source/wrapped+brane |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[brane]] [[worldvolume]] $\phi \colon \Sigma \longrightarrow X$ is said to _wrap_ a [[cycle]] $c \in H_{\dim(\Sigma)}(X)$ in [[spacetime]] $X$ when the pushforward $\phi_\ast [\Sigma] \in H_\bullet(X)$ of the [[fundamental class]] of $\Sigma$ is the class, $[c]$, of the given cycle in $X$. If the pushforward is a multiple of $[c]$, then the brane is said to wrap $c$ multiple times.
## Examples
Via the identification of [[D-brane charge]] in [[K-theory]], the K-theoretic [[McKay correspondence]] formalizes how D-branes wrap the fundamental cycles in the [[blow-up]] [[resolution of singularities|resultion]] of an [[ADE-singularity]] ([Gonzalez-Sprinberg & Verdier 83](#GSV83))
* [[membrane instanton]]
* [[M5-brane instanton]]
* [[fractional M2-brane]]
* [[3d-3d correspondence]]
* [[giant graviton]]
* in the [[Witten-Sakai-Sugimoto model]] (see there) for [[non-perturbative field theory|non-perturbative]] [[quantum chromodynamics]] [[baryons]] appear as wrapped [[D4-branes]]
<center>
<img src="https://ncatlab.org/nlab/files/BaryonsAsD4Branes.jpg" width="700">
</center>
> graphics grabbed from [Sugimoto 16](#Sugimoto16)
## Related concepts
* [[polarized brane]]
* [[brane intersection]]
* [[Dp-D(p+2)-brane bound state]]
* [[Dp-D(p+4)-brane bound state]]
## References
* {#GSV83} [[Gérard Gonzalez-Sprinberg]], [[Jean-Louis Verdier]], _Construction géométrique de la correspondance de McKay_, Ann. Sci. ́École Norm. Sup.16 (1983) 409–449. ([numdam](http://www.numdam.org/item?id=ASENS_1983_4_16_3_409_0))
* {#Sugimoto16} [[Shigeki Sugimoto]], _Skyrmion and String theory_, chapter 15 in [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))
[[!redirects wrapped branes]]
|
wreath | https://ncatlab.org/nlab/source/wreath |
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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Wreaths
* table of contents
{:toc}
## Idea
A **wreath** is a generalisation of a [[distributive law]] between two [[monads]] in a [[2-category]]. While a distributive law in a [[2-category]] $K$ can be seen as an object of $Mnd(Mnd(K))$, a wreath can be seen as an object of $EM(EM(K))$, where $EM$ denotes the completion of a 2-category under [[Eilenberg–Moore objects]]. Since $EM$ is a 2-monad, the multiplication $EM \circ EM \to EM$ produces from every wreath a **composite monad**.
## Related concepts
* [[distributive law]]
## Literature
* [[Steve Lack]], [[Ross Street]], _The formal theory of monads II_, Special volume celebrating the 70th birthday of Professor Max Kelly.
J. Pure Appl. Algebra 175 (2002), no. 1-3, 243--265.
[[!redirects wreaths]]
|
wreath product | https://ncatlab.org/nlab/source/wreath+product |
* [[wreath product of groups]]
* [[categorical wreath product]]
* [[weak wreath product]]
* [[wreath]] (in a 2-category)
[[!redirects wreath products]] |
wreath product of groups | https://ncatlab.org/nlab/source/wreath+product+of+groups | ## Definition ##
Let $H\leq Sym(X)$ and $G\leq Sym(Y)$ be permutation groups. Then their wreath product $H \wr G$ is defined as the [[semidirect product group|semidirect product]] $H^Y \rtimes G$ where $G$ operates on $H^Y$ by permuting the components. Note that this is itself a permutation group, acting on $X\times Y$ by letting $G$ act trivially on the first and naturally on the second factor and letting $H^Y$ act on $X\times Y$ such that the $y$-th component of $H^Y$ permutes $X\times\{y\}$ naturally and fixes everything else pointwise.
## Examples ##
* Every group $G$ is a permutation group on its underlying set $|G|$ (the regular representation aka the Cayley-Yoneda embedding $G\hookrightarrow Sym(|G|)$) and often only this special case is considered so that people speak of "**the** wreath product $H \wr G$" for abstract groups $G$ and $H$ without mentioning their permutation representations.
* $Dih(4)$ the dihedral group of order $2\cdot 4=8$ is the wreath product $C_2 \wr C_2$ (in this sense of regular representations).
* The sylow-$p$-subgroups $P_n$ of the symmetric groups $Sym(n)$ can be recursively described as the wreath product $C_p \wr P_a$ where $C_p$ is the cyclic group of order $p$ and $n=ap+r$ with $0\leq r \lneq p$.
* The sylow-$\ell$-subgroups of $GL_n(q)$ for $gcd(q,\ell)=1$ can be recursively described as a wreath product of the sylow-$\ell$-subgroup of a strictly smaller $GL_{m}(q)$ and a sylow-$\ell$-subgroups of a suitable symmetric group.
## Relation to the Borel construction
The homotopy quotient described in the [[Borel construction]],
$$
(X //G)_\bullet
\simeq_{iso}
X \times_G (E G)_\bullet
\,,
$$
has the wreath product $\pi_1(X)\wr G$ for fundamental group.
(...)
## References ##
(...) |
writhe | https://ncatlab.org/nlab/source/writhe |
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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
## Idea ##
**Writhe** is a measure of how much a knot or link _writhes_ around itself. As a 1-dimensional line cannot actually twist, this is not a [[knot invariant]] but is an invariant of [[framed knots]] (and links). Since a [[link diagram]] can be given a natural framing (the _blackboard framing_), it is possible to compute the writhe of a specific diagram. One place where this is used very neatly is to convert the [[Kauffman bracket]], which is an invariant of framed links, into the [[Jones polynomial]], being an invariant of ordinary links.
## Definition ##
Recall that a [[framed link]] can be thought of as a link together with a normal direction along each component, which we call the _framing direction_.
+-- {: .num_definition #writhe}
###### Definition
The **writhe** of a framed link is the [[linking number]] of the link with its infinitesimal displacement in the framing direction.
=--
For an oriented link diagram, the writhe is defined using the orientation of the crossings.
+-- {: .num_definition #writediag}
###### Definition
The **writhe** of an oriented [[link diagram]] is defined to be the sum of the orientations of its crossings.
=-- |
writing in the nLab | https://ncatlab.org/nlab/source/writing+in+the+nLab |
# Contents
* table of contents
{: toc}
## Introduction
Writing in the nLab should be a pleasant and rewarding experience. We are always glad for new people to join us in this collective effort, bringing their expertise to bear on the many topics touched on at this site. Many contributors put in a considerable amount of work to improving the nLab, and all such efforts are greatly appreciated.
However, the success of this project also depends in great degree on a certain ethos of cooperation and mutual respect that has evolved here. This page is meant to lay out some of the working principles that have helped to foster this atmosphere.
Before reading this page, you should have read or at least be aware of companion introductory nLab pages such as [[nPOV]], [[About]], and [[How to get started]] and [[HowTo]] (the last being especially useful for editing techniques). *This page* is meant to help smooth over possible misunderstandings that may arise about the nature of the project, and to help newcomers to nLab writing fit in more readily.
Perhaps the one overarching piece of advice we can impart to newcomers, which applies generally outside the nLab as well, can be condensed into a single principle (a form of the "prime directive"):
+-- {: .standout}
Let the mathematics, physics, and philosophy speak for itself; try to get your own self out of the way.
=--
The purpose of the community of nLab editors is to maximize insight, both for other members of that community and for others readers, and largely from a particular point of view called the [[nPOV]], as well as from the [[string theory]] point of view for physics related articles. When joining that community, your purpose should be the same. The existing community has found that this happens best in an atmosphere where we minimize distractions and help keep ourselves and our readers focused on the big picture. This advice may seem obvious, but there is a bit more to it than may appear at first glance, and so we elaborate on it below.
While scientific results are recorded or archived at the nLab, and undergo further polishing and revision there, the principal organ of communication we have is the [[nForum]]. A basic rule is that
+-- {: .standout}
With few exceptions, all edits to the nLab (either the creation of a new page or the revision of an extant one) should be announced at the nForum, in the "Latest Changes" category.
=--
The only real exceptions are very minor edits such as correction of spelling mistakes or obvious typos or indisputable grammatical errors. However, because of this rule there can at times be a large volume of Latest Changes posts; thus a corollary is that Latest Changes posts at the forum should generally be kept very short and to the point. They should also include a link to the nLab page in question (links at the nForum are created with the same `[[syntax]]` as on the nLab itself).
The nForum is also home to wide-ranging discussion of scientific matters related to this project, and we encourage newcomers to join these conversations as well and get to know the existing members of the community. We ask each other questions and suggest ideas; then when those ideas yield fruit, we often record the results back in the nLab.
The genesis of every article is recorded in a revision history. Thus every contributor gets some credit (the name or alias of each contributor is shown at the bottom of each of their revisions, and the changes introduced by each revision are also faithfully recorded). But the nLab belongs to us all. Speaking optimistically, we could liken the nLab to one of the great cathedrals of Europe, where unnamed architects and artisans worked together to build a monument to a glory far surpassing that of any one individual. Feeling oneself as part of such a collective project can be a great reward.
+-- {: .num_remark}
###### Remark
There is a uniform Joker name applied to those who don't perform an edit under their own names: "[[Anonymous]]". (This was originally "Anonymous Coward", a joke.) An edit attributed to "Anonymous" could be a case of someone who didn't submit under their real name because they forgot, or it could be someone who has good reason not to submit under their real name -- whatever. It's none of our business. We appreciate all the good edits made by the "Anonymous Cowards" out there just as we do those made by everyone else. However, we do encourage everyone to use their own name, or at least a consistent alias, if possible.
=--
## Some obvious truisms
There are some fairly obvious corollaries to be derived from the precept "get your own self out of the way" (with the specific implication "this is really not about you").
* The nLab is not a place to promote pet theories and "revolutionary ideas".
We've had a bunch of people pass through who want to talk about e.g., their revolutionary approach to point-set topology, or how to use categorical ideas to develop a private theory of music, or deep insights into how categorical concepts shed light on the very structures of consciousness, or their ruminations on mathematical aesthetics, or their ideas on how gravitation should be understood in terms of electromagnetic effects.
In all such cases, we've had to tell them to go away and leave us alone. Not that we deem all such people to be crackpots; some may actually be fine thinkers. But we have other things we need to be doing.
* The nLab is not a place where you just plop down notes indiscriminately.
People who have seen the description at [[nLab]] that this site is a "public place where people can make notes on stuff" may get the wrong idea. For example, students have sometimes set up shop and set about writing up stuff that is currently occupying them, without regard to what is going on in the rest of the nLab. They are usually told (politely) that they are misusing the nLab, and that such articles will unfortunately have to be removed.
A more accurate description, given at the article [[About]], is that the nLab is perhaps best conceived as a group lab book, with the idea that we scribble things down that are of common interest to other researchers here. That's closer, but it's a bit simplistic. What makes the nLab somewhat different from a traditional group lab book are the numerous hyperlinks between the parts of the nLab, meaning that the scribblings should (eventually) be well integrated with what is taking place elsewhere. The integration works best when editors become somewhat aware of related articles and what has gone into them.
Of course newcomers are not expected to know everything about what related articles already exist on the nLab; the established contributors are generally quite happy to help make connections between new and old material. In return, however, newcomers are asked to be sensitive and receptive to this integrative process. You could think of the nLab as already an evolved organism, with limbs and extensions being constantly grafted on, but in a controlled way so that the organism does not reject the graft as being of the wrong type. In Piagetian terms, new material might either be easily assimilated into the existing structure; or, some extra arrangements might need to be made to accommodate it comfortably, a process that may require patience and good will between the participants.
Again, the nForum is the place where we openly discuss such matters.
* The nLab is not a place to conduct literary experiments.
There is plenty of scope for different forms of personal expression; there is not a single uniform style throughout the nLab. That being said, idiosyncrasies in style are much the secondary consideration; getting the goods out there in useful efficient form for the reader (including one's self) is primary.
Unless you are supremely confident in your expository skills, the advice if you are a newcomer is generally to write straight mathematics (or physics, etc.), aiming for formal definitions and statements of results and the like. Not write imagined dialogues in the manner of Lakatos's Proofs and Refutations, or long essays, or stud the discussion with abundant scholarly footnotes and other marginalia. Such devices have a tendency to clutter or distract or call undue attention to themselves. Anything that distracts or deflects the reader's attention or any unnecessary digression is, in *mathematical writing*, generally bad.
Similarly, wordplay should be used very moderately, and the impulse to flaunt one's broad knowledge and culture (in the manner of David Foster Wallace, say) should almost never be indulged. A little levity can be a fine thing, but again, the overriding principle to remember is that what we want most of all is to convey scientific insight, and this is best accomplished when the English is *smooth and unobtrusive* -- doesn't call attention to itself. You may be the greatest wit since Oscar Wilde, but your privately amusing puns and learned allusions may be getting in the way here, and actually your readers might resent you for it. Just try to be sensitive to that, please. If you want a place to display your erudition, start your own blog.
To avoid any misunderstanding: we are not suggesting that one should write the mathematics (physics, philosophy) in a stuffy or formal or bureaucratic way. To the contrary, it can often be effective to adopt a light conversational tone, taking the reader into your confidence as it were; on the nLab we are often more conversational than is the traditional style in published papers. But the expression should be an unaffected one (being too chummy is as bad as being too precious). Just remember the prime directive, and you should be fine.
## Writing mathematics and physics
All nLab editors are expected to use the "iTeX" mathematics typesetting language of the instiki software, which displays in web browsers using MathML/MathJax. If you know TeX or LaTeX, this should not be very difficult to adapt to; most mathematical typesetting commands will be the same. Some of the more important (though still fairly minor) differences are documented in the [[FAQ]].
In the first place, to get started, writing a "stub" article (even just a sentence of two, just to get an idea down) is perfectly fine. Hopefully that will grow into something more polished, maybe with the input of others, but you shouldn't worry -- just make the stub intelligible enough for someone else to pick up the thread. Some people at this stage start a list of references to come back to, which can also be a good idea. If nothing else, a link to a relevant [Wikipedia](https://en.wikipedia.org/wiki/Portal:Mathematics)/[Encyclopedia of Mathematics](https://www.encyclopediaofmath.org/index.php/Main_Page)/[MathWorld](http://mathworld.wolfram.com/)/[SEP](https://plato.stanford.edu/) page --- which should have some references of its own --- can be the germ of a reference list.
### General organization
Most nLab articles are titled by a *concept* (usually to be formally defined in the article). There are also pages titled by naming a result (e.g., Tychonoff theorem), sometimes in synopsized form (e.g., closed projection characterization of compactness). Some articles are titled by a person's name, others by a book title (sometimes as referred to colloquially, e.g. Elephant). But nLab pages are not usually named with the sort of title you might see on a mathematics paper, such as "On a conjecture of Grothendieck".
If you look at a typical nLab entry which has grown up a little, you will frequently find it organized along the following lines. First comes an Idea or Introduction section, usually with language similar to language you might use during a hike with a colleague on mathematics, with few to no symbols and giving the general idea without getting too technical. Then you might see a Definition(s) section which gives one or several precise definitions, followed by a Properties section or Basic Results section, an Examples section, a Related Concepts section, and a References section. By no means is this (Wikipedia-like) organization rigidly adhered to, but it is certainly common. At the top (if the author(s) of the page didn't forget it), you will see a Table of Contents which serves as a broad outline of the contents (with sections, maybe some subsections, hopefully not *too* many sub-subsections -- more subs than that and it can get ridiculous).
Inside each section, the material is often arranged by making effective use of environments (definition, theorem, lemma, remark, etc.). How these are used is a matter of personal discretion. The best advice is to aim for a happy medium, between one extreme of erecting great walls of text (which can look like a sermon being preached, and where the reader can get lost), and another where the material is so finely diced into atomic environment units that the reader is in for a bumpy ride. Ideally the material is arranged somewhere in the middle, allowing for a smooth narrative flow, and where the environments are not too cluttered but act as well-placed signposts. Visually the arrangement should look appealingly smooth. Display lines (as in TeX math display mode) can often aid in that, breaking up big chunks of text and allowing for emphases where appropriate -- but too many of them and it backfires.
Choice of notation is largely up to the individual. It would be impossible to maintain a consistent choice of notation throughout the entirety of the nLab, because there are just too many authors and the nLab is too vast to keep track of it all. There is no Central Planning Committee for this type of thing. Thus, regarding notation, you should think of each article as a self-contained unit --- but do please try to maintain notational consistency within each article.
This means especially: try to be respectful of the notation others have already introduced. If they use a calligraphic font while discussing a concept, then it will help the reader if you use the same font for the same concept later in the article, even if you don't personally like calligraphic fonts that much. Above all: don't take it upon yourself to just change all the notation to suit your own liking. If you feel strongly about notation -- and many of us do -- then you can plead and argue your case at the nForum for changes you want made (within reason of course -- arguments shouldn't be overly long). Hey, you may be right, and the rest of the community will listen. But if a consensus or compromise is not reached, then please find it within yourself to adapt to what was there. (When you start an article, you can introduce your preferred notation, and then you'll be on the other side of that situation, right?)
For further words of wisdom about notation and sundry other expositional matters (many of which are being discussed without particular attribution in this article), please read [Halmos](#Halmos); it is strongly recommend as a great and helpful article written by an unquestionably excellent expositor.
### When in doubt, follow existing norms
The format you will see in a typical nLab article is somewhere between a Wikipedia article you might see on a topic in mathematics or mathematical physics, and an academic article in one of those areas. The organizational outline seen in the table of contents is very Wikipedia-like, but unlike Wikipedia, the development may be like that of an academic article with formal definition environments, theorem environments, etc., and with point of view that is not neutral with regard to where we tend to place emphasis: the [[nPOV]] and [[string theory]]. We also may conduct some original research, quite unlike Wikipedia.
But in most respects, there should not be too many surprises for people familiar with both types of articles. For example, under a References section we follow a standard type of bibliographic format (author(s), title in italics, journal with volume and number or publisher, date, page numbers), linking to online material wherever possible (always legally!!). In the text where one wants to refer (and link) to a bibliographic item, there is no official format for this, but the tendency is to use something simple like author-name-date in brackets, rather than putting a number in brackets. The page [[HowTo]] gives detailed instructions on how to do this.
Occasionally there is redundancy or repetition in the nLab; to a degree this is tolerated (a definition or proof may be recalled). However, we don't want *too* much of that. If you set out to create a new page, you may be told when you announce it at the nForum that we have that on some other page with a different title. Don't be discouraged: the material will of course not be an exact duplicate and there will surely be things which can be usefully incorporated into the extant article.
**But:** in that situation there may have to be a "redirect", where the term you used to title your article should now be a term which redirects to the extant article (or, it could go in the other direction, but usually not), and then the newly created article, after relevant material has been incorporated, is made "history". Please see [[HowTo]] for detailed instructions on how to perform this action. It's a slightly fiddly procedure though, and usually a more experienced editor will be happy to do it if needed.
### Anything I shouldn't do?
Specifically, there are some features enabled by the software, but which are somewhat deprecated or downplayed at this point in time in nLab culture.
One such feature is the "query box". This was once used so that someone reading an nLab article could insert a question or even have a running sequence of comments embedded right there within the article. But we found this didn't really work well, one reason being there it didn't notify author(s) that a question was being asked, so that you'd see the query only if you chanced to go back to the article. Some queries might sit for years before being noticed! And we're still trying to clean up query boxes.
So please don't use this the query box feature -- we ask that you ask your question or post your comment directly at the nForum. If you see an ancient query box, you or someone else should shunt it over to the nForum which serves an an archive as well.
Another thing we want to downplay (more recently) is the use of the footnote feature. It's really not a footnote at all; functionally it's more of an endnote that gets plopped slightly awkwardly at the bottom of the page in the same sized font as anything else, but without any html heading that announces: Footnotes. The bottom line is that it doesn't look very good and calls ungainly attention to itself, something that a footnote is not supposed to do.
Instead, if it's about something minor, consider using a link to another part of the nLab (or wherever) where the point is discussed. If it's less than minor, then maybe consider placing the remark in a Remark environment there in the text. In case of doubt, ask for advice at the nForum.
## Respecting the styles of other authors
This is perhaps the touchiest aspect of these recommendations, so we will say it as plainly as possible. The bottom line is this: the nLab can only function in an atmosphere of mutual consideration and respect; it will never work if there are overhanging attitudes of "my way or the highway". Somehow it *has* been at least moderately successful, we feel, despite the fact that there are already at least a few strong-willed personalities involved.
Perhaps the main reason it has worked is a prevailing ethos of cooperation, and in particular an unspoken obeisance to a "prime directive" (repeating ourselves somewhat):
+-- {: .standout}
Try to arrange nLab material so as to maximize scientific insight, letting the power of the [[nPOV]] shine through, in smooth and unobtrusive language. Try to get your own self and personal idiosyncrasies out of the way of this happening.
=--
The typical underlying tensions between
1. not wishing to step on the toes of others who have put some work into the nLab, and
1. wanting nevertheless to revise the nLab to make it better
are seemingly resolved by correctly interpreting and implementing that directive, through controlled discussion at the nForum. Of course, helping matters along is a semi-cohesive philosophy at the core of our (otherwise loosely connected) working group, of the importance of category theory and higher category theory to organize and simplify mathematics (and physics and in some respects philosophy). Those who do not share in that vision (or mission, whatever you wish to call it) might not be so easily assimilated into the nLab group -- although we are open to possibilities -- since such people may have very different ideas of what to emphasize in mathematics.
An example where a nontrivial rewriting of an article is welcome is where it makes the underlying principles clearer and the development more economical, or where it lays bare unifications with other subject areas, etc. In other words, where the *actual mathematics* is improved (from the [[nPOV]]). Sometimes entire articles may be revamped for such reasons.
Examples where rewrites are maybe not so welcome is where it comes to differences in style between otherwise reasonably competent writers. One writer may prefer a slightly more conversational tone (using 2nd person), and another a slightly more formal tone (using 3rd person). If one feels an impulse to change something in the style of another, the first question that should be asked is: is this *really* going to materially improve the reader's reception? Sometimes it might, oftentimes not really. But this is felt to be an important enough matter to make a revision, it should be discussed *beforehand* at the nForum -- don't shoot first and ask questions later. In all other cases, it's probably better to let it go. Concentrate on the mathematics.
Another small example which may be worth addressing: discrepancies between American and British (Commonwealth) spellings. One might see different spellings of the same word in the same article! Should alternate spellings (which are correct according to the author's dictionary) within the article be aligned to make them consistent? The nLab way seems to be to respect each other's differences in this (minor) respect: if one person writes 'neighborhood' and another 'neighbourhood', then that's okay: let it go. Similarly, we don't try to enforce consistent standards of punctuation and formatting, where conventions also vary regionally: e.g. do quotations use single or double quotes, does punctuation go inside or outside the quotation marks, is an apostrophe written as `don't` or `don\'t`, etc. This extends also to respecting each other's choice of whether to use `:` or `\colon` in mathematical notation.
Unlike the above advice about notation, such variations in spelling and punctuation may occur even within a single nLab page, and there is no need to try to make them uniform. Think of it as a token of respect we have for one another: we may come from all corners of the earth, but we come together to build the nLab. We can tolerate a small difference in cultural background such as this. (It was, however, agreed informally long ago to use American English spellings in page titles, although of course other spellings should be included as redirects.)
## Related pages
* [[nLab]]
* [[nForum]]
* [[nPOV]]
* [[About]]
* [[HowTo]]
* [[How to get started]]
## References
* Paul Halmos, *How To Write Mathematics*, Enseign. Math. (2) 16 (1970), 123-152. ([online pdf](http://www2.math.uu.se/~takis/ETC/Halmos_howToWriteMath.pdf))
[[!redirects writing in the nLab]]
[[!redirects writing on the nLab]]
|
Wronskian | https://ncatlab.org/nlab/source/Wronskian | Given a set of $n$ functions $f_1,\ldots,f_n$, one can define the matrix
$$
W(f_1,\ldots,f_n) = \left( \array{f_1 & f_2 & \cdots & f_n\\ f_1' & f_2' &\cdots & f_n'\\ \cdot &\cdot &\cdot &\cdots\\ f_1^{(n-1)} & f_2^{(n-1)} &\cdots &f_n^{(n-1)}}\right)
$$
The __Wronskian__ is its [[determinant]]. It is used in the study of linear independence of solution of differential equations and in mathematical physics.
* wikipedia [Wronskian](http://en.wikipedia.org/wiki/Wronskian) |
Wu class | https://ncatlab.org/nlab/source/Wu+class |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_Wu classes_ are a type of [[universal characteristic class]] in $\mathbb{Z}_2$-[[cohomology]] that refine the [[Stiefel-Whitney classes]].
## Definition
For $X$ a [[topological space]] equipped with a class $E : X \to B SO(n)$ (a real [[vector bundle]] of some [[rank]] $n$), write
$$
w_k \in H^k(X, \mathbb{Z}_2)
$$
for the [[Stiefel-Whitney classes]] of $X$. Moreover, write
$$
\cup : H^k(X, \mathbb{Z}_2) \times H^l(X, \mathbb{Z}_2)
\to
H^{k+l}(X, \mathbb{Z}_2)
$$
for the [[cup product]] on $\mathbb{Z}_2$-[[cohomology groups]] and write
$$
Sq^k(-) : H^l(X, \mathbb{Z}_2) \to H^{k+l}(X, \mathbb{Z}_2)
$$
for the [[Steenrod square]] operations.
+-- {: .num_defn #WuClassesBySteenrodSquares}
###### Definition
The **Wu class**
$$
\nu_k
\in
H^k(X,\mathbb{Z}_2)
$$
is defined to be the class that "represents" $Sq^k(-)$ under the cup product, in the sense that for all $x \in H^{n-k}(X, \mathbb{Z}_2)$ where $n$ is the dimension of $X$, we have
$$
Sq^k(x) = \nu_k \cup x
\,.
$$
=--
(e.g. [Milnor-Stasheff 74, p. 131-133](#MilnorStasheff74))
+-- {: .num_remark }
###### Remark
In other words this says that the lifts of Wu classes to [[integral cohomology]] ([[integral Wu structures]]) are _[[characteristic element of a bilinear form|characteristic elements]]_ of the [[intersection product]] on integral cohomology, inducing [[quadratic refinements]].
=--
## Properties
### Relation to Stiefel-Whitney classes
The total [[Stiefel-Whitney class]] $w$ is the total [[Steenrod square]] of the total Wu class $\nu$.
$$
w = Sq(\nu)
\,.
$$
Solving this for the components of $\nu$ in terms of the components of $w$, one finds the first few Wu classes as [[polynomials]] in the Stiefel-Whitney classes as follows
* $\nu_1 = w_1$;
* $\nu_2 = w_2 + w_1^2$
* $\nu_3 = w_1 w_2$
* $\nu_4 = w_4 + w_3 w_1 + w_2^2 + w_1^4$
* $\nu_5 = w_4 w_1 + w_3 w_1^2 + w_2^2 w_1 + w_2 w_1^3$
...
### Relation to Pontryagin classes
{#RelationToPontryaginClasses}
+-- {: .num_prop #InTermsOfPontryagin}
###### Proposition
Let $X$ be an [[orientation|oriented]] [[manifold]] $T X : X \to B SO(n)$ with [[spin structure]] $\hat T X : X \to B Spin(n)$. Then the following classes in [[integral cohomology]] of $X$, built from [[Pontryagin classes]], coincide with Wu-classes under mod-2-reduction $\mathbb{Z} \to \mathbb{Z}_2$:
* $\nu_4 = \frac{1}{2} p_1$
* $\nu_8 = \frac{1}{8}(11 p_1^2 - 20 p_2)$
* $\nu_{12} = \frac{1}{16}(37 p_1^3 - 100 p_1 p_2 + 80 p_3)$.
(all products are [[cup product|cup products]]).
=--
This is discussed in ([Hopkins-Singer, page 101](#HopkinsSinger)).
+-- {: .num_cor #DivisibilityOfCupSquare}
###### Corollary
Suppose $X$ is 8 dimensional. Then, for $G \in H^4(X, \mathbb{Z})$ any integral 4-class, the expression
$$
G \cup G - G \cup \frac{1}{2}p_1 \in H^4(X, \mathbb{Z})
$$
is always even (divisible by 2).
=--
+-- {: .proof}
###### Proof
By the basic properties of Steenrod squares, we have for the 4-class $G$ that
$$
G \cup G = Sq^4(G)
\,.
$$
By the definition \ref{WuClassesBySteenrodSquares} of Wu classes, the image of this integral class in $\mathbb{Z}_2$-coefficients equals the cup product with the Wu class
$$
G \cup G - G \cup \frac{1}{2}p_1
=
Sq^4(G) - G \cup \nu_4
=
0 \; mod \; 2.
\,,
$$
where the first step is by prop. \ref{InTermsOfPontryagin}.
=--
## Applications
### To higher dimensional Chern-Simons theory
+-- {: .num_remark}
###### Remark
The relation \ref{DivisibilityOfCupSquare} plays a central role in the definition of the [[higher dimensional Chern-Simons theory|7-dimensional Chern-Simons theory]] which is [[holographic principle|dual]] to the [[self-dual higher gauge theory]] on the [[M5-brane]]. In this context it was first pointed out in ([Witten 1996](#Witten)) and later elaborated on in ([Hopkins-Singer](#HopkinsSinger)).
Specifically, in this context $G$ is the 4-class of the [[circle n-bundle with connection|circle 3-bundle]] underlying the [[supergravity C-field]], subject to the quantization condition
$$
G_4 = \frac{1}{2}(\frac{1}{2}p_1) + a
\,,
$$
for some $a \in H^4(X, \mathbb{Z})$, which makes direct sense as an equation in $H^4(X, \mathbb{Z})$ if the [[spin structure]] on $X$ happens to be such $\frac{1}{2}p_1$ is further divisible by 2, and can be made sense of more generally in terms of [[twisted cohomology]] (which was suggested in ([Witten 1996](#Witten)) and made precise sense of in ([Hopkins-Singer](#HopkinsSinger)) ).
For simplicity, assume that $\frac{1}{2}p_1$ of $X$ is further divisible by 2 in the following. We then may consider direct refinements of the above ingredients to [[ordinary differential cohomology]] and so we consider [[circle n-bundle with connection|differential cocycles]] $\hat a, \hat G \in \hat H^4(X)$ with
\[
\hat G = \frac{1}{2}(\frac{1}{2}\hat \mathbf{p}_1) + \hat a
\in \hat H^4(X)
\,,
\label{DifferentialQuantizationCondition}
\]
where the differential refinement $\frac{1}{2}\hat \mathbf{p}_1$ is discussed in detail at _[[differential string structure]]_.
Now, after [[Kaluza-Klein mechanism|dimensional reduction]] on a 4-[[sphere]], the [[action functional]] of [[11-dimensional supergravity]] on the remaining 7-dimensional $X$ contains a [[higher dimensional Chern-Simons theory|higher Chern-Simons term]] which up to prefactors is of the form
$$
\hat G \mapsto
\exp i
\int_X
(
\hat G \cup \hat G -
(\frac{1}{4}\hat \mathbf{p}_1)^2
)
\,,
$$
where
* the cup product now is the differential [[Beilinson-Deligne cup product]] refinement of the integral cup product;
* the symbol $\exp(i \int_X (-))$ denotes [[fiber integration in ordinary differential cohomology]].
Using (eq:DifferentialQuantizationCondition) this is
$$
\cdots
=
\exp i
\int_X
\left(
\hat a \cup \hat a + \hat a \cup \frac{1}{2}\hat \mathbf{p}_1
\right)
\,.
$$
But by corollary \ref{DivisibilityOfCupSquare} this is further divisible by 2.
Hence the generator of the group of higher Chern-Simons action functionals is one half of this
$$
\hat G
\mapsto
\exp i
\int_X
\frac{1}{2}
(
\hat G \cup \hat G -
(\frac{1}{4}\hat \mathbf{p}_1)^2
)
\,.
$$
In ([Witten 1996](#Witten)) it is discussed that the space of [[states]] of this "fractional" functional over a 6-dimensional $\Sigma$ is the space of [[conformal blocks]] of the [[self-dual higher gauge theory]] on the [[M5-brane]].
=--
## Related concepts
* [[integral Wu structure]], [[twisted Wu structure]]
* [[shifted C-field flux quantization]]
* [[Pontryagin class]], [[Stiefel-Whitney class]], [[one-loop anomaly polynomial I8]]
* [[Euler class]]
## References
The original reference is
* [[Wen-Tsun Wu]], _On Pontrjagin classes: II_ Sientia Sinica 4 (1955) 455-490
See also around p. 228 of
* {#MilnorStasheff74} [[John Milnor]], [[Jim Stasheff]], _Characteristic classes_, Princeton University Press (1974)
and section 2 of
* Yanghyun Byun, _On vanishing of characteristic numbers in Poincaré complexes_, Transactions of the AMS, vol 348, number 8 (1996) ([pdf](http://www.ams.org/journals/tran/1996-348-08/S0002-9947-96-01495-X/S0002-9947-96-01495-X.pdf))
and
* [[Robert Stong]], Toshio Yoshida, _Wu classes_ Proceedings of the American Mathematical Society Vol. 100, No. 2, (1987) ([JSTOR](http://www.jstor.org/pss/2045970))
Details are reviewed in appendix E of
* [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_
{#HopkinsSinger}
This is based on or motivated from observations in
* [[Edward Witten]], _Five-Brane Effective Action In M-Theory_ ([arXiv:hep-th/9610234](http://arxiv.org/abs/hep-th/9610234))
{#Witten}
More discussion of Wu classes in this physical context is in
* [[Hisham Sati]], _Twisted topological structures related to M-branes II: Twisted $Wu$ and $Wu^c$ structures_ ([arXiv:1109.4461](http://arxiv.org/abs/1109.4461))
which also summarizes many standard properties of Wu classes.
[[!redirects Wu classes]] |
WZW term of QCD chiral perturbation theory -- references | https://ncatlab.org/nlab/source/WZW+term+of+QCD+chiral+perturbation+theory+--+references |
### The WZW term of QCD chiral perturbation theory
{#WZWTermOfChiralPerturbationTheoryReferences}
The [[gauged WZW model|gauged]] [[WZW term]] of [[chiral perturbation theory]]/[[quantum hadrodynamics]] which reproduces the [[chiral anomaly]] of [[QCD]] in the [[effective field theory]] of [[mesons]] and [[Skyrmions]]:
#### General
{#WZWTermOfChiralPerturbationTheoryReferencesGeneral}
The original articles:
* [[Julius Wess]], [[Bruno Zumino]], _Consequences of anomalous Ward identities_, Phys. Lett. B 37 (1971) 95-97 ([spire:67330](https://inspirehep.net/literature/67330), <a href="https://doi.org/10.1016/0370-2693(71)90582-X">doi:10.1016/0370-2693(71)90582-X</a>)
* {#Witten83a} [[Edward Witten]], _Global aspects of current algebra_, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 422-432 (<a href="https://doi.org/10.1016/0550-3213(83)90063-9">doi:10.1016/0550-3213(83)90063-9</a>)
See also:
* O. Kaymakcalan, S. Rajeev, J. Schechter, _Nonabelian Anomaly and Vector Meson Decays_, Phys. Rev. D 30 (1984) 594 ([spire:194756](https://inspirehep.net/literature/194756))
Corrections and streamlining of the computations:
* Chou Kuang-chao, Guo Han-ying, Wu Ke, Song Xing-kang, _On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action_, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 67-69 (<a href="https://doi.org/10.1016/0370-2693(84)90986-9">doi:10.1016/0370-2693(84)90986-9</a>))
* H. Kawai, S.-H. H. Tye,
_Chiral anomalies, effective lagrangians and differential geometry_, Physics Letters B Volume 140, Issues 5–6, 14 June 1984, Pages 403-407 (<a href="https://doi.org/10.1016/0370-2693(84)90780-9">doi:10.1016/0370-2693(84)90780-9</a>)
* J. L. Mañes, _Differential geometric construction of the gauged Wess-Zumino action_, Nuclear Physics B Volume 250, Issues 1–4, 1985, Pages 369-384 (<a href="https://doi.org/10.1016/0550-3213(85)90487-0">doi:10.1016/0550-3213(85)90487-0</a>)
* Tomáš Brauner, Helena Kolešová, _Gauged Wess-Zumino terms for a general coset space_, Nuclear Physics B Volume 945, August 2019, 114676 ([doi:10.1016/j.nuclphysb.2019.114676](https://doi.org/10.1016/j.nuclphysb.2019.114676))
See also
* Yasunori Lee, [[Kantaro Ohmori]], [[Yuji Tachikawa]], _Revisiting Wess-Zumino-Witten terms_ ([arXiv:2009.00033](https://arxiv.org/abs/2009.00033))
Interpretation as [[Skyrmion]]/[[baryon current]]:
* [[Jeffrey Goldstone]], [[Frank Wilczek]], _Fractional Quantum Numbers on Solitons_, Phys. Rev. Lett. 47, 986 (1981) ([doi:10.1103/PhysRevLett.47.986](https://doi.org/10.1103/PhysRevLett.47.986))
* {#Witten83b} [[Edward Witten]], _Current algebra, baryons, and quark confinement_, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 433-444 (<a href="https://doi.org/10.1016/0550-3213(83)90064-0">doi:10.1016/0550-3213(83)90064-0</a>)
* {#AdkinsNappi84} [[Gregory Adkins]], [[Chiara Nappi]], _Stabilization of Chiral Solitons via Vector Mesons_, Phys. Lett. 137B (1984) 251-256 ([spire:194727](http://inspirehep.net/record/194727), <a href="https://doi.org/10.1016/0370-2693(84)90239-9">doi:10.1016/0370-2693(84)90239-9</a>)
(beware that the two copies of the text at these two sources differ!)
* {#RhoEtAl16} [[Mannque Rho]] et al., _Introduction_, In: [[Mannque Rho]] et al. (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))
Concrete form for $N$-[[flavor (particle physics)|flavor]] [[quantum hadrodynamics]] in 2d:
* C. R. Lee, H. C. Yen, _A Derivation of The Wess-Zumino-Witten Action from Chiral Anomaly Using Homotopy Operators_, Chinese Journal of Physics, Vol 23 No. 1 (1985) ([spire:16389](https://inspirehep.net/literature/16389), [[LeeYenWZW85.pdf:file]])
Concrete form for 2 [[flavor physics|flavors]] in 4d:
* Masashi Wakamatsu, _On the electromagnetic hadron current derived from the gauged Wess-Zumino-Witten action_, ([arXiv:1108.1236](https://arxiv.org/abs/1108.1236), [spire:922302](https://inspirehep.net/literature/922302))
#### Including light vector mesons
{#WZWTermOfChiralPerturbationTheoryReferencesIncludingLightVectorMesons}
Concrete form for 2-[[flavor (particle physics)|flavor]] [[quantum hadrodynamics]] in 4d with [[light meson|light]] [[vector mesons]] included ([[omega-meson]] and [[rho-meson]]):
* {#MeissnerZahed86} [[Ulf-G. Meissner]], [[Ismail Zahed]], equation (6) in: _Skyrmions in the Presence of Vector Mesons_, Phys. Rev. Lett. 56, 1035 (1986) ([doi:10.1103/PhysRevLett.56.1035](https://doi.org/10.1103/PhysRevLett.56.1035))
* [[Ulf-G. Meissner]], [[Norbert Kaiser]], [[Wolfram Weise]], equation (2.18) in: _Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties_, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (<a href="https://doi.org/10.1016/0375-9474(87)90463-5">doi:10.1016/0375-9474(87)90463-5</a>)
* [[Ulf-G. Meissner]], equation (2.45) in: _Low-energy hadron physics from effective chiral Lagrangians with vector mesons_, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (<a href="https://doi.org/10.1016/0370-1573(88)90090-7">doi:10.1016/0370-1573(88)90090-7</a>)
* {#Kaiser00} Roland Kaiser, equation (12) in: _Anomalies and WZW-term of two-flavour QCD_, Phys. Rev. D63:076010, 2001 ([arXiv:hep-ph/0011377](https://arxiv.org/abs/hep-ph/0011377), [spire:537600](https://inspirehep.net/literature/537600))
#### Including heavy scalar mesons
{#WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyScalarMesons}
Including [[heavy mesons|heavy]] [[scalar mesons]]:
specifically [[kaons]]:
* [[Curtis Callan]], [[Igor Klebanov]], equation (4.1) in: _Bound-state approach to strangeness in the Skyrme model_, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (<a href="https://doi.org/10.1016/0550-3213(85)90292-5">doi10.1016/0550-3213(85)90292-5</a>)
* [[Igor Klebanov]], equation (99) of: _Strangeness in the Skyrme model_, in: D. Vauthrin, F. Lenz, J. W. Negele, _Hadrons and Hadronic Matter_, Plenum Press 1989 ([doi:10.1007/978-1-4684-1336-6](https://link.springer.com/book/10.1007/978-1-4684-1336-6))
* N. N. Scoccola, D. P. Min, H. Nadeau, [[Mannque Rho]], equation (2.20) in: _The strangeness problem: An $SU(3)$ skyrmion with vector mesons_, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (<a href="https://doi.org/10.1016/0375-9474(89)90029-8">doi:10.1016/0375-9474(89)90029-8</a>)
specifically [[D-mesons]]:
(...)
specifically [[B-mesons]]:
* [[Mannque Rho]], D. O. Riska, N. N. Scoccola, above (2.1) in: _The energy levels of the heavy flavour baryons in the topological soliton model_, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages343–352 (1992) ([doi:10.1007/BF01283544](https://doi.org/10.1007/BF01283544))
#### Including heavy vector mesons
{#WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyVectorMesons}
Inclusion of [[heavy mesons|heavy]] [[vector mesons]]:
specifically [[K*-mesons]]:
* S. Ozaki, H. Nagahiro, [[Atsushi Hosaka]], Equations (3) and (9) in: _Magnetic interaction induced by the anomaly in kaon-photoproductions_, Physics Letters B Volume 665, Issue 4, 24 July 2008, Pages 178-181 ([arXiv:0710.5581](https://arxiv.org/abs/0710.5581), [doi:10.1016/j.physletb.2008.06.020](https://doi.org/10.1016/j.physletb.2008.06.020))
#### Including electroweak interactions
{#WZWTermOfChiralPerturbationTheoryReferencesIncludingElectroweakInteractions}
Including [[electroweak fields]]:
* J. Bijnens, G. Ecker, A. Picha, _The chiral anomaly in non-leptonic weak interactions_, Physics Letters B Volume 286, Issues 3–4, 30 July 1992, Pages 341-347 (<a href="https://doi.org/10.1016/0370-2693(92)91785-8">doi:10.1016/0370-2693(92)91785-8</a>)
* [[Gerhard Ecker]], [[Helmut Neufeld]], [[Antonio Pich]], _Non-leptonic kaon decays and the chiral anomaly_, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (<a href="https://doi.org/10.1016/0550-3213(94)90623-8">doi:10.1016/0550-3213(94)90623-8</a>)
Discussion for the full [[standard model of particle physics]]:
* [[Jeffrey Harvey]], Christopher T. Hill, Richard J. Hill, _Standard Model Gauging of the WZW Term: Anomalies, Global Currents and pseudo-Chern-Simons Interactions_, Phys. Rev. D77:085017, 2008 ([arXiv:0712.1230](https://arxiv.org/abs/0712.1230))
|
WZW-type superstring field theory | https://ncatlab.org/nlab/source/WZW-type+superstring+field+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### $\infty$-Wess-Zumino-Witten theory
+--{: .hide}
[[!include infinity-Wess-Zumino-Witten theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[string field theory]] for the [[superstring]] whose [[action functional]] is of the form of the [[Wess-Zumino-Witten model]], with worldsheet fields replaced by [[second quantization|second-quantum]] string fields.
## References
### Definitions
The WZW-type action functional for open superstring field theory is due to
* [[Nathan Berkovits]], _Super-Poincaré invariant superstring field theory_ , Nucl. Phys. B450 (1995) 90, ([hep-th/9503099](http://arxiv.org/abs/hep-th/9503099))
For closed superstrings, specifically [[heterotic string]]s, a WZW-type action is due to
* [[Nathan Berkovits]], Yuji Okawa, [[Barton Zwiebach]], _WZW-like Action for Heterotic String Field Theory_ ([arXiv:hep-th/0409018](http://arxiv.org/abs/hep-th/0409018))
### Computations
Derivation of [[super Yang-Mills theory]] from open WZW-type superstring field theory is in
* [[Nathan Berkovits]], [[Martin Schnabl]], _Yang-Mills Action from Open Superstring Field Theory_ ([arXiv:hep-th/0307019](http://arxiv.org/abs/hep-th/0307019))
|
Xavier Bekaert | https://ncatlab.org/nlab/source/Xavier+Bekaert |
* [webpage](http://dept.phys.univ-tours.fr/147-bekaert)
## Selected writings
On [[Lie algebroids]] (motivated by [[higher spin gauge theory]]):
* [[Xavier Bekaert]], *Geometric tool kit for higher spin gravity (part II): An introduction to Lie algebroids and their enveloping algebras* [[arXiv:2308.00724](https://arxiv.org/abs/2308.00724)]
## Related entries
* [[self-dual higher gauge theory]]
* [[dual graviton]]
* [[higher spin gauge theory]]
category: people
|
Xena project | https://ncatlab.org/nlab/source/Xena+project |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A project bringing [[formal proof]] and [[proof assistants]] (particularly [[Lean]]) into the practice of undergraduate [[mathematics]].
## Related projects
[[!include proof assistants and formalization projects -- list]]
## References
* [[Kevin Buzzard]], _[Xena project](https://xenaproject.wordpress.com/2018/10/07/what-is-the-xena-project/)_
* {#Buzzard19} [[Kevin Buzzard]], _Using Lean with undergraduate mathematicians_, talk at _[Lean Together 2019 ](https://lean-forward.github.io/lean-together/2019/)_ ([recording](https://av-media.vu.nl/mediasite/Play/1979034f91b640229bffabdecd4eae5c1d?playFrom=76000))
* {#Buzzard19b} [[Kevin Buzzard]], _The future of Mathematics?_, public lecture at the 80th anniversary celebration of CNRS, May 2019 ([recording](https://youtu.be/aZHbnQlFOn4))
[[!redirects Xena]] |
Xenia de la Ossa | https://ncatlab.org/nlab/source/Xenia+de+la+Ossa |
Xenia de la Ossa is professor for [[mathematics]] at Oxford, working on questions in [[mathematical physics]] and [[string theory]].
## related $n$Lab entries
* [[Calabi-Yau varieties]] over [[finite fields]]
* [[Philip Candelas]], [[Xenia de la Ossa]], Fernando Rodriguez-Villegas, _Calabi-Yau Manifolds Over Finite Fields, I_ ([arXiv:hep-th/0012233](http://arxiv.org/abs/hep-th/0012233))
* [[Philip Candelas]], [[Xenia de la Ossa]], Fernando Rodriguez-Villegas, _Calabi-Yau Manifolds Over Finite Fields, II_ ([arXiv:hep-th/0402133](http://arxiv.org/abs/hep-th/0402133))
* [[string phenomenology]]
category: people |
Xi Dong | https://ncatlab.org/nlab/source/Xi+Dong |
* [webpage](http://web.physics.ucsb.edu/~xidong/)
## Selected writings
Interpretation of [[tensor networks]] encoding [[holographic entanglement entropy]] as [[quantum error correcting codes]]:
* {#ADH14} [[Ahmed Almheiri]], [[Xi Dong]], [[Daniel Harlow]], _Bulk Locality and Quantum Error Correction in AdS/CFT_, JHEP 1504:163,2015 ([arXiv:1411.7041](https://arxiv.org/abs/1411.7041))
Introducing [[holographic Renyi entropy]]:
* {#Dong16} [[Xi Dong]], *The Gravity Dual of Renyi Entropy*, Nature Communications 7, 12472 (2016) ([arXiv:1601.06788](https://arxiv.org/abs/1601.06788), [doi:10.1038/ncomms12472]( https://doi.org/10.1038/ncomms12472))
* [[Xi Dong]], *Holographic Renyi Entropy at High Energy Density*, Phys. Rev. Lett. 122, 041602 (2019) ([arXiv:1811.04081](https://arxiv.org/abs/1811.04081))
More on [[holographic entanglement entropy]]:
* Eugenia Colafranceschi, [[Xi Dong]], [[Donald Marolf]], Zhencheng Wang, *Algebras and Hilbert spaces from gravitational path integrals: Understanding Ryu-Takayanagi/HRT as entropy without invoking holography* [[arXiv:2310.02189](https://arxiv.org/abs/2310.02189)]
category: people |
Xi Yin | https://ncatlab.org/nlab/source/Xi+Yin |
* [webpage](http://www.people.fas.harvard.edu/~xiyin/Site/Home.html)
## Selected writings
On [[KK-compactification|dimensional reduction]] of [[black rings]] in [[D=5 gravity]] to [[black holes]] in [[D=4 gravity]]:
* [[Davide Gaiotto]], [[Andrew Strominger]], [[Xi Yin]], _5D Black Rings and 4D Black Holes_, JHEP 0602:023, 2006 ([arXiv:hep-th/0504126](https://arxiv.org/abs/hep-th/0504126))
On the [[M5-brane elliptic genus]]:
* [[Davide Gaiotto]], [[Andrew Strominger]], [[Xi Yin]], _The M5-Brane Elliptic Genus: Modularity and BPS States_, JHEP 0708:070, 2007 ([hep-th/0607010](https://arxiv.org/abs/hep-th/0607010))
* [[Davide Gaiotto]], [[Xi Yin]], _Examples of M5-Brane Elliptic Genera_, JHEP 0711:004, 2007 ([arXiv:hep-th/0702012](https://arxiv.org/abs/hep-th/0702012))
## Related $n$Lab entries
* [[black holes in string theory]]
category: people |
Xia Gu | https://ncatlab.org/nlab/source/Xia+Gu |
## Selected writings
On [[braid group representations]] for [[su(2)-anyon]]-[[anyon statistics|statistics]] from the [[monodromy]] of the [[Knizhnik-Zamolodchikov connection]] of bundles of [[conformal blocks]] over [[configuration spaces of points]]:
* [[Xia Gu]], [[Babak Haghighat]], [[Yihua Liu]], *Ising- and Fibonacci-Anyons from KZ-equations*, J. High Energ. Phys. **2022** 15 (2022) [[arXiv:2112.07195](https://arxiv.org/abs/2112.07195), <a href="https://doi.org/10.1007/JHEP09(2022)015">doi:10.1007/JHEP09(2022)015</a>]
On [[conformal blocks]] for [[Liouville theory]]:
* [[Xia Gu]], [[Babak Haghighat]], [[Kevin Loo]], *Irregular Fibonacci Conformal Blocks* [[arXiv:2311.13358](https://arxiv.org/abs/2311.13358)]
category: people
|
Xianzhe Dai | https://ncatlab.org/nlab/source/Xianzhe+Dai |
* [webpage](http://www.math.ucsb.edu/~dai/)
## related $n$Lab entries
* [[eta invariant]]
category: people |
Xiao-Gang Wen | https://ncatlab.org/nlab/source/Xiao-Gang+Wen |
Xiao-Gang Wen is the BMO Financial Group Isaac Newton Chair in Theoretical Physics at Perimeter Institute. His expertise is in [[condensed matter]] theory of [[strongly correlated system|strongly correlated]] electronic systems and [[topological phases of matter]]. He introduced and studied the concept of *[[topological order]]* and its *[[symmetry protected topological phase|symmetry proteced]]* versions.
* [Institute page](https://physics.mit.edu/faculty/xiao-gang-wen/) at MIT
* [Wikipedia entry](https://en.wikipedia.org/wiki/Xiao-Gang_Wen)
Interview on [[topological order]]:
* [[Philip Ball]], *Making the world from topological order*, National Science Review **6** 2 (2019) 227–230 $[$[doi:10.1093/nsr/nwy116](https://doi.org/10.1093/nsr/nwy116)$]$
## Selected writings
Introducing and developing the notion of [[topological order]]:
* {#Wen89} [[Xiao-Gang Wen]], *Vacuum degeneracy of chiral spin states in compactified space*, Phys. Rev. B **40** (1989) 7387(R) $[$[doi:10.1103/PhysRevB.40.7387](https://doi.org/10.1103/PhysRevB.40.7387)$]$
* {#WenNiu90} [[Xiao-Gang Wen]], Q. Niu, *Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces*, Phys. Rev. B **41** (1990) 9377 $[$[doi:10.1103/PhysRevB.41.9377](https://doi.org/10.1103/PhysRevB.41.9377)$]$
* {#Wen91} [[Xiao-Gang Wen]], *Non-Abelian statistics in the fractional quantum Hall states*, Phys. Rev. Lett. **66** (1991) 802 ([doi:10.1103/PhysRevLett.66.802](https://doi.org/10.1103/PhysRevLett.66.802))
* {#Wen93} [[Xiao-Gang Wen]], *Topological order and edge structure of $\nu = 1/2$ quantum Hall state*, Phys. Rev. Lett. **70** (1993) 355 $[$[doi:10.1103/PhysRevLett.70.355](https://doi.org/10.1103/PhysRevLett.70.355)$]$
* {#Wen95} [[Xiao-Gang Wen]], *Topological orders and Edge excitations in FQH states*, Advances in Physics **44** (1995) 405 $[$[arXiv:cond-mat/9506066v2](https://arxiv.org/abs/cond-mat/9506066v2), [doi:10.1080/00018739500101566](https://doi.org/10.1080/00018739500101566)$]$
Early review:
* {#Wen91Review} [[Xiao-Gang Wen]], *Topological orders and Chern-Simons theory in strongly correlated quantum liquid*, International Journal of Modern Physics B **05** 10 (1991) 1641-1648 [[doi:10.1142/S0217979291001541](https://doi.org/10.1142/S0217979291001541)]
On [[quantum spin liquids]] as exhibiting [[topological order]]:
* [[Xiao-Gang Wen]], [[Frank Wilczek]], [[Anthony Zee]], *Chiral spin states and superconductivity*, Phys. Rev. B **39** (1989) 11413 [[doi:10.1103/PhysRevB.39.11413](https://doi.org/10.1103/PhysRevB.39.11413)]
The additional requirement that the Berry connection be non-abelian (and/or the presence of anyons):
* {#GuWen09} [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], p. 2 of: *Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order*, Phys. Rev. B **80** 155131 (2009) $[$[arXiv:0903.1069](https://arxiv.org/abs/0903.1069), [doi:10.1103/PhysRevB.80.155131](https://doi.org/10.1103/PhysRevB.80.155131)$]$
On understanding generalized [[Laughlin wavefunctions]] as [[conformal blocks]]:
* B. Blok, [[Xiao-Gang Wen]], *Many-body systems with non-abelian statistics*, Nuclear Physics B **374** 3 (1992) 615-646 $[$<a href="https://doi.org/10.1016/0550-3213(92)90402-W">doi:10.1016/0550-3213(92)90402-W</a>$]$
* [[Xiao-Gang Wen]], Yong-Shi Wu, *Chiral operator product algebra hidden in certain fractional quantum Hall wave functions*, Nucl. Phys. B **419** (1994) 455-479 $[$<a href="https://doi.org/10.1016/0550-3213(94)90340-9">doi:10.1016/0550-3213(94)90340-9</a>$]$
Suggestion that [[topological order]] goes along with [[long-range entanglement]]:
* {#ChenGuWen2010} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], Section V of: _Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order_ Phys. Rev. B **82** 155138 (2010) $[$[arXiv:1004.3835](http://arxiv.org/abs/1004.3835)$]$
On the [[quantum adiabatic theorem]] in the context of [[topological phases of matter]]:
* [[Mathew B. Hastings]], [[Xiao-Gang Wen]], *Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance*, Phys. Rev. B **72** (2005) 045141 [[arXiv:cond-mat/0503554](https://arxiv.org/abs/cond-mat/0503554), [doi:10.1103/PhysRevB.72.045141](https://doi.org/10.1103/PhysRevB.72.045141)]
Introducing [[string-net models]] and the [[Levin-Wen model]]:
* [[Michael Levin]], [[Xiao-Gang Wen]], _String-net condensation: A physical mechanism for topological phases_, Phys.Rev. B71 (2005) 045110 ([arXiv:cond-mat/0404617](http://arxiv.org/abs/cond-mat/0404617))
Introducing the [[3d toric code]]:
* Alioscia Hamma, Paolo Zanardi, [[Xiao-Gang Wen]], *String and Membrane condensation on 3D lattices*, Phys. Rev. B72:035307, 2005 ([arXiv:cond-mat/0411752](https://arxiv.org/abs/cond-mat/0411752), [doi:10.1103/PhysRevB.72.035307](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.72.035307))
Introducing [[topological entanglement entropy]] in view of [[string-net models]]:
* [[Michael Levin]], [[Xiao-Gang Wen]], *Detecting topological order in a ground state wave function*, Phys. Rev. Lett., 96, 110405 (2006) ([arXiv:cond-mat/0510613](https://arxiv.org/abs/cond-mat/0510613))
Introducing the notion of [[symmetry protected topological phases]]:
* {#GuWen09} [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], *Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order*, Phys. Rev. B **80** 155131 (2009) $[$[arXiv:0903.1069](https://arxiv.org/abs/0903.1069), [doi:10.1103/PhysRevB.80.155131](https://doi.org/10.1103/PhysRevB.80.155131)$]$
On [[string-net models]] via [[tensor networks]]:
* [[Zheng-Cheng Gu]], [[Michael Levin]], [[Brian Swingle]], [[Xiao-Gang Wen]], *Tensor-product representations for string-net condensed states*, Phys. Rev. B **79** (2009) 085118 $[$[doi:10.1103/PhysRevB.79.085118](https://doi.org/10.1103/PhysRevB.79.085118)$]$
On the classification of free fermion [[SPTs]]:
* {#Wen12} [[Xiao-Gang Wen]], *Symmetry-protected topological phases in noninteracting fermion systems*, Phys. Rev. B **85** (2012) 085103 $[$[doi:10.1103/PhysRevB.85.085103](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.85.085103)$]$
On [[internal symmetry|internal]]-[[symmetry protected topological phases]]:
* {#YeWen13} [[Peng Ye]], [[Xiao-Gang Wen]], *Projective construction of two-dimensional symmetry-protected topological phases with $\mathrm{U}(1)$, $SO(3)$, or $SU(2)$ symmetries*, Phys. Rev. B **87** 195128 (2013) $[$[doi:10.1103/PhysRevB.87.195128](https://doi.org/10.1103/PhysRevB.87.195128), [arXiv:1212.2121](https://arxiv.org/abs/1212.2121)$]$
Claim that [[group cohomology]] classifies some [[symmetry protected topological phases]]:
* {#CGLW11} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], _Symmetry protected topological orders and the group cohomology of their symmetry group_, Phys. Rev. B **87** (2013) 155114 $[$[arXiv:1106.4772](http://arxiv.org/abs/1106.4772)$]$
* [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], *Symmetry protected topological orders and the group cohomology of their symmetry group*, Science **338** (2012) 1604-1606 ([arXiv:10.1103/PhysRevB.87.155114](https://doi.org/10.1103/PhysRevB.87.155114))
* {#GuWen14} [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], *Symmetry-protected topological orders for interacting fermions -- fermionic topological non-linear sigma-models and a group super-cohomology theory*, Phys. Rev. B **90** 115141 (2014) $[$[arXiv:1201.2648](http://arxiv.org/abs/1201.2648), [doi:10.1103/PhysRevB.90.115141](https://doi.org/10.1103/PhysRevB.90.115141)$]$
More on [[topological order]]/[[topological phases of matter]]:
* [[Xiao-Gang Wen]], _Topological orders and edge excitations in FQH states_, Advances in Physics __44__, 405 (1995). [cond-mat/9506066](http://arxiv.org/abs/cond-mat/9506066)
* Xiao-Gang Wen, _Vacuum degeneracy of chiral spin state in compactified spaces_, Phys. Rev. B, 40, 7387 (1989),
* Xiao-Gang Wen, _Topological orders in rigid states_, Int. J. Mod. Phys. B4, 239 (1990)
* Xiao-Gang Wen, Qian Niu, _Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces_, Phys. Rev. B41, 9377 (1990)
* E. Keski-Vakkuri, Xiao-Gang Wen, _Ground state structure of hierarchical QH states on torus and modular transformation_, Int. J. Mod. Phys. B7, 4227 (1993) [pdf](http://dao.mit.edu/~wen/pub/kw.pdf)
* Xiao-Gang Wen, _Mean field theory of spin liquid states with finite energy gap and topological orders_, Phys. Rev. B 44 2664 (1991).
* Xiao-Gang Wen, _Non-Abelian Statistics in the FQH states_, [pdf](http://dao.mit.edu/~wen/pub/nab.pdf)
Phys. Rev. Lett. 66, 802 (1991).
* Xiao-Gang Wen, Yong-Shi Wu, _Chiral operator product algebra hidden in certain FQH states_, Nucl. Phys. B419, 455 (1994), [pdf](http://dao.mit.edu/~wen/pub/nabdw.pdf)
* [[Xie Chen]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], _Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order_ Phys. Rev. B 82, 155138 (2010)
* [[Juven Wang]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], _Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond_, [arxiv:1405.7689](https://arxiv.org/abs/1405.7689), [Phys. Rev. Lett. 114, 031601 (2015)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.031601)
* [[Liang Kong]], Tian Lan, [[Xiao-Gang Wen]], Zhi-Hao Zhang, Hao Zheng, _Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry_ ([arXiv:2005.14178](https://arxiv.org/abs/2005.14178))
On [[quantum information theory]] ([[entanglement entropy]]) applied to [[topological phases of matter]]/[[topological order]]:
* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:
*[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$
## Related entries
* [[topological order]]
* [[symmetry protected trivial order]]
* [[Levin-Wen model]]
category: people |
Xiao-Song Lin | https://ncatlab.org/nlab/source/Xiao-Song+Lin |
* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=15130)
* [[Zhenghan Wang]], *[In Memory of X.-S. Lin](http://web.math.ucsb.edu/~zhenghwa/personal.php)*
## Selected writings
On the [[loop braid group]]:
* [[Xiao-Song Lin]], *The motion group of the unlink and its representations*, in:
*Xiao-Song Lin's Unpublished Papers* ([doi:10.1142/9789812819116_others01](https://doi.org/10.1142/9789812819116_others01)), Part B of:
[[Kevin Lin]], [[Zhenghan Wang]], [[Weiping Zhang]] (eds.) *Topology and Physics: Proceedings of the Nankai International Conference in Memory of Xiao-Song Lin*, Tianjin, China, 27-31 July 2007. World Scientific, 2008 ([doi:10.1142/6907](https://doi.org/10.1142/6907))
category: people
|
Xie Chen | https://ncatlab.org/nlab/source/Xie+Chen |
* [personal page](http://www.its.caltech.edu/~xcchen/)
## Selected writings
Suggestion that [[topological order]] goes along with [[long-range entanglement]]:
* {#ChenGuWen2010} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], _Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order_ Phys. Rev. B **82** 155138 (2010) $[$[arXiv:1004.3835](http://arxiv.org/abs/1004.3835)$]$
Claim that [[group cohomology]] classifies some [[symmetry protected topological phases]]:
* {#CGLW11} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], _Symmetry protected topological orders and the group cohomology of their symmetry group_, Phys. Rev. B **87** (2013) 155114 $[$[arXiv:1106.4772](http://arxiv.org/abs/1106.4772)$]$
* [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], *Symmetry protected topological orders and the group cohomology of their symmetry group*, Science **338** (2012) 1604-1606 ([arXiv:10.1103/PhysRevB.87.155114](https://doi.org/10.1103/PhysRevB.87.155114))
On [[quantum information theory]] ([[entanglement entropy]]) applied to [[topological phases of matter]]/[[topological order]]:
* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:
*[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$
category: people
|
Xin Wang | https://ncatlab.org/nlab/source/Xin+Wang |
* [arXiv page](https://arxiv.org/search/hep-th?searchtype=author&query=Wang,+X)
## Selected writings
On [[quantum Seiberg-Witten curves]] of [[E-string]]-theories in relation to [[D6-D8-brane bound states]]:
* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], [[Xin Wang]], *E-string Quantum Curve*, Nuclear Physics B **973** (2021) 115602 [[arXiv:2103.16996](https://arxiv.org/abs/2103.16996), [doi:10.1016/j.nuclphysb.2021.115602](https://doi.org/10.1016/j.nuclphysb.2021.115602)]
category: people
|
Xinwen Zhu | https://ncatlab.org/nlab/source/Xinwen+Zhu |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Xinwen_Zhu)
* [Institute page](https://www.pma.caltech.edu/people/xinwen-zhu)
## Selected writings
On the [[moduli stack of L-parameters]]:
* {#Zhu20} [[Xinwen Zhu]], _Coherent sheaves on the stack of Langlands parameters_ ([arXiv:2008.02998](https://arxiv.org/abs/2008.02998))
Relating the [[Pontrjagin algebra]] on [[loop groups]] of [[compact Lie groups]] to their [[Langlands dual groups]]:
* [[Zhiwei Yun]], [[Xinwen Zhu]], *Integral homology of loop groups via Langlands dual groups*, Represent. Theory **15** (2011) 347-369 [[arXiv:0909.5487](https://arxiv.org/abs/0909.5487), [doi:10.1090/S1088-4165-2011-00399-X](https://doi.org/10.1090/S1088-4165-2011-00399-X)]
category: people |
XTT | https://ncatlab.org/nlab/source/XTT | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
XTT is an [[set-level type theory|set-level]] [[cubical type theory]] developed by [[Jonathan Sterling]], [[Carlo Angiuli]], and [[Daniel Gratzer]]. The authors have stated that the name is not short for anything, but it is a pun on "eXtensional Type Theory".
XTT features [[boundary separation]], which implies [[UIP]] as a theorem of XTT, rather than an axiom as in [[Martin-Löf type theory]]. As a result, it is an example of a [[cubical type theory]] which is not a [[homotopy type theory]].
Additionally, XTT has [[regularity]], [[canonicity]], and certain [[higher inductive types]] like [[propositional truncation]]. It is an open question if it additionally has [[decidable]] [[type checking]] and [[normal form|normalization]].
## See also
* [[set-level type theory]]
* [[cubical type theory]]
## References
* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _Cubical syntax for reflection-free extensional equality_. In [[Herman Geuvers]] (ed.), _4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)_, volume 131 of _Leibniz International Proceedings in Informatics (LIPIcs)_, pages 31:1-31:25. ([arXiv:1904.08562](https://arxiv.org/abs/1904.08562), [doi:10.4230/LIPIcs.FCSD.2019.31](https://doi.org/10.4230/LIPIcs.FCSD.2019.31))
* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _A Cubical Language for Bishop Sets_, Logical Methods in Computer Science, 18 (1), 2022. ([arXiv:2003.01491](https://arxiv.org/abs/2003.01491)). |
xyz | https://ncatlab.org/nlab/source/xyz | |
XYZ particle | https://ncatlab.org/nlab/source/XYZ+particle |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_XYZ particles_ are [[charmonium]]-like [[exotic meson|exotic mesons]] which are increasingly being observed in [[experiment]] (typically in [[B-meson]]-experiments such as [[Belle collaboration|Belle]], [[LHCb experiment|LHCb]], , [[BaBar experiment|BaBar]]), but whch are hard to interpret as [[bound states]] in the traditional [[quark]] [[model (in theoretical physics)]] of [[quantum chromodynamics]].
## Related concepts
* [[tetraquark]]
* [[quantum hadrodynamics]]
## References
Review:
* Stephen Godfrey and Stephen L. Olsen, _The Exotic XYZ Charmonium-Like Mesons_, Annual Review of Nuclear and Particle Science Vol. 58:51-73 (Volume publication date 23 November 2008) ([arXiv:10.1146/annurev.nucl.58.110707.171145](https://doi.org/10.1146/annurev.nucl.58.110707.171145))
* Xiang Liu, _An overview of XYZ new particles_, Chin. Sci. Bull. 59, 3815-3830 (2014) ([arXiv:1312.7408](https://arxiv.org/abs/1312.7408))
* Stephen Lars Olsen, _The XYZ mesons: what they aren't_, EPJ Web of Conferences 202, 01003 (2019) ([doi:10.1051/epjconf/201920201003](https://doi.org/10.1051/epjconf/201920201003))
* Nora Brambilla, Simon Eidelman, Christoph Hanhart, Alexey Nefediev, Cheng-Ping Shen, Christopher E. Thomas, Antonio Vairo, Chang-Zheng Yuan, _The XYZ states: experimental and theoretical status and perspectives_, Physics Reports 2020 ([arXiv:1907.07583](https://arxiv.org/abs/1907.07583))
See also
* Wikipedia, *[XYZ particle](https://en.wikipedia.org/wiki/XYZ_particle)*
* Wikipedia, *[Exotic hadron](https://en.wikipedia.org/wiki/Exotic_hadron)*
For references specifically on [[tetraquarks]] see [there](tetraquark#References).
Discussion of exotic hadrons via [[hadron supersymmetry]]:
* [[Don Lichtenberg]], Renato Roncaglia, Enrico Predazzi, *Predicting exotic hadron masses from supersymmetry and a quark -- diquark model*, J. Phys. G: Nucl. Part. Phys. **23** 865 ([doi:10.1088/0954-3899/23/8/001](https://iopscience.iop.org/article/10.1088/0954-3899/23/8/001))
* [[Harry J. Lipkin]], *Exotic Hadrons in the Constituent Quark Model*, Prog. Theor. Phys. Suppl. **168** (2007) 15-22 ([arXiv:hep-ph/0703190](https://arxiv.org/abs/hep-ph/0703190), [doi:10.1143/PTPS.168.15](https://doi.org/10.1143/PTPS.168.15))
[[!redirects XYZ particles]]
[[!redirects XYZ meson]]
[[!redirects XYZ mesons]]
[[!redirects exotic hadron]]
[[!redirects exotic hadrons]]
|
Y-system | https://ncatlab.org/nlab/source/Y-system | __Y-system__ and __T-system__ are two related classes of algebraic relations associated with [[affine Lie algebra]]s and can be considered as encoding certain [[integrable system]]s. Y-system can be considered as a system of [[difference equation]]s for a finite set of commuting
variables $Y_i$, $i\in I$,
$$
Y_i(t+1)Y_i(t-1)=\prod_{j\neq i}\bigl(Y_j(t)+ 1\bigr)^{-a_{ij}}
$$
The elements of the T-system satisfy discrete [[Hirota equation]]s.
For reviews see
* Atsuo Kuniba, Tomoki Nakanishi, Junji Suzuki, _T-systems and Y-systems in integrable systems_, J. Phys. A44 103001 (2011) [doi](http://dx.doi.org/10.1088/1751-8113/44/10/103001); _T-systems and Y-systems for quantum affinizations
of quantum Kac-Moody algebras_, SIGMA 5 (2009), 108, 23 pages
Y-system has a remarkable connection to [[cluster algebra]]s:
* [[Bernhard Keller]], _Cluster algebras, quiver representations and triangulated categories_, arXiv:0807.1960
* [[Sergey Fomin]], [[Andrei Zelevinsky]], _Y-systems and generalized associahedra_, Ann. Math. __158__ 977-1018 (2003) [doi](https://doi.org/10.4007/annals.2003.158.977)
Y-systems are relevant for integrability phenomena in superstring theory and in relation to study of spectrum of N=4 SUSY Zang-Mills theory. See survey
* Stijn J. van Tongeren, _Integrability of the $AdS_5 \times S^5$ superstring and its deformations_, J. Phys. A: Math. Theor. 47 (2014) 433001 [arxiv/1310.4854](http://arxiv.org/abs/1310.4854)
[[!redirects T-system]]
[[!redirects Y-system and T-system]]
category: physics |
Yael Fregier | https://ncatlab.org/nlab/source/Yael+Fregier |
# Yaël Frégier
An Associate Professor (Maître de Conférence) at the Artois University with interests in "theoretical physics, algebraic topology, machine learning and its applications".
* [webpage](https://sites.google.com/view/homepagelml/home), [publications](https://sites.google.com/view/homepagelml/publications)
## Selected publications
* Martin Callies, [[Yaël Frégier]], [[Chris Rogers]], [[Marco Zambon]], _Homotopy moment maps_, Advances in Mathematics __303__ (2016) 954--1043 ([arXiv:1304.2051](https://arxiv.org/abs/1304.2051))
category: people
[[!redirects Y. Frégier]]
[[!redirects Yaël Frégier]]
|
Yael Karshon | https://ncatlab.org/nlab/source/Yael+Karshon |
* [webpage](http://www.math.toronto.edu/karshon/)
## Selected writings
On [[geometric quantization]]:
* [[Ana Cannas da Silva]], [[Yael Karshon]], [[Susan Tolman]], *Quantization of Presymplectic Manifolds and Circle Actions*, Transactions of the AMS **352** 2 (2000) 525-552 [[arXiv:dg-ga/9705008](http://arxiv.org/abs/dg-ga/9705008), [jstor:118052](https://www.jstor.org/stable/118052)]
On [[orbifolds]] regarded as naive local quotient [[diffeological spaces]]:
* {#IKZ10} [[Patrick Iglesias-Zemmour]], [[Yael Karshon]], Moshe Zadka, _Orbifolds as diffeologies_, Transactions of the American Mathematical Society 362 (2010), 2811-2831 ([arXiv:math/0501093](https://arxiv.org/abs/math/0501093))
## Related entries
* [[orbifold]]
* [[diffeological space]]
* [[Guillemin-Sternberg geometric quantization conjecture]]
category: people
|
Yakir Aharonov | https://ncatlab.org/nlab/source/Yakir+Aharonov |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Yakir_Aharonov)
* [Institute page](https://en-exact-sciences.tau.ac.il/profile/yakir)
* [GoogleScholar page](https://scholar.google.com/citations?user=bkVKG88AAAAJ&hl=en)
## Selected writings
Discussion of what came to be known as the [[Aharonov-Bohm effect]]:
* [[Yakir Aharonov]], [[David Bohm]], *Significance of Electromagnetic Potentials in the Quantum Theory*, Phys. Rev. **115** (1959) 485 [[doi:10.1103/PhysRev.115.485](https://doi.org/10.1103/PhysRev.115.485), [pdf](https://journals.aps.org/pr/pdf/10.1103/PhysRev.115.485)]
category: people |
Yakov Eliashberg | https://ncatlab.org/nlab/source/Yakov+Eliashberg | Yakov Eliashberg (Яков Матвеевич Элиашберг) is a Russian-American mathematician who has made contributions to [[contact manifold|contact]] and [[symplectic topology]]. He is currently the Herald L. and Caroline L. Ritch Professor of Mathematics at Stanford.
* [Stanford faculty page](https://mathematics.stanford.edu/people/yakov-eliashberg)
* [Wikipedia page](https://en.wikipedia.org/wiki/Yakov_Eliashberg)
## Selected writings
On [[contact manifold|contact]] 3-manifolds:
* [[Yakov Eliashberg]], _Classification of overtwisted contact structures on 3-manifolds_, Invent. Math. 98 (1989), 623-637. ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.465.8237&rep=rep1&type=pdf))
* [[Yakov Eliashberg]], _Contact 3-manifolds twenty years since J Martinet’s work_, Ann. Inst. Fourier 42 (1992), 165-192. ([pdf](http://www.numdam.org/item/10.5802/aif.1288.pdf))
On [[quantomorphism groups]]:
* [[Yakov Eliashberg]], [[Leonid Polterovich]], _Partially ordered groups and geometry of contact transformations_. Geom. Funct. Anal. 10 (2000), no. 6, 1448-1476. ([arXiv:math/9910065](https://arxiv.org/abs/math/9910065), [doi](https://doi.org/10.1007/PL00001656))
On [[symplectic field theory]]:
* [[Yakov Eliashberg]], [[Alexander Givental]], [[Helmut Hofer]], _Introduction to symplectic field theory_, in N. Alon, J. Bourgain, A. Connes, M. Gromov, V. Milman, eds., _Visions in Mathematics: GAFA 2000 Special volume, Part II_, 2010, pp 560-673. ([arXiv:math.SG/0010059](http://arxiv.org/abs/math/0010059), [book chapter](https://link.springer.com/chapter/10.1007/978-3-0346-0425-3_4))
On Gromov's [[h-principle|$h$-principle]]:
* [[Yakov Eliashberg]], Nikolai Mishachev, _Holonomic approximation and Gromov's h-principle_, [arXiv:math/0101196](https://arxiv.org/abs/math/0101196).
* [[Yakov Eliashberg]], Nikolai Mishachev, _Introduction to the h-principle_, Graduate Studies in Mathematics 48, Amer. Math. Soc., 2002. [Google Books](http://books.google.co.uk/books?isbn=0821832271)
## Related $n$Lab pages
* [[contact manifold]]
* [[h-principle]]
* [[quantomorphism group]]
* [[symplectic field theory]]
* [[Timeline of category theory and related mathematics]] |
Yakov Itin | https://ncatlab.org/nlab/source/Yakov+Itin |
* [institute page](http://www.math.huji.ac.il/~itin/)
* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=56436)
* [GoogleScholar page](https://scholar.google.com/citations?user=t6_RFl4AAAAJ&hl=en)
## Selected writings
On [[pre-metric electromagnetism]]:
* {#HehlItinObukhov16} [[Friedrich W. Hehl]], [[Yakov Itin]], [[Yuri N. Obukhov]], *On Kottler's path: origin and evolution of the premetric program in gravity and in electrodynamics*, International Journal of Modern Physics D **25** 11 (2016) 1640016 [[arXiv:1607.06159](https://arxiv.org/abs/1607.06159), [doi:10.1142/S0218271816400162](https://doi.org/10.1142/S0218271816400162)]
* [[Yakov Itin]], *Premetric representation of mechanics, electromagnetism and gravity*, International Journal of Geometric Methods in Modern Physics **15** supp01 (2018) [[doi:10.1142/S0219887818400029](https://doi.org/10.1142/S0219887818400029)]
category: people |
Yakov Kremnitzer | https://ncatlab.org/nlab/source/Yakov+Kremnitzer |
Yakov Kremnitzer is professor for pure [[mathematics]] at Oxford.
* [webpage](https://www.maths.ox.ac.uk/contact/details/kremnitzer)
## related $n$Lab entries
* ([[non-archimedean analytic geometry|non-archimedean]]) [[analytic geometry]]
category: people
[[!redirects Kobi Kremnitzer]]
[[!redirects Kobi Kremnizer]]
[[!redirects Yakov Kremnizer]] |
Yakov Shnir | https://ncatlab.org/nlab/source/Yakov+Shnir |
* [personal page](http://theor.jinr.ru/~shnir/)
## Selected writings
On [[magnetic monopoles]] ([[Dirac monopoles]], [[Yang-Mills monopoles]]):
* [[Yakov Shnir]], *Magnetic Monopoles*, Springer 2005 ([ISBN:978-3-540-29082-7](https://www.springer.com/gp/book/9783540252771))
category: people |
Yan Liu | https://ncatlab.org/nlab/source/Yan+Liu |
* [intitute page](http://sse.buaa.edu.cn/info/1009/1637.htm)
* [GoogleScholar page](https://scholar.google.com/citations?user=e0BQlXoAAAAJ&hl=en)
## Selected writings
On the [[AdS-CMT correspondence]]:
* {#ZaanenLiuSunSchalm15} [[Jan Zaanen]], [[Yan Liu]], Ya-Wen Sun, [[Koenraad Schalm]], *Holographic Duality in Condensed Matter Physics*, Cambridge University Press 2015 [[doi:10.1017/CBO9781139942492](https://doi.org/10.1017/CBO9781139942492)]
Introducing a holographic description of [[topological semimetals]] via the [[AdS-CMT correspondence]]:
* [[Karl Landsteiner]], [[Yan Liu]], *The holographic Weyl semi-metal*, Physics Letters B **753** (2016) 453-457 [[arXiv:1505.04772](https://arxiv.org/abs/1505.04772), [doi:10.1016/j.physletb.2015.12.052](https://doi.org/10.1016/j.physletb.2015.12.052)]
* [[Karl Landsteiner]], [[Yan Liu]], Ya-Wen Sun, *Quantum phase transition between a topological and a trivial semimetal from holography*, Phys. Rev. Lett. **116** 081602 (2016) [[arXiv:1511.05505](https://arxiv.org/abs/1511.05505), [doi:10.1103/PhysRevLett.116.081602](https://doi.org/10.1103/PhysRevLett.116.081602)]
* Ling-Long Gao, [[Yan Liu]], Hong-Da Lyu, *Black hole interiors in holographic topological semimetals* [[arXiv:2301.01468](https://arxiv.org/abs/2301.01468)]
category: people
|
Yan Soibelman | https://ncatlab.org/nlab/source/Yan+Soibelman |
* [website](http://www.math.ksu.edu/~soibel/)
## Selected writings
On [[string theory]] via [[2-spectral triples]] reduced to [[spectral triples]]:
* {#Soibelman11} [[Yan Soibelman]], _Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry_ ([pdf](http://www.math.ksu.edu/~soibel/nc-riem-3.pdf), [pdf](https://ncatlab.org/schreiber/files/SoibelmanCFTandRicciCurvature.pdf)), in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2001)
## Related $n$Lab entries
* [[deformation theory]]
* [[spectral triple]]
* [[2-spectral triple]]
category: people |
Yan Zhang | https://ncatlab.org/nlab/source/Yan+Zhang | [[!redirects Yan X Zhang]]
* [institute page](http://math.mit.edu/~yanzhang/)
* [personal page](http://math.mit.edu/~yanzhang/personal.html)
## Writings
On [[adinkras]]:
* {#Zhang11} _Adinkras for Mathematicians_ ([arXiv:1111.6055](https://arxiv.org/abs/1111.6055))
* {#Zhang13} _The combinatorics of Adinkras_, PhD thesis, MIT (2013) ([pdf](http://math.mit.edu/~yanzhang/math/thesis_adinkras.pdf))
## related $n$Lab entries
* [[adinkra]]
category: people
|
Yang monopole | https://ncatlab.org/nlab/source/Yang+monopole |
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#### Physics
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[[!include physicscontents]]
=--
#### $\infty$-Chern-Weil theory
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[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
## Idea
The _Yang monopole_ A [[monopole]] in [[Yang-Mills theory]].
The generalization of the [[Dirac monopole]] from 3+1 dimensional [[spacetime]] to 5+1 dimensional [[spacetime]].
## Definition
{#Definition}
### Recalling the Dirac monopole
For comparison, first note that the Dirac monopole is a [[circle group]] [[principal bundle]] with non-trivial [[first Chern class]] on a [[spacetime]] of the form
$$
X_4 \;\coloneqq\; (\mathbb{R}^3_{space} - \{0\}) \times \mathbb{R}_{time}
$$
witnessed (via [[Chern-Weil theory]]) by the [[magnetic charge]]
$$
Q = \int_{S^2} F_\nabla \in \mathbb{Z} \hookrightarrow \mathbb{R}
$$
which is the [[integration of differential forms|integration]] of the [[curvature]] [[differential 2-form]] $F_\nabla$ of any [[principal connection]] $\nabla$ on this bundle over any [[sphere]] wrapping the removed origin of $\mathbb{R}^3$. We may think of the Dirac monopole as being an effective [[magnetic monopole]] "[[particle]]" with [[worldline]] $\{0\} \times \mathbb{R}_{time}$. But in the above description we can just as well remove a little 3-[[ball]] $D^3_\epsilon$ from space, instead of just a point, and when viewed as such the Dirac monopole is a 2-[[brane]] with [[worldvolume]] $(\partial D_\epsilon) \times \mathbb{R}_{times}$.
### The Yang monopole
Now analogously, a _Yang monopole_ is a nontrivial [[special unitary group]]-[[principal bundle]] (for which there is no non-trivial [[first Chern class]]) with non-trivial _[[second Chern class]]_ on
$$
X_6 \coloneqq (\mathbb{R}^5_{space} - \{0\}) \times \mathbb{R}_{time}
$$
witnessed (via [[Chern-Weil theory]]) by the [[instanton number]]
$$
Q = \int_{S^4} \langle F_\nabla \wedge F_\nabla\rangle \in \mathbb{Z} \hookrightarrow \mathbb{R}
$$
which is the [[integration of differential forms|integration]] of the [[curvature]] [[differential 2-form]] $F_\nabla$ wedge-squared and evaluated in the canonical [[Killing form]] [[invariant polynomial]] to a [[differential 4-form]] of any [[principal connection]] $\nabla$ on this bundle over any 4-[[sphere]] wrapping the removed origin of $\mathbb{R}^5$.
As before, we may equivalently think of removing instead of just a point a small 5-[[ball]] $D_{\epsilon}^5$ and then the Yang monopole appears as a 4-[[brane]] with [[worldvolume]] $(\partial D_\epsilon^5) \times \mathbb{R}_{time}$. This is how the Yang monopole appears in [[string theory]]/[[M-theory]] ([Bergsgoeff-Gibbons-Townsend 06](BergsgoeffGibbonsTownsend06)).
In terms of [[higher gauge theory]] the Yang monopole is seen to be more directly analogous to the [[Dirac monopole]]: the 4-form $\langle F_\nabla \wedge F_\nabla\rangle$ is actually the [[curvature]] 4-form of a [[circle n-bundle with connection|circle 3-bundle with connection]] on [[spacetime]] which is induced from the given $SU(N)$-[[principal connection]], namely the [[Chern-Simons circle 3-bundle]] which is modulated by the [[universal characteristic class]] that is the differential refinement of the second Chern class:
$$
\mathbf{c}_2 \colon \mathbf{B}SU(N)_{conn} \to \mathbf{B}^3 U(1)_{conn}
$$
(this is discussed at _[[differential string structure]]_). Hence $\mathbf{c}_2(\nabla)$ is a [[circle n-bundle with connection|3-connection]] with [[curvature]]
$$
F_{\mathbf{c}_2(\nabla)} = \langle F_\nabla \wedge F_\nabla \rangle
\,.
$$
Now the analog of the [[first Chern class]] as one passes to such [[circle n-bundles]] is called the higher [[Dixmier-Douady class]], and the Yang monopole charge is just this 2-Dixmier-Douady class of the [[Chern-Simons circle 3-bundle]] induced by a $SU(N)$-[[principal connection]] with corresponding [[instanton number]]:
$$
Q = \int_{S^4} F_{\mathbf{c}_2(\nabla)} \in \mathbb{Z} \hookrightarrow \mathbb{R}
\,.
$$
## Examples
### In M-brane theory
The end-surface of an [[M5-brane]] ending on an [[M9-brane]] is a Yang-monopole in the M5 [[worldvolume]] ([Bergsgoeff-Gibbons-Townsend 06](#BergsgoeffGibbonsTownsend06)).
## References
### General
The Yang monopole was introduced as a generalization to 5+1 dimensional [[spacetime]] of the [[Dirac monopole]] in 3+1 dimensional spacetime in
* [[Chen Ning Yang]], _Generalization of Dirac's Monopole to $SU(2)$ Gauge Fields_, J. Math. Phys. 19, 320 (1978).
More general discussion is in
* Tigran Tchrakian, _Dirac-Yang monopoles and their regular counterparts_ ([arXiv:hep-th/0612249](http://arxiv.org/abs/hep-th/0612249))
See also
* Frederik Nørfjand, Nikolaj Thomas Zinner, _Non-existence theorems and solutions of the Wu-Yang monopole equation_ ([arxiv:1911.08140](https://arxiv.org/abs/1911.08140))
### In string theory
Appearance of Yang monopoles in [[string theory]] goes back to
* Adil Belhaj, Pablo Diaz, Antonio Segui, _On the Superstring Realization of the Yang Monopole_ ([arXiv:hep-th/0703255](http://arxiv.org/abs/hep-th/0703255))
The appearance of the Yang monopole as the [[boundary]] of an open [[M5-brane]] ending on an [[M9-brane]] is discussed in
* [[Eric Bergshoeff]], [[Gary Gibbons]], [[Paul Townsend]], _Open M5-branes_, Phys.Rev.Lett.97:231601 2006 ([arXiv:hep-th/0607193](http://arxiv.org/abs/hep-th/0607193))
{#BergsgoeffGibbonsTownsend06}
Based on this a Yang monopole is realized in [[type IIA string theory]] in section 2 of
* Adil Belhaj, Pablo Diaz, Antonio Segui, _The Yang Monopole in IIA Superstring: Multi-charge Disease and Enhancon Cure_ ([arXiv:1102.1538](http://arxiv.org/abs/1102.1538))
[[!redirects Yang monopoles]] |
Yang Yang | https://ncatlab.org/nlab/source/Yang+Yang |
* [institute page](https://grk1670.math.uni-hamburg.de/researchers/yang/)
## Selected writings
Constructing [[correlators]] in [[rational conformal field theory]] via [[string net models]]:
* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-Net Construction of RCFT Correlators*, Springer Briefs in Mathematical Physics **45**, Springer (2023) [[doi:10.1007/978-3-031-14682-4](https://doi.org/10.1007/978-3-031-14682-4), [arXiv:2112.12708](https://arxiv.org/abs/2112.12708)]
Constructing [[modular functors]] from [[pivotal category|pivotal]] [[bicategories]] using [[string net models]]:
* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-net models for pivotal bicategories* [[arXiv:2302.01468](https://arxiv.org/abs/2302.01468)]
Survey of [[algebra|algebraic]] [[structures]] in [[2d conformal field theory]] ([[vertex operator algebras]], [[conformal blocks]], [[modular tensor category|modular]] [[fusion categories]], [[modular functors]], [[FRS theorem]], [[string-net models]]):
* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], *Algebraic structures in two-dimensional conformal field theory*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]
category: people |
Yang Zhang | https://ncatlab.org/nlab/source/Yang+Zhang |
* [webpage](http://www.nbi.dk/~zhang/)
## Selected writings
On [[characteristic classes]]:
* {#Zhang11} [[Yang Zhang]], _A brief introduction to characteristic classes from the differentiable viewpoint_, 2011 ([pdf](http://staff.ustc.edu.cn/~yzhphy/notes/A%20brief%20introduction%20to%20characteristic%20classes%20from%20the%20differentiable%20viewpoint.pdf), [[Zhang_CharacteristicClasses.pdf:file]])
## Related $n$Lab entries
* [[characteristic class]]
* [[splitting principle]]
category: people |
Yang-Baxter equation | https://ncatlab.org/nlab/source/Yang-Baxter+equation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
(...)
Beware that the term _Yang-Baxter equation_ can mean (or be interpreted in the context of) any of several related but different concepts:
[[!include Yang-Baxter equations -- contents]]
## References
See also
* Wikipedia, _[Yang-Baxter equation](https://en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation)_
[[!redirects Yang-Baxter equations]]
|
Yang-Baxter equations -- contents | https://ncatlab.org/nlab/source/Yang-Baxter+equations+--+contents |
**[[Yang-Baxter equations]]**
* [[classical Yang-Baxter equation]]
* [[quantum Yang-Baxter equation]]
* [[dynamical quantum Yang-Baxter equation]]
* [[associative Yang-Baxter equation]]
* [[infinitesimal braid relation]]
* [[set theoretic Yang-Baxter equation]]
* [[quasitriangular bialgebra]] |
Yang-Hui He | https://ncatlab.org/nlab/source/Yang-Hui+He |
* [webpage](https://www.city.ac.uk/people/academics/yang-hui-he)
## Selected writings
On [[toric duality]]:
* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 ([arXiv:hep-th/0003085](https://arxiv.org/abs/hep-th/0003085))
* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _Phase structure of D-brane gauge theories and toric duality_ , J. High Energy Phys. 08 (2001) 040 ([hep-th/0104259](https://arxiv.org/abs/hep-th/0104259))
* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], [[Angel Uranga]], _Toric duality as Seiberg duality and brane diamonds, J. High Energy Phys. 12 (2001) 035 ([hep-th/0109063](https://arxiv.org/abs/hep-th/0109063))
* [[Bo Feng]], S. Franco, [[Amihay Hanany]], [[Yang-Hui He]], _Unhiggsing the del Pezzo_, J. High Energy Phys. 08 (2003) 058 ([hep-th/0209228](https://arxiv.org/abs/hep-th/0209228))
On [[discrete torsion]]:
* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], Nikolaos Prezas, _Discrete Torsion, Non-Abelian Orbifolds and the Schur Multiplier_, JHEP 0101:033, 2001 ([arXiv:hep-th/0010023](https://arxiv.org/abs/hep-th/0010023))
On [[quiver gauge theories]] and [[D-branes]]:
* [[Yang-Hui He]], _Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities_ ([arXiv:hep-th/0408142](https://arxiv.org/abs/hep-th/0408142))
* {#He18} [[Yang-Hui He]], _Quiver Gauge Theories: Finitude and Trichotomoty_, Mathematics 2018, 6(12), 291 ([doi:10.3390/math6120291](https://doi.org/10.3390/math6120291))
On [[quiver gauge theory]] and [[Donaldson-Thomas theory]] in [[heterotic string theory]]:
* {#HeLee12} [[Yang-Hui He]], Seung-Joo Lee, _Quiver Structure of Heterotic Moduli_, J. High Energ. Phys. (2012) 2012: 119 ([arXiv:1208.3004](https://arxiv.org/abs/1208.3004))
On [[NS5-branes]] and [[orientifolds]] with [[RR-field tadpole cancellation]]:
* {#FengHeKarchUranga01} Bo Feng, [[Yang-Hui He]], [[Andreas Karch]], [[Angel Uranga]], _Orientifold dual for stuck NS5 branes_, JHEP 0106:065, 2001 ([arXiv:hep-th/0103177](https://arxiv.org/abs/hep-th/0103177))
On [[exceptional structures]] ([[exceptional Lie algebras]] and [[sporadic finite simple groups]]) via [[del Pezzo surfaces]]:
* [[Yang-Hui He]], [[John McKay]], _Sporadic and Exceptional_ [[arXiv:1505.06742](https://arxiv.org/abs/1505.06742)]
On [[heterotic string theory]] [[string phenomenology]] and the [[landscape of string theory vacua]]:
* [[Lara Anderson]], [[Yang-Hui He]], [[Andre Lukas]], _Heterotic Compactification, An Algorithmic Approach_, JHEP 0707:049, 2007 ([arXiv:hep-th/0702210](https://arxiv.org/abs/hep-th/0702210))
* {#CHE18} [[Andrei Constantin]], [[Yang-Hui He]], [[Andre Lukas]], _Counting String Theory Standard Models_, Physics Letters B Volume 792, 10 May 2019, Pages 258-262 ([arXiv:1810.00444](https://arxiv.org/abs/1810.00444))
* {#He18a} [[Yang-Hui He]], _The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning_ ([arXiv:1812.02893](https://arxiv.org/abs/1812.02893))
* {#He18b} [[Yang-Hui He]], _Deep-learning the landscape_, talk at _[String and M-Theory: The new geometry of the 21st century](https://ims.nus.edu.sg/events/2018/wstring/wk.php)_ ([pdf slides](https://ims.nus.edu.sg/events/2018/wstring/files/yang.pdf), [video recording](https://www.youtube.com/watch?v=x3ThgBgkPlE))
* [[Yang-Hui He]], _Calabi-Yau Spaces in the String Landscape_ ([arXiv:2006.16623](https://arxiv.org/abs/2006.16623))
Relating the [[abc conjecture]] to [[D=4 N=4 super Yang-Mills theory]]:
* [[Yang-Hui He]], Zhi Hu, Malte Probst, James Read, _Yang-Mills Theory and the ABC Conjecture_, International Journal of Modern Physics A Vol. 33, No. 13, 1850053 (2018) ([arXiv:1602.01780](https://arxiv.org/abs/1602.01780), [doi:10.1142/S0217751X18500537](https://doi.org/10.1142/S0217751X18500537))
Application of [[machine learning]] to [[mathematical structures]] in [[algebraic geometry]], [[representation theory]], [[number theory]], [[combinatorics]] and [[string theory]]:
* [[Yang-Hui He]], *Machine-Learning Mathematical Structures*, International Journal of Data Science in the Mathematical Sciences [[arXiv:2101.06317](https://arxiv.org/abs/2101.06317), [doi:10.1142/S2810939222500010](https://doi.org/10.1142/S2810939222500010)]
reviewed in:
* [[Yang-Hui He]], *Universes as Bigdata*, [talk](M-Theory+and+Mathematics#He2020) at *[M-Theory and Mathematics 2020](M-Theory+and+Mathematics#2020)*, NYU Abu Dhabi (Jan 2020) [slides: [pdf](https://ncatlab.org/nlab/files/HeSlidesAtMTheoryAndMathematics2020.pdf), video: [YT](https://www.youtube.com/watch?v=i-kuoXjcKls)]
* [[Yang-Hui He]], *Machine-Learning Mathematical Structures*, [talk](/nlab/show/M-Theory+and+Mathematics#He2023) at *[M-Theory and Mathematics 2023](M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (Jan 2023) [slides: [[He-MTheoryAndMath2023.pdf:file]], video: [YT](https://www.youtube.com/watch?v=SlF2T4fxroQ)]
Discussion via [[machine learning]] of [[connection on a bundle|connections]] on [[heterotic line bundles]] over [[Calabi-Yau manifold|Calabi-Yau 3-folds]]:
* [[Anthony Ashmore]], Rehan Deen, [[Yang-Hui He]], [[Burt Ovrut]], *Machine Learning Line Bundle Connections* ([arXiv:2110.12483](https://arxiv.org/abs/2110.12483))
## Related entries
* [[quiver gauge theory]]
* [[string phenomenology]]
* [[heterotic string]]
* [[MSSM]]
* [[landscape of string theory vacua]]
category: people |
Yang-Mills equation | https://ncatlab.org/nlab/source/Yang-Mills+equation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Yang-Mills equations_ are the [[equations of motion]]/[[Euler-Lagrange equations]] of [[Yang-Mills theory]]. They generalize [[Maxwell's equations]].
## Related concepts
* [[Yang-Mills instanton]]
* [[Yang-Mills monopole]]
## References
(For full list of references see at _[[Yang-Mills theory]]_)
### General
* {#Uhlenbeck12} [[Karen Uhlenbeck]], notes by [[Laura Fredrickson]], _Equations of Gauge Theory_, lecture at Temple University, 2012 ([pdf](https://web.stanford.edu/~ljfred4/Attachments/TempleLectures.pdf), [[UhlenbeckGaugeTheory.pdf:file]])
* DispersiveWiki, _[Yang-Mills equations](http://wiki.math.toronto.edu/DispersiveWiki/index.php/Yang-Mills_equations)_
* [TP.SE](http://theoreticalphysics.stackexchange.com/), _[Which exact solutions of the classical Yang-Mills equations are known?](http://theoreticalphysics.stackexchange.com/questions/317/which-exact-solutions-of-the-classical-yang-mills-equations-are-known)_
### Solutions
#### General
Wu and Yang (1968) found a static solution to the sourceless $SU(2)$ [[Yang-Mills equations]]. Recent references include
* J. A. O. Marinho, O. Oliveira, B. V. Carlson, T. Frederico, _Revisiting the Wu-Yang Monopole: classical solutions and conformal invariance_
There is an old review,
* Alfred Actor, _Classical solutions of $SU(2)$ Yang—Mills theories_, Rev. Mod. Phys. 51, 461–525 (1979),
that provides some of the known solutions of $SU(2)$ gauge theory in [[Minkowski spacetime|Minkowski]] ([[monopoles]], plane waves, etc) and [[Euclidean space]] ([[instantons]] and their cousins). For general [[gauge groups]] one can get solutions by embedding $SU(2)$'s.
#### Instantons and monopoles
For [[Yang-Mills instantons]] the most general solution is known, first worked out by
* [[Michael Atiyah]], [[Nigel Hitchin]], [[Vladimir Drinfeld]], [[Yuri Manin]], _Construction of instantons_, Physics Letters 65 A, 3, 185--187 (1978) [pdf](http://www.new.ox.ac.uk/system/files/ADHM.pdf)
for the [[classical groups]] [[special unitary group|SU]], [[special orthogonal group|SO]] , [[symplectic group|Sp]], and then by
* C. Bernard, N. Christ, A. Guth, E. Weinberg, _Pseudoparticle Parameters for Arbitrary Gauge Groups_, Phys. Rev. __D16__, 2977 (1977)
for [[exceptional Lie groups]]. The latest twist on the [[Yang-Mills instanton]] story is the construction of solutions with non-trivial [[holonomy]]:
* Thomas C. Kraan, Pierre van Baal, _Periodic instantons with nontrivial holonomy_, Nucl.Phys. B533 (1998) 627-659, [hep-th/9805168](http://arxiv.org/abs/hep-th/9805168)
There is a nice set of lecture notes
* David Tong, _TASI Lectures on Solitons_ ([hep-th/0509216](http://arxiv.org/abs/hep-th/0509216)),
on topological solutions with different co-dimension ([[Yang-Mills instantons]], [[Yang-Mills monopoles]], vortices, domain walls). Note, however, that except for instantons these solutions typically require extra scalars and broken U(1)'s, as one may find in [[super Yang-Mills theories]].
Some of the material used here has been taken from
* [TP.SE](http://theoreticalphysics.stackexchange.com/), _[Which exact solutions of the classical Yang-Mills equations are known?](http://theoreticalphysics.stackexchange.com/questions/317/which-exact-solutions-of-the-classical-yang-mills-equations-are-known)_
Another model featuring Yang-Mills fields has been proposed by Curci and Ferrari, see [[Curci-Ferrari model]].
[[!redirects Yang-Mills equations]]
|
Yang-Mills field | https://ncatlab.org/nlab/source/Yang-Mills+field |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Yang--Mills [[field (physics)|field]]_ is the [[gauge theory|gauge field]] of [[Yang-Mills theory]].
It is modeled by a cocycle $\hat F \in \mathbf{H}(X, \mathbf{B}U(n)_{conn})$ in differential [[nonabelian cohomology]]. Here $\mathbf{B} U(n)_{conn}$ is the [[moduli stack]] of $U(n)$-[[principal connections]], the [[stackification]] of the [[groupoid of Lie-algebra valued forms]], regarded as a [[groupoid]] [[internal category|internal to]] [[smooth spaces]].
This is usually represented by a [[vector bundle]] [[connection on a bundle|with connection]].
As a [[nonabelian cohomology|nonabelian]] [[Čech cohomology|Čech cocycle]] the Yang-Mills field on a space $X$ is represented by
* a [[cover]] $\{U_i \to X\}$
* a collection of $Lie(U(n))$-valued 1-forms $(A_i \in \Omega^1(U_i, Lie(U(n))))$;
* a collection of $U(n)$-valued smooth functions $(g_{i j} \in C^\infty(U_{i j}, U(n)))$;
* such that on double overlaps
$$
A_j = Ad_{g_{i j}} \circ A_i + g_{i j} g g_{i j}^{-1}
\,,
$$
* and such that on triple overlaps
$$
g_{i j} g_{j k} = g_{i k}
\,.
$$
# Examples #
* For $U(n) = U(1)$ this is the [[electromagnetic field]].
* For $U(n) = SU(2) \times U(1)$ this is the "electroweak field";
* For $U(n) = SU(3) $ this is the strong nuclear force field.
[[!redirects Yang--Mills field]]
[[!redirects Yang–Mills field]]
[[!redirects Yang-Mills fields]] |
Yang-Mills instanton | https://ncatlab.org/nlab/source/Yang-Mills+instanton | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
#### Chern-Weil theory
+-- {: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In $SU(n)$-[[Yang-Mills theory]] an _[[instanton]]_ is a field configuration with non-vanishing second [[Chern class]] that minimizes the Yang-Mills energy.
## Definition
Let $(X,g)$ be a [[compact space|compact]] [[Riemannian manifold]] of [[dimension]] 4. Let $G$ be a compact [[Lie group]].
A [[configuration space|field configuration]] of $G$-[[Yang-Mills theory]] on $(X,g)$ is a $G$-[[principal bundle]] $P \to X$ with [[connection on a bundle|connection]] $\nabla$.
For $G = SU(n)$ the [[special unitary group]], there is canonically an [[associated bundle|associated]] complex [[vector bundle]] $E = P \times_G \mathbb{C}^n$.
Write $F_\nabla \in \Omega^2(X,End(E))$ for the [[curvature]] [[differential form|2-form]] of $\nabla$.
One says that $\nabla$ is an **instanton configuration** if $F_\nabla$ is [[Hodge star operator|Hodge]]-self dual
$$
\star F_\nabla = F_\nabla
\,,
$$
where $\star : \Omega^k(X) \to \Omega^{4-k}(X)$ is the [[Hodge star operator]] induced by the [[Riemannian metric]] $g$.
The second [[Chern class]] of $P$, which by the [[Chern-Weil homomorphism]] is given by
$$
c_2(E) = \int_X Tr(F_\nabla \wedge F_\nabla)
=
k
\in H^4(X, \mathbb{Z})
$$
is called the **instanton number** or the **instanton sector** of $\nabla$.
Notice that therefore any connection, even if not self-dual, is in some instanton sector, as its underlying bundle has some second Chern class, meaning that it can be obtained from shifting a self-dual connection. The self-dual connections are a convenient choice of "base point" in each instanton sector.
## Examples
### $SU(2)$-instantons from the correct maths to the traditional physics story
{#FromTheMathsToThePhysicsStory}
[[!include SU2-instantons from the correct maths to the traditional physics story]]
## Properties
### As gradient flows between flat connections.
We discuss how Yang-Mills instantons may be understood as trajectories of the [[gradient flow]] of the [[Chern-Simons theory]] [[action functional]].
Let $(\Sigma,g_\Sigma)$ be a [[compact space|compact]] 3-[[dimensional]] [[Riemannian manifold]] .
Let the [[cartesian product]]
$$
X = \Sigma \times \mathbb{R}
$$
of $\Sigma$ with the [[real line]] be equipped with the product metric of $g$ with the canonical metric on $\mathbb{R}$.
Consider field configurations $\nabla$ of [[Yang-Mills theory]] over $\Sigma \times \mathbb{R}$ with finite Yang-Mills action
$$
S_{YM}(\nabla) = \int_{\Sigma \times \mathbb{R}} F_\nabla \wedge \star F_\nabla
\,\,\lt \infty
\,.
$$
These must be such that there is $t_1 \lt t_2 \in \mathbb{R}$ such that $F_\nabla(t \lt t_1) = 0$ and $F_\nabla(T \gt t_2) = 0$, hence these must be solutions interpolating between two [[curvature|flat]] connections $\nabla_{t_1}$ and $\nabla_{t_2}$.
For $A \in \Omega^1(U\times \mathbb{R}, \mathfrak{g})$ the [[Lie algebra valued 1-form]] corresponding to $\nabla$, we can always find a [[gauge transformation]] such that $A_{\partial_t} = 0$ ("[[temporal gauge]]"). In this gauge we may hence equivalently think of $A$ as a 1-parameter family
$$
t \mapsto A(t) \in \Omega^1(\Sigma, \mathfrak{g})
$$
of connections on $\Sigma$. Then the self-duality condition on a Yang-Mills instanton
$$
F_\nabla = - \star F_\nabla
$$
reads equivalently
$$
\frac{d}{d t} A = -\star_{g} F_A
\,\,\,
\in
\Omega^1(\Sigma, \mathfrak{g})
\,.
$$
+-- {: .num_defn #HodgeInnerProduct}
###### Definition
On the linear [[configuration space]] $\Omega^1(\Sigma, \mathfrak{g})$ of [[Lie algebra valued forms]] on $\Sigma$ define the [[Hodge inner product]] [[metric]]
$$
G(\alpha, \beta) := \int_{\Sigma} \langle \alpha \wedge \star_g \beta \rangle
\,,
$$
where $\langle-,-\rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}$ is the [[Killing form]] [[invariant polynomial]] on the [[Lie algebra]] $\mathfrak{g}$.
=--
+-- {: .num_prop}
###### Proposition
The instanton equation
$$
\frac{d}{d t} A =
-\star_{g} F_A
$$
is the equation characterizing trajectories of the [[gradient flow]] of the [[Chern-Simons action functional]]
$$
S_{CS} : \Omega^1(\Sigma, \mathfrak{g}) \to \mathbb{R}
$$
$$
A \mapsto \int_\Sigma CS(A)
$$
with respect to the [Hodge inner product metric](#HodgeInnerProduct) on $\Omega^1(\Sigma,\mathfrak{g})$.
=--
+-- {: .proof}
###### Proof
The [[variational calculus|variation]] of the Chern-Simons action is
$$
\delta S_{CS}(A) = \int_\Sigma \langle \delta A \wedge F_A\rangle
$$
(see [[Chern-Simons theory]] for details).
In other words, we have the 1-form on $\Omega^1(\Sigma,\mathfrak{g})$:
$$
\delta S_{CS}(-)_A
=
\int_\Sigma \langle - \wedge F_A \rangle
\,.
$$
The corresponding [[gradient vector field]]
$$
\nabla S_{CS}
:=
G^{-1} \delta S_{CS}
$$
is uniquely defined by the equation
$$
\begin{aligned}
\delta S_{CS}(-)
& = G(-,\nabla S_{CS})
\\
\int_\Sigma \langle - , \star \nabla S_{CS}\rangle
\end{aligned}
\,.
$$
With the formula (see [[Hodge star operator]])
$$
\star \star A = (-1)^{1(3+1)} A = A
$$
we find therefore
$$
\nabla S_{CS} = \star_g F_A
\,.
$$
Hence the [[gradient flow]] equation
$$
\frac{d}{d t} A + \nabla S_{CS}_A = 0
$$
is indeed
$$
\frac{d}{d t} A = - \star_g F_A
\,.
$$
=--
Since [[curvature|flat]] connections are the [[critical loci]] of $S_{CS}$ this says that a finite-action Yang-Mills instanton on $\Sigma \times \mathbb{R}$ is a gradient flow trajectory between two _Chern-Simons theory [[vacuum|vacua]]_ .
Often this is interpreted as saying that "a Yang-Mills instanton describes the [[tunneling]] between two [[Chern-Simons theory]] [[vacua]]".
### As Dp-D(p+4)-brane bound states
{#AsDpDpPlus4BraneBoundStates}
Due to the [[higher WZW term]] $\propto \int_{D_{p+4}} C_{p+1} \wedge \langle F \wedge F \rangle$ in the [[Green-Schwarz sigma model]] for [[D-brane|D(p+4)-branes]], [[Yang-Mills instantons]] in the [[Chan-Paton gauge field]] on $D (p+4)$-branes are equivalently [[Dp-D(p+4)-brane bound states]] (see e.g. [Polchinski 96, 5.4](#Polchinski96), [Tong 05, 1.4](#Tong05)).
The lift to [[M-theory]] as [[M5-MO9 brane bound states]] is due to [Strominger 90](#Strominger90), [Witten 96](#Witten96).
[[!include Dp-D(p+4)-brane bound states -- contents]]
## Examples
* In $SU(2)$-YM theory: see _[[BPST instanton]]_ .
* In $SU(3)$-YM theory, [[QCD]]/[[strong nuclear force]]: see _[[instanton in QCD]]_
## Related concepts
* [[theta vacuum]], [[instanton sea]]
* [[instanton]]
* [[Bogomolny equation]], [[Nahm transform]]
* [[small instanton]]
* [[caloron]]
* [[non-perturbative effect]]
* [[instanton Floer homology]]
* [[self-dual higher gauge field]]
* [[magnetic monopole]]
[[Dirac monopole]], [[Yang monopole]]
* [[fiber bundles in physics]]
* [[Dirac charge quantization]]
* [[gravitational instanton]]
## References
The [[ADHM construction]]:
* {#ADHM} [[Michael Atiyah]], [[Nigel Hitchin]], [[Vladimir Drinfeld]], [[Yuri Manin]], *Construction of instantons*, Physics Letters A **65** 3 (1978) 185-187 [[pdf](http://www.new.ox.ac.uk/system/files/ADHM.pdf), <a href="https://doi.org/10.1016/0375-9601(78)90141-X">doi:10.1016/0375-9601(78)90141-X</a>]
### General
Introductions and surveys:
* J. Zinn-Justin, _The principles of instanton calculus_, Les Houches (1984)
* [[Mikhail Shifman]] et al., _ABC of instantons_, Fortschr. Phys. **32** 11 (1984) 585
* [[Tohru Eguchi]], [[Peter Gilkey]], [[Andrew Hanson]], Section 10.2 of: _Gravitation, gauge theories and differential geometry_, Physics Reports Volume 66, Issue 6, December 1980, Pages 213-393 (<a href="https://doi.org/10.1016/0370-1573(80)90130-1">doi:10.1016/0370-1573(80)90130-1</a>)
* {#Tong05} [[David Tong]], _TASI Lectures on Solitons_ ([arXiv:hep-th/0509216](https://arxiv.org/abs/hep-th/0509216)), _Lecture 1: Instantons_ ([pdf](http://www.damtp.cam.ac.uk/user/tong/tasi/instanton.pdf))
* [[Bert Lindenhovius]], *Instantons and the ADHM construction*, Amsterdam (2011) [[[Lindenhovius-Instantons.pdf:file]]]
* Nick Dorey, Timothy J. Hollowood, Valentin V. Khoze, Michael P. Mattis, *The Calculus of Many Instantons*, Phys. Rept. **371** (2002) 231-459 [[arXiv:hep-th/0206063](https://arxiv.org/abs/hep-th/0206063), <a href="https://doi.org/10.1016/S0370-1573(02)00301-0">doi:10.1016/S0370-1573(02)00301-0</a>]
* [[Taro Kimura]], *Instanton Counting and Localization*, Chapter 1 in: *Instanton Counting, Quantum Geometry and Algebra*, Mathematical Physics Studies, Springer (2021) [[doi:10.1007/978-3-030-76190-5_4](https://doi.org/10.1007/978-3-030-76190-5_4)]
A survey in view of the [[asymptotic series|asymptotic]] nature of the [[Feynman perturbation series]] is in
* {#Suslov05} [[Igor Suslov]], section 4.5 of _Divergent perturbation series_, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J.Exp.Theor.Phys. 100 (2005) 1188 ([arXiv:hep-ph/0510142](https://arxiv.org/abs/hep-ph/0510142))
For a fairly comprehensive list of literature see the bibliography of
* Marcus Hutter, _Instantons in QCD: Theory and Application of the Instanton Liquid Model_ ([arXiv:hep-ph/0107098](http://arxiv.org/abs/hep-ph/0107098))
Identification of the [[moduli space]] of instantons with that of [[holomorphic maps]] from the [[Riemann sphere]] to the [[loop group]] of the [[gauge group]]:
* {#Atiyah84} [[Michael Atiyah]], *Instantons in two and four dimensions*, Commun. Math. Phys. 93, 437–451 (1984) ([doi:10.1007/BF01212288](https://doi.org/10.1007/BF01212288))
Detailed argument for the [[theta-vacuum]] structure from [[chiral symmetry breaking]] is offered in
* [[Curtis Callan]], R.F. Dashen, [[David Gross]], _The Structure of the Gauge Theory Vacuum_, Phys.Lett. 63B (1976) 334-340 ([spire](http://inspirehep.net/record/3673?ln=en))
* G. Morchio, [[Franco Strocchi]], _Chiral symmetry breaking and theta vacuum structure in QCD_, Annals Phys.324:2236-2254, 2009 ([arXiv:0907.2522](https://arxiv.org/abs/0907.2522))
The multi-instantons in $SU(2)$-Yang-Mills theory ([[BPST instantons]]) were discovered in
* A. A. Belavin, A.M. Polyakov, A.S. Schwartz, Yu.S. Tyupkin, _Pseudoparticle solutions of the Yang-Mills equations_, Phys. Lett. B 59 (1), 85-87 (1975) <a href="http://dx.doi.org/10.1016/0370-2693(75)90163-X">doi</a>
* A. A. Belavin, V.A. Fateev, A.S. Schwarz, Yu.S. Tyupkin, _Quantum fluctuations of multi-instanton solutions_, Phys. Lett. B 83 (3-4), 317-320 (1979) <a href="http://dx.doi.org/10.1016/0370-2693(79)91117-1">doi</a>
See also
* [[Michael Atiyah]], [[Nigel Hitchin]], J. M. Singer, _Deformations of instantons_, Proc. Nat. Acad. Sci. U.S. __74__, 2662 (1977)
* [[Edward Witten]], _Some comments on the recent twistor space constructions_, Complex manifold techniques in theoretical physics (Proc. Workshop, Lawrence, Kan., 1978), pp. 207–218, Res. Notes in Math., 32, Pitman, Boston, Mass.-London, 1979.
Methods of algebraic geometry were introduced in
* M. F. Atiyah, R. S. Ward, _Instantons and algebraic geometry_, Comm. Math. Phys. __55__, n. 2 (1977), 117-124, [MR0494098](http://www.ams.org/mathscinet-getitem?mr=0494098), [euclid](http://projecteuclid.org/euclid.cmp/1103900980)
The more general [[ADHM construction]] in terms of linear algebra of vector bundles on projective varieties is proposed in
* M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld, Yu.I. Manin, _Construction of instantons_, Physics Letters 65 A, 3, 185--187 (1978) [pdf](http://www.new.ox.ac.uk/system/files/ADHM.pdf)
Monographs with the standard material include
* [[Dan Freed]], [[Karen Uhlenbeck]], _Instantons and four-manifolds_, Springer-Verlag, (1991)
* [[Robbert Dijkgraaf]], _Topological gauge theories and group cohomology_ ([ps](staff.science.uva.nl/~rhd/papers/group.ps))
* Nicholas Manton, Paul M. Sutcliffe, _Topological solitons_, Cambridge Monographs on Math. Physics, [gBooks](http://books.google.com/books?id=e2tPhFdSUf8C)
Yang-Mills instantons on spaces other than just spheres are explicitly discussed in
* [[Gabor Kunstatter]], _Yang-mills theory in a multiply connected three space_, Mathematical Problems in Theoretical Physics: Proceedings of the VIth International Conference on Mathematical Physics Berlin (West), August 11-20,1981. Editor: R. Schrader, R. Seiler, D. A. Uhlenbrock, Lecture Notes in Physics, vol. 153, p.118-122 ([web](http://adsabs.harvard.edu/abs/1982LNP...153..118K))
based on
* [[Chris Isham]], [[Gabor Kunstatter]], Phys. Letts. v.102B, p.417, 1981. ([doi](http://dx.doi.org/10.1016/0370-2693%2881%2991244-2))
* [[Chris Isham]] [[Gabor Kunstatter]], J. Math. Phys. v.23, p.1668, 1982. ([doi](http://dx.doi.org/10.1063/1.525552))
In
* Henrique N. Sá Earp, _Instantons on $G_2$−manifolds_ PhD thesis (2009) ([pdf](http://www.ime.unicamp.br/~hqsaearp/index_files/HENRIQUE_SA_EARP_THESIS_UPDATED.PDF))
is a discussion of Yang-Mills instantons on a 7-dimensional [[manifold with special holonomy]].
* [[Michael Atiyah]], [[R. Bott]], _The Yang-Mills equations over Riemann surfaces_, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615, [MR85k:14006](http://www.ams.org/mathscinet-getitem?mr=702806), [doi](http://dx.doi.org/10.1098/rsta.1983.0017).
### As Dp-D(p+4)-brane bound states
The argument that [[Yang-Mills instantons]] in the [[Chan-Paton gauge field]] on a [[D-brane|D(p+4)-brane]] are equivalent to [[Dp-D(p+4) brane bound states]] goes back to
* {#Witten96} [[Edward Witten]], _Small Instantons in String Theory_, Nucl. Phys. B460:541-559, 1996 ([arXiv:hep-th/9511030](https://arxiv.org/abs/hep-th/9511030))
* [[Michael Douglas]], _Branes within Branes_, In: Baulieu L., Di Francesco P., Douglas M., Kazakov V., Picco M., Windey P. (eds.) _[Strings, Branes and Dualities](https://link.springer.com/book/10.1007/978-94-011-4730-9)_ NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 520. Springer, Dordrecht ([arxiv:hep-th/9512077](https://arxiv.org/abs/hep-th/9512077), [doi:10.1007/978-94-011-4730-9_10](https://doi.org/10.1007/978-94-011-4730-9_10))
* [[Michael Douglas]], _Gauge Fields and D-branes_, J. Geom. Phys. 28 (1998) 255-262 ([arXiv:hep-th/9604198](https://arxiv.org/abs/hep-th/9604198))
following
* {#Strominger90} [[Andrew Strominger]], _Heterotic solitons_, Nucl. Phys. B343 (1990) 167-184 (<a href="https://doi.org/10.1016/0550-3213(90)90599-9">doi:10.1016/0550-3213(90)90599-9</a>) Erratum: Nucl. Phys. B353 (1991) 565-565 (<a href="https://doi.org/10.1016/0550-3213(91)90349-3">doi:10.1016/0550-3213(91)90349-3</a>) ([spire:27900](http://inspirehep.net/record/27900))
Review is in:
* {#Polchinski96} [[Joseph Polchinski]], Section 5.4 of: _TASI Lectures on D-Branes_ ([arXiv:hep-th/9611050](https://arxiv.org/abs/hep-th/9611050))
* {#Tong05} [[David Tong]], Section 1.4 of _TASI Lectures on Solitons_ ([hep-th/0509216](https://arxiv.org/abs/hep-th/0509216))
Discussion specifically of [[D0-D4-brane bound states]]:
* [[Cumrun Vafa]], _Instantons on D-branes_, Nucl. Phys. B463 (1996) 435-442 ([arXiv:hep-th/9512078](https://arxiv.org/abs/hep-th/9512078))
with emphasis to the resulting [[configuration spaces of points]], as in
* [[Cumrun Vafa]], [[Edward Witten]], Section 4.1 of: _A Strong Coupling Test of S-Duality_, Nucl. Phys. B431:3-77, 1994 ([arXiv:hep-th/9408074](https://arxiv.org/abs/hep-th/9408074))
Discussion specifically of [[D1-D5-brane bound states]]
* [[Neil Lambert]], _D-brane Bound States and the Generalised ADHM Construction_, Nucl. Phys. B519 (1998) 214-224 ([arXiv:hep-th/9707156](https://arxiv.org/abs/hep-th/9707156))
Discussion specifically of [[D4-D8-brane bound states]]:
In the [[Witten-Sakai-Sugimoto model]] [[geometric engineering of QFT|geometrically engineering]] [[QCD]], where the [[D4-branes]] get interpreted as [[baryons]]:
* {#SakaiSugimoto04} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], Section 5.7 of: _Low energy hadron physics in holographic QCD_, Prog. Theor. Phys.113:843-882, 2005 ([arXiv:hep-th/0412141](https://arxiv.org/abs/hep-th/0412141))
[[!redirects Yang-Mills instantons]]
[[!redirects instanton number]]
[[!redirects instanton numbers]]
|
Yang-Mills instanton Floer homology | https://ncatlab.org/nlab/source/Yang-Mills+instanton+Floer+homology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_Instanton Floer homology_ is a variant of [[Floer homology]] which applies to 3-dimensional [[manifold]]s. It is effectively the [[Morse homology]] of the [[Chern-Simons theory]] [[action functional]].
For $\Sigma$ a 3-[[dimension]]al [[compact space|compact]] [[smooth manifold]] and $G$ a [[simply connected]] [[compact space|compact]] [[Lie group]] let $[\Sigma,\mathbf{B}G_{conn}]$ be the [[space]] of $G$-[[connection on a bundle|connections]] on $\Sigma$, which is equivalently the [[groupoid of Lie algebra valued forms]] on $\Sigma$ in this case.
The _instanton Floer homology_ groups of $\Sigma$ are something like the "mid-dimensional" [[singular homology]] groups of the [[configuration space]] $[\Sigma,\mathbf{B}G_{conn}]$.
More precisely, there is canonically the [[Chern-Simons action functional]]
$$
S_{CS} : [\Sigma,\mathbf{B}G_{conn}] \to U(1)
$$
on this space of connections, and one can form the corresponding [[Morse homology]].
The [[critical locus]] of $S_{CS}$ is the space of flat $G$-connections (those with vanishing [[curvature]]), whereas the [[flow line]]s of $S_{CS}$ correspond to the [[Yang-Mills instanton]]s on $\Sigma \times [0,1]$.
## References
The original reference is
* [[Andreas Floer]], _An instanton-invariant for 3-manifolds_ , Comm. Math. Phys. __118__ (1988), no. 2, 215–240, [euclid](http://projecteuclid.org/euclid.cmp/1104161987)
Reviews:
* [[Simon Donaldson]], _Floer homology groups in Yang-Mills theory_ Cambridge Tracts in Mathematics __147__ (2002), [pdf](http://catdir.loc.gov/catdir/samples/cam031/2001035888.pdf)
* Tomasz S. Mrowka, _Introduction to Instanton Floer Homology_ at _Introductory Workshop: Homology Theories of Knots and Links_ , MSRI ([video](http://www.msri.org/web/msri/online-videos/-/video/showVideo/4015))
Generalizations to 3-manifolds with [[boundary]] are discussed in
* Dietmar Salamon, Katrin Wehrheim, _Instanton Floer homology with Lagrangian boundary conditions_ ([arXiv:0607318](http://arxiv.org/abs/math/0607318))
[[!redirects instanton Floer homology]]
[[!redirects instanton homology]] |
Yang-Mills monopoles as rational maps -- references | https://ncatlab.org/nlab/source/Yang-Mills+monopoles+as+rational+maps+--+references |
### Identification of Yang-Mills monopoles with rational maps
{#ReferencesIdentificationOfYangMillsMonopolesWithRationalMaps}
The following lists references concerned with the identification of the (extended) [[moduli space of Yang-Mills monopoles]] (in the [[BPS state|BPS limit]], i.e. for vanishing [[Higgs field|Higgs]] potential) with a [[mapping space]] of [[complex manifold|complex]] [[rational maps]] from the [[complex plane]], equivalently [[holomorphic maps]] from the [[Riemann sphere]] $\mathbb{C}P^1$ (at infinity in $\mathbb{R}^3$) to itself (for [[gauge group]] [[SU(2)]]) or generally to a complex [[flag variety]] such as (see [Ionnadou & Sutcliffe 1999a](#IonnadouSutcliffe99a) for review) to a [[coset space]] by the [[maximal torus]] (for maximal [[symmetry breaking]]) or to [[complex projective space]] $\mathbb{C}P^{n-1}$ (for [[gauge group]] [[SU(n)]] and minimal symmetry breaking).
The identification was [[conjecture|conjectured]] (following an analogous result for [[Yang-Mills instantons]]) in:
* {#Atiyah84} [[Michael Atiyah]], Section 5 of: *Instantons in two and four dimensions*, Commun. Math. Phys. 93, 437–451 (1984) ([doi:10.1007/BF01212288](https://doi.org/10.1007/BF01212288))
Full understanding of the rational map involved as "[[scattering]] data" of the monopole is due to:
* [[Jacques Hurtubise]], *Monopoles and rational maps: a note on a theorem of Donaldson*, Comm. Math. Phys. 100(2): 191-196 (1985) ([euclid:cmp/1103943443](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-100/issue-2/Monopoles-and-rational-maps--a-note-on-a-theorem/cmp/1103943443.full))
The identification with ([[pointed object|pointed]]) [[holomorphic functions]] out of [[Riemann sphere|$\mathbb{C}P^1$]] was proven...
...for the case of [[gauge group]] [[SU(2)|$SU(2)$]] (maps to [[Riemann sphere|$\mathbb{C}P^1$]] itself) in
* {#Donaldson84} [[Simon Donaldson]], _Nahm's Equations and the Classification of Monopoles_, Comm. Math. Phys., Volume 96, Number 3 (1984), 387-407, ([euclid:cmp.1103941858](https://projecteuclid.org/euclid.cmp/1103941858))
...for the more general case of [[classical Lie group|classical]] [[gauge group]] with maximal symmetry breaking (maps to the [[coset space]] by the [[maximal torus]]) in:
* [[Jacques Hurtubise]], *The classification of monopoles for the classical groups*, Commun. Math. Phys. 120, 613–641 (1989) ([doi:10.1007/BF01260389](https://doi.org/10.1007/BF01260389))
* [[Jacques Hurtubise]], [[Michael K. Murray]], *On the construction of monopoles for the classical groups*, Comm. Math. Phys. 122(1): 35-89 (1989) ([euclid:cmp/1104178316](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-122/issue-1/On-the-construction-of-monopoles-for-the-classical-groups/cmp/1104178316.full))
* [[Michael Murray]], *Stratifying monopoles and rational maps*, Commun. Math. Phys. 125, 661–674 (1989) ([doi:10.1007/BF01228347](https://doi.org/10.1007/BF01228347))
* [[Jacques Hurtubise]], [[Michael K. Murray]], *Monopoles and their spectral data*, Comm. Math. Phys. 133(3): 487-508 (1990) ([euclid:cmp/1104201504](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-133/issue-3/Monopoles-and-their-spectral-data/cmp/1104201504.full))
... for the fully general case of [[semisimple Lie groups|semisimple]] [[gauge groups]] with any [[symmetry breaking]] (maps to any [[flag varieties]]) in
* [[Stuart Jarvis]], *Euclidian Monopoles and Rational Maps*, Proceedings of the London Mathematical Society **77** 1 (1998) 170-192 ([doi:10.1112/S0024611598000434](https://doi.org/10.1112/S0024611598000434))
* [[Stuart Jarvis]], *Construction of Euclidian Monopoles*, Proceedings of the London Mathematical Society, **77** 1 (1998) ([doi:10.1112/S0024611598000446](https://doi.org/10.1112/S0024611598000446))
and for un-pointed maps in
* [[Stuart Jarvis]], *A rational map of Euclidean monopoles via radial scattering*, J. Reine angew. Math. **524** (2000) 17-41([doi:10.1515/crll.2000.055](https://doi.org/10.1515/crll.2000.055))
Further discussion:
* [[Charles P. Boyer]], [[B. M. Mann]], *Monopoles, non-linear $\sigma$-models, and two-fold loop spaces*, Commun. Math. Phys. 115, 571–594 (1988) ([arXiv:10.1007/BF01224128](https://doi.org/10.1007/BF01224128))
* {#IonnadouSutcliffe99a} [[Theodora Ioannidou]], [[Paul Sutcliffe]], *Monopoles and Harmonic Maps*, J. Math. Phys. 40:5440-5455 (1999) ([arXiv:hep-th/9903183](https://arxiv.org/abs/hep-th/9903183))
* {#IonnadouSutcliffe99b} [[Theodora Ioannidou]], [[Paul Sutcliffe]], *Monopoles from Rational Maps*, Phys. Lett. B457 (1999) 133-138 ([arXiv:hep-th/9905066](https://arxiv.org/abs/hep-th/9905066))
* Max Schult, *Nahm's Equations and Rational Maps from $\mathbb{C}P^1$ to $\mathbb{C}P^n$* \[<a href="https://arxiv.org/abs/2310.18058">arXiv:2310.18058</a>\]
Review:
* [[Alexander B. Atanasov]], *Magnetic monopoles and the equations of Bogomolny and Nahm* ([pdf](http://abatanasov.com/Files/3D%20Monopoles.pdf)), chapter 5 in: *Magnetic Monopoles, 't Hooft Lines, and the Geometric Langlands Correspondence*, 2018 ([pdf](http://abatanasov.com/Files/Thesis.pdf), [slides](http://abatanasov.com/Files/Thesis%20Presentation.pdf))
On the relevant [[homotopy of rational maps]] (see there for more references):
* {#Segal79} [[Graeme Segal]], _The topology of spaces of rational functions_, Acta Math. Volume 143 (1979), 39-72 ([euclid:1485890033](https://projecteuclid.org/euclid.acta/1485890033))
|
Yang-Mills theory | https://ncatlab.org/nlab/source/Yang-Mills+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
Yang--Mills theory is a [[gauge theory]] on a given 4-[[dimensional]] ([[pseudo-Riemannian metric|pseudo]]-)[[Riemannian manifold|Riemannian]] [[manifold]] $X$ whose field is the [[Yang–Mills field]] -- a cocycle $\nabla \in \mathbf{H}(X,\bar \mathbf{B}U(n))$ in differential [[nonabelian cohomology]] represented by a [[vector bundle]] [[connection on a bundle|with connection]] -- and whose [[action functional]] is
$$
\nabla \mapsto
\frac{1}{g^2 }\int_X tr(F_\nabla \wedge \star F_\nabla) \;+\; i \theta \int_X tr(F_\nabla \wedge F_\nabla)
$$
for
* $F_\nabla$ the [[field strength]], locally the [[curvature]] $\mathfrak{u}(n)$-[[Lie algebra valued differential form]] on $X$ ( with $\mathfrak{u}(n)$ the [[Lie algebra]] of the [[unitary group]] $U(n)$);
* $\star$ the [[Hodge star]] operator of the metric $g$;
* $\frac{1}{g^2}$ the _Yang-Mills [[coupling constant]]_ and $\theta$ the _[[theta angle]]_, some [[real numbers]] (see at _[[S-duality]]_).
(See [this example](A+first+idea+of+quantum+field+theory#YangMillsLagrangian) at _[[A first idea of quantum field theory]]_.)
## Properties
### Classification of solutions
* [[Narasimhan-Seshadri theorem]]
* [[Donaldson-Uhlenbeck-Yau theorem]]
### Quantization
Despite its fundamental role in the [[standard model of particle physics]], various details of the [[quantization]] of Yang-Mills theory are still open. See at _[[quantization of Yang-Mills theory]]_.
## Applications
All gauge fields in the [[standard model of particle physics]] as well as in [[GUT]] models are Yang--Mills fields.
The matter fields in the standard model are spinors charged under the Yang-Mills field. See
* [[spinors in Yang-Mills theory]]
## History
{#History}
From [Jaffe-Witten](#JaffeWitten):
> By the 1950s, when Yang–Mills theory was discovered, it was already known that the [[quantum field theory|quantum version]] of [[electromagnetism|Maxwell theory]] -- known as [[quantum electrodynamics|Quantum Electrodynamics]] or QED -- gives an extremely accurate account of [[electromagnetic fields]] and [[Lorentz force|forces]]. In fact, QED improved the accuracy for certain earlier quantum theory predictions by several orders of magnitude, as well as predicting new splittings of energy levels.
> So it was natural to inquire whether [[non-abelian group|non-abelian]] [[gauge theory]] described other [[forces]] in [[observable universe|nature]], notably the [[weak nuclear force|weak force]] (responsible among other things for certain forms of radioactivity) and the [[strong nuclear force|strong or nuclear force]] (responsible among other things for the binding of [[protons]] and [[neutrons]] into nuclei). The massless nature of [[classical field theory|classical]] Yang–Mills waves was a serious obstacle to applying Yang–Mills theory to the other forces, for the [[weak nuclear force|weak]] and [[strong nuclear force|nuclear forces]] are short range and many of the particles are massive. Hence these phenomena did not appear to be associated with long-range fields describing massless particles.
> In the 1960s and 1970s, physicists overcame these obstacles to the physical interpretation of non-abelian gauge theory. In the case of the weak force, this was accomplished by the Glashow–Salam–Weinberg [[electroweak theory]] with gauge group $H = $ [[special unitary group|SU(2)]] $\times$ [[circle group|U(1)]]. By elaborating the theory with an additional "[[Higgs field]]", one avoided the massless nature of [[classical field theory|classical]] Yang–Mills waves. The [[Higgs field]] transforms in a two-dimensional [[representation]] of $H$; its non-zero and approximately constant value in the [[vacuum state]] reduces the [[structure group]] from $H$ to a $U(1)$ [[subgroup]] ([[diagonal|diagonally]] embedded in $SU(2) \times U(1)$. This [[theory (physics)|theory]] describes both the [[electromagnetism|electromagnetic]] and [[weak nuclear force|weak forces]], in a more or less unified way; because of the reduction of the structure group to $U(1)$, the long-range fields are those of [[electromagnetism]] only, in accord with what we see in [[observable universe|nature]].
> The solution to the problem of massless Yang–Mills fields for the [[strong nuclear force|strong interactions]] has a completely different nature. That solution did not come from adding fields to Yang–Mills theory, but by discovering a remarkable property of the quantum Yang–Mills theory itself, that is, of the quantum theory whose [[classical field theory|classical]] [[Lagrangian density|Lagrangian]] has been given $[$above$]$. This property is called "[[asymptotic freedom]]". Roughly this means that at short distances the field displays [[quantum physics|quantum]] behavior very similar to its classical behavior; yet at long distances the [[classical field theory|classical theory]] is no longer a good guide to the quantum behavior of the [[field (physics)|field]].
> [[asymptotic freedom|Asymptotic freedom]], together with other experimental and theoretical discoveries made in the 1960s and 1970s, made it possible to describe the [[strong nuclear force|nuclear force]] by a [[non-abelian group|non-abelian]] [[gauge theory]] in which the [[gauge group]] is $G = $ [[special unitary group|SU(3)]]. The additional [[field (physics)|fields]] describe, at the [[classical field theory|classical level]], "[[quarks]]," which are [[spinor|spin 1/2 objects]] somewhat analogous to the [[electron]], but transforming in the [[fundamental representation]] of $SU(3)$. The non-abelian gauge theory of the strong force is called [[quantum chromodynamics|Quantum Chromodynamics]] (QCD).
> The use of [[QCD]] to describe the [[strong nuclear force|strong force]] was motivated by a whole series of experimental and theoretical discoveries made in the 1960s and 1970s, involving the [[symmetries]] and high-energy behavior of the strong interactions. But [[classical field theory|classical]] [[non-abelian group|non-abelian]] [[gauge theory]] is very different from the [[observable universe|observed world]] of [[strong nuclear force|strong interactions]]; for [[QCD]] to describe the strong force successfully, it must have at the quantum level the following three properties, each of which is dramatically different from the behavior of the classical theory:
> (1) It must have a "[[mass gap]];" namely there must be some constant $\Delta \gt 0$ such that every excitation of the [[vacuum]] has [[energy]] at least $\Delta$.
> (2) It must have "[[quark]] [[confinement]]," that is, even though the theory is described in terms of elementary [[field (physics)|fields]], such as the quark fields, that transform non-trivially under [[special unitary group|SU(3)]], the physical particle states -- such as the [[proton]], [[neutron]], and [[pion]] --are [[special unitary group|SU(3)]]-[[invariant]].
> (3) It must have "[[chiral symmetry breaking]]," which means that the vacuum is potentially invariant (in the [[limit of a sequence|limit]], that the quark-bare masses vanish) only under a certain [[subgroup]] of the full [[symmetry group]] that [[action|acts]] on the [[quark]] [[field (physics)|fields]].
> The first point is necessary to explain why the [[strong nuclear force|nuclear force]] is strong but short-ranged; the second is needed to explain why we never see individual [[quarks]]; and the third is needed to account for the "[[current algebra]]" theory of soft [[pions]] that was developed in the 1960s.
> Both [[experiment]] -- since [[QCD]] has numerous successes in confrontation with experiment -- and [[lattice gauge theory|computer simulations]], carried out since the late 1970s, have given strong encouragement that QCD does have the properties cited above. These properties can be seen, to some extent, in theoretical calculations carried out in a variety of highly oversimplified models (like strongly coupled [[lattice gauge theory]]). But they are not fully understood theoretically; there does not exist a convincing, whether or not mathematically complete, theoretical computation demonstrating any of the three properties in QCD, as opposed to a severely simplified truncation of it.
This is the problem of [[non-perturbative quantum field theory|non-perturbative]] [[quantization of Yang-Mills theory]]. See there for more.
## Related concepts
* [[Gauss law]]
* [[D=5 Yang-Mills theory]]
* [[D=2 Yang-Mills theory]]
* [[massive Yang-Mills theory]]
* [[self-dual Yang-Mills theory]]
* [[super Yang-Mills theory]]
* [[minimal coupling]]
* [['t Hooft double line notation]]
* [[Einstein-Yang-Mills theory]]
* [[Einstein-Maxwell theory]]
* [[Einstein-Yang-Mills-Dirac theory]]
* [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]]
* [[Yang-Mills equation]]
* [[standard model of particle physics]]
* [[electromagnetism]]
* [[spinors in Yang-Mills theory]]
* [[QED]], [[QCD]],
* [[electroweak field]]
* [[Yang monopole]], [['t Hooft-Polyakov monopole]]
* [[S-duality]], [[Montonen-Olive duality]]
* [[electric-magnetic duality]]
* [[geometric Langlands duality]]
* [[Chern-Simons theory]]
* [[Yang-Mills instanton]]
* [[confinement]]
* [[asymptotic freedom]]
* [[uncertainty of fluxes]]
## References
### General
Yang-Mills theory is named after the article
* [[Chen Ning Yang]], [[Robert Mills]], _Conservation of Isotopic Spin and Isotopic Gauge Invariance_. Physical Review 96 (1): 191– 195. (1954) ([web](http://prola.aps.org/abstract/PR/v96/i1/p191_1))
which was the first to generalize the principle of [[electromagnetism]] to a [[non-abelian group|non-abalian]] [[gauge group]]. This became accepted as formulation of [[QCD]] and [[weak interactions]] (only) after [[spontaneous symmetry breaking]] (the [[Higgs mechanism]]) was understood in the 1960s.
The identification of Yang-Mills [[gauge potentials]] with [[connections]] on [[fiber bundles]] is due to:
* {#WuYang75} [[Tai Tsun Wu]], [[Chen Ning Yang]], *Concept of nonintegrable phase factors and global formulation of gauge fields*, Phys. Rev. D **12** (1975) 3845 [[doi:10.1103/PhysRevD.12.3845](https://doi.org/10.1103/PhysRevD.12.3845)]
On the historical origins:
* A. C. T. Wu, [[Chen Ning Yang]], *Evolution of the concept of vector potential in the description of the fundamental interactions*, International Journal of Modern Physics A **21** 16 (2006) 3235-3277 [[doi:10.1142/S0217751X06033143](https://doi.org/10.1142/S0217751X06033143)]
* [[Chen Ning Yang]], *The conceptual origins of Maxwell’s equations and gauge theory*, Phyics Today **67** 11 (2014) [[doi:10.1063/PT.3.2585](https://doi.org/10.1063/PT.3.2585), [pdf](http://home.ustc.edu.cn/~lxsphys/2021-3-18/The%20conceptual%20origins%20of%20Maxwell%27s%20equations%20and%20gauge%20theory.pdf)]
Review of the basics:
* {#JaffeWitten00} [[Arthur Jaffe]], [[Edward Witten]], _Quantum Yang-Mills theory_ (2000) [[pdf](https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf), [pdf](https://www.arthurjaffe.com/Assets/pdf/QuantumYangMillsWebRevised.pdf), [[JaffeWitten-QuantumYangMills.pdf:file]]]
> (in the context of the [[mass gap problem]])
* [[Mikio Nakahara]], Section 10.5.4 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>]
* {#Donaldson05} [[Simon Donaldson]], _Yang-Mills theory and geometry_ (2005) [[pdf](https://www.ma.imperial.ac.uk/~skdona/YMILLS.PDF), [[Donaldson-YangMillsGeometry.pdf:file]]]
* {#Donaldson06} [[Simon Donaldson]], _Gauge Theory: Mathematical Applications_, Encyclopedia of Mathematical Physics, Academic Press (2006) 468-481 [[doi:10.1016/B0-12-512666-2/00075-4](https://doi.org/10.1016/B0-12-512666-2/00075-4), [author pdf](http://wwwf.imperial.ac.uk/~skdona/EMP.PDF), [[DonaldsonGaugeTheory.pdf:file]]]
* [[José Figueroa-O'Farrill]], *Gauge theory*, lecture notes (2006) [[web](http://empg.maths.ed.ac.uk/Activities/GT/index.html)]
* {#Uhlenbeck12} [[Karen Uhlenbeck]], notes by [[Laura Fredrickson]], _Equations of Gauge Theory_, lecture at Temple University, 2012 ([pdf](https://web.stanford.edu/~ljfred4/Attachments/TempleLectures.pdf), [[UhlenbeckGaugeTheory.pdf:file]])
* [[Gerd Rudolph]], [[Matthias Schmidt]], Chapters 7-9 of: *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer (2017) [[doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8)]
See also the references at _[[QCD]]_, _[[gauge theory]]_, _[[Yang-Mills monopole]]_, _[[Yang-Mills instanton]]_ and at _[[super Yang-Mills theory]]_.
Classical discussion of YM-theory over [[Riemann surfaces]] (which is closely related to [[Chern-Simons theory]], see also at _[[moduli space of flat connections]]_) is in
* {#AtiyahBott83} [[Michael Atiyah]], [[Raoul Bott]], _The Yang-Mills equations over Riemann surfaces_, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 ([jstor](http://www.jstor.org/stable/37156), [lighning summary](http://math.stackexchange.com/a/295505/58526))
which is reviewed in the lecture notes
* [[Jonathan Evans]], _Aspects of Yang-Mills theory_, ([web](http://www.homepages.ucl.ac.uk/~ucahjde/yangmills.htm))
For the relation to [[instanton Floer homology]] see also
* [[Simon Donaldson]], _Floer homology groups in Yang-Mills theory_ Cambridge University Press (2002) ([pdf](http://catdir.loc.gov/catdir/samples/cam031/2001035888.pdf))
For the relation to [[Tamagawa numbers]] see
* Aravind Asok, Brent Doran, Frances Kirwan, _Yang-Mills theory and Tamagawa numbers_ ([arXiv:0801.4733](http://arxiv.org/abs/arXiv:0801.4733))
### Classical solutions
Wu and Yang (1968) found a static solution to the sourceless $SU(2)$ [[Yang-Mills equations]]. Recent references include
* J. A. O. Marinho, O. Oliveira, B. V. Carlson, T. Frederico, _Revisiting the Wu-Yang Monopole: classical solutions and conformal invariance_
There is an old review,
* Alfred Actor, _Classical solutions of $SU(2)$ Yang—Mills theories_, Rev. Mod. Phys. 51, 461–525 (1979),
that provides some of the known solutions of $SU(2)$ gauge theory in [[Minkowski spacetime|Minkowski]] ([[monopoles]], plane waves, etc) and [[Euclidean space]] ([[instantons]] and their cousins). For general [[gauge groups]] one can get solutions by embedding $SU(2)$'s.
For [[Yang-Mills instantons]] the most general solution is known, first worked out by
* [[Michael Atiyah]], [[Nigel Hitchin]], [[Vladimir Drinfeld]], [[Yuri Manin]], _Construction of instantons_, Physics Letters 65 A, 3, 185--187 (1978) [pdf](http://www.new.ox.ac.uk/system/files/ADHM.pdf)
for the [[classical groups]] [[special unitary group|SU]], [[special orthogonal group|SO]] , [[symplectic group|Sp]], and then by
* C. Bernard, N. Christ, A. Guth, E. Weinberg, _Pseudoparticle Parameters for Arbitrary Gauge Groups_, Phys. Rev. __D16__, 2977 (1977)
for [[exceptional Lie groups]]. The latest twist on the [[Yang-Mills instanton]] story is the construction of solutions with non-trivial [[holonomy]]:
* Thomas C. Kraan, Pierre van Baal, _Periodic instantons with nontrivial holonomy_, Nucl.Phys. B533 (1998) 627-659, [hep-th/9805168](http://arxiv.org/abs/hep-th/9805168)
There is a nice set of lecture notes
* David Tong, _TASI Lectures on Solitons_ ([hep-th/0509216](http://arxiv.org/abs/hep-th/0509216)),
on topological solutions with different co-dimension (instantons, monopoles, vortices, domain walls). Note, however, that except for instantons these solutions typically require extra scalars and broken U(1)'s, as one may find in [[super Yang-Mills theories]].
Some of the material used here has been taken from
* [TP.SE](http://theoreticalphysics.stackexchange.com/), _[Which exact solutions of the classical Yang-Mills equations are known?](http://theoreticalphysics.stackexchange.com/questions/317/which-exact-solutions-of-the-classical-yang-mills-equations-are-known)_
Another model featuring Yang-Mills fields has been proposed by Curci and Ferrari, see [[Curci-Ferrari model]].
See also
* DispersiveWiki, _[Yang-Mills equations](http://wiki.math.toronto.edu/DispersiveWiki/index.php/Yang-Mills_equations)_
### Phase space and canonical quantization
{#ReferencesPhaseSpaceAndCanonicalQuantization}
On the [[phase space]], [[Poisson brackets]] and their [[quantization]] in Yang-Mills theory:
* Taichiro Kugo, Izumi Ojima, *Manifestly Covariant Canonical Formulation of the Yang-Mills Field Theories. I: General Formalism*, Progress of Theoretical Physics **60** 6 (1978) 1869–1889 [[doi:10.1143/PTP.60.1869](https://doi.org/10.1143/PTP.60.1869)]
* {#FriedmanPapastamatiou83} [[John L. Friedman]], Nicholas J. Papastamatiou, *On the canonical quantization of Yang-Mills theories*, Nuclear Physics B **219** 1 (1983) 125-142 \[<a href="https://doi.org/10.1016/0550-3213(83)90431-5">doi:10.1016/0550-3213(83)90431-5</a>\]
* {#BassettoLazzizzeraSoldati84} A. Bassetto, I. Lazzizzera, R. Soldati, *Yang-Mills theories in space-like axial and planar gauges*, Nuclear Physics B **236** 2 (1984) 319-335 \[<a href="https://doi.org/10.1016/0550-3213(84)90538-8">doi:10.1016/0550-3213(84)90538-8</a>]
* D. M. Gitman, S. L. Lyakhovich & I. V. Tyutin, *Canonical quantization of the Yang-Mills Lagrangian with higher derivatives*, Soviet Physics Journal **28** (1985) 554–556 [[doi:10.1007/BF00896182](https://doi.org/10.1007/BF00896182)]
* Kurt Haller, *Yang-Mills theory and quantum chromodynamics in the temporal gauge*, Phys. Rev. D **36** (1987) 1839 [[doi:10.1103/PhysRevD.36.1839](https://doi.org/10.1103/PhysRevD.36.1839)]
* {#Haagensen93} P. E. Haagensen, *On The Exact Implementation Of Gauss' Law In Yang-Mills Theory* [[arXiv:hep-ph/9307319](https://arxiv.org/abs/hep-ph/9307319)]
* [[Sarada G. Rajeev]], O. T. Turgut, *Poisson Algebra of Wilson Loops in Four-Dimensional Yang-Mills Theory*, Int. J. Mod. Phys. A **10** (1995) 2479 [[arXiv:hep-th/9410053](https://arxiv.org/abs/hep-th/9410053), [doi:10.1142/S0217751X95001194](https://doi.org/10.1142/S0217751X95001194)]
> (in [[light-front formalism]])
* [[Jonathan Dimock]], *Canonical Quantization of Yang-Millson a circle*, Reviews in Mathematical Physics **08** 01 (1996) 85-102 [[doi:10.1142/S0129055X96000044](https://doi.org/10.1142/S0129055X96000044)]
* {#BlaschkeGieres21} [[Daniel N. Blaschke]], [[François Gieres]], *On the canonical formulation of gauge field theories and Poincaré transformations*, Nuclear Physics B **965** (2021) 115366 [[doi:10.1016/j.nuclphysb.2021.115366](https://doi.org/10.1016/j.nuclphysb.2021.115366), [arXiv:2004.14406](https://arxiv.org/abs/2004.14406)]
* {#Riello21} Aldo Riello, *Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back*, SciPost Phys. **10** 125 (2021) [[doi:10.21468/SciPostPhys.10.6.125](https://scipost.org/SciPostPhys.10.6.125)]
[[!redirects Yang-Mills theories]]
[[!redirects Yang--Mills theory]]
[[!redirects Yang–Mills theory]]
[[!redirects Yang-Mills action]]
[[!redirects Yang-Mills action functional]]
|
Yangian | https://ncatlab.org/nlab/source/Yangian |
#Contents#
* table of contents
{:toc}
## Idea
A _Yangian_ is a certain [[quantum group]] that arises naturally in [[integrable systems]] in [[quantum field theory]], as well as in _[[semi-holomorphic 4d Chern-Simons theory]]_.
## Related Concepts
* [[quantum affine algebra]]
## References
* Wikipedia, _[Yangian](http://en.wikipedia.org/wiki/Yangian)_
* A. I. Molev, _Yangians and their applications_, in "Handbook of Algebra" vol. __3__ (M. Hazewinkel, Ed.), Elsevier 2003, 907-959 [math.QA/0211288](https://arxiv.org/abs/math/0211288)
* A. I. Molev, _Yangians and classical Lie algebras_, AMS Math. Surv. Monog. __143__, 2007; 400 pp; Russian edition:
_Янгианы и классические алгебры Ли_, МЦНМО, Москва, 2009
* N. J. Mackay, _Introduction to Yangian symmetry in integrable field theory_ ([arXiv:hep-th/0409183](https://arxiv.org/abs/hep-th/0409183))
* [[Vassili Gorbounov]], R. Rimanyi, V. Tarasov, A. Varchenko, _Cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra_, [arXiv:1204.5138](http://arxiv.org/abs/1204.5138)
* V. G. [[Drinfeld]], _Degenerate affine Hecke algebras and Yangians_, Funct. Anal. Appl. 20 (1986), 58–60.
* Denis Uglov, _Symmetric functions and the Yangian decomposition of the Fock and basic modules of the affine Lie algebra $\mathfrak{sl}^N$_, Math. Soc. Japan Memoirs __1__, 1998, 183-241 [euclid](http://projecteuclid.org/euclid.msjm/1389985795) [doi](http://dx.doi.org/10.2969/msjmemoirs/00101C030)
* A. N. Kirillov, N. Y. Reshetikhin, _The Yangians, Bethe Ansatz and combinatorics_, Lett. Math. Phys. __12__, 199 (1986)
* Sachin Gautam, Valerio Toledano-Laredo, _Yangians and quantum loop algebras_, Selecta Mathematica __19__ (2013), 271-336 [arXiv:1012.3687](https://arxiv.org/abs/1012.3687); _Yangians, quantum loop algebras and abelian difference equations_ (Formerly: Yangians and quantum loop algebras II. Equivalence of categories via abelian difference equations) J. Amer. Math. Soc. __29__ (2016) 775--824 [arXiv:1310.7318](https://arxiv.org/abs/1310.7318); _III. Meromorphic tensor equivalence for Yangians and quantum loop algebras_, Publ.math. IHES __125__, 267--337 (2017). [doi](https://doi.org/10.1007/s10240-017-0089-9) [arXiv:1403.5251](https://arxiv.org/abs/1403.5251)
* Dmitry Galakhov, [[Alexei Morozov]], Nikita Tselousov, *Towards the theory of Yangians* [[arXiv:2311.00760](https://arxiv.org/abs/2311.00760)]
Braided Yangians and Yangians associated to R-matrices:
* Dmitri I. Gurevich, Pavel A. Saponov, _Generalized Yangians and their Poisson counterparts_, Theor. Math. Phys. __192__, 1243--1257 (2017) [doi](https://doi.org/10.1134/S004057791709001X)
Yangians for quivers and relation to quantum equivariant cohomology of Nakajima's quiver varieties:
* Davesh Maulik, Andrei Okounkov, _Quantum groups and quantum cohomology_, Astérisque 408, SMF 2019 [arXiv:1211.1287](https://arxiv.org/abs/1211.1287)
Review in the context of [[AdS-CFT]] includes
* [[Alessandro Torrielli]], _Yangians, S-matrices and AdS/CFT_, J.Phys.A44:263001,2011 ([arXiv:1104.2474](http://arxiv.org/abs/1104.2474))
Quiver Yangians appearing in description of Hall algebras of $\omega$-semistable compactly supported sheaves with fixed slope on resolutions of Kleinian singularities:
* [[Duiliu-Emanuel Diaconescu]], [[Mauro Porta]], Francesco Sala, _McKay correspondence, cohomological Hall algebras and categorification_, [arXiv:2004.13685](https://arxiv.org/abs/2004.13685)
In
* [[Kevin Costello]], _Supersymmetric gauge theory and the Yangian_ ([arXiv:1303.2632](http://arxiv.org/abs/1303.2632))
is discussed that the holomorphically [[topological twist|twisted]] [[N=1 D=4 super Yang-Mills theory]] is controled by the [[Yangian]] in analogy to how [[Chern-Simons theory]] is controled by a [[quantum group]]. See at _[[semi-holomorphic 4d Chern-Simons theory]]_.
[[!redirects Yangians]] |
Yannick Bertrand | https://ncatlab.org/nlab/source/Yannick+Bertrand |
* [institute page](http://www.ens-lyon.fr/PHYSIQUE/presentation/anciens/bertrand-yannick)
## Selected writings
On [[D=6 supergravity]] with [[number of supersymmetries|$\mathcal{N} = (4,0)$]]:
* [[Yannick Bertrand]], [[Stefan Hohenegger]], [[Olaf Hohm]], [[Henning Samtleben]], *Toward Exotic 6D Supergravities*, Phys. Rev. D **103** (2021) 046002 [[arXiv:2007.11644](https://arxiv.org/abs/2007.11644), [doi:10.1103/PhysRevD.103.046002](https://doi.org/10.1103/PhysRevD.103.046002)]
* [[Yannick Bertrand]], [[Stefan Hohenegger]], [[Olaf Hohm]], [[Henning Samtleben]], *Supersymmetric action for 6D $(4,0)$ supergravity*, JHEP (2022) [[arXiv:2206.04100](https://arxiv.org/abs/2206.04100)]
category: people
|
Yaron | https://ncatlab.org/nlab/source/Yaron | An occasional browser in and contributor to the nLab |
Yaron Oz | https://ncatlab.org/nlab/source/Yaron+Oz |
* [webpage](http://www2.tau.ac.il/Person/exact/physics/researcher.asp?id=agklghgle)
## Selected writings
On [[D=3 N=2 super Yang-Mills theory]]:
* [[Jan de Boer]], [[Kentaro Hori]], [[Yaron Oz]], _Dynamics of N=2 Supersymmetric Gauge Theories in Three Dimensions_, Nucl. Phys. B500:163-191, 1997 ([arXiv:hep-th/9703100](https://arxiv.org/abs/hep-th/9703100))
On [[mirror symmetry]] between [[Higgs branches]]/[[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]] (with emphasis of [[Hilbert schemes of points]]):
* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], _Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes_, Nucl. Phys. B493:101-147, 1997 ([arXiv:hep-th/9611063](https://arxiv.org/abs/hep-th/9611063))
* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Zheng Yin, _Mirror Symmetry in Three-Dimensional Gauge Theories, $SL(2,\mathbb{Z})$ and D-Brane Moduli Spaces_, Nucl. Phys. B493:148-176, 1997 ([arXiv:hep-th/9612131](https://arxiv.org/abs/hep-th/9612131))
On a [[matrix model]] for the [[3-brane in 6d]] arising as the [[brane intersection|intersection]] of two [[M5-branes]]:
* {#KachruOzYin98} [[Shamit Kachru]], [[Yaron Oz]], [[Zheng Yin]], *Matrix Description of Intersecting M5 Branes* JHEP 9811:004, (1998) ([arXiv:hep-th/9803050](https://arxiv.org/abs/hep-th/9803050))
On [[geometric engineering of QFT]] on [[D4-D6 brane intersections]] subject to the [[s-rule]]:
* [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Section 3 of: _Strong Coupling Dynamics of Four-Dimensional N=1 Gauge Theories from M Theory Fivebrane_, Adv. Theor. Math. Phys.1:1-52, 1998 ([arXiv:hep-th/9706082](https://arxiv.org/abs/hep-th/9706082))
On the [[AdS-CFT correspondence]]:
* {#AharonyGubserMaldacenaOoguriOz99} [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], _Large $N$ Field Theories, String Theory and Gravity_, Phys. Rept. 323:183-386, 2000 ([arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111))
On [[D4-D8 brane bound states]] as [[black branes]]:
* {#BrandhuberOz99} [[Andreas Brandhuber]], [[Yaron Oz]], _The D4-D8 Brane System and Five Dimensional Fixed Points_, Phys.Lett. B460:307-312, 1999 ([arXiv:hep-th/9905148](https://arxiv.org/abs/hep-th/9905148))
On the [[CP^1 sigma-model|$\mathbb{C}P^1$ sigma-model]]:
* [[Shmuel Elitzur]], [[Yaron Oz]], [[Eliezer Rabinovici]], [[Johannes Walcher]], *Open/Closed Topological $\mathbb{C}P^1$ Sigma Model Revisited*, J. High Energ. Phys. **2012** 101 (2012) [[arXiv:1106.2967](https://arxiv.org/abs/1106.2967)]
category: people |
Yasuhiko Yamada | https://ncatlab.org/nlab/source/Yasuhiko+Yamada |
* [webpage](http://www.math.kobe-u.ac.jp/home-j/yamaday-e.html)
## Selected writings
On the [[sl(2)|$\mathfrak{sl}(2)$]]-[[WZW model]] [[2d CFT]] at fractional [[level (Chern-Simons theory)|level]] (cf. also [as logarithmic CFTs](logarithmic+CFT#FractionalLevelWZWModelReferences)):
* [[Hidetoshi Awata]], [[Yasuhiko Yamada]], *Fusion rules for the fractional level $\widehat{\mathfrak{sl}(2)}$ algebra*, Mod. Phys. Lett. A **7** (1992) 1185-1196 $[$[spire:332974](https://inspirehep.net/literature/332974), [doi:10.1142/S0217732392003645](https://doi.org/10.1142/S0217732392003645)$]$
On [[elliptic genera]] as [[partition functions]] of [[2d SCFTs]]:
* [[Toshiya Kawai]], [[Yasuhiko Yamada]], Sung-Kil Yang, _Elliptic Genera and $N=2$ Superconformal Field Theory_, Nucl. Phys. B **414** (1994) 191-212 [[arXiv:hep-th/9306096](http://arxiv.org/abs/hep-th/9306096), <a href="https://doi.org/10.1016/0550-3213(94)90428-6">doi:10.1016/0550-3213(94)90428-6</a>]
category: people |
Yasunori Lee | https://ncatlab.org/nlab/source/Yasunori+Lee |
* [institute page](https://db.ipmu.jp/member/personal/5321en.html)
## Selected writings
On [[anomaly cancellation]] via the [[Green-Schwarz mechanism]] from the point of view of [[higher gauge theory]]:
* [[Yasunori Lee]], [[Kantaro Ohmori]], [[Yuji Tachikawa]], *Matching higher symmetries across Intriligator-Seiberg duality* ([arXiv:2108.05369](https://arxiv.org/abs/2108.05369))
category: people |
Yasuyuki Kawahigashi | https://ncatlab.org/nlab/source/Yasuyuki+Kawahigashi |
* [website](http://www.ms.u-tokyo.ac.jp/~yasuyuki/index-e.html)
## Selected writings
Relating [[vertex operator algebras]] with [[conformal nets]]:
* {#CarpiKawahigahshiLongoWeiner15} [[Sebastiano Carpi]], [[Yasuyuki Kawahigashi]], [[Roberto Longo]], Mihály Weiner, _From vertex operator algebras to conformal nets and back_, Memoirs of the American Mathematical Society **254** 1213 (2018) [[doi:10.1090/memo/1213](https://doi.org/10.1090/memo/1213), [arXiv:1503.01260](http://arxiv.org/abs/1503.01260)]
(...)
category: people |
Yde Venema | https://ncatlab.org/nlab/source/Yde+Venema |
Yde Venema is an associate professor at the Institute for Logic, Language and Computation at the Universiteit van Amsterdam.
He is interested in [[modal logic]] and its (mathematical) foundations and, in particular in algebraic and coalgebraic aspects of modal logic.
[website](http://staff.science.uva.nl/~yde/)
## Selected writings
On [[modal logic]]:
* {#BlackburnDeRijkeVenema01} [[Patrick Blackburn]], [[Maarten de Rijke]], [[Yde Venema]], *Modal Logic*, Cambridge Tracts in Theoretical Computer Science **53** (2001) [[doi:10.1017/CBO9781107050884](https://doi.org/10.1017/CBO9781107050884)]
category: people
[[!redirects Yde Venema]]
[[!redirects Venema]]
|
year | https://ncatlab.org/nlab/source/year |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _year_ is a [[physical unit]] of [[time]].
## Related concepts
* [[second]]
## References
* Wikipedia, _[Year](https://en.wikipedia.org/wiki/Year)_
[[!redirects years]]
|
Yehonatan Sella | https://ncatlab.org/nlab/source/Yehonatan+Sella |
* [webpage](http://www.math.ucla.edu/~ysella/)
## related $n$Lab entries
* [[singular cohomology]]
category: people |
Yemon Choi | https://ncatlab.org/nlab/source/Yemon+Choi | Interloper/dilettante. "Functional analyst", who sometimes wonders if the term meant something different in Britain that it does in North America. Keeps finding category-theoretic things all over the place in *old* analysis, never mind the brave new world of NCG, and is starting to feel that they should be collated somewhere.
* [website](http://maths.lancs.ac.uk/~choiy1/)
(Last properly active in 2014, but might be returning in 2021...)
### To do list ###
Some things I am considering trying to write about, or contribute to. (This is mainly meant as a list to remind or nag me to write things down.)
* [[projective Banach space|projective, injective and flat Banach spaces]] (both in the metric and topological categories of [[Banach space]]s)
* RKHS
* the bidualization monad on [[Banach space]]s
* Kaijser and Wick-Pelletier's work on interpolation
* Arens-Eells spaces, a.k.a. the adjunction between Banach spaces and metric spaces.
* Richard Arens's paper using "phyla", or: where Banach algebras and [[symmetric monoidal categories]] should have met. (There is now some stuff on [[Banach algebra#arens products|Arens products]] on the [[Banach algebra|Banach algebras]] page.)
* Categori(c)al basics of operator spaces: MAX, MIN, and various tensor products
* functorial aspects of the group von Neumann algebra construction
* dual Banach algebras
### A survey paper which is more rewarding than this page ###
J.M.F. Castillo. The hitchhiker guide to categorical Banach space theory. Part I.
Extracta Math. 25(2) (2010), 103--149.
(Currently freely available on the [journal's website](http://www.eweb.unex.es/eweb/extracta/) )
category: people
[[!redirects YemonChoi]] |
Yetter model | https://ncatlab.org/nlab/source/Yetter+model |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Yetter model_ is a [[4d TQFT]] [[sigma-model]] [[quantum field theory]] whose target space is a [[discrete infinity-groupoid|discrete]] [[2-groupoid]] and whose [[background gauge field]] is a [[circle n-bundle|circle 4-bundle]].
Together with the [[Dijkgraaf-Witten theory|Dijkgraaf-Witten model]] these form the first two steps in filtering of target spaces by [[homotopy type]] [[truncated|truncation]] of [[schreiber:∞-Chern-Simons theory]] <a href="http://nlab.mathforge.org/schreiber/show/infinity-Chern-Simons+theory#DiscreteTargets">with discrete target spaces</a>. It is hence also an example of a [[4d Chern-Simons theory]].
The Yetter model is not the same as the [[Crane-Yetter model]].
## Definition
Fix
* $G$ a [[discrete infinity-groupoid|discrete]] [[2-group]]; write $\mathbf{B}G$ for its [[delooping]] [[2-groupoid]];
* $\alpha : \mathbf{B}G \to \mathbf{B}^4 U(1)$ a [[characteristic class]] with coefficients in the [[circle n-group|circle 4-group]]. This is equivalently a cocycle in degree $4$ [[group cohomology]]
$$
[\alpha] \in H_{Grpd}^4(G, U(1))
\,.
$$
The **Yetter-model** is the <a href="http://nlab.mathforge.org/schreiber/show/infinity-Chern-Simons+theory#DiscreteTargets">∞-Dijkgraaf-Witten theory</a> induced by this data.
## Related concepts
* [[schreiber:∞-Chern-Simons theory]]
* [[higher dimensional Chern-Simons theory]]
* [[1d Chern-Simons theory]]
* [[2d Chern-Simons theory]]
* [[3d Chern-Simons theory]]
* [[4d Chern-Simons theory]]
* [[5d Chern-Simons theory]]
* [[6d Chern-Simons theory]]
* [[7d Chern-Simons theory]]
* [[infinite-dimensional Chern-Simons theory]]
* [[AKSZ sigma-model]]
## References
The model without a [[background gauge field]]/cocycle was considered in
* [[David Yetter]], _TQFTs from homotopy 2-types_ , Journal of Knot Theory and its Ramifications 2 (1993), 113-123.
The effect of having a nontrivial [[group cohomology|group 4-cocycle]] was considered (but now only on a 1-group) in
* D. Birmingham, M. Rakowski, _On Dijkgraaf-Witten Type Invariants_, Lett. Math. Phys. 37 (1996), 363.
* [[Marco Mackaay]], _Spherical 2-categories and 4-manifold invariants_, Adv. Math. 153 (2000), no. 2, 353–390. ([arXiv:math/9805030](http://arxiv.org/abs/math/9805030))
{#Mackay} .
The reinterpretation of the "state sum" equation used in the above publications as giving [[homomorphisms]] of [[simplicial sets]]/[[topological spaces]] is given in
* [[Tim Porter]], _Interpretations of Yetter's notion of $G$-coloring : simplicial fibre bundles and non-abelian cohomology_, Journal of Knot Theory and its Ramifications 5 (1996) 687-720,
and then extended to colorings in [[homotopy n-types]] in
* [[Tim Porter]], _Topological Quantum Field Theories from Homotopy n-types_, Journal of the London Math. Soc. (2) 58 (1998) 723-732.
See also
* [[João Faria Martins]] and [[Tim Porter]], _On Yetter's invariants and an extension of the Dijkgraaf-Witten invariant to categorical groups_, Theory and Applications of Categories, Vol. 18, 2007, No. 4, pp 118-150.
([TAC](http://www.tac.mta.ca/tac/volumes/18/4/18-04abs.html))
which has some remarks about higher (2-)group cocycles towards the end.
[[!redirects Yetter models]]
|
Yetter-Drinfeld module | https://ncatlab.org/nlab/source/Yetter-Drinfeld+module |
# Yetter--Drinfeld modules
* table of contents
{: toc}
## Definition
A __Yetter--Drinfeld module__ over a $k$-[[bialgebra]] $B=(B,\Delta,\epsilon)$, (with [[Sweedler notation]] $\Delta(b) = \sum b_{(1)}\otimes b_{(2)}$), is a $k$-module which is simultaneously a $B$-module and a $B$-[[comodule]] with certain compatibility -- also called Yetter-Drinfeld condition -- between the $B$-action and $B$-coaction.
+-- {: .un_defn}
###### Compatibility for left-right YD Modules
The compatibility for a left $B$-module $B\otimes M\to M$, $b\otimes m\mapsto b\blacktriangleright m$, which is a right $B$-comodule
with respect to the coaction $\rho:M\to M\otimes B$, $\rho(m) = \sum m_{[0]}\otimes m_{[1]}$,
is the following
$$
\sum (b_{(1)}\blacktriangleright m_{[0]})\otimes b_{(2)} m_{[1]}
= \sum (b_{(2)}\blacktriangleright m)_{[0]}
\otimes (b_{(2)}\blacktriangleright m)_{[1]} b_{(1)}
$$
or equivalently, if $B$ is a Hopf algebra with invertible antipode $S$ (or instead just with the
[[skew-antipode]] denoted $S^{-1}$)
$$
\sum (b_{(2)}\blacktriangleright m_{[0]})\otimes b_{(3)} m_{[1]} S^{-1}(b_{(1)})
= \sum (b\blacktriangleright m)_{[0]}
\otimes (b\blacktriangleright m)_{[1]}
$$
=--
+-- {: .un_defn}
###### Compatibility for left-left YD Modules
$$
b_{(1)} m_{[-1]}\otimes (b_{(2)}\blacktriangleright m_{[0]})
=
(b_{(1)}\blacktriangleright m)_{[-1]} b_{(2)} \otimes
(b_{(1)}\blacktriangleright m)_{[0]}
$$
=--
+-- {: .un_defn}
###### Compatibility for right-left YD Modules
$$
m_{[-1]}b_{(1)}\otimes (m_{[0]}\blacktriangleleft b_{(2)})
=
b_{(2)} (m\blacktriangleleft b_{(1)})_{[-1]} \otimes (m\blacktriangleleft b_{(1)})_{[0]}
$$
=--
+-- {: .un_defn}
###### Compatibility for right-right YD Modules
$$
m_{[0]}\blacktriangleleft b_{(1)}\otimes m_{[1]} b_{(2)}
=
(m\blacktriangleleft b_{(2)})_{[0]}\otimes
b_{(1)} (m\blacktriangleleft b_{(2)})_{[1]}
$$
=--
## The category of Yetter--Drinfeld modules
Morphisms of YD $B$-modules are morphisms of underlying $B$-modules which are also the morphisms of underlying
$B$-comodules.
The _category of left-right YD modules_ over a bialgebra $B$ is denoted by ${}_B \mathcal{Y D}^B$; the category is rarely alternatively called the (left-right) Yetter--Drinfeld category
and it can be presented as the category of entwined modules for certain special entwining structure.
${}_B \mathcal{Y D}^B$ is a monoidal category: if $V$ and $W$ are left-right YD modules, $V\otimes W$ is the tensor product of underlying vector spaces equipped with left $B$-action
$$
b\blacktriangleright (v\otimes w) = (b_{(1)}\blacktriangleright v)\otimes (b_{(2)}\blacktriangleright w)
$$
and right $B$-coaction
$$
v\otimes w\mapsto v_{[0]}\otimes w_{[0]}\otimes w_{[1]}v_{[1]}
$$
Note the order within the rightmost tensor factor! One checks directly that this tensor product indeed satisfies the Yetter-Drinfeld condition.
Radford and Towber prefer slightly different monoidal structure: in above formulas use the opposite product and coopposite coproduct on $B$. (They mention, however, both structures.)
Monoidal category ${}_B \mathcal{Y D}^B$ is equipped
with "pre-braiding" morphisms
$$
R_{V,W}: V\otimes W\to W\otimes V,\,\,\,\,\,\,\,\,
v\otimes w\mapsto w_{[0]} \otimes (w_{[1]}\blacktriangleright v).
$$
In Radford-Towber convention the pre-braiding is $v\otimes w\mapsto (v_{[1]}\blacktriangleright w)\otimes v_{[0]}$.
Prebraidings satisfy all conditions for a [[braided monoidal category|braiding]] except for invertibility
of $R_{V,W}$
which is fullfilled for all $V,W$ iff $B$ is a Hopf algebra. $R_{V,W}$ is always fullfilled if both $V$ and $W$ are finite dimensional. In particular, $R_{V,V}$ satisfies the Yang-Baxter equation.
If $A$ is a commutative algebra in
${}_B\mathcal{Y D}^B$ then the [[smash product algebra]] $A\sharp B$
is an associative [[bialgebroid]], said to be the extension of scalars
from the bialgebra $B$ along $k\hookrightarrow A$. If $B=H$
is a Hopf algebra with bijective antipode then this bialgebroid is in fact
a [[Hopf algebroid]], both in the sense of Lu and in the sense of Bohm.
If $B=H$ is a finite-dimensional [[Hopf algebra]], then the category
${}_H \mathcal{Y D}^H$ is equivalent to the category of ${}_{D(H)}\mathcal{M}$ of left $D(H)$-modules, where $D(H)$ is the [[Drinfeld double]] of $H$, which in turn is equivalent to the center
of the monoidal category ${}_H\mathcal{H}$ of left $H$-modules.
The commutative algebras in the center of a monoidal category, play
role in the [[dynamical extension of a monoidal category]]. Hence the commutative algebras in ${}_H\mathcal{Y D}^H$ provide such examples. An important example, is the dual $H^*$ when $H$ is finite-dimensional.
The smash product algebra is in that case the [[Heisenberg double]], hence
it inherits a Hopf algebroid structure.
If $F$ is a counital 2-cocycle for a bialgebra $H$, the Drinfeld twist $H^F$ of $F$ is also a bialgebra and there is a monoidal equivalence ${}_H\mathcal{M}\cong {}_{H^F}\mathcal{M}$. In Section 2 of [Škoda-Stojić2023](#Škoda-Stojić2023) it is shown how this monoidal equivalence lifts to a braided monoidal equivalence between the categories of Yetter-Drinfeld modules
${}_H\mathcal{M}^H\cong {}_{H^F}\mathcal{M}^{H^F}$.
* [[Zoran Škoda]], Martina Stojić, _Comment on "Twisted bialgebroids versus bialgebroids from Drinfeld twist"_, [arXiv:2308.05083](https://arxiv.org/abs/2308.05083)
## Yetter-Drinfeld module algebras
A __left-right Yetter-Drinfeld module algebra__ is a monoid $(A,\mu)$ in ${}_B\mathcal{Y D}^B$. Let its multiplication map be denoted $\mu:a\otimes c\mapsto a\cdot c$. Let us unwind the requirements that $\mu:A\otimes A\to A$ is a morphism in ${}_B\mathcal{Y D}^B$.
Requirement that $\mu$ is a map of $B$-modules is, for $a,c\in A$
$$
b\blacktriangleright (a\cdot c) = (b_{(1)}\blacktriangleright a)\cdot (b_{(2)}\blacktriangleright c),
$$
which, together with compatibility of unit $b\blacktriangleright 1 = \epsilon(b) 1$, means that the action is Hopf ($A$ is a left $B$-module algebra).
Requirement that $\mu$ is a map of $B$-comodules is
$$\rho\circ\mu = (\mu\otimes id)\rho_{A\otimes A}$$
$$
\rho(a\cdot c)
= (\mu\otimes id)(a_{[0]}\otimes c_{[0]}\otimes c_{[1]} a_{[1]})
= a_{[0]}\cdot c_{[0]}\otimes c_{[1]} a_{[1]},
$$
that is (along with the counit condition), $A$ is right $B^\op$-comodule algebra.
A left-right Yetter-Drinfeld module algebra $A$ is __braided-commutative__ if
$$
\mu\circ R_{A,A} = \mu.
$$
In explicit terms, for all $a,c\in A$,
$$
c_{[0]}\cdot (c_{[1]}\blacktriangleright a) = a\cdot c.
$$
## Anti Yetter--Drinfeld modules
The most general coefficients for Hopf cyclic cohomology are called stable anti-Yetter-Drinfeld modules. These kind of modules appeared for the first time in different name in B. Rangipour's PhD thesis under supervision of M. Khalkhali. Later on it was generalized by
P.M. Hajac, M. Khalkhali, B. Rangipour, and Y. Sommerhaeuser. The category of AYD modules is not monodical but product of an AYD module with a YD module results in an AYD module. By the work of Rangipour--Sutlu one knows that there is such category over Lie algebras and there is a one-to-one correspondence between AYD modules over a Lie algebra and those over the universal enveloping algebra of the Lie algebra. This correspondence is extended by the same authors for bicrossed product Hopf algebras.
The true meaning of the AYD modules in non commutative geometry is not known yet. There are some attempts by A. Kaygun--M. Khalkhali to relate them to the curvature of flat connections similar to the work of T. Brzeziński on YD modules, however their identification are not restricted to AYD and works for a wide variety of YD type modules.
## Literature
* Susan Montgomery, _Hopf algebras and their actions on rings_,
CBMS Lecture Notes __82__, AMS 1993, 240p.
* A.M. Semikhatov, _Yetter--Drinfeld structures on Heisenberg doubles and chains_, [arXiv:0908.3105](https://arxiv.org/abs/0908.3105)
* wikipedia [Yetter--Drinfeld category](http://en.wikipedia.org/wiki/Yetter%E2%80%93Drinfeld_category)
* Peter Schauenburg, _Hopf Modules and Yetter--Drinfel′d Modules_,
Journal of Algebra __169__:3 (1994) 874--890 [doi](https://doi.org/10.1006/jabr.1994.1314); _Hopf modules and the double of a quasi-Hopf algebra_, Trans. Amer. Math. Soc. 354 (2002), 3349--3378 [doi](https://doi.org/10.1090/S0002-9947-02-02980-X) [pdf](http://www.ams.org/journals/tran/2002-354-08/S0002-9947-02-02980-X/S0002-9947-02-02980-X.pdf); _Actions of monoidal categories, and generalized Hopf smash products_, Journal of Algebra __270__ (2003) 521--563, [doi](http://dx.doi.org/10.1016/S0021-8693%2803%2900403-4) [ps](http://www.mathematik.uni-muenchen.de/%7Eschauen/papers/amcghsp.ps)
* V. G. Drinfel'd, _Quantum groups_, Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 798--820, AMS 1987, [djvu:1.3M](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.djvu), [pdf:2.5M](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.pdf)
* David N. Yetter, _Quantum groups and representations of monoidal categories_, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 2, 261--290 [MR91k:16028](https://www.ams.org/mathscinet-getitem?mr=1074714) [doi](http://dx.doi.org/10.1017/S0305004100069139)
* [[Gabriella Bohm|Gabriella Böhm]], Dragos Stefan, _(Co)cyclic (co)homology of bialgebroids: An approach via (co)monads_, Comm. Math. Phys. 282 (2008), no.1, 239--286, [arxiv/0705.3190](http://arxiv.org/abs/0705.3190); _A categorical approach to cyclic duality_, J. Noncommutative Geometry __6__ (2012), no. 3, 481--538, [arxiv/0910.4622](http://arxiv.org/abs/0910.4622)
* Atabey Kaygun, Masoud Khalkhali, _Hopf modules and noncommutative differential geometry_, Lett. in Math. Physics __76:1__, pp 77--91 (2006) [arxiv/math.QA/0512031](http://arxiv.org/abs/math/0512031), [doi](http://dx.doi.org/10.1007/s11005-006-0062-x)
* [[T. Brzeziński]], _Flat connections and (co)modules_, [in:] New Techniques in Hopf Algebras and Graded Ring Theory, S Caenepeel and F Van Oystaeyen (eds), Universa Press, Wetteren, 2007 pp. 35--52 [arXiv:math.QA/0608170](https://arxiv.org/abs/math.QA/0608170)
* P.M. Hajac, M. Khalkhali, B. Rangipour, Y. Sommerhaeuser, _Hopf-cyclic homology and cohomology with coefficients_, C. R. Math. Acad. Sci. Paris __338__(9), 667--672 (2004) [math.KT/0306288](https://arxiv.org/abs/math/0306288); _Stable anti-Yetter--Drinfeld modules_. C. R. Math Acad. Sci. Paris __338__(8), 587--590 (2004)
* B. Rangipour, Serkan Sütlü, _Characteristic classes of foliations via SAYD-twisted cocycles_, [arXiv:1210.5969](https://arxiv.org/abs/1210.5969);
_SAYD modules over Lie--Hopf algebras_, [arXiv:1108.6101](https://arXiv.org/abs/1108.6101); _Cyclic cohomology of Lie algebras_, [arXiv:1108.2806](https://arxiv.org/abs/1108.2806)
* Florin Panaite, Mihai D. Staic, _Generalized (anti) Yetter--Drinfeld modules as components of a braided T-category_, arXiv:[math.QA/0503413](https://arxiv.org/abs/math/0503413)
* D. Bulacu, S. Caenepeel, F. Panaite, _Doi--Hopf modules and Yetter--Drinfeld categories for quasi-Hopf algebras_, Communications in Algebra, 34 (9), 3413--3449 (2006) [math.QA/0311379](https://arxiv.org/abs/math/0311379)
* Florin Panaite, Dragos Stefan, _Deformation cohomology for Yetter--Drinfel'd modules and Hopf (bi)modules_, [math.QA/0006048](https://arxiv.org/abs/math/0006048)
* Nicolás Andruskiewitsch, István Heckenberger, Hans-Jürgen Schneider, _The Nichols algebra of a semisimple Yetter--Drinfeld module_, American J. of Math. __132__:6, (2010) 1493--1547 [doi](https://doi.org/10.1353/ajm.2010.0019)
* M. Cohen, D. Fischman, S. Montgomery, _On Yetter–Drinfeld categories and $H$-commutativity_, Commun. Algebra __27__ (1999) 1321--1345
* Yukio Doi, _Hopf modules in Yetter--Drinfeld categories_, Commun. Alg. __26__:9, 3057--3070 (1998) [doi](https://doi.org/10.1080/00927879808826327)
* I. Heckenberger, H.-J. Schneider, _Yetter--Drinfeld modules over bosonizations of dually paired Hopf algebras_, [arXiv:1111.4673](https://arxiv.org/abs/1111.4673)
* V. Ulm, _Actions of Hopf algebras in categories of Yetter--Drinfeld modules_, Comm. Alg. __31__:6, 2719--2743
* L. A. Lambe, D. E. Radford, Algebraic aspects of the quantum Yang–Baxter equation, J. Algebra 154 (1992) 228--288 [doi](https://doi.org/10.1006/jabr.1993.1014)
* D.E. Radford, J. Towber, _Yetter--Drinfel'd categories associated to an arbitrary bialgebra_, J. Pure Appl. Algebra __87__ (1993), 259--279 [MR94f:16060](http://www.ams.org/mathscinet-getitem?mr=1228157) [doi](https://doi.org/10.1016/0022-4049%2893%2990114-9)
* [[Georgia Benkart]], Mariana Pereira, Sarah Witherspoon, _Yetter--Drinfeld modules under cocycle twists_, J. Algebra 324:11 (2010) 2990--3006 [arxiv:0908.1563](https://arxiv.org/abs/0908.1563)
* [[Shahn Majid]], Robert Oeckl, _Twisting of quantum differentials and the Planck scale Hopf algebra_, Commun. Math. Phys. __205__, 617--655 (1999)
* Huixiang Chen, Yinhuo Zhang, _Cocycle deformations and Brauer groups_, Comm. Alg. 35:2 (2007) 399--433 [doi](https://doi.org/10.1080/00927870601052422); arXiv v. Cocycle deformation and Brauer group isomorphisms, [arXiv:math/0505003](https://arxiv.org/abs/math/0505003)
category: algebra
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|
Yevsey Nisnevich | https://ncatlab.org/nlab/source/Yevsey+Nisnevich |
* [Mathematics Genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=18827)
## related $n$Lab pages
* [[Nisnevich site]]
category: people |
Yi Zhang | https://ncatlab.org/nlab/source/Yi+Zhang |
* [Research Group page](https://sites.google.com/view/zhangyiphysics/)
## Selected writings
On characterizing [[anyon]] [[braiding]] / [[modular transformations]] on [[topological order|topologically ordered]] [[ground states]] by analysis of ([[topological entanglement entropy|topological]]) [[entanglement entropy]] of subregions:
* [[Yi Zhang]], [[Tarun Grover]], [[Ari M. Turner]], [[Masaki Oshikawa]], [[Ashvin Vishwanath]], *Quasiparticle statistics and braiding from ground-state entanglement*, Phys. Rev. B **85** (2012) 235151 $[$[doi:10.1103/PhysRevB.85.235151](https://doi.org/10.1103/PhysRevB.85.235151)$]$
* [[Yi Zhang]], [[Tarun Grover]], [[Ashvin Vishwanath]], *General procedure for determining braiding and statistics of anyons using entanglement interferometry*, Phys. Rev. B **91** (2015) 035127 $[$[arXiv:1412.0677](https://arxiv.org/abs/1412.0677), [doi:10.1103/PhysRevB.91.035127](https://doi.org/10.1103/PhysRevB.91.035127)$]$
category: people |
Yi-Hong Chen | https://ncatlab.org/nlab/source/Yi-Hong+Chen |
* [Inspire page](https://inspirehep.net/authors/1013842)
## Selected writings
On the [[fictitious gauge field]]-[[model (in theoretical physics)|model]] for [[anyon statistics]] and the resulting [[superfluidity]]/[[superconductivity]] *of* [[anyons]]:
* *Induced quantum numbers and anyon superconductivity*, PhD thesis (1990) $[$[spire:306797](https://inspirehep.net/literature/306797)$]$
* [[Yi-Hong Chen]], [[Frank Wilczek]], [[Edward Witten]], [[Bertrand Halperin]], *On Anyon Superconductivity*, International Journal of Modern Physics B **03** 07 (1989) 1001-1067 (reprinted in [Wilczek 1990](#Wilczek90)) $[$[doi:10.1142/S0217979289000725](https://doi.org/10.1142/S0217979289000725), [[CWWH-AnyonSuperfluidity.pdf:file]]$]$
category:people |
Yi-Nan Wang | https://ncatlab.org/nlab/source/Yi-Nan+Wang |
* [ResearchGate page](https://www.researchgate.net/profile/Yi-Nan-Wang)
## Selected writings
On [[QFT with defects|defects]] in the [[KK-compactification]] of the [[D=6 N=(2,0) SCFT]] on [[4-manifolds]]:
* [[Jin Chen]], [[Wei Cui]], [[Babak Haghighat]], [[Yi-Nan Wang]], *SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds*, JHEP **2023** 208 (2023) [[arXiv:2305.09734](https://arxiv.org/abs/2305.09734), <a href="https://doi.org/10.1007/JHEP11(2023)208">doi:10.1007/JHEP11(2023)208</a>]
category: people
[[!redirects Yinan Wang]]
|
Yi-Zhi Huang | https://ncatlab.org/nlab/source/Yi-Zhi+Huang |
* [webpage](http://www.rci.rutgers.edu/~yzhuang/)
## Selected writings
On [[vertex operator algebras]] and [[2d conformal field theory]]:
* {#FrenkenHuangLepowsky} [[Igor Frenkel]], [[Yi-Zhi Huang]], [[James Lepowsky]], _On Axiomatic approaches to Vertex Operator Algebras and Modules_, Memoirs of the AMS Vol 104, No 494 (1993)
As algebras over the holomorphic punctured sphere operad
* {#Huang91} [[Yi-Zhi Huang]], _Geometric interpretation of vertex operator algebras_, Proc. Natl. Acad. Sci. USA __88__ (1991) pp. 9964-9968 ([doi:10.1073/pnas.88.22.9964](https://doi.org/10.1073/pnas.88.22.9964))
* {#HuangCFT} [[Yi-Zhi Huang]], _Two-dimensional conformal geometry and vertex operator algebras_, Progr. in Math. Birkhauser 1997, [gbooks](http://books.google.hr/books?isbn=0817638296)
On the [[representation categories]] of (rational) [[vertex operator algebras]] as ([[modular tensor categories|modular]]) [[fusion categories]]:
* {#Huang05} [[Yi-Zhi Huang]], *Vertex operator algebras, the Verlinde conjecture and modular tensor categories*, Proc. Nat. Acad. Sci. **102** (2005) 5352-5356 $[$[arXiv:math/0412261](http://arxiv.org/abs/math/0412261), [doi:10.1073/pnas.0409901102](https://doi.org/10.1073/pnas.0409901102)$]$
* {#Huang08} [[Yi-Zhi Huang]], *Rigidity and modularity of vertex tensor categories*, Communications in Contemporary Mathematics **10** supp01 (2008) 871-911 $[$[arXiv:math/0502533](https://arxiv.org/abs/math/0502533), [doi:10.1142/S0219199708003083](https://doi.org/10.1142/S0219199708003083)$]$
* {#HuangLepowskiZhang14} [[Yi-Zhi Huang]], [[James Lepowsky]], Lin Zhang, *Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules*, In: Bai, Fuchs, Huang, Kong, Runkel, Schweigert (eds.), *Conformal Field Theories and Tensor Categories* Mathematical Lectures from Peking University. Springer (2014) $[$[arXiv:1012.4193](https://arxiv.org/abs/1012.4193), [doi:10.1007/978-3-642-39383-9_5](https://doi.org/10.1007/978-3-642-39383-9_5)$]$
Relation to [[2d conformal field theory]]:
* [[Yi-Zhi Huang]], _Two-dimensional conformal geometry and vertex operator algebras_ Birkhäuser (1997) ([doi:10.1007/978-1-4612-4276-5](https://link.springer.com/book/10.1007/978-1-4612-4276-5))
On [[vertex operator algebras]], their associated [[modular tensor categories]] and a [[proof]] of the [[Verlinde formula]]:
* {#Huang} [[Yi-Zhi Huang]], *Vertex operator algebras, the Verlinde conjecture and modular tensor categories*, Proc. Nat. Acad. Sci. **102** (2005) 5352-5356 $[$[arXiv:math/0412261](http://arxiv.org/abs/math/0412261), [doi:10.1073/pnas.0409901102](https://doi.org/10.1073/pnas.0409901102)$]$
On [[full field algebra]] for [[2d conformal field theory]]:
* {#HuangKong05} [[Yi-Zhi Huang]], [[Liang Kong]], _Full field algebras_, Commun. Math. Phys. **272** (2007) 345-396 [[arXiv:0511328](http://arxiv.org/abs/math/0511328), [doi:10.1007/s00220-007-0224-4](https://doi.org/10.1007/s00220-007-0224-4)]
On [[braided fusion categories]] formed by [[affine Lie algebra]]-[[representations]] at admissible fractional [[level (Chern-Simons theory)|level]]:
* [[Thomas Creutzig]], [[Yi-Zhi Huang]], [[Jinwei Yang]], *Braided tensor categories of admissible modules for affine Lie algebras*, Commun. Math. Phys. **362** (2018) 827–854 [[arXiv:1709.01865](https://arxiv.org/abs/1709.01865)]
On [[vertex operator algebras]] for [[orbifold]] [[CFT]]s:
* [[Yi-Zhi Huang]], _Representation theory of vertex operator algebras and orbifold conformal field theory_ ([arXiv:2004.01172](https://arxiv.org/abs/2004.01172))
## Related entries
* [[vertex operator algebra]]
* [[2d conformal field theory]]
category: people |
Yiannis Sakellaridis | https://ncatlab.org/nlab/source/Yiannis+Sakellaridis |
* [website](https://math.jhu.edu/~sakellar/)
### Writings
* [[Yiannis Sakellaridis]], _Spherical Varieties, Functoriality, and Quantization_ ([arXiv:2111.03004](https://arxiv.org/abs/2111.03004))
* [[David Ben-Zvi]], [[Yiannis Sakellaridis]], [[Akshay Venkatesh]], _Relative Langlands duality_ ([pdf](https://www.math.ias.edu/~akshay/research/BZSVpaperV1.pdf)).
category: people |
Yiannis Vlassopoulos | https://ncatlab.org/nlab/source/Yiannis+Vlassopoulos |
**Yiannis Vlassopoulos** (Βλασσόπουλος Γιάννης, also spelled Ioannis and also Yannis) is a mathematician interested in mathematical physics, geometry and more recently also in artificial intelligence. He was a student of [[David Morrison]]. Part of his thesis is reflected in
* Yiannis Vlassopoulos, _Quantum cohomology and Morse theory on the loop space of toric varieties_, [math.AG/0203083](https://arxiv.org/abs/math.AG/0203083)
On [[tensor networks]] in [[machine learning]] of [[natural language]]:
* [[Vasily Pestun]], Yiannis Vlassopoulos, _Tensor network language model_ [[arXiv:1710.10248](https://arxiv.org/abs/1710.10248)]
See also:
* [[Maxim Kontsevich]], Alex Takeda, Yiannis Vlassopoulos, _Pre-Calabi-Yau algebras and topological quantum field theories_, [arXiv:2112.14667](https://arxiv.org/abs/2112.14667)
* Maxim Kontsevich, Yannis Vlassopoulos, Natalia Iyudu, _Pre-Calabi-Yau algebras and [[double Poisson bracket]]s_, [arXiv:1906.07134](https://arxiv.org/abs/1906.07134)
* Maxim Kontsevich, Yannis Vlassopoulos, Natalia Iyudu, _[[pre-Calabi-Yau algebra|Pre-Calabi-Yau algebra]]s as noncommutative Poisson structures_, J. Algebra _567_ (2021) 63--90 [doi](https://doi.org/10.1016/j.jalgebra.2020.08.029)
The following is an interesting excerpt from his research project he had earlier at IHES:
> Givental has conjectured, that the U(1)-equivariant Floer cohomology of the universal covering of the loop space, of asymplectic manifold, should have the structure of a [[D-module]], over the Heisenberg algebra of first order differential operators on a complex torus and that this should be the same as the quantum cohomology D-module of the manifold. I intent to study this conjectured equality and its implications in computing the quantum D-module. This implies also computation of the quantum ring, as the later is the semi-classical limit of the former. There are three concrete directions of research. First, note that there is a "Fourier transform" of equivariant cycles that transforms relations in the D-module to differential operators. If we could compute this transform, then we could compute the D-module. Because of the infinite dimensionality of the loop space though, there are problems with computing the integral involved I have managed to compute it in the case of positive toric manifolds, but this method relies on the Fourier expansion and doesn't seem to generalize to non-toric manifolds. For the case of general semi-positive symplectic manifolds, I propose a totally different method, which relies on using localization techniques and a certain exact sequence arising from [[Morse theory]] of the simplistic action functional, in order to regularize the ratios of relevant equivariant [[Euler class]]es. The second program I propose, is to use the model of Getzler, Jones and Petrack ([pdf](https://bpb-us-e1.wpmucdn.com/sites.northwestern.edu/dist/c/2278/files/2019/08/cyclic1.pdf)), for the [[equivariant cohomology]] of the [[loop space]]. They identify it with a version of the cyclic bar complex, involving Connes's operator B and this could be used to compute the relevant "Fourier transform".
A talk on ideas on language semantics (some category theoretic ideas involved!)
* _Modelling language semantics and probability densities of text continuations_, QNLP 2022 [youtube](https://www.youtube.com/watch?v=c3IOe3JPfq8)
Related notions include [[pre-Calabi-Yau algebra]].
category: people
[[!redirects Ioannis Vlassopoulos]]
[[!redirects Ioánnis Vlassópoulos]]
[[!redirects Yannis Vlassopoulos]] |
Yihua Liu | https://ncatlab.org/nlab/source/Yihua+Liu |
* [GoogleScholar page](https://scholar.google.com/citations?user=oGk19_4AAAAJ&hl=en)
## Selected writings
On [[braid group representations]] for [[su(2)-anyon]]-[[anyon statistics|statistics]] from the [[monodromy]] of the [[Knizhnik-Zamolodchikov connection]] of bundles of [[conformal blocks]] over [[configuration spaces of points]]:
* [[Xia Gu]], [[Babak Haghighat]], [[Yihua Liu]], *Ising- and Fibonacci-Anyons from KZ-equations*, J. High Energ. Phys. **2022** 15 (2022) [[arXiv:2112.07195](https://arxiv.org/abs/2112.07195), <a href="https://doi.org/10.1007/JHEP09(2022)015">doi:10.1007/JHEP09(2022)015</a>]
On [[conformal blocks]] for [[Liouville theory]]:
* [[Babak Haghighat]], [[Yihua Liu]], [[Nicolai Reshetikhin]], *Flat Connections from Irregular Conformal Blocks* [[arXiv:2311.07960](https://arxiv.org/abs/2311.07960)]
category: people
|
Yimin Yang | https://ncatlab.org/nlab/source/Yimin+Yang |
* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=9369)
## Selected writings
On [[equivariant K-theory]]:
* {#Yang95} [[Yimin Yang]], _On the Coefficient Groups of Equivariant K-Theory_, Transactions of the American Mathematical Society
Vol. 347, No. 1 (Jan., 1995), pp. 77-98 ([jstor:2154789](https://www.jstor.org/stable/2154789))
category: people |
Yitzhak Frishman | https://ncatlab.org/nlab/source/Yitzhak+Frishman |
* [institute page](https://www.weizmann.ac.il/particle/frishman/home)
* [InSpire page](https://inspirehep.net/authors/1009396)
## Selected writings
On [[non-perturbative quantum field theory]] (from [[2d CFT]] to [[QCD]]):
* [[Yitzhak Frishman]], [[Jacob Sonnenschein]], *Non-Perturbative Field Theory -- From Two Dimensional Conformal Field Theory to QCD in Four Dimensions*, Cambridge University Press (2010) [[doi:10.1017/CBO9780511770838](https://doi.org/10.1017/CBO9780511770838), summary: [arXiv:1004.4859](https://arxiv.org/abs/1004.4859)] open access (2023) [[doi:10.1017/9781009401654](https://doi.org/10.1017/9781009401654)]
category: people |
Yoichi Ando | https://ncatlab.org/nlab/source/Yoichi+Ando |
* [GoogleScholar page](https://scholar.google.com/citations?user=4f0O4R0AAAAJ&hl=en)
## Selected writings
On [[topological crystalline insulators]]:
* [[Liang Fu]], *Topological Crystalline Insulators*, Phys. Rev. Lett. **106** (2011) 106802 ([doi:10.1103/PhysRevLett.106.106802](https://doi.org/10.1103/PhysRevLett.106.106802))
* [[Yoichi Ando]] and [[Liang Fu]], *Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials*, Annual Review of Condensed Matter Physics, **6** (2015) 361-381 $[$[doi:10.1146/annurev-conmatphys-031214-014501](https://doi.org/10.1146/annurev-conmatphys-031214-014501)$]$
* Y. Tanaka, Zhi Ren, T. Sato, K. Nakayama, S. Souma, T. Takahashi, Kouji Segawa, [[Yoichi Ando]], *Experimental realization of a topological crystalline insulator in SnTe*, Nature Physics **8** (2012) 800–803 $[$[doi:10.1038/nphys2442](https://doi.org/10.1038/nphys2442)$]$
* Timothy H Hsieh, Hsin Lin, Junwei Liu, Wenhui Duan, Arun Bansil, [[Liang Fu]], *Topological crystalline insulators in the SnTe material class*,
Nature Communications **3** (2012) 982 $[$[doi:10.1038/ncomms1969](https://doi.org/10.1038/ncomms1969)$]$
category: people
|
Yoichi Kazama | https://ncatlab.org/nlab/source/Yoichi+Kazama |
* [weboage](https://academictree.org/physics/inst.php?locid=452)
## Selected writings
Introducing [[Kazama-Suzuki models]]:
* [[Yoichi Kazama]], [[Hisao Suzuki]], _$N = 2$ superconformal field theories and superstring compactification_, Nucl. Phys. B 321 (1989), pp. 232-268 ([http://inspirehep.net/record/263185)](http://inspirehep.net/record/263185), <a href="https://doi.org/10.1016/0550-3213(89)90250-2">doi:10.1016/0550-3213(89)90250-2</a>)
category: people |
Yoichiro Nambu | https://ncatlab.org/nlab/source/Yoichiro+Nambu |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Yoichiro_Nambu)
* [obituary by Madhusree Mukerjee](http://www.huffingtonpost.com/madhusree-mukerjee/the-passing-of-a-gentle-g_b_7827966.html)
* [Interview with Yoichiro Nambu by Babak Ashrafi, July 2004](https://www.aip.org/history-programs/niels-bohr-library/oral-histories/30538)
* Christopher T. Hill, *A personal recollection of Yoichiro Nambu*, Section VI in: *Nambu and Compositeness* [[arXiv:2401.08716](https://arxiv.org/abs/2401.08716)]
On his contributions to [[string theory]]:
* [[Hiroshi Itoyama]], *Birth of String Theory*, Progress of Theoretical and Experimental Physics **2016** 6 (2016) 06A103 [[arXiv:1604.03701](http://arxiv.org/abs/1604.03701), [doi:10.1093/ptep/ptw063](https://doi.org/10.1093/ptep/ptw063)]
## Selected writings
Introducing discussion of what came to be called the *[[Nambu-Goto action]]* for [[strings]] (then motivated as explanation for the "dual resonance model" for [[hadrons]], cf. *[[Polyakov gauge-string duality]]*):
* {#Nambu70} [[Yoichiro Nambu]], *Duality and Hadrodynamics*, Notes prepared for the Copenhagen High Energy Symposium (1970) [[doi:10.1142/9789812795823_0026](https://doi.org/10.1142/9789812795823_0026), [[Nambu-DualityAndHadrodynamics.pdf:file]]]
Prediction of the [[omega-meson]]:
* [[Yoichiro Nambu]], _Possible Existence of a Heavy Neutral Meson_, Phys. Rev. 106, 1366 (1957) ([doi:10.1103/PhysRev.106.1366](https://doi.org/10.1103/PhysRev.106.1366))
## Related $n$Lab entries
* [[Nambu-Goto action]]
* [[Nambu bracket]], [[Nambu mechanics]]
category: people
[[!redirects Nambu]] |
Yolanda Lozano | https://ncatlab.org/nlab/source/Yolanda+Lozano |
* [webpage](https://www.unioviedo.es/hepth/people/Yolanda/yolanda.html)
## Selected writings
On [[massive type IIA string theory]] and its embedding in [[M-theory]] via [[M9-branes]]:
* [[Eric Bergshoeff]], [[Yolanda Lozano]], [[Tomas Ortin]], *Massive Branes*, Nucl. Phys. B **518** (1998) 363-423 [[arXiv:hep-th/9712115](https://arxiv.org/abs/hep-th/9712115), <a href="https://doi.org/10.1016/S0550-3213(98)00045-5">doi:10.1016/S0550-3213(98)00045-5</a>]
On the [[KK-monopole]] in massive [[D=11 supergravity]], [[KK-reduction|reducing]] to the [[D6-brane]] in [[massive type IIA string theory]]:
* [[Eric Bergshoeff]], [[Eduardo Eyras]], [[Yolanda Lozano]], *The massive Kaluza-Klein monopole*, Physics Letters B
**430** 1–2 (1998) 77-86 [<a href="https://doi.org/10.1016/S0370-2693(98)00501-2">doi:10.1016/S0370-2693(98)00501-2</a>, [arXiv:hep-th/9802199](https://arxiv.org/abs/hep-th/9802199)]
* [[Eduardo Eyras]], [[Yolanda Lozano]], *The Kaluza-Klein Monopole in a Massive IIA Background*, Nucl. Phys. B**546** (1999) 197-218 [[arXiv:hep-th/9812188](https://arxiv.org/abs/hep-th/9812188), <a href="https://doi.org/10.1016/S0550-3213(99)00098-X">doi:10.1016/S0550-3213(99)00098-X</a>]
On [[M-branes]]:
* [[Laurent Houart]], [[Yolanda Lozano]], _Brane Descent Relations in M-theory_, Phys.Lett. B **479** (2000) 299-307 [[arXiv:hep-th/0001170](https://arxiv.org/abs/hep-th/0001170), <a href="https://doi.org/10.1016/S0370-2693(00)00317-8">doi:10.1016/S0370-2693(00)00317-8</a>]
On [[M-brane polarization]]:
* [[Yolanda Lozano]], _Non-commutative Branes from M-theory_, Phys. Rev. D64 (2001) 106011 ([arXiv:hep-th/0012137](https://arxiv.org/abs/hep-th/0012137))
On [[giant gravitons]]:
* [[Yolanda Lozano]], Jeff Murugan, [[Andrea Prinsloo]], _A giant graviton genealogy_, JHEP 08 (2013) 109 ([arXiv:1305.6932](https://arxiv.org/abs/1305.6932))
On the [[BMN matrix model]] and [[nonabelian T-duality]]:
* [[Yolanda Lozano]], [[Carlos Nunez]], Salomon Zacarias, _BMN Vacua, Superstars and Non-Abelian T-duality_, JHEP 09 (2017) 008 ([arXiv:1703.00417](https://arxiv.org/abs/1703.00417))
On [[black brane|black]]$\;$[[D6-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D4-brane bound states]] inside them realizing [[AdS3-CFT2]] as [[defect field theory]] "inside" [[AdS7-CFT6]]:
* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], $1/4$ BPS $AdS_3/CFT_2$ ([arxiv:1909.09636](https://arxiv.org/abs/1909.09636))
* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], _Two dimensional $N=(0,4)$ quivers dual to $AdS_3$ solutions in massive IIA_ ([arxiv:1909.10510](https://arxiv.org/abs/1909.10510))
* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], _$AdS_3$ solutions in massive IIA, defect CFTs and T-duality_ ([arxiv:1909.11669](https://arxiv.org/abs/1909.11669))
Similarly for [[D2-D6-NS5 brane bound states]]:
* Federico Faedo, [[Yolanda Lozano]], [[Nicolo Petri]], _Searching for surface defect CFTs within $AdS_3$_, JHEP 11 (2020) 052 ([arXiv:2007.16167](https://arxiv.org/abs/2007.16167))
On [[AdS2/CFT1]] on [[D1-D3 brane intersections]]:
* [[Giuseppe Dibitetto]], [[Yolanda Lozano]], [[Nicolò Petri]], [[Anayeli Ramirez]], _Holographic Description of M-branes via $AdS_2$_ ([arXiv:1912.09932](https://arxiv.org/abs/1912.09932))
* [[Yolanda Lozano]], [[Carlos Nunez]], [[Anayeli Ramirez]], Stefano Speziali, _New $AdS_2$ backgrounds and $\mathcal{N}=4$ Conformal Quantum Mechanics_ ([arXiv:2011.00005](https://arxiv.org/abs/2011.00005))
category: people |
Yonatan Harpaz | https://ncatlab.org/nlab/source/Yonatan+Harpaz |
* [webpage](https://sites.google.com/site/yonatanharpaz/)
* [thesis](https://www.math.univ-paris13.fr/~harpaz/thesis.pdf)
* [HDR-Thesis](https://www.math.univ-paris13.fr/~harpaz/hdr.pdf)
###Papers
* with [[Matan Prasma]], _The Grothendieck construction for model categories_
, Advances in Mathematics, 281, 2015, p. 1306-1363
category: people
|
Yoneda embedding | https://ncatlab.org/nlab/source/Yoneda+embedding |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
For $C$ a [[locally small category]], every [[object]] $X$ of $C$ induces a [[presheaf]] on $C$: the [[representable presheaf]] $h_X$ represented by $X$. This assignment extends to a [[functor]] $C \to [C^{op}, Set]$ from $C$ to its [[category of presheaves]].
The [[Yoneda lemma]] implies that this functor is [[full and faithful functor|full and faithful]] and hence realizes $C$ as a [[full subcategory]] inside its category of presheaves.
Recall from the discussion at [[representable presheaf]] that the presheaf represented by an object $X$ of $C$ is the functor $h_X :C^{op} \to Set$ whose assignment is illustrated by
<img src="http://ncatlab.org/ericforgy/files/hxalpha3.jpg" width = "400"/>
which sends each object $U$ to $Hom_C(U,X)$ and each morphism $\alpha:U'\to U$ to the function
$$h_X\alpha: Hom_C(U,X)\to Hom_C(U',X).$$
Moreover, for $f : X \to Y$ a [[morphism]] in $C$, this induces a [[natural transformation]] $h_f : h_X \to h_Y$, whose component on $U$ in $X$ is illustrated by
<img src="http://ncatlab.org/ericforgy/files/hf.jpg" width = "400"/>
For this to be a natural transformation, we need to have the commuting diagram
$$\array{
h_X U & \stackrel{h_f U}{\rightarrow} & h_Y U \\
\mathllap{h_X\alpha\quad}{\downarrow} & {} & \mathrlap{\downarrow}{\quad h_Y\alpha} \\
h_X U' & \stackrel{h_f U'}{\rightarrow} & h_Y U'
}$$
but this simply means that it doesn't matter if we first "comb" the strands back to $U'$ and then comb the strands forward to $Y$, or comb the strands forward to $Y$ first and then comb the strands back to $U'$
<img src="http://ncatlab.org/ericforgy/files/nattrans.jpg" width = "600"/>
which follows from associativity of composition of morphisms in $C$.
## Definition
The **Yoneda embedding** for $C$ a [[locally small category]] is the [[functor]]
$$
Y : C \to [C^{op}, Set]
$$
from $C$ to the [[category of presheaves]] over $C$
which is the image of the [[hom-functor]]
$$
Hom : C^{op} \times C \to Set
$$
under the Hom [[adjunction]]
$$
Hom(C^{op} \times C , Set) \simeq
Hom(C, [C^{op}, Set])
$$
in the [[closed monoidal category|closed]] [[monoidal category|symmetric monoidal category]] [[Cat]].
Hence $Y$ sends any [[object]] $c \in C$ to the [[representable presheaf]] which assigns to any other object $d$ of $C$ the [[hom-set]] of [[morphism]]s from $d$ into $c$:
$$
Y(c)
\;\colon\;
C^{op} \stackrel{C(-,c)}{\to} Set
\,.
$$
The Yoneda embedding is sometimes denoted by よ, the [hiragana](https://en.wikipedia.org/wiki/Yo_(kana%29) for "Yo"; see the references [below](#ReferencesNotation).
## Remarks
We can also curry the Hom functor in the other variable, thus obtaining a contravariant functor
$$
C^{op} \to [C, Set]
$$
which is explicitly given by $c \mapsto C(c,-)$. This is sometimes jokingly called the **contravariant Yoneda embedding**.
However, since $C^{op}(-,c)=C(c,-)$, it is easy to see that the contravariant Yoneda embedding is just the Yoneda embedding $Y: C^{\op} \to [(C^{op})^{op}, Set]=[C, Set]$ of $C^{op}$, and hence does not require special treatment.
## Properties
+-- {: .num_prop #YonedaEmbeddingIsFullyFaithful}
###### Proposition
**(Yoneda embedding is a [[fully faithful functor]])**
For $\mathcal{C}$ any [[category]], the [[functor]]
$$
\array{
\mathcal{C}
&\overset{Y}{\hookrightarrow}&
[\mathcal{C}^{op}, Set]
\\
c &\mapsto& Hom_{\mathcal{C}}(-,c)
}
$$
is [[fully faithful functor|fully faithful]].
=--
+-- {: .proof}
###### Proof
We need to show that for $c_1, c_2 \in \mathcal{C}$ any two [[objects]], we have that every [[morphism]] of [[presheaves]] between their [[representable functor|represented presheaves]]
$$
Hom_{\mathcal{C}}(-,c_1)
\overset{\phi}{\longrightarrow}
Hom_{\mathcal{C}}(-,c_2)
$$
is of the form
$$
\phi \;=\; Hom_{\mathcal{C}}(-,f)
$$
for a unique morphism
$$
f \;\colon\; c_1 \to c_2
$$
in $\mathcal{C}$. This follows by the [[Yoneda lemma]], which says that morphisms $\phi$ as above are identified with the elements in
$$
Hom_{\mathcal{C}}(-,c_2)(c_1)
\;=\;
Hom_{\mathcal{C}}(c_1,c_2)
\,.
$$
=--
It is also [[limit]] preserving (= [[continuous functor]]), but does in general not preserve [[colimit]]s.
The Yoneda embedding of a [[small category]] $S$ into the category of [[presheaf|presheaves]] on $S$ gives a [[free cocompletion]] of $S$.
If the Yoneda embedding of a category has a [[left adjoint]], then that category is called a _[[total category]]_ .
## Related concepts
* A [[category]] is a _[[total category]]_ if its Yoneda embedding has a [[left adjoint]].
* [[restricted Yoneda embedding]]
* [[(infinity,1)-Yoneda embedding]]
* [[singleton]] injection, the Yoneda embedding for 0-category theory.
* [[Yoneda lemma for bicategories]]
## References
### General
Early accounts:
* [[Alexander Grothendieck]], Section A.1 of: *Technique de descente et théorèmes d'existence en géométrie algébriques. II. Le théorème d'existence en théorie formelle des modules*, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195, 22 p. ([numdam:SB_1958-1960__5__369_0](http://www.numdam.org/item/?id=SB_1958-1960__5__369_0))
For more see at _[[Yoneda lemma]]_ the list of references given [there](Yoneda+lemma#References).
### Notation
{#ReferencesNotation}
It seems that the notation "よ" for the Yoneda embedding (the [hiragana](https://en.wikipedia.org/wiki/Yo_(kana%29) for "Yo") was first used in
* [[Theo Johnson-Freyd]], Claudia Scheimbauer, p. 33 of _(Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories_, ([arxiv:1502.06526](https://arxiv.org/pdf/1502.06526.pdf))
Their Latex code for the command reads as follows:
`\usepackage[utf8]{inputenc}`
`\DeclareFontFamily{U}{min}{}`
`\DeclareFontShape{U}{min}{m}{n}{<-> udmj30}{}`
`\newcommand\yo{\!\text{\usefont{U}{min}{m}{n}\symbol{'207}}\!}`
Subsequent references that use this notation include:
* [[Emily Riehl]], [[Dominic Verity]], p. 10 of _Elements of $\infty$-category theory_ ([web](http://www.math.jhu.edu/~eriehl/elements.pdf#page=10))
* {#Li-Bland15} David Li-Bland, p. 5 of _The stack of higher internal categories and stacks of iterated spans_ ([arXiv:1506.08870](https://arxiv.org/pdf/1506.08870.pdf#page=5))
* [[Fosco Loregian]], p. 4 of _This is the (co)end, my only (co)friend_ ([arXiv:1501.02503](https://arxiv.org/pdf/1501.02503.pdf#page=4))
* [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], p. 53 of _Equivariant stable homotopy theory and the Kervaire invariant problem_, ([web](https://web.math.rochester.edu/people/faculty/doug/mybooks/esht.pdf#page=53))
[[!redirects Yoneda embeddings]] |
Yoneda extension | https://ncatlab.org/nlab/source/Yoneda+extension |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
##Idea##
The _Yoneda extension_ of a [[functor]] $F : C \to D$ is a universal [[extension]] ([[Kan extension]]) along the [[Yoneda embedding]] $Y \colon C \to [C^{op},Set]$ of its [[domain]] category to a functor
$$
\tilde F \colon [C^{op}, Set] \to D
\,.
$$
The Yoneda extension exhibits the [[presheaf]] category $PSh(C)$ as the [[free cocompletion]] of $C$.
##Definition##
For $C$ a [[small category]] and $F \colon C \to D$ a [[functor]], its **Yoneda extension**
$$
\tilde F \;\colon\; [C^{op},Set] \to D
$$
is the left [[Kan extension]] $Lan_Y F \;\colon\; [C^{op}, Set] \to D$ of $F$ along the [[Yoneda embedding]] $Y$:
$$
\tilde F \;\coloneqq\; Lan_Y F
\,.
$$
### Remarks ###
Often it is of interest to Yoneda extend not $F \;\colon\; C \to D$
itself, but the composition $Y \circ F \;\colon\; C \to [D^{op}, Set]$ to get a functor entirely between presheaf categories
$$
\hat F \;\coloneqq\; \tilde{Y \circ F}
\;\colon\;
[C^{op},Set] \to [D^{op}, Set]
\,.
$$
This is in fact a [[left adjoint]] to the restriction functor $F^\ast \;\colon \; [D^{op}, Set] \to [C^{op}, Set]$ which maps $H \mapsto H \circ F$. This is relevant, for instance, to [[restriction and extension of sheaves]].
### Formula ###
Recalling the general formula for the left [[Kan extension]] of a functor $F \;\colon\; C \to D$ through a functor $p : C \to C'$
$$
(Lan F)(c') \simeq \colim_{(p(c) \to c') \in (p,c')} F(c)
$$
one finds for the Yoneda extension the formula
$$
\begin{aligned}
\tilde F (A)
& \;\coloneqq\; (Lan F)(A)
\\
& \simeq
\colim_{(Y(U) \to A) \in (Y,A) } F(U)
\end{aligned}
\,.
$$
(Recall the notation for the [[comma category]] $(Y,A) \;\coloneqq\; (Y, const_A)$ whose objects are pairs $(U \in C, (Y(U) \to A) \in [C^{op}, Set] )$.
For the full extension $\hat F : [C^{op}, Set] \to [D^{op}. Set]$ this yields
$$
\begin{aligned}
\hat F(A)(V)
&=
(\colim_{(Y(U) \to A) \in (Y,A) } F(U))(V)
\\
&\simeq
\colim_{(Y(U) \to A) \in (Y,A) } F(U)(V)
\\
&\simeq
\colim_{(Y(U) \to A) \in (Y,A) } Hom_{D}(V,F(U))
\end{aligned}
\,.
$$
Here the first step is from above, the second uses that colimits in presheaf categories are computed objectwise and the last one is again using the [[Yoneda lemma]].
## Properties ##
* The restriction of the Yoneda extension to $C$ coincides with the original functor:
$
\tilde F \circ Y \simeq F
$.
* The Yoneda extension commutes with small colimits in $C$, i.e. for $\alpha : A \to C$ a [[diagram]], we have $\tilde F (colim (Y \circ \alpha)) \simeq colim F \circ \alpha$ .
* Moreover, $\tilde F$ is defined up to [[isomorphism]] by these two properties.
## Generalizations ##
* [[(infinity,1)-Yoneda extension]]
|
Yoneda lemma | https://ncatlab.org/nlab/source/Yoneda+lemma |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **Yoneda lemma** says that the [[set]] of [[morphisms]] from a [[representable presheaf]] $y(c)$ into an arbitrary [[presheaf]] $X$ is in [[natural bijection]] with the set $X(c)$ assigned by $X$ to the representing [[object]] $c$.
The Yoneda lemma is an elementary but deep and central result in [[category theory]] and in particular in [[sheaf and topos theory]]. It is essential background behind the central concepts of _[[representable functors]]_, _[[universal constructions]]_, and _[[universal elements]]_.
## Statement and proof
{#StatementOfYonedaLemma}
### Classical
+-- {: .num_defn #FunctorUnderlyingTheYonedaEmbedding}
###### Definition
**([[functor]] underlying the [[Yoneda embedding]])
For $\mathcal{C}$ a [[locally small category]] we write
$$
[C^{op}, Set] \coloneqq Func(C^{op}, Set)
$$
for the [[functor category]] out of the [[opposite category]] of $\mathcal{C}$ into [[Set]].
This is also called the _[[category of presheaves]]_ on $\mathcal{C}$.
Other notation used for it includes $Set^{C^{op}}$ or $Hom(C^{op},Set))$.
There is a [[functor]]
$$
\array{
C &\overset{y}{\longrightarrow}& [C^op,Set]
\\
c &\mapsto& Hom_{\mathcal{C}}(-,c)
}
$$
(called the _[[Yoneda embedding]]_ for reasons explained below) from $\mathcal{C}$ to its [[category of presheaves]], which sends each [[object]] to the [[hom-functor]] into that object, also called the [[representable presheaf|presheaf represented]] by $c$.
=--
+-- {: .num_remark}
###### Remark
**([[Yoneda embedding]] is [[adjunct]] of [[hom-functor]])**
The Yoneda embedding functor $y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set]$ from Def. \ref{FunctorUnderlyingTheYonedaEmbedding} is equivalently the [[adjunct]] of the [[hom-functor]]
$$
Hom_{\mathcal{C}}
\;\colon\;
\mathcal{C}^{op} \times \mathcal{C}
\longrightarrow
Set
$$
under the [[product category]]/[[functor category]] [[adjoint functor|adjunction]]
$$
Hom(C^{op} \times C, Set)
\stackrel{\simeq}{\to}
Hom(C, [C^{op}, Set])
$$
in the [[closed monoidal category|closed]] [[monoidal category|symmetric monoidal category]] of categories.
=--
+-- {: .num_prop #YonedaLemma}
###### Proposition
**([[Yoneda lemma]])**
Let $\mathcal{C}$ be a [[locally small category]], with [[category of presheaves]] denoted $[\mathcal{C}^{op},Set]$, according to Def. \ref{FunctorUnderlyingTheYonedaEmbedding}.
For $X \in [\mathcal{C}^{op}, Set]$ any [[presheaf]], there is a canonical [[isomorphism]]
$$
Hom_{[C^op,Set]}(y(c),X) \;\simeq\; X(c)
$$
between the [[hom-set]] of [[presheaf]] [[homomorphisms]] from the [[representable presheaf]] $y(c)$ to $X$, and the value of $X$ at $c$.
=--
This is the standard notation used mostly in pure [[category theory]] and [[enriched category theory]]. In other parts of the literature it is customary to denote the presheaf represented by $c$ as $h_c$. In that case the above is often written
$$
Hom(h_c, X) \simeq X(c)
$$
or
$$
Nat(h_c, X) \simeq X(c)
$$
to emphasize that the morphisms of presheaves are [[natural transformation|natural transformations]] of the corresponding functors.
+-- {: .proof}
###### Proof
The proof is by chasing the element $Id_c \in C(c, c)$ around both legs of a [[naturality square]] for a [[natural transformation]] $\eta: C(-, c) \to X$ (hence a homomorphism of presheaves):
$$
\array{
C(c, c)
& \stackrel{\eta_c}{\to} &
X(c) & & & & Id_c & \mapsto & \eta_c(Id_c) & \stackrel{def}{=} & \xi \\
_\mathllap{C(f, c)} \downarrow & & \downarrow _\mathrlap{X(f)} & & & & \downarrow & & \downarrow _\mathrlap{X(f)} & & \\
C(b, c) & \underset{\eta_b}{\to} & X(b) & & & & f & \mapsto & \eta_b(f) & &
}
$$
What this diagram shows is that the entire transformation $\eta: C(-, c) \to X$ is completely determined from the single value $\xi \coloneqq \eta_c(Id_c) \in X(c)$, because for each object $b$ of $C$, the component $\eta_b: C(b, c) \to X(b)$ must take an element $f \in C(b, c)$ (i.e., a morphism $f: b \to c$) to $X(f)(\xi)$, according to the commutativity of this diagram.
The crucial point is that the naturality condition on any [[natural transformation]] $\eta : C(-,c) \Rightarrow X$ is sufficient to ensure that $\eta$ is already entirely fixed by the value $\eta_c(Id_c) \in X(c)$ of its component $\eta_c : C(c,c) \to X(c)$ on the [[identity morphism]] $Id_c$.
And every such value extends to a natural transformation $\eta$.
More in detail, the bijection is established by the map
$$
[C^{op}, Set](C(-,c),X)
\stackrel{|_{c}}{\to}
Set(C(c,c), X(c))
\stackrel{ev_{Id_c}}{\to}
X(c)
$$
where the first step is taking the component of a [[natural transformation]] at $c \in C$ and the second step is [[evaluation]] at $Id_c \in C(c,c)$.
The inverse of this map takes $f \in X(c)$ to the natural transformation $\eta^f$ with components
$$
\eta^f_d := X(-)(f) : C(d,c) \to X(d)
\,.
$$
=--
### In homotopy type theory
Discussion in [[homotopy type theory]].
Note: the [[HoTT book]] calls a [[internal category in HoTT]] a "precategory" and a [[univalent category]] a "category", but here we shall refer to the standard terminology of "category" and "univalent category" respectively.
By Lemma 9.5.3 in the HoTT book (see [[product category]]), we have an induced functor $\mathbf{y} : A \to \mathit{Set}^{A^{op}}$ which we call the **yoneda embedding**.
**Theorem 9.5.4 (The Yoneda Lemma)**
For any [[category]] $A$, any $a:A$, and any functor $F: \mathit{Set}^{A^{op}}$, we have an [[isomorphism]]
$$hom_{\mathit{Set}^{A^{op}}}(\mathbf{y}a,F) \cong F a \qquad \qquad(9.5.5)$$
Moreover this is natural in both $a$ and $F$.
**Proof.** Given a [[natural transformation]] $\alpha : \mathbf{y}a \to F$, we can consider the component $\alpha_a : \mathbf{y}a(a) \to F a$. Since $\mathbf{y} a(a)\equiv hom_A(a,a)$, we have $1_a:\mathbf{y}a(a)$, so that $\alpha_a(1_a):F a$. This gives a function $\alpha \mapsto \alpha_a(1_a)$ from left to right in (9.5.5).
In the other direction, given $x: F a$, we define $\alpha : \mathbf{y} a \to F$ by
$$\alpha_{a'}(f) \equiv F_{a,a'}(f)(x)$$
Naturality is easy to check, so this gives a function from right to left in (9.5.5).
To show that these are inverses, first suppose given $x: F a$. Then with $\alpha$ defined as above, we have $\alpha : \mathbf{y}a \to F$ and define $x$ as above, then for any $f:hom_A(a',a)$ we have
$$
\begin{aligned}
\alpha_{a'}(f) &= \alpha_{a'}(\mathbf{y} a_{a,a'}(f)(1_a))\\
&= (\alpha_{a'} \circ \mathbf{y}a_{a,a'}(f))(1_a)\\
&= (F_{a,a'}(f) \circ \alpha_a)(1_a)\\
&= F_{a,a'}(f_(\alpha_a(1_a))\\
&= F_{a,a'}(f)(x).
\end{aligned}
$$
Thus, both composites are equal to identities. The proof of naturality follows from this. $\square$
**Corollary 9.5.6**
The Yoneda embedding $\mathbf{y} : A \to \mathit{Set}^{A^{op}}$ is [[fully faithful]].
**Proof.** By the Yoneda lemma, we have
$$hom_{\mathit{Set}^{A^{op}}}(\mathbf{y}a,\mathbf{y}b) \cong \mathbf{y} b(a) \equiv hom_A(a,b)$$
It is easy to check that this isomorphism is in fact the action of $\mathbf{y}$ on hom-sets. $\square$
## Corollaries {#YonedaCorollaries}
The Yoneda lemma has the following direct consequences. As the Yoneda lemma itself, these are as easily established as they are useful and important.
### corollary I: Yoneda embedding
The Yoneda lemma implies that the [[Yoneda embedding]] functor $y \colon C \to [C^op,Set]$ really is an _embedding_ in that it is a [[full and faithful functor]], because for $c,d \in C$ it naturally induces the isomorphism of Hom-sets.
$$
[C^{op},Set](C(-,c),C(-,d)) \simeq (C(-,d))(c) = C(c,d)
$$
### corollary II: uniqueness of representing objects
Since the [[Yoneda embedding]] is a [[full and faithful functor]], an [[isomorphism]] of [[representable functor|representable presheaves]] $y(c) \simeq y(d)$ must come from an [[isomorphism]] of the representing objects $c \simeq d$:
$$
y(c) \simeq y(d) \;\; \Leftrightarrow \;\;
c \simeq d
$$
### corollary III: universality of representing objects
A [[presheaf|presheaf]] $X \colon C^{op} \to Set$ is [[representable functor|representable]] precisely if the [[comma category|comma category]] $(y,const_X)$ has a [[terminal object]]. If a [[terminal object]] is $(d, g : y(d) \to X) \simeq (d, g \in X(d))$ then $X \simeq y(d)$.
This follows from unwrapping the definition of [[morphisms]] in the [[comma category]] $(y,const_X)$ and applying the Yoneda lemma to find
$$
(y,const_X)((c,f \in X(c)), (d, g \in X(d)))
\simeq
\{
u \in C(c,d) : X(u)(g) = f
\}
\,.
$$
Hence $(y,const_X)((c,f \in X(c)), (d, g \in X(d))) \simeq pt$ says precisely that $X(-)(f) \colon C(c,d) \to X(c)$ is a bijection.
### Interpretation
For emphasis, here is the interpretation of these three corollaries in words:
* **corollary I** says that the interpretation of presheaves on $C$ as generalized objects probeable by objects $c$ of $C$ is consistent: the probes of $X$ by $c$ are indeed the maps of generalized objects from $c$ into $X$;
* **corollary II** says that probes by objects of $C$ are sufficient to distinguish objects of $C$: two objects of $C$ are the same if they have the same probes by other objects of $C$.
* **corollary III** characterizes [[representable functor|representable functors]] by a [[universal property]] and is hence the bridge between the notion of [[representable functor]] and [[universal construction|universal constructions]].
## Generalizations
The Yoneda lemma tends to carry over to all important generalizations of the context of [[locally small category|categories]]:
* There is an analog of the Yoneda lemma in [[enriched category theory]]. See [[enriched Yoneda lemma]].
* In the context of [[module|modules]] (see also [[Day convolution]]) the Yoneda lemma becomes the important statement of [[Yoneda reduction]], which identifies the bimodule $\hom_C(-, -)$ as a unit bimodule.
* There is a [[Yoneda lemma for bicategories]].
* There is a [[Yoneda lemma for tricategories]].
* There is a [[Yoneda lemma for (∞,1)-categories]].
* In [[functional programming]], the Yoneda embedding is related to the [[continuation passing style]] transform.
* Formulation of the lemma in [[dependent type theory]]: [A type theoretical Yoneda lemma](http://homotopytypetheory.org/2012/05/02/a-type-theoretical-yoneda-lemma/) at [homotopytypetheory.org](http://homotopytypetheory.org)
## Necessity of naturality
The assumption of naturality is necessary for the Yoneda lemma to hold. A simple counter-example is given by a category with two objects $A$ and $B$, in which $Hom(A,A) = Hom(A,B) = Hom(B,B) = \mathbb{Z}_{\geq 0}$, the set of integers greater than or equal to $0$, in which $Hom(B,A) = \mathbb{Z}_{\geq 1}$, the set of integers greater than or equal to $1$, and in which composition is addition. Here it is certainly the case that $Hom(A,-)$ is isomorphic to $Hom(B,-)$ for any choice of $-$, but $A$ and $B$ are not isomorphic (composition with any arrow $B \rightarrow A$ is greater than or equal to $1$, so cannot have an inverse, since $0$ is the identity on $A$ and $B$).
A finite counter-example is given by the category with two objects $A$ and $B$, in which $Hom(A,A) = Hom(A,B) = Hom(B,B) = \{0, 1\}$, in which $Hom(B,A) = \{0, 2\}$, and composition is multiplication modulo 2. Here, again, it is certainly the case that $Hom(A,-)$ is isomorphic to $Hom(B,-)$ for any choice of $-$, but $A$ and $B$ are not isomorphic (composition with any arrow $B \rightarrow A$ is $0$, so cannot have an inverse, since $1$ is the identity on $A$ and $B$).
On the other hand, there have been examples of locally finite categories where naturality is not necessary. For example, ([Lovász, Theorem 3.6 (iv)](#Lovasz)) states precisely that finite relational structures $A$ and $B$ are isomorphic if, and only if, $Hom(C,A) \cong Hom(C,B)$ for every finite relational structure $C$. Later ([Pultr, Theorem 2.2](#Pultr)) generalised the result to finitely well-powered, locally finite categories with (extremal epi, mono) [[factorization system]].
## The Yoneda lemma in semicategories
An interesting phenomenon arises in the case of [[semicategory|semicategories]] i.e. "categories" (possibly) lacking [[identity morphisms]]:
the Yoneda lemma fails in general, since its validity in a semicategory $\mathcal{G}$ implies that $\mathcal{G}$ is in fact already a category because the Yoneda lemma permits to embed $\mathcal{G}$ into $PrSh(\mathcal{G})$ and the latter is always a category, the embedding then implying that $\mathcal{G}$ is itself a category!
But for [[regular semicategories]] $\mathcal{R}$ there is a [[unity of opposites]] in the category of all [[semipresheaves]] on $\mathcal{R}$ between the so called regular presheaves that are [[colimits]] of [[representable presheaf|representables]] and presheaves satisfying the Yoneda lemma, whence _the Yoneda lemma holds dialectically for regular presheaves!_
For some of the details see at _[[regular semicategory]]_ and the references therein.
## Applications
* The Yoneda lemma is the or a central ingredient in various [[reconstruction theorem|reconstruction theorems]], such as those of [[Tannaka duality]]. See there for a detailed account.
* In its incarnations as [[Yoneda reduction]] the Yoneda lemma governs the algebra of [[end|ends]] and [[coend|coends]] and hence that of [[bimodule|bimodules]] and [[profunctor|profunctors]].
* The Yoneda lemma is effectively the reason that [[Isbell conjugation]] exists. This is a fundamental duality that relates [[geometry]] and [[algebra]] in large part of mathematics.
## Related entries
* [[Yoneda reduction]]
* [[co-Yoneda lemma]]
* [[Yoneda structure]]
* [[Brown representability theorem]]
* [[continuation-passing style]]
* [[regular semicategory]]
* [[Yoneda lemma for (∞,1)-categories]]
\linebreak
## References
{#References}
For general references see any text on [[category theory]], as listed in the references [there](category+theory#References).
The term _Yoneda lemma_ originated in an interview of [[Nobuo Yoneda]] by [[Saunders Mac Lane]] at Paris Gare du Nord:
* Yoshiki Kinoshita, *Nobuo Yoneda (1930-1996)*
&
[[Saunders MacLane]], *The Yoneda Lemma*,
Math. Japonica, **47**, No. 1 (1998) 155-156
([pdf](https://dmitripavlov.org/scans/yoneda.pdf) [[MathJaponica_YonedaObituary.pdf:file]], also: [posting to catlist in 1996](http://www.mta.ca/~cat-dist/catlist/1999/yoneda))
In _[[Categories for the Working Mathematician]]_ MacLane writes that this happened in 1954.
<img src="http://ncatlab.org/nlab/files/YonedaObituary.jpg">
Review and exposition:
* [Using Yoneda rather than J to present the identity type](https://www.cs.bham.ac.uk/~mhe/yoneda/yoneda.html)
* [[Alexander Grothendieck]], Section A.1 of: *Technique de descente et théorèmes d'existence en géométrie algébriques. II. Le théorème d'existence en théorie formelle des modules*, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195, 22 p. ([numdam:SB_1958-1960__5__369_0](http://www.numdam.org/item/?id=SB_1958-1960__5__369_0))
* [[Saunders MacLane]], §III.2 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]
* [[Tom Leinster]], _[[LeinsterYoneda.ps:file]]_
* [[Emily Riehl]], _Category Theory in Context. Chapter 2. Universal Properties, Representability, and the Yoneda Lemma_ [pdf](http://www.math.jhu.edu/~eriehl/context.pdf)
* {#RRZ2004} Marie La Palme Reyes, Gonzalo E. Reyes, and Houman Zolfaghari, _Generic figures and their glueings: A constructive approach to functor categories_, Polimetrica sas, 2004 ([author page](https://reyes-reyes.com/2004/06/01/generic-figures-and-their-glueings-a-constructive-approach-to-functor-categories/),[pdf](https://marieetgonzalo.files.wordpress.com/2004/06/generic-figures.pdf)).
* [[Paolo Perrone]], _Notes on Category Theory with examples from basic mathematics_, Chapter 2. ([arXiv](http://arxiv.org/abs/1912.10642))
A discussion of the Yoneda lemma from the point of view of [[universal algebra]] is in
* [[Vaughan Pratt]], _The Yoneda lemma without category theory: algebra and applications_ ([pdf](http://boole.stanford.edu/pub/yon.pdf)).
A treatment of the Yoneda lemma for [[internal category in an (infinity,1)-category|categories internal to an (∞,1)-topos]] is in
* Louis Martini, _Yoneda's lemma for internal higher categories_, ([arXiv:2103.17141](https://arxiv.org/abs/2103.17141))
Early Lovász-Type results include
* {#Lovasz} László Lovász, _Operations with structures_, Acta Mathematica Academiae Scientiarum Hungarica 18.3-4 (1967): 321-328.
* {#Pultr} Aleš Pultr. _Isomorphism types of objects in categories determined by numbers of morphisms_, Acta Scientiarum Mathematicarum, 35:155–160, 1973.
[[!redirects yoneda lemma]]
[[!redirects Yoneda Lemma]]
[[!redirects Yoneda embedding lemma]]
[[!redirects Yoneda imbedding lemma]] |
Yoneda lemma - contents | https://ncatlab.org/nlab/source/Yoneda+lemma+-+contents |
**[[Yoneda lemma]]**
## Ingredients
* [[category]]
* [[functor]]
* [[natural transformation]]
* [[presheaf]]
* [[category of presheaves]]
* [[representable presheaf]]
* [[Yoneda embedding]]
## Incarnations
* [[Yoneda lemma]]
* [[enriched Yoneda lemma]]
* [[co-Yoneda lemma]]
* [[Yoneda reduction]]
## Properties
* [[free cocompletion]]
* [[Yoneda extension]]
## Universal aspects
* [[representable functor]]
* [[universal construction]]
* [[universal element]]
## Classification
* [[classifying space]], [[classifying stack]]
* [[moduli space]], [[moduli stack]], [[derived moduli space]]
* [[classifying topos]]
* [[subobject classifier]]
* [[universal principal bundle]], [[universal principal ∞-bundle]]
* [[classifying morphism]]
## Induced theorems
* [[Tannaka duality]]
...
## In higher category theory
[[Yoneda lemma for higher categories]]:
* [[Yoneda lemma for (∞,1)-categories]]
* [[Yoneda lemma for bicategories]]
* [[Yoneda lemma for tricategories]]
<div markdown="1">[Edit this sidebar](/nlab/edit/Yoneda+lemma+-+contents)</div> |
Yoneda lemma for (infinity,1)-categories | https://ncatlab.org/nlab/source/Yoneda+lemma+for+%28infinity%2C1%29-categories |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
# Contents#
* table of contents
{: toc}
## Idea
The statement of the [[Yoneda lemma]] generalizes from [[categories]] to [[(∞,1)-categories]].
## Yoneda embedding
{#YonedaEmbedding}
\begin{definition}
For $C$ an [[(∞,1)-category]] and $PSh(C)$ its [[(∞,1)-category of (∞,1)-presheaves]], the **$(\infty,1)$-Yoneda embedding** is the [[(∞,1)-functor]]
$$
y \colon C \to PSh(C)
$$
given by $y(X) \colon U \mapsto C(U,X)$.
\end{definition}
Seen under the [[(infinity,1)-Grothendieck construction|$\infty$-Grothendieck construction]] this is [Riehl & Verity 2018, Def. 6.2.3](#RiehlVerity18).
## Properties
### Yoneda lemma
{#YonedaLemma}
\begin{prop}
**$(\infty,1)$-Yoneda embedding**
Let $C$ be an [[(∞,1)-category]] and $PSh(C) \coloneqq Func(C^\op, \infty Grpd)$ be the corresponding [[(∞,1)-category of (∞,1)-presheaves]]. Then the canonical [[(∞,1)-functor]]
$$
Y \colon C \to PSh(C)
$$
is a [[full and faithful (∞,1)-functor]].
\end{prop}
For small $\infty$-categories this is [[Higher Topos Theory|HTT, prop. 5.1.3.1]]. For possibly large $\infty$-categories see [Riehl & Verity 2018, Thm. 7.2.22](#RiehlVerity18) (which considers $\infty$-presheaves regarded under the [[(infinity,1)-Grothendieck construction|$\infty$-Grothendieck construction]]) and [[Kerodon]], [Thm. 8.2.5.4](https://kerodon.net/tag/03NJ).
+-- {: .num_prop }
###### Proposition
**$(\infty,1)$-Yoneda theorem**
For $C$ a small $(\infty,1)$-category and $F \colon C^{op} \to \infty Grpd$ an $(\infty,1)$-functor, the composite
$$
C^{op}
\to PSh_{(\infty,1)}(C)^{op} \stackrel{Hom(-,F)}{\to}
\infty Grpd
$$
is equivalent to $F$.
=--
For small $\infty$-sites this is [[Higher Topos Theory|HTT, Lemma 5.5.2.1]]. For possibly large $\infty$-sites see [[Kerodon]], [Prop. 8.2.1.3](https://kerodon.net/tag/03M5).
+-- {: .proof}
###### Proof
For small $\infty$-sites, the statement may be obtained as a consequence of the [[sSet]]-[[enriched category theory|enriched]] [[Yoneda lemma]] by using the fact that the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C)$ is modeled by the [[enriched functor category]] $[C^{op}, sSet]_{proj}$ with $C$ regarded as a [[simplicially enriched category]] and using the global [[model structure on simplicial presheaves]].
=--
### Naturality
{#Naturality}
+-- {: .num_prop }
###### Proposition
$PSh$ can be extended to a functor $PSh \colon (\infty,1)Cat \to (\infty,1)\widehat{Cat}$ so that the yoneda embedding $C \to PSh(C)$ is a natural transformation.
Here, $(\infty,1)\widehat{Cat}$ is the (∞,1)-category of large (∞,1)-categories.
=--
This follows from ([[Higher Topos Theory|HTT, prop. 5.3.6.10]]), together with the identification of $PSh(C)$ with the category obtained by freely adjoining small colimits to $C$. This functor is locally left adjoint to the contravariant functor $C \mapsto Func(C^\op, \infty Grpd)$.
### Preservation of limits
+-- {: .num_prop }
###### Proposition
The $(\infty,1)$-Yoneda embedding $y : C \to PSh(C)$ preserves all [[(∞,1)-limit]]s that exist in $C$.
=--
([[Higher Topos Theory|HTT, prop. 5.1.3.2]])
### Local Yoneda embedding
{#LocalYonedaEmbedding}
+-- {: .num_prop }
###### Proposition
For $C$ an [[(∞,1)-site]] and $\mathcal{X}$ an [[(∞,1)-topos]], [[(∞,1)-geometric morphism]]s $(f^* \dashv f_*) Sh(C) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{X}$ from the [[(∞,1)-sheaf (∞,1)-topos]] $Sh(C)$ to $\mathcal{X}$ correspond to the **local** [[(∞,1)-functor]]s $f^* : C \to \mathcal{X}$, those that
* are left [[exact (∞,1)-functor]]s;
* send [[covering]] families $\{U_i \to X\}$ in $\mathcal{G}$ to [[effective epimorphism]]
$$
\coprod_i f^*(U_i) \to f^*(X)
\,.
$$
More preseicely, the [[(∞,1)-functor]]
$$
Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G}))
\stackrel{L}{\to}
Topos(\mathcal{X}, PSh_{(\infty,1)}(\mathcal{G}))
\stackrel{y}{\to}
Func(\mathcal{X}, \mathcal{G})
$$
given by precomposition of [[inverse image]] functors by [[∞-stackification]] and by the [(∞,1)-Yoneda embedding](#YonedaEmbedding)
is a [[full and faithful (∞,1)-functor]] and its essential image is spanned by these local morphisms.
=--
([[Higher Topos Theory|HTT, prop. 6.2.3.20]])
## Related concepts
* [[Brown representability theorem]]
* [[Yoneda Lemma for higher categories]]
## References
* {#LurieHTT} [[Jacob Lurie]], Prop. 5.1.3.1 and Lemma 5.5.2.1 in: _[[Higher Topos Theory]]_ (2009)
* {#RiehlVerity18} [[Emily Riehl]], [[Dominic Verity]], Def. 6.2.3 and Thm. 7.2.22 in: *The comprehension construction*, Higher Structures **2** 1 (2018) ([arXiv:1706.10023](https://arxiv.org/abs/1706.10023), [hs:39](http://137.111.162.45/index.php/higher_structures/article/view/39))
* MathOverflow, _The Yoneda Lemma for $(\infty,1)$-categories?_ ([MO:9737/381](https://mathoverflow.net/q/9737/381))
* {#Kerodon} [[Kerodon]], Part 2, Chapter 8: *The Yoneda Embedding* $[$[kerodon:03JA](https://kerodon.net/tag/03JA), esp. [Prop. 8.2.1.3](https://kerodon.net/tag/03M5) and [Thm. 8.2.5.4](https://kerodon.net/tag/03NJ)$]$
Discussion in the context of an [[∞-cosmos]]:
* {#RiehlVerity17} [[Emily Riehl]], [[Dominic Verity]], Section 6 of: _Fibrations and Yoneda's lemma in an $\infty$-cosmos_, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 ([arXiv:1506.05500](https://arxiv.org/abs/1506.05500), [doi:10.1016/j.jpaa.2016.07.003](https://doi.org/10.1016/j.jpaa.2016.07.003))
Discussion [[category internal to an (infinity,1)-topos|internal to]] any [[(∞,1)-topos]]:
* [[Louis Martini]], *Yoneda's lemma for internal higher categories*, [[arXiv:2103.17141](https://arxiv.org/abs/2103.17141)]
Formalization of the $(\infty,1)$-Yoneda lemma via [[simplicial homotopy type theory]] (in [[Rzk]]):
* [[Nikolai Kudasov]], [[Emily Riehl]], [[Jonathan Weinberger]]. *Formalizing the $\infty$-categorical Yoneda lemma* (2023) [[arXiv:2309.08340](https://arxiv.org/abs/2309.08340)]
[[!redirects Yoneda lemma for (∞,1)-categories]]
[[!redirects (infinity,1)-Yoneda lemma]]
[[!redirects (∞,1)-Yoneda lemma]]
[[!redirects Yoneda embedding for (∞,1)-categories]]
[[!redirects (∞,1)-Yoneda embedding]]
[[!redirects (infinity,1)-Yoneda embedding]]
[[!redirects Yoneda lemma for infinity-categories]]
[[!redirects Yoneda lemma for infinity1-categories]]
|
Yoneda lemma for bicategories | https://ncatlab.org/nlab/source/Yoneda+lemma+for+bicategories | +--{: .rightHandSide}
+--{: .toc .clickDown tabindex="0"}
### Context
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Statement
The **Yoneda lemma for bicategories** is a version of the [[Yoneda lemma]] that applies to [[bicategories]], the most common algebraic sort of weak [[2-category]]. It says that for any bicategory $C$, any object $x\in C$, and any [[pseudofunctor]] $F\colon C^{op}\to Cat$, there is an equivalence of categories
$$ [C^{op},Cat](C(-,x), F) \simeq F(x) $$
which is [[pseudonatural transformation|pseudonatural]] in $x$ and $F$, and which is given by evaluation at $1_x$, i.e. $\alpha\colon C(-,x)\to F$ maps to $\alpha_x(1_x)$.
For bicategories $A$ and $B$, $[A,B]$ denotes the bicategory of [[pseudofunctors]], [[pseudonatural transformations]], and [[modifications]] from $A$ to $B$. Note that it is a strict 2-category as soon as $B$ is.
## Implications
In particular, the Yoneda lemma for bicategories implies that there is a [[Yoneda embedding]] for bicategories $C\to [C^{op},Cat]$ which is [[2-fully-faithful functor|2-fully-faithful]], i.e. an equivalence on hom-categories. Therefore, $C$ is [[equivalence|equivalent]] to a sub-bicategory of $[C^{op},Cat]$. Since [[Cat]] is a [[strict 2-category]], it follows that $C$ is equivalent to a strict 2-category, which is one form of the [[coherence theorem for bicategories]]. (Conversely, another form of the coherence theorem can be used to prove the Yoneda lemma; see below.)
## Proof
A detailed proof of the bicategorical Yoneda lemma is given in ([Johnson & Yau 20, Chap. 8](#JohnsonYau)).
### Explicit proof
An explicit proof, involving many diagrams, has been written up by [[Igor Baković]] and can be found [here](http://www.irb.hr/korisnici/ibakovic/yoneda.pdf).
### High-technology proof
We will take it for granted that $[C^{op},Cat]$ is a well-defined bicategory; this is a basic fact having nothing to do with the Yoneda lemma. We also take it as given that "evaluation at $1_x$" functor
$$ [C^{op},Cat](C(-,x), F) \to F(x) $$
is well-defined and pseudonatural in $F$ and $x$; our goal is to prove that it is an equivalence. (Granted, these basic facts require a fair amount of verification as well.)
We will use part of the [[coherence theorem for pseudoalgebras]], which says that for a suitably well-behaved strict [[2-monad]] $T$, the inclusion $T$-$Alg_{strict} \hookrightarrow Ps$-$T$-$Alg$ of the 2-category of strict $T$-algebras and strict $T$-morphisms into the 2-category of pseudo $T$-algebras and pseudo $T$-morphisms has a left adjoint, usually written as $(-)'$. Moreover, for any pseudo $T$-algebra $A$, the unit $A\to A'$ is an equivalence in $Ps$-$T$-$Alg$.
First, there is a 2-monad $T$ such that strict $T$-algebras are strict 2-categories, strict $T$-morphisms are strict 2-functors, pseudo $T$-algebras are [[biased|unbiased bicategories]], and pseudo $T$-morphisms are [[pseudofunctors]]. By Mac Lane's coherence theorem for bicategories, any ordinary bicategory can equally well be considered as an unbiased one. Thus, since $Cat$ is a strict 2-category, for any bicategory $C$ there is a strict 2-category $C'$ such that pseudofunctors $C\to Cat$ are in bijection with strict 2-functors $C'\to Cat$.
Now note that a pseudonatural transformation between two pseudofunctors (resp. strict 2-functors) $C\to D$ is the same as a single pseudofunctor (resp. strict 2-functor) $C\to Cyl(D)$, where $Cyl(D)$ is the bicategory whose objects are the 1-cells of $D$, whose 1-cells are squares in $D$ commuting up to isomorphism, and whose 2-cells are "cylinders" in $D$. Likewise, a modification between two such transformations is the same as a single functor (of whichever sort) $C\to 2Cyl(D)$, where the objects of $2Cyl(D)$ are the 2-cells of $D$, and so on. Therefore, $C'$ classifies not only pseudofunctors out of $C$, but transformations and modifications between them; thus we have an isomorphism
$$[C^{op},Cat] \cong [(C')^{op},Cat]_{strict,pseudo}$$
where $[A,B]_{strict,pseudo}$ denotes the 2-category of strict 2-functors, pseudonatural transformations, and modifications between two strict 2-categories. Thus we can equally well analyze the functor
$$ [(C')^{op},Cat]_{strict,pseudo}(\overline{C(-,x)}, \overline{F}) \to \overline{F}(x) = F(x) $$
given by evaluation at $1_x$. Here $\overline{C(-,x)}$ and $\overline{F}$ denote the strict 2-functors $(C')^{op}\to Cat$ corresponding to the pseudofunctors $C(-,x)$ and $F$ under the $(-)'$ adjunction. However, we also have a strict 2-functor $C'(-,x)$, and the equivalence $C\simeq C'$ induces an equivalence $C'(-,x)\simeq \overline{C(-,x)}$. Therefore, it suffices to analyze the functor
$$ [(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F}) \to \overline{F}(x). $$
Now for any $A$ and $B$, we have an inclusion functor $[A,B]_{strict,strict} \to [A,B]_{strict,pseudo}$ where $[A,B]_{strict,strict}$ denotes the 2-category of strict 2-functors, strict 2-natural transformations, and modifications. This functor is [[bijective on objects functor|bijective on objects]] and [[locally fully faithful 2-functor|locally fully faithful]]. Moreover, the composite
$$ [(C')^{op},Cat]_{strict,strict}(C'(-,x), \overline{F}) \to [(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F}) \to \overline{F}(x). $$
is an *isomorphism*, by the [[enriched Yoneda lemma]], in the special case of $Cat$-enrichment. Since
$$[(C')^{op},Cat]_{strict,strict}(C'(-,x), \overline{F}) \to [(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F})$$
is fully faithful, if we can show that it is essentially surjective, then the [[2-out-of-3 property]] for equivalences of categories will imply that the desired functor is an equivalence.
Here we at last descend to something concrete. Given $\alpha\colon C'(-,x)\to \overline{F}$, we have an obvious choice for a strict transformation for it to be equivalent to, namely $\beta$ whose components $\beta_y\colon C'(y,x)\to \overline{F}(y)$ is given by $f \mapsto \overline{F}(f)(a)$ where $a = \alpha_x(1_x)\in \overline{F}(x)$. Since $\alpha$ is pseudonatural, for any $f\colon y\to x$ in $C'$ we have an isomorphism
$$\alpha_y(f) = \alpha_y(f\circ 1_x) \cong \overline{F}(f)(\alpha_x(1_x)) = \overline{F}(f)(a) = \beta_y(f).$$
We then simply verify that these isomorphisms are the components of an (invertible) modification $\alpha\cong \beta$. This completes the proof.
##Related entries
* [[Yoneda lemma for tricategories]]
## References
Review:
* [[Angelo Vistoli]], §3.6.2 in: *Grothendieck topologies, fibered categories and descent theory*, in: *[[Fundamental algebraic geometry -- Grothendieck's FGA explained]]*, Mathematical Surveys and Monographs **123**, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [math.AG/0412512](http://arxiv.org/abs/math/0412512)]
See also:
* {#JohnsonYau} [[Niles Johnson]], [[Donald Yau]], Chap. 8: _2-Dimensional Categories_, ([arXiv:2002.06055](https://arxiv.org/abs/2002.06055))
An account of [[Morita equivalence]] as a corollary of the Yoneda lemma for bicategories is in
* [[Niles Johnson]], _Morita Theory For Derived Categories: A Bicategorical Perspective_ ([arXiv:0805.3673](http://arxiv.org/abs/0805.3673))
The stricter case of 2-categories is detailed in
* Jonas Hedman, _2-Categories and Yoneda lemma_ ([pdf](http://www.diva-portal.org/smash/get/diva2:1064822/FULLTEXT01.pdf))
* Max Kelly, _Basic Concepts of Enriched Category Theory_ ([TAC](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf))
[[!redirects bicategorical Yoneda lemma]]
[[!redirects 2-Yoneda lemma]]
[[!redirects Yoneda embedding for bicategories]]
[[!redirects bicategorical Yoneda embedding]]
[[!redirects 2-Yoneda embedding]]
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Yoneda lemma for higher categories | https://ncatlab.org/nlab/source/Yoneda+lemma+for+higher+categories | [[!redirects Yoneda Lemma for higher categories]]
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###Context###
#### Yoneda lemma
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[[!include Yoneda lemma - contents]]
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#### Higher category theory
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[[!include higher category theory - contents]]
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# Contents#
* table of contents
{: toc}
## Idea
One expects the [[Yoneda lemma]] to generalize to essentially every flavor of [[higher category theory]]. Various special cases have been (defined and) proven, such as the:
* [[Yoneda lemma for (∞,1)-categories]]
* [[Yoneda lemma for bicategories]]
* [[Yoneda lemma for tricategories]]
## Yoneda embedding
{#YonedaEmbedding}
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###### Definition
For $C$ an [[(∞,n)-category]] and $PSh(C)\coloneqq Func(C^\op, (\infty,(n-1)) Cat)$ its [[(∞,n)-category of (∞,n)-presheaves]], the **$(\infty,n)$-Yoneda embedding** is the [[(∞,n)-functor]]
$$
y : C \to PSh(C)
$$
given by $y(X) : U \mapsto C(U,X)$.
=--
## Properties
### Yoneda lemma
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###### Proposition
**$(\infty,n)$-Yoneda embedding**
Let $C$ be an [[(∞,n)-category]] and $PSh(C)$ be the corresponding [[(∞,n)-category of (∞,n)-presheaves]]. Then the canonical [[(∞,n)-functor]]
$$
Y : C \to PSh(C)
$$
is a [[full and faithful (∞,n)-functor]].
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+-- {: .num_prop }
###### Proposition
**$(\infty,n)$-Yoneda theorem**
For $C$ a small $(\infty,n)$-category and $F : C^{op} \to (\infty,(n-1)) Cat$ an $(\infty,n)$-functor, the composite
$$
C^{op} \to PSh_{(\infty,1)}(C)^{op} \stackrel{Hom(-,F)}{\to}
(\infty,(n-1)) Cat$$
is equivalent to $F$.
=--
### Preservation of limits
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###### Proposition
The $(\infty,n)$-Yoneda embedding $y : C \to PSh(C)$ preserves all [[(∞,n)-limit]]s that exist in $C$.
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## Related concepts
* [[higher topos theory]]
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