Yang-Mills instanton

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### Context
#### Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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#### Topological physics
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[[!include topological physics -- contents]]
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#### Chern-Weil theory
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[[!include infinity-Chern-Weil theory - contents]]
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#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), doi:10.1016/0375-9601(78)90141-X]

### 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 (doi:10.1016/0370-1573(80)90130-1)

* {#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), doi:10.1016/S0370-1573(02)00301-0]

* [[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) doi

* 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) doi

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 (doi:10.1016/0550-3213(90)90599-9) Erratum: Nucl. Phys. B353 (1991) 565-565 (doi:10.1016/0550-3213(91)90349-3) ([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

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### 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

### 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 $\mathbb{C}P^n$* \[arXiv:2310.18058\]

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

+-- {: .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), pdf]

* {#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 \[doi:10.1016/0550-3213(83)90431-5\]

* {#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 \[doi:10.1016/0550-3213(84)90538-8]

* 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

#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

* [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

An occasional browser in and contributor to the nLab


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

* [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), doi:10.1016/0550-3213(94)90428-6]

category: people


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

* [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

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

+-- {: .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

* [webpage](http://www.math.ucla.edu/~ysella/)

## related $n$Lab entries

* [[singular cohomology]]

category: people


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

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[[!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]] with discrete target spaces. 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 ∞-Dijkgraaf-Witten theory 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

# 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

[[!redirects Yetter-Drinfeld module]]
[[!redirects Yetter-Drinfeld modules]]
[[!redirects Yetter–Drinfeld module]]
[[!redirects Yetter–Drinfeld modules]]
[[!redirects Yetter--Drinfeld module]]
[[!redirects Yetter--Drinfeld modules]]

[[!redirects Yetter-Drinfeld category]]
[[!redirects Yetter-Drinfeld categories]]
[[!redirects Yetter–Drinfeld category]]
[[!redirects Yetter–Drinfeld categories]]
[[!redirects Yetter--Drinfeld category]]
[[!redirects Yetter--Drinfeld categories]]


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

* [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

* [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

* [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), doi:10.1007/JHEP11(2023)208]

category: people

[[!redirects Yinan Wang]]


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

* [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

**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

* [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), doi:10.1007/JHEP09(2022)015]

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

* [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

* [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

* [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

* [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), doi:10.1016/0550-3213(89)90250-2)

category: people


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

* [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), doi:10.1016/S0550-3213(98)00045-5]

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 [doi:10.1016/S0370-2693(98)00501-2, [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), doi:10.1016/S0550-3213(99)00098-X]

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), doi:10.1016/S0370-2693(00)00317-8]

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

* [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

+-- {: .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

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

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 $U'$

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]]