Guillemin-Sternberg geometric quantization conjecture

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#### Geometric quantization
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[[!include geometric quantization - contents]]
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#### Physics
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[[!include physicscontents]]
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#### Symplectic geometry
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[[!include symplectic geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Guillemin-Sternberg conjecture_ states that, given a [[symplectic manifold]] equipped with a [[Hamiltonian action]] by some group, under suitable conditions the processes of [[geometric quantization]] and of [[symplectic reduction]] can be interchanged without changing the result (a [[representation]] of the [[group]] on the [[space of quantum states]]). This is usually abbreviated to:

* _quantization commutes with reduction_.

Since [[symplectic reduction]] also means restricting a [[mechanical system]] to a constraint surface, one may also read this as saying:

* _quantization commutes with imposing constraints_.

The original article ([Guillemin-Sternberg 82](#GuilleminSternberg82)) proved this statement for the case of [[compact Lie groups]] acting on [[compact topological space|compact]] [[Kähler manifolds]] and for [[geometric quantization]] via [[Kähler polarizations]].

For the more powerful and more flexible [[geometric quantization by push-forward]], the corresponding statement is widely known as the _Guillemin-Sternberg conjecture_.  Since its formulation, this, too, has been proven in fair generality, see the [References](#References) below.

## Related concepts

* [[symplectic reduction]], [[presymplectic reduction]] ([[gauge reduction]]), [[Poisson reduction]], [[BV-BRST formalism]]

* [[equivariant cohomology]]

[[!include symplectic reduction - table]]

* [[light-cone gauge quantization]]

## References
 {#References}

The original statement is due to

* [[Victor Guillemin]], [[Shlomo Sternberg]], _Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538.
 {#GuilleminSternberg82}

The first proof in [[geometric quantization by push-forward]] for compact groups acting on compact manifolds appears in

* [[Eckhard Meinrenken]], _Symplectic surgery and the $Spin^c$-Dirac operator_, Adv. Math. 134 (1998), 240-277.

A survey is in

* [[Michèle Vergne]], _Quantification géométrique et r´éduction symplectique, Séminaire Bourbaki, Vol. 2000/2001. Astérisque No. 282 (2002), Exp. No. 888, viii, 249–278.

Discussion for [[Kähler polarizations]] of mildly singular phase spaces is in

* [[Johannes Huebschmann]], _Kähler quantization and reduction_, J. reine angew. Math. 591 (2006), 75-109 ([arXiv:math/0207166](http://arxiv.org/abs/math/0207166))

The generalization to non-compact groups acting on non-compact spaces (but with compact quotients, hence "co-compactly") by passing to [[KK-theory]] and defining the push-forward by the Baum-Connes [[analytic assembly map]] was proposed in

* [[Klaas Landsman]] _Functorial quantization and the Guillemin-Sternberg conjecture_. In S. Ali, G. Emch, A. Odzijewicz, M. Schlichenmaier, and S. Woronowicz (eds.), _Twenty years of Bialowieza: a mathematical anthology_, pages 23–45, Singapore, 2005. World Scientific. ([arXiv:math-ph/0307059](http://arxiv.org/abs/math-ph/0307059))

and proven for various cases in

* [[Peter Hochs]], [[Klaas Landsman]], _The Guillemin-Sternberg conjecture for noncompact groups and spaces_ ([arXiv:math-ph/0512022](http://arxiv.org/abs/math-ph/0512022))

based on

* [[Peter Hochs]], _Quantisation commutes with reduction for cocompact Hamiltonian group actions_ ([pdf](http://www.math.ru.nl/~landsman/ThesisPeterHochs.pdf))

Discussion taking into account the [[metaplectic correction]] is in

* Brian Hall, William Kirwin, _Unitarity in "quantization commutes with reduction"_, Comm. Math. Phys. 275 (2007), no. 3, pages 401 -- 442 ([arXiv:math/0610005](http://arxiv.org/abs/math/0610005))

The full statement for this general case was then proven in

* {#MathaiZhang} [[Varghese Mathai]], [[Weiping Zhang]] (appendix by [[Ulrich Bunke]]), _Geometric quantization for proper actions_, Advances in Mathematics
Volume 225, Issue 3, 20 October 2010, Pages 1224–1247 ([arXiv:0806.3138](http://arxiv.org/abs/0806.3138))

Lecture notes on this:

* [[Weiping Zhang]], _Analytic aspects of geometric quantization_, 2009 ([pdf](http://www.math.ac.cn/PPT/PPT-hlg-lecture/ppt-hlg-2008/zhangweiping-hlg.pdf), [[ZhangAnalyticGeometricQuantization.pdf:file]])

Further generalizations are discussed in

* [[Peter Hochs]], [[Varghese Mathai]], _Quantizing tame actions_ ([arXiv:1309.6760](http://arxiv.org/abs/1309.6760))

* Jord Boeijink, [[Klaas Landsman]], [[Walter van Suijlekom]], _Quantization commutes with singular reduction: cotangent bundles of compact Lie groups_ ([arXiv:1508.06763](http://arxiv.org/abs/1508.06763))

Similar discussion is in

* [[Paul-Emile Paradan]], _Spin-quantization commutes with reduction_ ([arXiv:0911.1067](http://arxiv.org/abs/0911.1067))

based on

* [[Paul-Emile Paradan]], _Localization of the Riemann-Roch character_ ([arXiv:math/9911024](http://arxiv.org/abs/math/9911024))

Discussion specifically for [[circle group]] [[actions]] is in

*  [[Ana Cannas da Silva]], [[Yael Karshon]], [[Susan Tolman]], *Quantization of Presymplectic Manifolds and Circle Actions*, Transactions of the AMS **352** 2 (2000) 525-552  [[arXiv:dg-ga/9705008](http://arxiv.org/abs/dg-ga/9705008), [jstor:118052](https://www.jstor.org/stable/118052)]

[[!redirects geometric quantization conjecture]]
[[!redirects quantization commutes with reduction]]
[[!redirects reduction commutes with quantization]]


Guillermo Cortiñas

* [webpage](http://mate.dm.uba.ar/~gcorti/)

category: people

[[!redirects Guillermo Cortinas]]


Guiseppe Longo

Giuseppe Longo is a computer scientist based at the ENS Paris, where he heads the team: Complexité et information morphologiques.

* [homepage](http://www.di.ens.fr/~longo/)

category:people


Guiseppe Peano > history

see at *[[Giuseppe Peano]]*


Gunnar Carlsson

* [webpage](http://math.stanford.edu/~gunnar/)

## Selected writings

On [[cyclic loop spaces]]:

* [[Gunnar Carlsson]], [[Ralph Cohen]], *The cyclic groups and the free loop space*, Commentarii Mathematici Helvetici **62** (1987) 423–449  ([doi:10.1007/BF02564455](https://doi.org/10.1007/BF02564455), [dml:140092](https://eudml.org/doc/140092))

On an [[algorithm]] for [[topological data analysis]] by a variant of [[Reeb graphs]]:

* [[Gurjeet Singh]], [[Facundo Mémoli]], [[Gunnar Carlsson]], *Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition*, in: M. Botsch, R. Pajarola (eds.), *Eurographics Symposium on Point-Based Graphics* (2007) $[$[doi:10.2312/SPBG/SPBG07/091-100](http://dx.doi.org/10.2312/SPBG/SPBG07/091-100), [pdf](http://diglib.eg.org/bitstream/handle/10.2312/SPBG.SPBG07.091-100/091-100.pdf?sequence=1&isAllowed=y)$]$

On [[topological data analysis]]:

* [[Gunnar Carlsson]], _Topology and data_, Bull. Amer. Math. Soc. 46 (2009), 255-308 ([doi:10.1090/S0273-0979-09-01249-X](https://doi.org/10.1090/S0273-0979-09-01249-X))

and a flashy exposition for commercial use:

* [[Gunnar Carlsson]], *Professor Gunnar Carlsson Introduces Topological Data Analysis* $[$[video](https://www.ayasdi.com/resources/professor-gunnar-carlsson-introduces-topological-data-analysis/)$]$

On [[persistent homology]]:

* A. Zomorodian, [[Gunnar Carlsson]], _Computing persistent homology_, Discrete Comput. Geom. __33__, 249–274 (2005)

* [[Gunnar Carlsson]], V. de Silva, _Zigzag persistence_, [arXiv:0812.0197](http://arxiv.org/abs/0812.0197)

* [[Gunnar Carlsson]], _Persistent Homology and Applied Homotopy Theory_, in: [[Handbook of Homotopy Theory]], CRC Press, 2019 ([arXiv:2004.00738](https://arxiv.org/abs/2004.00738))

Introducing [[zigzag persistence]], connecting [[persistent homology]] to [[quiver representation]]-theory, re-proving [[Gabriel's theorem]] for the case of [[ADE classification|A-type]] [[quivers]] and introducing the [[diamond principle]]:

* {#CarlssonDeSilva10} [[Gunnar Carlsson]], [[Vin de Silva]], *Zigzag Persistence*, Found Comput Math **10** (2010) 367–405 $[$[arXiv:0812.0197](https://arxiv.org/abs/0812.0197), [doi:10.1007/s10208-010-9066-0](https://doi.org/10.1007/s10208-010-9066-0)$]$

and applied to [[level sets]]:

* [[Gunnar Carlsson]], [[Vin de Silva]], [[Dmitriy Morozov]], *Zigzag persistent homology and real-valued functions*, in: *SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry (2009) 247–256 $[$[doi:10.1145/1542362.1542408](https://doi.org/10.1145/1542362.1542408)$]$

* [[Gunnar Carlsson]], [[Vin de Silva]], [[Sara Kališnik]], [[Dmitriy Morozov]], *Parametrized Homology via Zigzag Persistence*, Algebr. Geom. Topol. **19** (2019) 657-700 $[$[arXiv:1604.03596](https://arxiv.org/abs/1604.03596), [doi:10.2140/agt.2019.19.657](https://doi.org/10.2140/agt.2019.19.657)$]$

## Related $n$Lab entries

* [[equivariant stable homotopy theory]]

* [[Segal conjecture]], [[Sullivan conjecture]]

  [[Burnside ring]], [[stable cohomotopy]], [[equivariant stable cohomotopy]]

* [[persistent homology]]

category: people

[[!redirects G. Carlsson]]


Guo Chuan Thiang

* [personal page](https://bicmr.pku.edu.cn/~guochuanthiang/)

## Selected writings

On the classification by [[topological K-theory]] of [[topological insulators]] ([[K-theory classification of topological phases of matter]]):

* [[Guo Chuan Thiang]], _On the K-theoretic classification of topological phases of matter_, Annales Henri Poincare 17(4) 757-794 (2016) ([arXiv:1406.7366](https://arxiv.org/abs/1406.7366))

* [[Guo Chuan Thiang]], _Topological phases: isomorphism, homotopy and K-theory_, Int. J. Geom. Methods Mod. Phys. 12 1550098 (2015) ([arXiv:1412.4191](https://arxiv.org/abs/1412.4191))

Aspects of the classification of [[semi-metals]] seen in [[ordinary cohomology]]:

* [[Varghese Mathai]], [[Guo Chuan Thiang]], *Global topology of Weyl semimetals and Fermi arcs*, J. Phys. A: Math. Theor. **50** (2017) 11LT01 ([arXiv:1607.02242](https://arxiv.org/abs/1607.02242),  [doi:10.1088/1751-8121/aa59b2](https://doi.org/10.1088/1751-8121/aa59b2))

* [[Varghese Mathai]], [[Guo Chuan Thiang]], *Differential Topology of Semimetals*,
Commun. Math. Phys. **355** (2017) 561–602, ([arXiv:1611.08961](https://arxiv.org/abs/1611.08961), [doi:10.1007/s00220-017-2965-z](https://doi.org/10.1007/s00220-017-2965-z))

On [[T-duality]] in the [[K-theory classification of topological phases of matter]]:

* [[Kiyonori Gomi]], [[Guo Chuan Thiang]], *Crystallographic T-duality*. J.Geom.Phys **139** (2019) 50-77 [[arXiv:1806.11385](https://arxiv.org/abs/1806.11385), [doi:10.1016/j.geomphys.2019.01.002](https://doi.org/10.1016/j.geomphys.2019.01.002)]

category: people


Guo-Qiang Zhang

* [webpage at Case Western](http://newton.case.edu/)

category:people


Guozhen Wang

* [webpage](https://pouiyter.github.io/)

## Selected writings

On [[homotopy groups of spheres]] and the framed [[bordism ring]] ([[MFr]]):

* {#WangXu10} [[Guozhen Wang]], [[Zhouli Xu]], Section 2.6 of: _A survey of computations of homotopy groups of Spheres and Cobordisms_, 2010 ([pdf](http://math.mit.edu/~guozhen/homotopy%20groups.pdf), [[WangXuHomotopyGroupsOfSpheres2010.pdf:file]])

On computation of [[stable homotopy groups of spheres]] via the [[Adams spectral sequence]]:

* [[Daniel Isaksen]], [[Guozhen Wang]], [[Zhouli Xu]], _Stable homotopy groups of spheres_, PNAS October 6, 2020 117 (40) 24757-2476 ([arXiv:2001.04247](https://arxiv.org/abs/2001.04247), [doi:10.1073/pnas.2012335117](https://doi.org/10.1073/pnas.2012335117))

* [[Daniel Isaksen]], [[Guozhen Wang]], [[Zhouli Xu]], _More stable stems_ ([arXiv:2001.04511](https://arxiv.org/abs/2001.04511))

## Related $n$Lab entries

* [[cobordism ring]]

* [[homotopy groups of spheres]]

category: people

[[!redirects Wang Guozhen]]


Gurjeet Singh

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=127089)

## Selected writings

On an [[algorithm]] for [[topological data analysis]] by a variant of [[Reeb graphs]]:

* [[Gurjeet Singh]], [[Facundo Mémoli]], [[Gunnar Carlsson]], *Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition*, in: M. Botsch, R. Pajarola (eds.), *Eurographics Symposium on Point-Based Graphics* (2007) $[$[doi:10.2312/SPBG/SPBG07/091-100](http://dx.doi.org/10.2312/SPBG/SPBG07/091-100), [pdf](http://diglib.eg.org/bitstream/handle/10.2312/SPBG.SPBG07.091-100/091-100.pdf?sequence=1&isAllowed=y)$]$

category: people


Gustav Lejeune Dirichlet

[[!redirects Gustav Dirichlet]]

* Wikipedia, [Peter Gustav Lejeune Dirichlet](https://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlet)

## Selected writings

In which the notion of a [[field]] was first introduced, but only for subfields of the [[complex numbers]]:

* [[Gustav Lejeune Dirichlet]] (1871), [[Richard Dedekind]] (ed.), Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German), vol. 1 (2nd ed.), Braunschweig, Germany: Friedrich Vieweg und Sohn ([Google Books](https://books.google.com/books?id=SRJTAAAAcAAJ&pg=PA424))

category: people

[[!redirects Peter Dirichlet]]
[[!redirects Peter Lejeune Dirichlet]]
[[!redirects Gustav Dirichlet]]
[[!redirects Gustav Lejeune Dirichlet]]
[[!redirects Peter Gustav Lejeune Dirichlet]]
[[!redirects Johann Peter Gustav Lejeune Dirichlet]]


Gustavo Granja

* [webpage](https://www.math.tecnico.ulisboa.pt/~ggranja/)

## Selected writings

On [[quaternionic line bundles]]:

* [[Gustavo Granja]], _On quaternionic line bundles_, MIT 1999 ([dspace:1721.1/85302](https://dspace.mit.edu/handle/1721.1/85302), [pdf](https://dspace.mit.edu/bitstream/handle/1721.1/85302/43854510-MIT.pdf?sequence=2&isAllowed=y))

category: people


Gustavo Rigolin

* [personal page](https://sites.google.com/site/rigolin/)

* [GoogleScholar page](https://scholar.google.com/citations?user=80TisMAAAAAJ&hl=en)

## Selected writings

On the [[quantum adiabatic theorem]] in the generality of non-abelian [[Berry phases]] such as needed for [[anyon]]-[[braiding]]:

* [[Gustavo Rigolin]], [[Gerardo Ortiz]], *The Adiabatic Theorem for Quantum Systems with Spectral Degeneracy*, Phys. Rev. A **85** 062111 (2012) [[arXiv:1111.5333](https://arxiv.org/abs/1111.5333), [doi:10.1103/PhysRevA.85.062111](https://doi.org/10.1103/PhysRevA.85.062111)]

category: people


GUT

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###Context###
#### Fields and quanta
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[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General idea

The [[standard model of particle physics]] asserts that the fundamental quantum [[field (physics)|physical fields]] and [[particles]] are modeled as [[sections]] of and [[connection on a bundle|connections]] on a [[vector bundle]] that is [[associated bundle|associated]] to a $G$-[[principal bundle]], where the [[Lie group]] $G$ -- called the [[gauge group]] -- is  the [[product]] of ([[special unitary group|special]]) [[unitary groups]] $G = SU(3) \times SU(2) \times U(1)$ (or rather a [[quotient]] of this by the [[cyclic group]] $Z/6$, see [there](standard+model+of+particle+physics#GaugeGroup)) and where the [[representation]] of $G$ used to form the [[associated bundle|associated]] [[vector bundle]] looks fairly ad hoc on first sight.

A **grand unified theory** ("GUT" for short) in this context is an attempt to realize the standard model as sitting inside a conceptually simpler [[model (in particle physics)|model]], in particular one for which the [[gauge group]] is a bigger but _simpler_ group $\hat{G}$, preferably a _[[simple Lie group]]_ in the technical sense, which contains $G$ as a [[subgroup]]. Such a grand unified theory would be [[phenomenology|phenomenologically]] viable if a process of [[spontaneous symmetry breaking]] at some high [[energy]] scale -- the "GUT scale" -- would reduce the model back to the [[standard model of particle physics]] without adding spurious extra effects that would not be in agreement with existing observations in [[experiment]].

The terminology "grand unified" here refers to the fact that such a single simple group $\hat{G}$ would unify the fundamental [[forces]] of [[electromagnetism]], the [[weak nuclear force]] and the [[strong nuclear force]] in a way that generalizes the way in which the [[electroweak field]] already unifies the [[weak nuclear force]] and [[electromagnetism]], and electromagnetism already unifies, as the word says, electricity and magnetism.

Since no GUT model has been fully validated yet (but see for instance [Fong-Meloni 14](#FongMeloni14)), GUTs are [[model (physics)|models]] "beyond the [[standard model of particle physics|standard model]]". Often quantum physics "beyond the standard model" is expected to also say something sensible about [[quantum gravity]] and hence unify not just the three gauge forces but also the fourth known fundamental force, which is [[gravity]]. Models that aim to do all of this would then "unify" the entire content of the [[standard model of particle physics]] plus the [[standard model of cosmology]], hence "everything that is known about fundamental physics" to date. Therefore such theories are then sometimes called a _[[theory of everything]]_.

(Here it is important to remember the context, both "grand unified" and "of everything" refers to aspects of presently available models of fundamental physics, and not to deeper philosophical questions of [[ontology]].)

\linebreak

[[length scales]] in the [[observable universe]] (from [[cosmological constant|cosmic]] scales, over [[fundamental particle]]-[[masses]] around the [[electroweak symmetry breaking]] to [[GUT scale]] and [[Planck scale]]):

> graphics grabbed from [Zupan 19](flavour+in+particle+physics#Zupan19)

### The $SU(5)$-GUT (Georgi-Glashow)
  {#IdeaofSU5GUT}

The argument for the hypothesis of "grand unification" is fairly compelling if one asks for _[[simple objects|simple algebraic structures]]_ in the technical sense ([[simple Lie groups]] and their [[irreducible representations]]):

The _exact_ gauge group of the [[standard model of particle physics]] is really a [[quotient group]]

$$
  G_{SM}
  \;=\;
  \big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6
  \,,
$$

where the [[cyclic group]] $\mathbb{Z}_6$ [[free action|acts freely]], hence exhibiting a subtle global twist in the gauge structure. This seemingly ad hoc group turns out to be [[isomorphism|isomorphic]] to the [[subgroup]]

$$
  \underset{
    \simeq \big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6  }{
  \underbrace{
    S \big( U(2) \times U(3)\big)
  }}
  \;\subset\;
  SU(5)
$$

of [[special unitary group|SU(5)]] (see [Baez-Huerta 09, p. 33-36](#BaezHuerta09)). The latter happens to be a [[simple Lie group]], thus exhibiting the exact standard model Lie group as being "simply" a "(2+3)-breaking" of a [[simple Lie group]].

Moreover, the [[gauge group]]-[[representation]] $V_{SM}$ that captures one [[generation of fundamental particles]] of the [[standard model of particle physics]], which looks fairly ad hoc as a representation of $G_{SM}$ (e.g. [Baez-Huerta 09, table 1](#BaezHuerta09)), but as a representation of $SU(5)$ it is simply

$$
  V_{SM} \simeq \Lambda \mathbb{C}^5
$$

(see [Baez-Huerta 09, p. 36-41](#BaezHuerta09)).

This leads to the $SU(5)$-GUT due to [Georgi-Glashow 74](#GeorgiGlashow74)

### D-Series GUTs

#### The $Spin(10)$-GUT (known as "$SO(10)$")
 {#TheGUTKnownAsSO10}

There is a further inclusion $SU(5) \hookrightarrow $ [[Spin(10)]] into the [[spin group]] in 10 (Euclidean) dimensions (still a [[simple Lie group]]), and one [[generation of fundamental particles]] regarded as an $SU(5)$-[[representation]] $\Lambda \mathbb{C}^5$ as [above](#IdeaofSU5GUT) extends to a [[spin representation]] (see [Baez-Huerta 09, theorem 2](#BaezHuerta09)). This has the aesthetically pleasing effect that under this identification the 1-[[generation of fermions|generation]] rep $V_{SM}$ is now identified as

$$
  V_{SM} \;\simeq\; \mathbf{16} \oplus \overline{\mathbf{16}}
$$

namely as the [[direct sum]] of [[generalized the|the]] two (complex) [[irreducible representations]] of [[Spin(10)]], together being the [[Dirac representation]] of [[Spin(10)]].

The exact [[gauge group]] of the [[standard model of particle physics]] (see [there](standard+model+of+particle+physics#GaugeGroup)) is [[isomorphism|isomorphic]] to the [[subgroup]] of [[Spin(9)]] $\subset$ [[Spin(10)]] which respects a splitting $\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3$ ([Krasnov 19](Spin9#Krasnov19)).

Again, this means that under the embedding of the gauge group $G_{SM}$ all the way into the [[simple Lie group]] [[Spin(10)]], its ingredients become _simpler_, not just in a naive sense, but in the technical mathematical sense of _[[simple objects|simple algebraic objects]]_.

Discussion of [[SO(10)]] (i.e. [[Spin(10)]]) GUT-models with realistic [[phenomenology]] is in [BLM 09](#BLM09) [Malinský 09](#Malinsky09), [Lavoura-Wolfenstein 10](#LavouraWolfenstein10) [Altarelli-Meloni 13](#AltarelliMeloni13) [Dueck-Rodejohann 13](#DueckRodejohann13) [Ohlsson-Pernow 19](#OhlssonPernow19) [CPS 19](#CPS19).

\begin{imagefromfile}
  "file_name": "MinimalNonSusySO10GUTMalinsky.jpg",
  "width": 500
\end{imagefromfile}

> slide grabbed from [Malinský 09](#Malinsky09)

Discussion of [[leptoquarks]] in $SO(10)$-models possibly explaining the [[flavour anomalies]]: [AMM 19](#AMM19)

\linebreak

#### The $Spin(11)$-GUT (known as "$SO(11)$")
 {#TheGUTKnownAsSO11}

Models with [[Spin(11)]] ("[[SO(11)]]") GUT group.

Specifically with [[gauge-Higgs unification]] in a [[Randall-Sundrum model]]-like 6d spacetime: [Hosotani-Yamatsu 15](#HosotaniYamatsu15), [Furui-Hosotani-Yamatsu 16](#FuruiHosotaniYamatsu16), [Hosotani 17](#Hosotani17), [Hosotani-Yamatsu 17](#HosotaniYamatsu17)

See the references [below](#ReferencesSpin11).

\linebreak

#### The $Spin(12)$-GUT (known as "$SO(12)$")
 {#TheGUTKnownAsSO12}

Models with [[Spin(12)]] ("[[SO(12)]]") GUT group.

Specifically with [[gauge-Higgs unification]] in a [[Randall-Sundrum model]]-like 6d spacetime: [Nomura-Sato 08](#NomuraSato08), [Nomura 09](#Nomura09), [Chiang-Nomura-Sato 11](#ChiangNomuraSato11))

See the references [below](#ReferencesSpin12).

\linebreak

#### The $Spin(16), Spin(18)$-GUT (known as "$SO(16), SO(18)$")
 {#TheGUTKnownAsSO16}

Models with [[Spin(16)]] ("[[SO(16)]]") GUT group.

[Wilczek-Zee 82](#WilczekZee82), [Senjanovic-Wilczek-Zee 84](#SenjanovicWilczekZee84), [Martínez-Melfo-Nesti-Senjanovic 11](#MartínezMelfoNestiSenjanovic11)

See also [di Lucio 11, p. 44](#diLucio11) and see the references [below](#ReferencesSpin16).

Predicts [[fourth generation of fermions]]...

### E-series GUTs

The most studied choices of GUT-groups $G$ are [[special unitary group|SU(5)]], [[spin group|Spin(10)]] (in the physics literature often referred to as [[special orthogonal group|SO(10)]]) and [[E6]] (review includes [Witten 86, sections 1 and 2](#Witten86)).

It so happens that, mathematically, the sequence [[special unitary groups|SU(5)]], [[spin group|Spin(10)]], [[E6]] naturally continues (each step by consecutively adding a node to the [[Dynkin diagrams]]) with the [[exceptional Lie groups]] [[E7]], [[E8]] that naturally appear in [[heterotic string theory|heterotic]] [[string phenomenology]] (exposition is in [Witten 02a](#Witten02a)) and conjecturally further via the [[U-duality]] [[Kac-Moody groups]] [[E9]], [[E10]], [[E11]] that are being argued to underly [[M-theory]]. In the context of [[F-theory]] model building, also properties of the observes [[Yukawa couplings]] may point to exceptional GUT groups ([Zoccarato 14, slide 11](#Zoccarato14), [Vafa 15, slide 11](#Vafa15)).

#### The $E_6$-GUT

(...)

review in [Britto 17](#Britto17)

(...)

#### The $E_7$-GUT (?)

(...)

#### The $E_8$-GUT (?)

(...)

## Properties

### Relation to proton decay
  {#RelationToProtonDecay}

Many GUT models imply that the [[proton]] -- which in the [[standard model of particle physics]] is a stable [[bound state]] (of [[quarks]]) -- is in fact unstable, albeit with an extremely long mean liftetime, and hence may decay (e.g. [KM 14](#KM14)). [[experiment|Experimental]] searches for such _[[proton decay]]_ (see there for more) put strong bounds on this effect and hence heavily constrain or rule out many [[GUT]] [[model (physics)|models]].

But in recent years it is claimed that there are in fact realistic $SU(5)$ [[GUT]] [[model (physics)|models]] that do not imply any [[proton decay]], quite generically so for [[MSSM]]-models ([Mütter-Ratz-Vaudrvange 16](#MuetterRatzVaudrvange16)), but also for non-supersymmetric models  ( [Fornal-Grinstein 17](#FornalGrinstein17), [Fornal-Grinstein 18](#FornalGrinstein18), in particular in [[gauge-Higgs grand unification]] such as [[Spin(11)]]- ("[[SO(11)]]"-) and [[Spin(12)]]- ("[[SO(12)]]"-) models: ([Hosotani-Yamatsu 15](#HosotaniYamatsu15), [Furui-Hosotani-Yamatsu 16, Sec. 2.6](#FuruiHosotaniYamatsu16) [Hosotani 17, Section 6](#Hosotani17)).

### Relation to neutrino masses
 {#RelationToNeutrinoMasses}

The high energy scale required by a [[seesaw mechanism]] to produce the experimentally observer [[neutrino]] masses happens to be about the conventional [[GUT scale]], for review see for instance ([Mohapatra 06](#Mohapatra06)).

> I also noted at the same time that interactions between a pair of lepton doublets and a pair of scalar doublets can generate a neutrino mass, which is suppressed only by a factor $M^{-1}$, and that therefore with a reasonable estimate of $M$ could produce observable neutrino oscillations. The subsequent confirmation of neutrino oscillations lends support to the view of the Standard Model as an effective field theory, with M somewhere in the neighborhood of $10^{16} GeV$. ([Weinberg 09, p. 15](neutrino#Weinberg09))

Detailed matching of parameters of non-supersymmetric $Spin(10)$-GUT to [[neutrino]] [[masses]] is discussed in [Ohlsson-Pernow 19](#OhlssonPernow19)

### Relation to leptoquarks
 {#RelationToLeptoquarks}

Generically, GUT-theories predict the existence of [[leptoquarks]] ([Murayama-Yanagida 92](#MurayamaYanagida92)), possibly related to the apparently observed

* [[flavour anomalies]] ([BDFKFS 18](#BDFKFS18), [AMM 19](#AMM19), [Heek-Teresi 18](#HeekTeresi18), [Heek-Teresi 19](#HeekTeresi19))

* [anomalies](anomalous+magnetic+moment#Anomalies) in [[anomalous magnetic moment]] of [[muon]] and [[electron]]

### Occurrence in string phenomenology
 {#InStringPhenomenology}

Discussion of [[string phenomenology]] of [[intersecting D-brane models]] [[KK-compactification|KK-compactified]] with non-geometric [[fibers]] such that the would-be string [[sigma-models]] with these [[target spaces]] are in fact [[Gepner models]] (in the sense of _[Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_) is in ([Dijkstra-Huiszoon-Schellekens 04a](#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b)):

>  A plot of [[standard model of particle physics|standard model]]-like [[coupling constants]] in a computer scan of [[Gepner model]]-[[KK-compactification]] of [[intersecting D-brane models]] according to [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b).

>  The blue dot indicates the couplings in $SU(5)$-[[GUT]] theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.

## Related concepts

* [[theory of everything]]

* [[gauge coupling unification]]

* [[supersymmetry]]

* [[Connes-Lott-Chamseddine-Barrett model]]

[[!include fundamental scales -- contents]]

## References
 {#References}

### General

Original articles include

* {#PatiSalam74} [[Jogesh Pati]], [[Abdus Salam]], _Lepton number as the fourth "color"_, Phys. Rev. D 10, 275 – Published 1 July 1974 ([doi:10.1103/PhysRevD.10.275](https://doi.org/10.1103/PhysRevD.10.275))

* {#GeorgiGlashow74} [[Howard Georgi]], [[Sheldon Glashow]], _Unity of All Elementary-Particle Forces_, Phys. Rev. Lett. **32** (1974) 438 [[doi:10.1103/PhysRevLett.32.438](https://doi.org/10.1103/PhysRevLett.32.438)]

Textbook accounts:

* [[Robert E. Marshak]], Chapter 8 of: *Conceptual Foundations of Modern Particle Physics*, World Scientific 1993 ([doi:10.1142/1767](https://doi.org/10.1142/1767))

* [[Howard Georgi]], §18 in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

See also

* Wikipedia, _[Pati-Salam model](https://en.wikipedia.org/wiki/Pati%E2%80%93Salam_model)_

* Wikipedia, _[Grand unification energy](https://en.wikipedia.org/wiki/Grand_unification_energy)_

Discussion with an eye towards [[supergravity]] unification:

* {#GellMann83} [[Murray Gell-Mann]], introductory talk at _[Shelter Island II](https://en.wikipedia.org/wiki/Shelter_Island_Conference)_, 1983 ([[Gell-Mann_ShelterIslandII_1983.pdf:file]])

  in: _Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics_. MIT Press. pp. 301–343. ISBN 0-262-10031-2.

* [[Murray Gell-Mann]], [[Pierre Ramond]], Richard Slansky, _Complex Spinors and Unified Theories_, Supergravity, [[Peter van Nieuwenhuizen]] and D.Z. Freedman, eds, North Holland Publishing Co, 1979, (reprinted as [arXiv:1306.4669](http://arxiv.org/abs/1306.4669))

A basic textbook account is in

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 1.2 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

and a detailed one is in

* A. Hebecker, J. Hisano, _Grand Unified Theories_, chapter 114. in Particle Data Group's _[The Review of Particle Physics](http://pdg.lbl.gov/2018/)_ 2018 ([pdf](http://pdg.lbl.gov/2018/reviews/rpp2018-rev-guts.pdf))

See also

* {#diLucio11} Luca di Lucio, _Aspects of Symmetry Breaking inGrand Unified Theories_, 2011 ([pdf](https://www.sissa.it/tpp/phdsection/AlumniThesis/Luca%20Di%20Luzio.pdf))

Survey of arguments for the hypothesis of grand unification includes

* [[Michael Peskin]], _Beyond the Standard Model_ ([arXiv:hep-ph/9705479](http://arxiv.org/abs/hep-ph/9705479))

* [[Jogesh Pati]], _Discovery of Proton Decay: A Must for Theory, a Challenge for Experiment_ ([arXiv:hep-ph/0005095](http://arxiv.org/abs/hep-ph/0005095))

* {#Witten02a} [[Edward Witten]], _Quest For Unification_, Heinrich Hertz lecture at [SUSY 2002](http://www.desy.de/susy02/) at DESY, Hamburg ([arXiv:hep-ph/0207124](http://arxiv.org/abs/hep-ph/0207124))

Introductions to GUTs aimed more at mathematicians:

* {#Witten86} [[Edward Witten]], section 1 and 2 of _Physics and geometry_, Proceedings of the international congress of mathematicians, 1986 ([pdf](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0267.0306.ocr.pdf))

* {#BaezHuerta09} [[John Baez]], [[John Huerta]], _The algebra of grand unified theories_, Bull. Am. Math. Soc. 47:483-552,2010 ([arXiv:0904.1556](http://arxiv.org/abs/0904.1556))

* {#Britto17} Vivian Anthony Britto, _A Mathematical Construction of an $E_6$ Grand Unified Theory_, Munich 2017 ([pdf](https://www.theorie.physik.uni-muenchen.de/TMP/theses/thesisbritto.pdf), [[BrittoGUT.pdf:file]])

On [[division algebra|division-algebraic]] relations in GUT-[[broken symmetry|symmetry breaking]]:

* [[Nichol Furey]], *An Algebraic Roadmap of Particle Theories, Part I: General construction* [[arXiv:2312.12377](https://arxiv.org/abs/2312.12377)]

* [[Nichol Furey]], *An Algebraic Roadmap of Particle Theories, Part II: Theoretical checkpoints* [[arXiv:2312.12799](https://arxiv.org/abs/2312.12799)]

* [[Nichol Furey]], *An Algebraic Roadmap of Particle Theories, Part III: Intersections* [[arXiv:2312.14207](https://arxiv.org/abs/2312.14207)]

### Proton (non-)decay
  {#ReferencesProtonNonDecay}

Discussion of experimental bounds on [[proton decay]] in GUTs includes

* {#KM14} Helena Kolešová, [[Michal Malinský]], _Proton lifetime in the minimal $SO(10)$ GUT and its implications for the LHC_, Phys. Rev. D 90, 115001 (2014) ([arXiv:1409.4961](http://arxiv.org/abs/1409.4961))

Claim that [[proton decay]] may be entirely avoided:

* {#MuetterRatzVaudrvange16} Andreas Mütter, Michael Ratz, Patrick K.S. Vaudrevange, _Grand Unification without Proton Decay_ ([arXiv:1606.02303](https://arxiv.org/abs/1606.02303))

  (claims that many [[string theory]] and [[supergravity]] models have this property)

* {#FornalGrinstein17} Bartosz Fornal, Benjamin Grinstein, _$SU(5)$ Unification without Proton Decay_, Physics Review Letters ([arXiv:1706.08535](https://arxiv.org/abs/1706.08535))

* {#FornalGrinstein18} Bartosz Fornal, Benjamin Grinstein, _Grand Unified Theory with a Stable Proton_, Int. J. Mod. Phys. A 33 (2018) 1844013 ([arXiv:1808.00953](https://arxiv.org/abs/1808.00953))

Claim that [[threshold corrections]] can considerably alter (decrease) proton decay rate predictions in non-supersymmetric GUTs:

* Joydeep Chakrabortty, Stephen F. King, Rinku Maji, _Unification, Proton Decay and Topological Defects in non-SUSY GUTs with Thresholds_ ([arXiv:1901.05867](https://arxiv.org/abs/1901.05867))

### Realistic models and phenomenology
 {#ReferencesRealisticModels}

Discussion of [[phenomenology|phenomenologically]] viable GUT-[[model (in theoretical physics)|models]] (compatible with [[experiment]] and the [[standard model of particle physics]]):

#### $SU(5)$-GUT

More on the original $SU(5)$-approach:

* Ilja Doršner, *Towards a minimal $SU(5)$ model* $[$[arXiv:2206.06036](https://arxiv.org/abs/2206.06036)$]$

On quasi-realistic "flipped" [[SU(5)|$SU(5)$]]-[[GUT]],  modeled in 4d [[heterotic string theory]] and subsuming realistic [[Starobinsky model of cosmic inflation|Starobinsky-type cosmic inflation]]:

* [[Ignatios Antoniadis]], [[John Ellis]], [[John S. Hagelin]], [[Dimitri Nanopoulos]], *The flipped $SU(5) \times U(1)$ string model revamped*, Physics Letters B **231** 1–2 (1989) 65-74 [doi:10.1016/0370-2693(89)90115-9]

* {#AntoniadisNanopoulosRizos21a} [[Ignatios Antoniadis]], [[Dimitri Nanopoulos]], [[John Rizos]], *Cosmology of the string derived flipped $SU(5)$*, J. Cosm. Astrop. Phys **03** (2021) 017 [[arXiv:2011.09396](https://arxiv.org/abs/2011.09396), [doi:10.1088/1475-7516/2021/03/017](https://doi.org/10.1088/1475-7516/2021/03/017)]

* {#AntoniadisNanopoulosRizos21a} [[Ignatios Antoniadis]], [[Dimitri Nanopoulos]], [[John Rizos]], *Particle physics and cosmology of the string derived no-scale flipped $SU(5)$*,  Eur. Phys. J. C **82** (2022) 377  [[arXiv:2112.01211](https://arxiv.org/abs/2112.01211), [doi:10.1140/epjc/s10052-022-10353-6](https://doi.org/10.1140/epjc/s10052-022-10353-6)]

#### $SO(10)$-GUT

The idea of "SO(10)" ([[Spin(10)]]) GUT originates with

* [[Harald Fritzsch]], [[Peter Minkowski]], _Unified interactions of leptons and hadrons_, Annals of Physics Volume 93, Issues 1–2, 5 September 1975, Pages 193-266 (doi:10.1016/0003-4916(75)90211-0)

Review:

* {#Malinsky09} [[Michal Malinský]], _35 years of GUTs - where do we stand?_, 2009 ([pdf](https://www.mpi-hd.mpg.de/lin/seminar_theory/talks/Malinsky.pdf))

* {#BLM09} Stefano Bertolini, Luca Di Luzio, [[Michal Malinský]], _Intermediate mass scales in the non-supersymmetric SO(10) grand unification: a reappraisal_, Phys. Rev. D80:015013, 2009 ([arXiv:0903.4049](https://arxiv.org/abs/0903.4049))

for non-superymmetric [[model (physics)|models]]:

* {#LavouraWolfenstein10} L. Lavoura and Lincoln Wolfenstein, _Resuscitation of minimal $SO(10)$ grand unification_, Phys. Rev. D 48, 264 ([doi:10.1103/PhysRevD.48.264](https://doi.org/10.1103/PhysRevD.48.264))

* {#AltarelliMeloni13} Guido Altarelli, Davide Meloni, _A non Supersymmetric SO(10) Grand Unified Model for All the Physics below $M_{GUT}$_ ([arXiv:1305.1001](https://arxiv.org/abs/1305.1001))

* {#DueckRodejohann13} Alexander Dueck, Werner Rodejohann, _Fits to $SO(10)$ Grand Unified Models_ ([arXiv:1306.4468](http://arxiv.org/abs/1306.4468))

* {#FongMeloni14} Chee Sheng Fong, Davide Meloni, Aurora Meroni, Enrico Nardi, _Leptogenesis in $SO(10)$_ ([arXiv:1412.4776](http://arxiv.org/abs/1412.4776))

  (in view of [[leptogenesis]])

* {#OhlssonPernow19} Tommy Ohlsson, Marcus Pernow, _Fits to Non-Supersymmetric SO(10) Models with Type I and II Seesaw Mechanisms Using Renormalization Group Evolution_ ([arXiv:1903.08241](https://arxiv.org/abs/1903.08241))

* {#CPS19} Mainak Chakraborty, M.K. Parida, Biswonath Sahoo, _Triplet Leptogenesis, Type-II Seesaw Dominance, Intrinsic Dark Matter, Vacuum Stability and Proton Decay in Minimal SO(10) Breakings_ ([arXiv:1906.05601](https://arxiv.org/abs/1906.05601))

  > Results indicating non-SUSY $SO(10)$ as self sufficient theory for neutrino masses, baryon asymmetry, dark matter, vacuum stability of SM scalar potential, origin of three gauge forces, and observed proton stability.

* Nobuchika Okada, Digesh Raut, Qaisar Shafi, _Inflation, Proton Decay, and Higgs-Portal Dark Matter in $SO(10) \times U(1)_\pri$_ ([arXiv:1906.06869](https://arxiv.org/abs/1906.06869))

* V. Suryanarayana Mummidi, Ketan M. Patel, *Leptogenesis and fermion mass fit in a renormalizable $SO(10)$ model* ([arXiv:2109.04050](https://arxiv.org/abs/2109.04050))

* George Lazarides, Rinku Maji, Rishav Roshan, Qaisar Shafi, *A Predictive $SO(10)$ Model* [[arXiv:2210.03710](https://arxiv.org/abs/2210.03710)]

with possible relation to [[flavour anomalies]]:

* Purushottam Sahu, Aishwarya Bhatta, Rukmani Mohanta, Shivaramakrishna Singirala, Sudhanwa Patra, *Flavour anomalies and dark matter assisted unification in $SO(10)$ GUT* [[arXiv:2204.06392](https://arxiv.org/abs/2204.06392)]

for [[supersymmetry|supersymmetric]] [[model (physics)|models]]:

* Archana Anandakrishnan, B. Charles Bryant, Stuart Raby, _LHC Phenomenology of $SO(10)$ Models with Yukawa Unification II_ ([arXiv:1404.5628](http://arxiv.org/abs/1404.5628))

* Ila Garg, _New minimal supersymmetric $SO(10)$ GUT phenomenology and its cosmological implications_ ([arXiv:1506.05204](http://arxiv.org/abs/1506.05204))

On the [[symmetry breaking]] from [[Spin(10)]] to the [[standard model of particle physics|standard model]] [[gauge group]] in terms of [[real normed division algebras]] ([[octonions]], [[quaternions]]):

* [[Kirill Krasnov]], *Geometry of $Spin(10)$ Symmetry Breaking* [[arXiv:2209.05088](https://arxiv.org/abs/2209.05088)]

* [[Nichol Furey]], M. J. Hughes, *Division algebraic symmetry breaking* [[arXiv:2210.10126](https://arxiv.org/abs/2210.10126)]

#### $SO(11)$-GUT
 {#ReferencesSpin11}

Discussion for [[Spin(11)]] GUT group ("[[SO(11)]]"):

* {#HosotaniYamatsu15} [[Yutaka Hosotani]], Naoki Yamatsu, _Gauge–Higgs grand unification_, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 11, November 2015 ([doi:10.1093/ptep/ptv153](https://doi.org/10.1093/ptep/ptv153), [doi:10.1093/ptep/ptw116](https://doi.org/10.1093/ptep/ptw116))

* {#FuruiHosotaniYamatsu16} Atsushi Furui, [[Yutaka Hosotani]], Naoki Yamatsu, _Toward Realistic Gauge-Higgs Grand Unification_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 9, September 2016, 093B01 ([arXiv:1606.07222](https://arxiv.org/abs/1606.07222))

* {#Hosotoni16} [[Yutaka Hosotani]], _Gauge-Higgs EW and Grand Unification_, International Journal of Modern Physics AVol. 31, No. 20n21, 1630031 (2016)  ([arXiv:1606.08108](https://arxiv.org/abs/1606.08108))

* {#Hosotani17} [[Yutaka Hosotani]], _New dimensions from gauge-Higgs unification_ ([arXiv:1702.08161](https://arxiv.org/abs/1702.08161))

* {#HosotaniYamatsu17} Yutaka Hosotani, Naoki Yamatsu, _Electroweak Symmetry Breaking and Mass Spectra in Six-Dimensional Gauge-Higgs Grand Unification_ ([arXiv:1710.04811](https://arxiv.org/abs/1710.04811))

#### $SO(12)$-GUT
 {#ReferencesSpin12}

Discussion for [[Spin(12)]] GUT group ("[[SO(12)]]"):

* S. Rajpoot and P. Sithikong, _Implications of the $SO(12)$ gauge symmetry for grand unification_, Phys. Rev. D 23, 1649 (1981) ([doi:10.1103/PhysRevD.23.1649](https://doi.org/10.1103/PhysRevD.23.1649))

* {#NomuraSato08} Takaaki Nomura, Joe Sato, _Standard(-like) Model from an $SO(12)$ Grand Unified Theory in six-dimensions with $S^2$ extra-space_, Nucl.Phys.B811:109-122, 2009 ([arXiv:0810.0898](https://arxiv.org/abs/0810.0898))

* {#Nomura09} Takaaki Nomura, _Physics beyond the standard model with $S^2$ extra-space_, 2009 ([pdf](http://krishna.th.phy.saitama-u.ac.jp/joe/doctor/nomura-doctor.pdf), [[NomuraSO12GUT.pdf:file]])

* {#ChiangNomuraSato11} Cheng-Wei Chiang, Takaaki Nomura, Joe Sato, _Gauge-Higgs unification models in six dimensions with $S^2/\mathbb{Z}_2$ extra space and GUT gauge symmetry_ ([arXiv:1109.5835](https://arxiv.org/abs/1109.5835))

#### $SO(16)$- and $Spin(18)$-GUT
 {#ReferencesSpin16}

Discussion for [[Spin(16)]] and [[Spin(18)]] GUT group ("[[SO(16)]]" and "[[SO(18)]]"):

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

* {#SenjanovicWilczekZee84} Goran Senjanović, [[Frank Wilczek]], [[Anthony Zee]], _Reflections on mirror fermions_,  Physics Letters B Volume 141, Issues 5–6, 5 July 1984, Pages 389-394 Physics Letters B (doi:10.1016/0370-2693(84)90269-7)

* {#MartínezMelfoNestiSenjanovic11} Homero Martínez, Alejandra Melfo, Fabrizio Nesti, Goran Senjanović, _Three Extra Mirror or Sequential Families: a Case for Heavy Higgs and Inert Doublet_, Phys. Rev. Lett.106:191802, 2011 ([arXiv:1101.3796](https://arxiv.org/abs/1101.3796))

* {#Guigan19} Michael McGuigan, _Dark Horse, Dark Matter: Revisiting the $SO(16) \times SO(16)'$ Nonsupersymmetric Model in the LHC and Dark Energy Era_ ([arXiv:1907.01944](https://arxiv.org/abs/1907.01944))

#### $E_8$-GUT

See at _[heterotic string -- phenomenology](heterotic+string+theory#ReferencesPhenomenology)_

See also:

* Alfredo Aranda, Francisco J. de Anda, Stephen F. King, _Exceptional Unification of Families and Forces_ ([arXiv:2005.03048](https://arxiv.org/abs/2005.03048))

* Alfredo Aranda, Francisco J. de Anda, _Complete $E_8$ Unification in 10 Dimensions_ ([arXiv:2007.13248](https://arxiv.org/abs/2007.13248))

* George Manolakos, Gregory Patellis, George Zoupanos, _$\mathcal{N}=1$ trinification from dimensional reduction of $N=1$, 10D $E_8$ over $SU(3)/U(1)\times U(1) \times \mathbb{Z}_3$ and its phenomenological consequences_ ([arXiv:2009.07059](https://arxiv.org/abs/2009.07059))

* Francisco J. de Anda, Alfredo Aranda, António P. Morais, Roman Pasechnik, _Gauge couplings evolution from the Standard Model, through Pati-Salam theory, into $E_8$ unification of families and forces_ ([arXiv:2011.13902](https://arxiv.org/abs/2011.13902))

* Alfredo Aranda, Francisco J. de Anda, António P. Morais, Roman Pasechnik, *A Different Take on $E_8$ Unification* ([arXiv:2107.05421](https://arxiv.org/abs/2107.05421))

* Alfredo Aranda, Francisco J. de Anda, António P. Morais, Roman Pasechnik, *Sculpting the Standard Model from low-scale Gauge-Higgs-Matter $E_8$ Grand Unification in ten dimensions* ([arXiv:2107.05495](https://arxiv.org/abs/2107.05495))

[[!include heterotic string phenomenology -- references]]

### In type II string theory

Computer scan of [[Gepner model]]-compactifications in relation to GUT-models is in

* {#DijkstraHuiszoonSchellekens04a} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models_, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th/0403196](https://arxiv.org/abs/hep-th/0403196))

* {#DijkstraHuiszoonSchellekens04b} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Supersymmetric Standard Model Spectra from RCFT orientifolds_, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th/0411129](https://arxiv.org/abs/hep-th/0411129))

Realization of GUTs in the context of [[M-theory on G2-manifolds]] and possible resolution of the [[doublet-triplet splitting problem]] is discussed in

* {#Witten02b} [[Edward Witten]], _Deconstruction, $G_2$ Holonomy, and Doublet-Triplet Splitting_, ([arXiv:hep-ph/0201018](http://arxiv.org/abs/hep-ph/0201018))

* [[Bobby Acharya]], Krzysztof Bozek, Miguel Crispim Romao, Stephen F. King, Chakrit Pongkitivanichkul, _$SO(10)$ Grand Unification in M theory on a $G_2$ manifold_ ([arXiv:1502.01727](http://arxiv.org/abs/1502.01727))

Discussion of GUTs in [[F-theory]] includes

* [[Chris Beasley]], [[Jonathan Heckman]], [[Cumrun Vafa]], _GUTs and Exceptional Branes in F-theory - I_ ([arxiv:0802.3391](http://arxiv.org/abs/0802.3391)), _II: Experimental Predictions_ ([arxiv:0806.0102](http://arxiv.org/abs/0806.0102))

* [[Chris Beasley]], [[Jonathan Heckman]], [[Cumrun Vafa]], _GUTs and Exceptional Branes in F-theory - I_, JHEP 0901:058,2009 ([arXiv:0802.3391](http://arxiv.org/abs/0802.3391))

* {#Zoccarato14} Gianluca Zoccarato, _Yukawa couplings at the point
of $E_8$ in F-theory_, 2014 ([pdf](http://stringpheno2014.ictp.it/parallels/tuesday/F-theory(B)/zoccarato.pdf))

* {#Vafa15} [[Cumrun Vafa]], _Reflections on F-theory_, 2015 (pdf)

### In Connes-Lott models

Discussion of GUTs within [[Connes-Lott models]]:

* {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Quanta of Geometry: Noncommutative Aspects_, Phys. Rev. Lett. 114 (2015) 9, 091302 ([arXiv:1409.2471](https://arxiv.org/abs/1409.2471))

* {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Geometry and the Quantum: Basics_, JHEP 12 (2014) 098 ([arXiv:1411.0977](https://arxiv.org/abs/1411.0977))

* {#Connes17} [[Alain Connes]], section 4 of _Geometry and the Quantum_, in _Foundations of Mathematics and Physics One Century After Hilbert_, Springer 2018. 159-196 ([arXiv:1703.02470](https://arxiv.org/abs/1703.02470), [doi:10.1007/978-3-319-64813-2](https://www.springer.com/gp/book/9783319648125))

### Exotica: Leptoquarks, $Z'$-bosons, etc.

Topological defects can play considerable role to constrain the non-SUSY and SUSY GUTs:

* Joydeep Chakrabortty, Rinku Maji, Sunando Kumar Patra, Tripurari Srivastava, Subhendra Mohanty, _Roadmap of left-right models based on GUTs_
([arXiv:1711.11391] (https://arxiv.org/abs/1711.11391), Phys.Rev. D97 (2018) no.9, 095010.)

Relation to [[Z'-bosons]]:

* {#Sahoo06} S. Sahoo, _The prediction of mass of $Z'$-boson in an $SO(10)$-based model_, Indian J. Phys. 80 (2), 191-194 (2006) ([pdf](http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/6019/1/The%20Prediction%20of%20Mass%20of%20Z%27-Boson_By%20S%20Sahoo.pdf))

Relation to [[leptoquarks]] and [[flavour anomalies]]:

* {#MurayamaYanagida92} H. Murayama, T. Yanagida, _A viable $SU(5)$ GUT with light leptoquark bosons_, Mod.Phys.Lett. A7 (1992) 147-152 ([arXiv:315898](inspirehep.net/record/315898), [doi:10.1142/S0217732392000070](https://doi.org/10.1142/S0217732392000070))

* {#BDFKFS18} Damir Bečirević, Ilja Doršner, Svjetlana Fajfer, Nejc Košnik, Darius A. Faroughy, Olcyr Sumensari, _Scalar leptoquarks from GUT to accommodate the $B$-physics anomalies_, Phys. Rev. D 98, 055003 (2018) ([arXiv:1806.05689](https://arxiv.org/abs/1806.05689))

* {#AMM19} Ufuk Aydemir, Tanumoy Mandal, Subhadip Mitra, _A single TeV-scale scalar leptoquark in SO(10) grand unification and B-decay anomalies_ ([arXiv:1902.08108](https://arxiv.org/abs/1902.08108))

* {#HeekTeresi18} Julian Heeck, Daniele Teresi, _Pati-Salam explanations of the B-meson anomalies_, JHEP 12 (2018) 103 ([arXiv:1808.07492](https://arxiv.org/abs/1808.07492))

* {#HeekTeresi19} Julian Heeck, Daniele Teresi, _Pati-Salam and lepton universality in B decays_ ([arXiv:1905.05211](https://arxiv.org/abs/1905.05211))

* {#Malinsky19} [[Michal Malinský]], _Lepton non-universality in $B$-decays in the minimal leptoquark gauge model_ ([arXiv:1906.09174](https://arxiv.org/abs/1906.09174))

* Shyam Balaji, Michael A. Schmidt, _A unified $SU(4)$ theory for the $R_D^{(\ast)}$ and $R_K^{(\ast)}$ anomalies_ ([arXiv:1911.08873](https://arxiv.org/abs/1911.08873))

Modeled [[intersecting D-brane model|on branes]]:

* [[Javier Fuentes-Martin]], [[Gino Isidori]], Julie Pagès, Ben A. Stefanek,
*Flavor Non-universal Pati-Salam Unification and Neutrino Masses*, Physics Letters B Volume 820, 10 September 2021, 136484 ([arXiv:2012.10492](https://arxiv.org/abs/2012.10492), [talk recording](https://www.youtube.com/watch?v=ko4R8S8XYV4&t=2s))

[[!redirects GUTs]]
[[!redirects grand unified theory]]
[[!redirects grand unified theories]]

[[!redirects grand unified field theory]]
[[!redirects grand unified field theories]]

[[!redirects Grand Unified Theory]]

[[!redirects grand unification]]
[[!redirects grand unifications]]

[[!redirects Pati-Salam model]]
[[!redirects Pati-Salam models]]

[[!redirects Pati-Salam GUT]]
[[!redirects Pati-Salam GUT model]]
[[!redirects Pati-Salam GUT models]]

[[!redirects GUT scale]]

[[!redirects GUT model]]
[[!redirects GUT models]]


Guy de Téramond

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

## Selected writings

On the [[running coupling constant]] in [[QCD]]:

* [[Alexandre Deur]], [[Stanley Brodsky]], [[Guy de Teramond]], *The QCD Running Coupling*, Prog. Part. Nuc. Phys. 90 1 (2016) ([arXiv:1604.08082](https://arxiv.org/abs/1604.08082), [doi:10.1016/j.ppnp.2016.04.003](https://doi.org/10.1016/j.ppnp.2016.04.003))

Introducing [[holographic light front QCD]]:

* [[Guy de Teramond]], [[Stanley Brodsky]], _Light-Front Holography: A First Approximation to QCD_, Phys. Rev. Lett. 102:081601, 2009 ([arXiv:0809.4899](https://arxiv.org/abs/0809.4899))

* [[Stanley Brodsky]], [[Guy de Teramond]], _Light-Front Hadron Dynamics and AdS/CFT Correspondence_, Phys. Lett. B582:211-221, 2004 ([arXiv:hep-th/0310227](https://arxiv.org/abs/hep-th/0310227))

Review:

* [[Stanley Brodsky]], [[Guy de Teramond]],  [[Hans Günter Dosch]], [[Joshua Erlich]],
_Light-Front Holographic QCD and Emerging Confinement_, Physics Reports Volume 584, 8 July 2015, Pages 1-105 ([arXiv:1407.8131](https://arxiv.org/abs/1407.8131))

* [[Stanley Brodsky]], [[Guy de Teramond]], [[Hans Günter Dosch]], _Light-Front Holography and Supersymmetric Conformal Algebra: A Novel Approach to Hadron Spectroscopy, Structure, and Dynamics_ ([arXiv:2004.07756](https://arxiv.org/abs/2004.07756))

* [[Guy de Téramond]], *Emergent phenomena in QCD: The holographic perspective* [[arXiv:2212.14028](https://arxiv.org/abs/2212.14028)]

On [[hadron supersymmetry]] via [[light cone gauge|light cone]] [[supersymmetric quantum mechanics]] in [[holographic light front QCD]];

* {#dTDB14}  [[Guy de Teramond]], [[Hans Günter Dosch]], [[Stanley Brodsky]], _Baryon Spectrum from Superconformal Quantum Mechanics and its Light-Front Holographic Embedding_, Phys. Rev. D 91, 045040 (2015) ([arXiv:1411.5243](https://arxiv.org/abs/1411.5243))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Superconformal Baryon-Meson Symmetry and Light-Front Holographic QCD_, Phys. Rev. D 91, 085016 (2015) ([arXiv:1501.00959](https://arxiv.org/abs/1501.00959))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum_, Phys. Rev. D 92, 074010 (2015) ([arXiv:1504.05112](https://arxiv.org/abs/1504.05112))

* {#BTDL16}  [[Stanley Brodsky]], [[Guy de Téramond]], [[Hans Günter Dosch]], Cédric Lorcé, _Meson/Baryon/Tetraquark Supersymmetry from Superconformal Algebra and Light-Front Holography_, International Journal of Modern Physics AVol. 31, No. 19, 1630029 (2016) ([arXiv:1606.04638](https://arxiv.org/abs/1606.04638))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum II_, Phys. Rev. D 95, 034016 (2017) ([arXiv:1612.02370](https://arxiv.org/abs/1612.02370))

* {#NielsenBrodsky18} [[Marina Nielsen]], [[Stanley Brodsky]], _Hadronic Superpartners from Superconformal and Supersymmetric Algebra_, Phys. Rev. D 97, 114001 (2018) ([arXiv:1802.09652](https://arxiv.org/abs/1802.09652))

* [[Marina Nielsen]], [[Stanley Brodsky]], [[Guy de Téramond]], [[Hans Günter Dosch]], Fernando S. Navarra, [[Liping Zou]], _Supersymmetry in the Double-Heavy Hadronic Spectrum_, Phys. Rev. D 98, 034002 (2018) ([arXiv:1805.11567](https://arxiv.org/abs/1805.11567))

Review:

* [[Guy de Téramond]], _Holographic QCD in light-front quantization and superconformal algebra: An overview_, SCGP, Stonybrook 2020 [pdf](http://media.scgp.stonybrook.edu/presentations/2020/20200820_de%20Teramond.pdf) [video](https://scgp.stonybrook.edu/video_portal/video.php?id=4554)

In view of [[tetraquarks]]:

* [[Hans Günter Dosch]], [[Stanley Brodsky]], [[Guy de Teramond]], [[Marina Nielsen]], [[Liping Zou]], _Exotic States in a Holographic Theory_ ([arXiv:2012.02496](https://arxiv.org/abs/2012.02496))

category: people

[[!redirects Guy de Teramond]]

[[!redirects Guy F. de Téramond]]
[[!redirects Guy F. de Teramond]]


GW/DT correspondence

#Contents#
* table of contents
{:toc}

## Idea

A correspondence between [[Gromov-Witten invariants]] and [[Donaldson-Thomas invariants]].

## Related concepts

* [[Gromov-Witten theory]]

* [[Donaldson-Thomas theory]]

* [[Pandharipande-Thomas theory]]

## References

Original articles:

* [[Davesh Maulik]], [[Nikita Nekrasov]], [[Andrei Okounkov]], [[Rahul Pandharipande]], _Gromov–Witten theory and Donaldson–Thomas theory, I_. Compositio Mathematica. 142 (5): 1263–1285. (2003) ([arXiv:math/0312059](https://arxiv.org/abs/math/0312059). [doi:10.1112/S0010437X06002302](http://www.journals.cambridge.org/abstract_S0010437X06002302))

* [[Davesh Maulik]], [[Alexei Oblomkov]], [[Andrei Okounkov]], [[Rahul Pandharipande]], _Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds_ ([arXiv:0809.3976](https://arxiv.org/abs/0809.3976v1))

* [[Dustin Ross]], _On GW/DT and Ruan's Conjecture in All Genus for Calabi-Yau 3-Orbifolds_, [arXiv:1409.7015](https://arxiv.org/abs/1409.7015v3)

Exposition:

* Henry Liu, Melissa Chiu-Chu Liu, _Enumerative Geometry Seminar_, Columbia University, Fall 2018, [webpage](http://math.columbia.edu/~hliu/seminars/f18-enumerative-geometry.html), [notes](http://math.columbia.edu/~hliu/classes/f18-sem-gw-dt.pdf)

[[!redirects Gromov-Witten/Donaldson-Thomas correspondence]]


Gysin map

## Related concepts

[[!include generalized fiber integration synonyms - table]]

[[!redirects Gysin maps]]
[[!redirects Gysin morphism]]
[[!redirects Gysin morphisms]]
[[!redirects Gysin homomorphism]]
[[!redirects Gysin homomorphisms]]


Gábor Pete

* [webpage](http://www.math.bme.hu/~gabor/)

## Related entries

* [[Morse theory]]

category: people

[[!redirects Gabor Pete]]


Gérard Gonzalez-Sprinberg

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=57482)

## related $n$Lab entries

* [[McKay correspondence]]

* [[fractional D-brane]]

category: reference

[[!redirects Gerard Gonzalez-Sprinberg]]


Gérard Huet

* [home page](http://sanskrit.inria.fr/huet/)

* [Wikipedia entry](http://en.wikipedia.org/wiki/G%C3%A9rard_Huet)

## Related entries

* [[calculus of constructions]]

category: people

[[!redirects Gerard Huet]]


Gérard Laumon

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

## Selected writings

On [[algebraic stacks]]:

* {#LaumontMoret-Bailly} [[Gérard Laumon]], [[Laurent Moret-Bailly]], _Champs algébriques_, Ergebn. der Mathematik und ihrer Grenzgebiete **39**, Springer (2000) [[doi:10.1007/978-3-540-24899-6](https://doi.org/10.1007/978-3-540-24899-6)]

## Related entries

* [[number theory]]

* [[geometric Langlands correspondence]]

* [[Artin stack]]

category: people


Gödel logic

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related entries

* [[fuzzy logic]]

* [[linear logic]]

* [[hypersequent calculus]]

## References

See also:

* Wikipedia, *Gödel logic

[[!redirects Gödel logics]]

[[!redirects Goedel logic]]
[[!redirects Goedel logics]]

[[!redirects Godel logic]]
[[!redirects Godel logics]]


Gödel numbering

## Related concepts

* [[incompleteness theorem]]

## References

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

[[!redirects Gödel numbering]]
[[!redirects Gödel numberings]]
[[!redirects Goedel numbering]]
[[!redirects Goedel numberings]]
[[!redirects Godel numbering]]
[[!redirects Godel numberings]]

[[!redirects Gödel coding]]
[[!redirects Gödel codings]]
[[!redirects Goedel coding]]
[[!redirects Goedel codings]]
[[!redirects Godel coding]]
[[!redirects Godel codings]]


Gödel-numbering

See *[[Gödel numbering]]*


Göran Lindblad

* [Wikipedia entry](https://en.wikipedia.org/wiki/G%C3%B6ran_Lindblad_(physicist))

* [insitute page](https://www.physics.kth.se/condensed/goran-lindblad-1.1211657)

## Selected writings

Early discussion of the [environmental representation](quantum+channel#QuantumChannelsAndDecoherence) of [[completely positive maps]] ([[quantum channels]]):

* {#Lindblad75} [[Göran Lindblad]], *Completely positive maps and entropy inequalities*,  Commun. Math. Phys. **40** (1975) 147–151 [[doi:10.1007/BF01609396](https://doi.org/10.1007/BF01609396)]

category: people

[[!redirects Goeran Lindblad]]
[[!redirects Goran Lindblad]]


Göran Sundholm

A philosopher based in Leiden in The Netherlands. Research interests: Philosophy of Logic, Mathematics, and Language: exploring the possibilities of Mathematical Constructivism (especially in the form of Martin-Löf's Constructive Type Theory) outside mathematics. Modern History of Logic: one focus is on Bolzano, Frege, and Wittgenstein's Tractatus; another field of interest is the Metalogical Turn - Logic as Language versus Logic as Calculus.

* [website](https://www.universiteitleiden.nl/en/staffmembers/goran-sundholm)

## Related pages

* [[Per Martin-Löf]]

* [[Martin-Löf dependent type theory]]

* [[dependent type theoretic methods in natural language semantics]]

[[!redirects Goeran Sundholm]]
[[!redirects Göran Sundholm]]


Götz Pfeiffer

* [webpage](http://schmidt.ucg.ie/~goetz/)

## related $n$Lab entries

* [[table of marks]]

category: people

[[!redirects Goetz Pfeiffer]]


Günter Hotz

Günter Hotz (born 16.11.1931) is a german mathematician and computer scientist and a professor emeritus of the University of the Saarland, Saarbrücken, Germany.

* [Wikipedia (english)](https://en.wikipedia.org/wiki/G%C3%BCnter_Hotz)
* [Wikipedia (german)](https://de.wikipedia.org/wiki/G%C3%BCnter_Hotz)
* [Personal homepage (outdated)](http://www-hotz.cs.uni-sb.de/home/hotz/welcome_english.html)
* [ResearchGate](https://www.researchgate.net/profile/Guenter_Hotz)
* [Mathematics Genealogy Project](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=21668)

Hotz is one of the "pioneers" of computer science in Germany and has been strongly involved in establishing computer science as a scientific discipline in Germany in the early 70s.

His scientific contributions include automata theory, formal languages, VLSI design, information theory, topology and more.

Hotz has a mathematical background in algebraic topology. He wrote his dissertation thesis in 1958 in Göttingen under supervision of [Kurt Reidemeister](https://en.wikipedia.org/wiki/Kurt_Reidemeister) about a topic from Knot theory. In 2008, he published an efficient algorithm for [deciding the equivalence of Knots](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.578.691).

In his 1965 habilitation thesis "Eine Algebraisierung des Syntheseproblems von Schaltkreisen" (EIK, Bd. 1, (185-205), Bd, 2, (209-231), English title would be "An algebraisation of the synthesis problem of switching circuits) he used the theory of monoidal categories for describing the syntax and semantics of switching circuits.

* [Part I  (in German)](https://www.magentacloud.de/lnk/LiPMlYfh)

* [Part II (in German)](https://www.magentacloud.de/lnk/YivslUWJ)

* compressed and merged: [[HotzSchaltkreise.pdf:file]]

In this thesis, he introduced [string diagrams](https://ncatlab.org/nlab/show/string+diagram) (under the name "plane nets") as a topological representation of the morphisms in free monoidal categories. This probably is the first formal definition of string diagrams in the literature.

In his 1966 publication "Eindeutigkeit und Mehrdeutigkeit formaler Sprachen", EIK, Bd. 2 (235-246), (in German, English title would be "Unambiguity and ambiguity of formal languages"), he uses the theory of monoidal categories and string diagrams to give an exact mathematical definition for the derivations in Semi-Thue systems and Chomsky grammars.

In the early 80s, monoidal categories have been generalized to bicategories of [logic-topological nets](http://www.academia.edu/5888548/Hierarchical_design_based_on_a_calculus_of_nets) and used as the mathematical foundation of the VLSI design system "CADIC".


Günter Scharf

* [webpage](http://phys-merger.physik.unibas.ch/~aste/scharf.html)

## Selected writings

Scharf pioneered the application of [[causal perturbation theory]] to the rigorous construction of [[perturbative quantum field theories]]:

{#Textbooks} In his book

* _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Springer 1995

he discusses [[quantum electrodynamics]] this way, and in

* _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

he discusses [[Yang-Mills theory]], hence [[electroweak theory]] and [[quantum chromodynamics]] as well as [[perturbative quantum gravity]].

On [[vacua]]/[[ground states]] of the [[free field|free]] [[Dirac field]] in a [[electromagnetic field|Coulomb]] [[background field]]:

* {#KlausScharf77} [[Martin Klaus]], [[Günter Scharf]], *The regular external field problem in quantum electrodynamics*, Helv. Phys. Acta **50** (1977) $[$[doi:10.5169/seals-114890](http://doi.org/10.5169/seals-114890)$]$

## related $n$Lab entries

* [[causal perturbation theory]]

* [[locally covariant perturbative quantum field theory]]

category: people

[[!redirects Gunter Scharf]]
[[!redirects Guenter Scharf]]


Günter Tamme

* [Wikipedia entry](http://de.wikipedia.org/wiki/Günter_Tamme)

## Books

* _[[Introduction to Étale Cohomology]]_

category: people


Günther Trautmann

* [webpage](https://www.mathematik.uni-kl.de/trm/seite)

## Selected writings

On [[intersection theory]]:

* [[Günther Trautmann]], *Introduction to intersection theory* (2007) $[$[[IntersTh.pdf:file]]$]$

category: people


H-category

[[!redirects H-categories]]
[[!redirects A3-spaceoid]]
[[!redirects A3-spaceoids]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

## Contents ##
* table of contents
{:toc}

## Idea ##

The [[oidification]] of an [[A3-space]]/[[H-monoid]], or equivalently, a [[category]] internal to $Ho(Top)_*$.

## Definition ##

### Classically ###

A **H-category** or **A3-spaceoid** is a [[category]] [[internalization|internal to]] the [[classical homotopy category]] of [[topological spaces]] [[Ho(Top)]], or in the homotopy category $Ho(Top)_*$ of [[pointed topological spaces]].

### Internally in an (infinity,1)-category ###

For the time being see [[category object in an (infinity,1)-category#HCategoryTypes]].

### In homotopy type theory ###

In [[homotopy type theory]], a **H-category** or **A3-spaceoid** is a [[type]] $A$ with

* For each $a,b:A$, a type $hom_A(a,b)$, whose elements are called **arrows** or **morphisms**.

* For each $a:A$, a morphism $1_a:hom_A(a,a)$, called the **identity morphism**.

* For each $a,b,c:A$, a function
$$hom_A(b,c) \to hom_A(a,b) \to hom_A(a,c)$$
called composition, and denoted infix by $g \mapsto f \mapsto g \circ f$, or sometimes $gf$.

* For each $a,b:A$ and $f:hom_A(a,b)$, we have the **left unitor** $\lambda_f:f=1_b \circ f$ and the **right unitor** $\rho_f:f=f\circ 1_a$.

* For each $a,b,c,d:A$,
$$f:hom_A(a,b),\ g:hom_A(b,c),\ h:hom_A(c,d)$$
we have the **associator** $\alpha_{f, g, h}:h\circ (g\circ f)=(h\circ g)\circ f$.

## See also ##

* [[category]]
* [[H-spaceoid]]
* [[A3-space]]
* [[homotopy category]]


h-cobordism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _h-cobordism_ is a [[cobordism]] such that the two boundary inclusions$\Sigma_1, \Sigma_2 \hookrightarrow \Sigma$ are [[homotopy equivalences]].

## Properties

### h-Cobordism theorem

The _[[h-cobordism theorem]]_ gives sufficient conditions for an h-cobordism to be equivalent to a [[cylinder]].

## References

See also

* Wikiedia, _[h-cobordism](https://en.wikipedia.org/wiki/H-cobordism)_

[[!redirects h-cobordisms]]


h-cobordism theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _h-cobordism theorem_ gives sufficient conditions for an [[h-cobordism]] to be equivalent to a [[cylinder]].

## Related entries

* [[exotic smooth structure]]

## References

Due to

* {#Smale} [[Stephen Smale]],  (1962), _On the structure of manifolds_, Amer. J. of Math. 84 : 387-399

Review includes

* {#Milnor1965a} [[John Milnor]] (1965), _Lectures on the h-cobordism theorem_ (Princeton Univ. Press, Princeton)

See also

* Wikipedia, _[h-cobordism](https://en.wikipedia.org/wiki/H-cobordism)_

Generalization to [[equivariant cobordism]]:

* [[Shôrô Araki]], *Equivariant $s$-cobordism theorems*, J. Math. Soc.  Japan **40** 2 (1988) [[pdf](file:///C:/Users/us13/Downloads/04020349-2.pdf)]

[[!redirects h cobordism theorem]]


h-cofibration

The term “h-cofibration” can refer to two closely related, but different notions:

* [[Hurewicz cofibrations]];

* the dual notion to [[sharp maps]].

In this article, we concentrate on the latter.

## Definition

A map $f\colon X\to Y$ in a [[relative category]] $C$ is an __h-cofibration__
if the [[cobase change]] functor $X/C\to Y/C$ is a [[relative functor]], i.e., preserves weak equivalences.

## Terminology

Another name for such morphisms is “flat map”, used by Hill–Hopkins–Ravenel (Appendix B.2).
This choice of terminology conflicts with flat maps used to define flat [[monoidal model categories]].

## Properties

A [[model category]] is [[left proper]] if and only if all cofibrations are h-cofibrations.

In a [[left proper]] [[model category]], [[cobase changes]] along h-cofibrations are [[homotopy cobase changes]].

The notion of h-cofibrations is most useful in the left proper case,
and one can argue that in the non-left proper case, the above property
should be taken as the definition instead.

## Related concepts

* The dual notion is known as a “[[sharp map]]” or “h-fibration”.

## References

* [[Michael Batanin]], [[Clemens Berger]], _Homotopy theory for algebras over polynomial monads_, Theory and Applications of Categories 32:6 (2017), 148–253.  [arXiv](http://arxiv.org/abs/1305.0086).

[[!redirects h-cofibrations]]


H-cohomology

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #HCohomology}
###### Definition
**(H-cohomology)**

Given a [[smooth manifold]] or [[formal smooth manifold]] $X$, and given a [[differential 3-form]] $H \in \Omega^3(X)$, then forming the [[wedge product]] with $H$ is a nilpotent operation, and hence defines the [[differential]] of a [[cochain complex]] whose entries are the [[vector spaces]] $\Omega^{even/odd}(X)$ of all odd-graded or all even-graded [[differential forms]] on $X$:

$$
  \cdots
    \to
  \Omega^{even}(X)
    \overset{H\wedge(-)}{\longrightarrow}
  \Omega^{odd}(X)
    \overset{H\wedge(-)}{\longrightarrow}
  \Omega^{even}(X)
    \overset{H\wedge(-)}{\longrightarrow}
  \Omega^{odd}(X)
    \to
   \cdots
  \,.
$$

The [[cochain cohomology]] of this [[cochain complex]]

$$
  H^{even/odd}_{H\wedge}(X)
  \coloneqq;
  \frac{
    ker
    \left(
      \Omega^{even/odd}(X)
       \overset{H\wedge(-)}{\longrightarrow}
      \Omega^{odd/even}(X)
     \right)
   }{
    im
    \left(
      \Omega^{odd/even}(X)
       \overset{H\wedge(-)}{\longrightarrow}
      \Omega^{even/odd}(X)
     \right)
   }
$$

is also called the _H-cohomology_ of $X$.

=--

([Cavalcanti 03, p. 19](#Cavalcanti03))

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

Often it is assumed that the 3-form $H$ in def. \ref{HCohomology} is [[closed differential form|closed]], because in this case H-cohomology serves as an approximation to the corresponding $H$-[[twisted de Rham cohomology]] (prop. \ref{TwistedDeRhamCohomologyCoincidesWithHCohomologyOnInfinitesimllayThickenedPoint} below). But of course for def. \ref{HCohomology} to make sense in itself, $H$ need not be closed.

=--

It is immediate to consider this more generally:

+-- {: .num_defn #HCohomologyInGradedCommutativeAlgebra}
###### Definition
**(H-cohomology in [[graded-commutative algebra]])**

Let $\mathcal{A}$ be a [[graded commutative algebra]] in [[characteristic zero]] and let $H \in \mathcal{A}$ be an odd-graded element. This implies that consecutive multiplication with $H$

$$
  \cdots
    \to
  \mathcal{A}
    \overset{H \cdot(-)}{\longrightarrow}
  \mathcal{A}
    \overset{H \cdot (-)}{\longrightarrow}
  \mathcal{A}
    \overset{H \cdot (-)}{\longrightarrow}
  \mathcal{A}
    \to
   \cdots
$$

is a [[chain complex]].

The [[cochain cohomology]] of this [[cochain complex]]

$$
  H^{even/odd}_{H\cdot}(\mathcal{A})
  \coloneqq;
  \frac{
    ker
    \left(
      \mathcal{A}_{even/odd}
       \overset{H \cdot(-)}{\longrightarrow}
      \mathcal{A}_{odd/even}
     \right)
   }{
    im
    \left(
      \mathcal{A}_{odd/even}
       \overset{H\cdot(-)}{\longrightarrow}
      \mathcal{A}_{even/odd}
     \right)
   }
$$

is the _H-cohomology_ of $\mathcal{A}$.

=--

## Properties

### Detection of decomposability

+-- {: .num_prop #HCohomologyNeverVanishes}
###### Proposition
**(H-cohomology of weakly [[decomposable forms]] is non-vanishing)**

Let $\mathcal{A}$ be a $\mathbb{N}$-[[graded-commutative algebra]].
and let $H \in \mathcal{A}$ be of odd degree $deg(H) = 2 n +1$

If $H$ is weakly [[decomposable differential form|decomposable]] in that it is in the ideal generated by the elements of degree 1

$$
  H
    \;=\;
  \underoverset{i = 1}{k}{\sum}
  \underset{deg = 1}{\underbrace{\alpha_i}}
    \wedge
  \underset{deg = 2n}{\underbrace{\beta_i}}
$$

then its H-cohomology (def. \ref{HCohomologyInGradedCommutativeAlgebra}) does not vanish.

=--

This statement and the following proof are due to [Sackel 18](#Sackel18).

+-- {: .proof}
###### Proof

First consider the following:

_Claim:_ Assuming that the H-cohomology of $H$ did vanish, then for every sequence of [[natural numbers]] $1 \leq i_1 \lt \ldots \lt i_{\ell} \leq k$, there would exist elements $\gamma_{i_1\cdots i_{\ell}} \in \mathcal{A}$ of degree $deg(\gamma_{i_1\cdots i_{\ell}}) = k-3-\ell$  (vanishing if $\ell \gt k-3$) such that the products of $\alpha$-s with indices _not_ contained in this sequence

$$
  \label{ProductsOfAlphas}
  \alpha_{\widehat{i_1\cdots i_{\ell}}}
  \;\coloneqq\;
  \left\{
    \array{
      1 &\vert& \ell = k
      \\
      \alpha_1
        \wedge
        \cdots
        \wedge
      \widehat{\alpha_{i_1}}
        \wedge
        \cdots
        \wedge
      \widehat{\alpha_{i_{\ell}}}
        \wedge
        \cdots
        \wedge
      \alpha_k
      &\vert&
      1 \lt \ell \lt k
      \\
      \alpha_1 \wedge \cdots \wedge \alpha_k
      &\vert&
      \ell =0
    }
  \right.
$$

(hats indicate that the corresponding element is omitted) would satisfy

$$
  \label{InductiveDecompositionOfAlpha}
  \alpha_{\widehat{i_1\cdots i_{\ell}}}
  \;=\;
   (-1)^{i_1+\cdots+i_{\ell}+\ell}
   \left(
      H
        \wedge
     \gamma_{i_1\cdots i_{\ell}}
        +
     \sum_{j=1}^{\ell}(-1)^{j+1}
       \beta_{i_j}
       \wedge
      \gamma_{i_1\cdots\widehat{i_j}\cdots i_{\ell}}
   \right)
   \,.
$$

Observe that this claim immediately implies the statement [[proof by contradiction|by contradiction]]: For the case $\ell = k$ the left hand side of (eq:InductiveDecompositionOfAlpha) is $1$ by definition (eq:ProductsOfAlphas), while its right-hand side would vanish, because the corresponding elements $\gamma$ are in negative degree. This is a contradiction, implying that the assumption  of vanishing H-cohomology must be wrong.

Hence it is now sufficient to prove the above claim. We proceed by [[induction]] on $\ell$:

For $\ell = 0$ we have

$$
  H
    \wedge
  (\alpha_1 \wedge \cdots \wedge \alpha_k) = 0
  \,,
$$

and so by the assumption of vanishing $H$-cohomology, there exists $\gamma_{\emptyset}$ with

$$
  \alpha_{\hat \emptyset}
  \coloneqq
  \alpha_1 \wedge \cdots \wedge \alpha_k
  =
  H \wedge \gamma_{\emptyset}
  \,.
$$

This is (eq:InductiveDecompositionOfAlpha) for $\ell = 0$.

So assume now that there is $1 \leq \ell \leq k$ such that (eq:InductiveDecompositionOfAlpha) holds for all $\ell \lt \ell_0$. Then

$$
  \begin{aligned}
   H
   \wedge
  \alpha_{\widehat{i_1\cdots i_{\ell_0}}}
   &=  \sum_{i=1}^{k} \beta_i \wedge \alpha_i \wedge \alpha_{\widehat{i_1\cdots i_{\ell_0}}}
   \\
   &= \sum_{j=1}^{\ell_0} (-1)^{i_j+j}\beta_{i_j} \wedge \alpha_{\widehat{i_1\cdots \widehat{i_j}\cdots i_{\ell_0}}}
   \\
   &= \sum_{j=1}^{\ell_0} (-1)^{i_1 + \cdots + i_{\ell_0}+\ell_0+j+1}\beta_{i_j} \wedge
   \left(H \wedge \gamma_{i_1\cdots \widehat{i_j}\cdots i_{\ell_0}}+\sum_{k=1}^{j-1}(-1)^{k+1}\beta_{i_k}\wedge \gamma_{i_1\cdots \widehat{i_k} \cdots \widehat{i_j} \cdots i_{\ell_0}} + \sum_{k=j+1}^{\ell_0}(-1)^k\beta_{i_k}\wedge \gamma_{i_1\cdots \widehat{i_j} \cdots \widehat{i_k} \cdots i_{\ell_0}}\right)
  \\
  &=
  (-1)^{i_1 + \cdots + i_{\ell_0}+\ell_0}\sum_{j=1}^{\ell_0} H \wedge (-1)^{j+1}\beta_{i_j} \wedge \gamma_{i_1\cdots \widehat{i_j}\cdots i_{\ell_0}}
  \,,
  \end{aligned}
$$

where in the third line we used the induction assumption.

By vanishing of $H$-cohomology, this implies (eq:InductiveDecompositionOfAlpha) for $\ell = \ell_0$, thus finishing the proof of the claim, by [[induction]].

=--

## Examples

### Odd symplectic forms

+-- {: .num_example #HCohomoloyOfOddSymplecticForm}
###### Example
**(H-cohomology of odd symplectic form)**

Let $X$ be a [[finite number|finite]] [[dimension|dimensional]] [[graded manifold]] in degrees 0 and 1 and equipped with a [[symplectic form]] $\omega$ of odd coordinate degree (e.g. the symplectic form on a [[Poisson Lie algebroid]] when regarded as a [[symplectic Lie n-algebroid]]).

Then every H-cohomology-class (def. \ref{HCohomologyInGradedCommutativeAlgebra}) for $H = \omega$ has a unique representative which on any [[local coordinate chart]] of [[Darboux coordinates]] $(x^i, p_i)_{i \in \{1, \cdots, n\}}$ is of the form

$$
  f(x,p) d x^1 \wedge \cdots \wedge d x^n
  \,.
$$

=--

([Ševera 05, p. 1](#Severa05))

## References

The terminology "$H$-cohomology" is used in

* {#Cavalcanti03} [[Gil Cavalcanti]], _New aspects of the $d d^c$-lemma_ ([arXiv:math/0501406](https://arxiv.org/abs/math/0501406))

The case of H-cohomology for $H = \omega$ an odd symplectic form (as on a [[Poisson Lie algebroid]] regarded as a  [[symplectic Lie n-algebroid]]) is considered in

* {#Severa05} [[Pavol Ševera]], p. 1 of _On the origin of the BV operator on odd symplectic supermanifolds_, Lett Math Phys (2006) 78: 55. ([arXiv:0506331](https://arxiv.org/abs/math/0506331))

The general non-vanishing of H-cohomology is pointed out in

* {#Sackel18} [[Kevin Sackel]], [MO reply Feb 22 2018](https://mathoverflow.net/a/293565/381)

[[!redirects H-cohomology group]]
[[!redirects H-cohomology groups]]


H-groupoid

[[!redirects H-spatial groupoid]]
+-- {: .rightHandSide}
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
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#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Binary operations
* table of contents
{: toc}

## Idea

In [[homotopy type theory]], a [[1-type|1-truncated]] [[H-space]]/[[An-space|$A_2$-space]]. In [[category theory]], the [[groupoidal categorification]] of a [[unital magma]].

## Definitions

### In type theory

In [[intensional type theory]] with [[function extensionality]], a [[groupoid]] is a [[n-truncated|1-truncated]] [[type]] and a **H-spatial groupoid**, **$A_2$-spatial groupoid**, **unital magmoidal groupoid**, or **magmoidal groupoid with unit** is a [[groupoid]] that is also an [[H-space]]/[[An-space|$A_2$-space]].

### In category theory

In [[category theory]], a **unital magmoidal groupoid** or **magmoidal groupoid with unit** is a [[magmoidal groupoid]] $(G, \otimes)$ with a unit $I$ and natural unitary isomorphisms $\lambda_A \colon A\otimes I \simeq^\dagger A$ and $\rho_A \colon I\otimes A \simeq^\dagger A$.

## See also

* [[unital magma]]

* [[H-space]]

* [[An-space]]

* [[k-tuply associative n-groupoid]]

* [[groupoid]]

* [[magmoidal groupoid]]

* [[H-monoidal groupoid]]

[[!redirects H-groupoids]]

[[!redirects A2-groupoid]]
[[!redirects A2-groupoids]]

[[!redirects unital magmoidal groupoid]]
[[!redirects unital magmoidal groupoids]]

[[!redirects magmoidal groupoid with unit]]
[[!redirects magmoidals groupoid with unit]]


h-groupoid

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

#Contents#
* table of contents
{:toc}

## Definition

In [[homotopy type theory]]: a [[homotopy n-type|homotopy 1-type]], a [[type]] of [[h-level]] 3.

The [[categorical semantics]] of which is a [[1-truncated]] object in an [[(infinity,1)-category]].

## Related concepts

[[!include homotopy n-types - table]]

[[!include types and logic - table]]

[[!redirects h-groupoids]]


H-infinity ring spectrum

#Contents#
* table of contents
{:toc}

## Idea

An _$H_\infty$-spectrum_ ([Bruner-May-McLure.Steinberger 86](#BrunerMayMcLureSteinberger86)) is a [[ring spectrum]] (i.e. a [[monoid object]] in the [[stable homotopy category]]) equipped with extra structure modelling [[extended power]] operations.

## References

* {#BrunerMayMcLureSteinberger86} [[Robert Bruner]], [[Peter May]], [[James McClure]], [[Mark Steinberger]], *$H_\infty$-Ring Spectra and their Applications*, Lecture Notes in Mathematics **1176**, Springer (1986) [[doi:10.1007/BFb0075405](https://link.springer.com/book/10.1007/BFb0075405), [pdf](http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf)]

* [[Justin Noel]], _H-infinity is not E-infinity_, Proceedings of the 4th Arolla Conference ([arXiv:0910.3566](https://arxiv.org/abs/0910.3566))

[[!redirects H-infinity ring spectra]]

[[!redirects H-∞ ring spectrum]]
[[!redirects H-∞ ring spectra]]


h-initial object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

Let $\mathcal{C}$ be an [[(∞,1)-category]]. An [[object]] $x$ in $\mathcal{C}$ is said to be **$h$-initial** if it defines an [[initial object]] in the [[homotopy category of an (infinity,1)-category|homotopy category]] $Ho(\mathcal{C})$.

Dually, an object is called **$h$-terminal** if it defines a [[terminal object]] in $Ho(\mathcal{C})$.

To be distinguished from the concept of _[[initial object in an (∞,1)-category]]_.

[[!redirects h-initial objects]]


H-magmoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

## Contents ##
* table of contents
{:toc}

## Idea ##

The [[oidification]] of an "H-magma", which is to [[magmas]] what [[H-spaces]] are to [[unital magmas]].

## Definition ##

### Classically ###

A **H-magmoid** is a [[magmoid]] [[internalization|internal to]] the [[classical homotopy category]] of [[topological spaces]] [[Ho(Top)]], or in the homotopy category $Ho(Top)_*$ of [[pointed topological spaces]],  which has a [[unit]] up to [[homotopy]].

### In homotopy type theory ###

In [[homotopy type theory]], an H-magmoid $A$ consists of the following.

* A type $A_0$, whose elements are called objects. Typically $A$ is coerced to $A_0$ in order to write $x:A$ for $x:A_0$.

* For each $a,b:A$, a type $hom_A(a,b)$, whose elements are called **arrows** or **morphisms**.

* For each $a,b,c:A$, a function
$$hom_A(b,c) \to hom_A(a,b) \to hom_A(a,c)$$
called composition, and denoted infix by $g \mapsto f \mapsto g \circ f$, or sometimes $gf$.

## See also ##

* [[magmoid]]
* [[H-spaceoid]]
* [[H-category]] (category object internal to a homotopy category)
* [[homotopy category]]

[[!redirects H-magmoids]]


H-monoidal groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[homotopy type theory]], a [[1-type|1-truncated]] [[H-monoid]]/[[An-space|$A_3$-space]]. In [[category theory]], a [[magmoidal groupoid with unit]] with an [[associator]] but no higher [[coherence laws]].

## Definitions

### In type theory

In [[intensional type theory]] with [[function extensionality]], a [[groupoid]] is a [[n-truncated|1-truncated]] [[type]] and a **H-monoidal groupoid** or **$A_3$-spatial groupoid** is a [[groupoid]] that is also an [[H-monoid]]/[[An-space|$A_3$-space]].

### In category theory
 {#InCategoryTheory}

In [[category theory]], a **H-monoidal groupoid** or **$A_3$-spatial groupoid** is a [[magmoidal groupoid with unit]] $(G, \otimes, I, \lambda_{(-)}, \rho_{(-)})$ with [[unital]] [[associator]] [[natural isomorphisms]] $\alpha_{A, B, C} \colon A \otimes (B \otimes C) \simeq^\dagger (A \otimes B) \otimes C$.

## See also

* [[H-space]]

* [[An-space]]

* [[groupoid]]

* [[k-tuply associative n-groupoid]]

* [[H-groupoid]]

[[!redirects H-monoidal groupoid]]
[[!redirects H-monoidal groupoids]]
[[!redirects A3-spatial groupoid]]
[[!redirects A3-spatial groupoids]]


h-principle

#Gromov's homotopy principle or h-principle
* table of contents
{:toc}

## Definition

Let $p \colon X\to B$ be a smooth [[fiber bundle]] and write $X^{(r)}$ for the $r$-[[jets]]  of [[sections]] of $p$; there is an induced [[projection]] $X^{(r)} \to B$ which makes it into a smooth bundle.

One considers a fixed partial differential relation (see [[differential equation]]) which is a subset $R\subset X^{(r)}$. One is interested in finding [[formal geometry|formal]] sections $s$ of $X^{(r)}\to B$ whose image is in $R$, and which are, in addition, __holonomic sections__, i.e. equal to the $r$-jet $J_f^r$ of a $C^r$-section $f$ of $p:X\to B$.

One says that $R$ satisfies the __h-principle__ if every formal section of $R$ is [[homotopy|homotopic]] to a holonomic section within the appropriate topological space of formal sections.

##  Applications and examples

For example, in the case of holomorphic fiber bundles $R$ may be the Cauchy-Riemann relation, thus the question is if there is a deformation of a continuous section into a holomorphic one. In other words, one wants to reduce the problem of existence of maps of certain type to the analogous topological problem. Applications include results on the spaces of immersions, submersions, k-mersions, holomorphic maps, symplectic and isometric embeddings, contact structures and so on.

[[Mikhail Gromov]] demonstrated the result that if $B$ is an open $n$-manifold, and if $R$ is open as a subspace of $X^{(r)}$ and is $Diff(B)$-invariant, then $R$ satisfies the $h$-principle ([EliasMisha 01, sec 2.1](#EliasMisha01), [Haefliger 71, p. 128](#Haefliger71))

##  General methods

Gromov introduced many techniques of proving the h-principle including the method of microflexible/continuous sheaves, the methods of convex integration and the removal of singularities.

## Related concepts

* [[microflexible sheaf]]

* [[c-principle]]

* [[Oka principle]]

* [[formal immersion of smooth manifolds]]

## References

Review:

* Y. Eliashberg, N. Mishachev, _Introduction to the h-principle_, Graduate Studies in Mathematics 48. Amer. Math. Soc. 2002 [gBooks](http://books.google.co.uk/books?isbn=0821832271)

* Andrés Angel, [[Johannes Ebert]], _Gromov's h-principle and its applications research seminar_, 2010, outline [pdf](http://wwwmath.uni-muenster.de/u/jeber_02/frueherelehre/hprinciple.pdf) (with useful bibliography)

* [[eom]]: [H-principle](http://www.encyclopediaofmath.org/index.php/H-principle), [convex integration](http://www.encyclopediaofmath.org/index.php/Convex_integration);

Wikipedia: [homotopy principle](https://en.wikipedia.org/wiki/Homotopy_principle)

* MathOverflow: [h-principle-and-pdes](http://mathoverflow.net/questions/88513/h-principle-and-pdes)

Original articles:

* [[Mikhail Gromov]], _Partial differential relations_, Ergebn. Math. Grenzgeb. (3), 9, Springer (1986) ([doi:10.1007/978-3-662-02267-2](https://link.springer.com/book/10.1007/978-3-662-02267-2))

* [[Mikhail Gromov]], _A topological technique for the construction of solutions of differential equations and inequalities_, Proc. Int. Congress Math. Nice 1970, vol. 2, 221-225 ([djvu w OCR](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0221.0226.ocr.djvu), [pdf](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0221.0226.ocr.pdf))

* [[Mikhail Gromov]], Papers on h-principle: geometric methods for solving partial differential equations/inequalities and the homotopy structure in the spaces of their solutions, [list](http://www.ihes.fr/~gromov/topics/topic2.html) (with some links)

* {#Haefliger71} [[André Haefliger]], _Lectures on the Theorem of Gromov_, Proceedings of Liverpool Singularities Symposium, II (1969/1970), p. 128-141. Lecture Notes in Math. 209, Springer, Berlin, 1971 ([pdf](https://link.springer.com/content/pdf/10.1007/BFb0068900.pdf)).
* [[John Francis]], _[The h-principle in topology](https://sites.math.northwestern.edu/~jnkf/classes/hprin/)_ (overview [pdf](https://sites.math.northwestern.edu/~jnkf/classes/hprin/1overview.pdf))

See also:

* {#EliasMisha01} Y. Eliashberg, N. Mishachev, _Holonomic approximation and Gromov's h-principle_, [arXiv:math/0101196](https://arxiv.org/abs/math/0101196)

* [London h-principle learning seminar](https://www.mathematik.hu-berlin.de/~wendl/h-principle.html)

* [[Konrad Voelkel]], Helene Sigloch, _Homotopy sheaves and h-principles_, ([pdf](https://www.konradvoelkel.com/wp-content/uploads/homotopy-sheaves.pdf))

* [[Alexander Kupers]], *Three applications of delooping to H-principles*, Geom Dedicata **202** (2019) 103–151  ([arXiv:1701.06788](https://arxiv.org/abs/1701.06788), [doi:10.1007/s10711-018-0405-7](https://doi.org/10.1007/s10711-018-0405-7))

[[!redirects homotopy principle]]


h-set

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

# h-Sets
* table of contents
{: toc}

## Idea
 {#Idea}

In [[homotopy type theory]] an _h-set_ is a [[type]] $X$ -- hence a [[homotopy type]] -- with the special property that any two of its [[terms]] $x,y : X$ are [[equality|equal]] ([[equivalence|equivalent]]) in an at most essentially unique way, hence that the [[identity type]] $(x = y) : Type$ is an [[h-proposition]].

The notion of h-set is an [[internalization]] of the notion of [[0-truncated]] object into homotopy type theory, essentially an internalization of the notion of _[[set]]_ (or possibly of _[[preset]]_). See below in _[Relation to internal sets](#RelationToInternalSets)_ for more on this.

h-Sets can also be regarded as a way of embedding [[extensional type theory]] into [[intensional type theory]].

## Definition

Let $A$ be a [[type]] in [[dependent type theory]] with [[dependent sums]], [[dependent products]], and [[identity types]].  We define a new type $isSet(A)$ as follows:

$$isSet(A) \coloneqq \prod_{x\colon A} \prod_{y\colon A} isProp(\mathrm{Id}_{A}(x, y))$$

(using any equivalent definition of the [[predicate]] [[isProp]] for [[h-propositions]]; and where "$\prod$" denotes [[dependent product]] types and "$\mathrm{Id}_A$" denotes [[identity types]] on $A$).

In other words, the only relationship between two elements of an h-set is whether they are equal; there is no room for more than one path between them.  By [[beta reduction|beta-reducing]] this definition, we can express it as
$$isSet(A) \coloneqq \prod_{x,y\colon A} \prod_{p,q\colon \mathrm{Id}_A(x, y)} \mathrm{Id}_{\mathrm{Id}_A(x, y)}(p, q)$$
In other words, any two parallel paths in $A$ are equal.

A provably [[equivalence in homotopy type theory|equivalent]] definition is
$$isSet(A) \coloneqq \prod_{x\colon A} \prod_{p\colon \mathrm{Id}_A(x, x)} \mathrm{Id}_{\mathrm{Id}_A(x, x)}(p, \mathrm{refl}_A(x))$$
This says that a version of Streicher's "[[axiom K]]" holds for h-sets. (See also at _[[axiom UIP]]_.)

Another definition of a h-set is an $S^1$-[[localization of a type|local type]], a type for which the canonical function $\mathrm{const}_{A, S^1}:A \to (S^1 \to A)$, which takes elements of $A$ to [[constant functions]] $S^1 \to A$ from the [[circle type]] $S^1$ to $A$, is an [[equivalence of types]]:

$$\mathrm{isSet}(A) \coloneqq \prod_{f:S^1 \to A} \sum_{y:\sum_{x:A} \mathrm{Id}_{S^1 \to A}(\mathrm{const}_{A, S^1}(x), f)} \prod_{z:\sum_{x:A} \mathrm{Id}_{S^1 \to A}(\mathrm{const}_{A, S^1}(x), f)} \mathrm{Id}_{\sum_{x:A} \mathrm{Id}_{S^1 \to A}(\mathrm{const}_{A, S^1}(x), f)}(y, z)$$

## Examples

* Most (non-[[higher inductive type|higher]]) [[inductive types]] are h-sets (assuming that all their parameters and indices are so).  In particular, the type of [[natural numbers]] is an h-set.  This can be proven from Theorem \ref{DecidableIsSet} below.

* In a [[set-level type theory]], *all* types are h-sets.

* The [[univalence axiom]] is the only well-known property that implies that *not* all types are h-sets.  In particular, it implies that the [[universe type]] is not an h-set, and that many [[higher inductive types]] are not h-sets.

## Properties

### Equivalent characterizations

There are some equivalent characterizations of a type $A$ being an h-set:

* A type $A$ is an h-set if and only if all its identity types $x=_A y$ have [[split support]], i.e. $\prod_{(x,y:A)} \Vert x=y\Vert \to (x=y)$.  This is proven in [(KECA)](#KECA).

* More generally, $A$ is an h-set if and only if there is some $R:A\to A\to Prop$ which is reflexive (i.e. $\prod_{(x:A)} R(x,x)$) and such that $\prod_{(x,y:A)} R(x,y) \to (x=y)$.  This is Theorem 7.2.2 in the [[HoTT Book]].

### Hedberg's theorem

[[Hedberg's theorem]] states that

\begin{theorem}
Suppose that $A$ is a [[type]] with [[untruncated decidable equality]], a type such that for every element $a:A$ and $b:A$, there is an element in the sum type of the identity type $a =_A b$ and the type of functions from $a =_A b$ to the [[empty type]].

$$a:A, b:A \vdash \delta(a, b):(a =_A b) + (a =_A b) \to \emptyset$$

Then $A$ is a h-set.
\end{theorem}

A proof of this theorem could be found in [[Hedberg's theorem]].

### Relation to internal sets
 {#RelationToInternalSets}

When using [[homotopy type theory]] as the ambient [[foundations]], h-sets generally play the role of the [[sets]].  When homotopy type theory is the [[internal logic]] of some [[(∞,1)-category]], then the h-sets are the "internal sets" in this internal logic.  (Not to be confused with the other meaning of [[internal set]].)

Note, though, that this notion of "internal set" is of a different sort from the usual notions of [[internal category]] or [[internal groupoid]].  If an internal set is an h-set, then an "internal groupoid" should mean a 1-truncated type, whereas an internal groupoid usually means some kind of [[groupoid object in an (∞,1)-category]].  Conversely, the usual meaning of "internal groupoid" suggests that the meaning of "internal set" should be something more like a [[setoid]], with the h-sets being more like [[presets]].  This latter meaning is how "sets" are more often defined by constructive type theorists.

The point is that to be worthy of the name "set", a notion ought to come with "[[quotients]] of [[equivalence relations]]".  If we start with a notion which does *not* have quotients, such as the types in ordinary [[Martin-Löf dependent type theory]], then in order to get a good notion of "set" we need to "freely add quotients", which semantically means passing to the [[exact completion]] whose objects are [[setoids]].  But if we start with a notion that does have quotients, then this is unnecessary.  In homotopy type theory, h-sets do have quotients, which can be constructed using [[higher inductive types]]; thus it makes sense to call them "sets" rather than "presets".

A good way to reconcile these seemingly clashing terminologies is to talk about [[exact completions]] of [[unary sites]] or [[(∞,1)-sites]].  The presence of a [[Grothendieck topology]] allows us to "remember" to what extent our given notion has well-behaved quotients: if we have no quotients, then we use the [[trivial topology]], whereas if we have quotients, we can use the [[regular topology]].  And the exact completion builds in quotients "freely" but preserving those which the topology asserts to already exist.  In particular, if we start with quotients (an [[exact category]] or $(\infty,1)$-category), then the exact completion of the regular topology is idempotent, whereas if we start with a trivial topology, then the exact completion gives a category of setoids.  Thus, in general, the good notion of "internal set" in a unary site is "object of the exact completion".

### Pretopos of hsets
 {#PretoposOfHsets}

The h-sets in [[HoTT]] form a [[ΠW-pretopos]] ([Rijke-Spitters 13](#RijkeSpitters13)). See also at _[[structural set theory]]_.

## Related concepts

[[!include homotopy n-types - table]]

[[!include types and logic - table]]

## References

* {#Hedberg} Michael Hedberg, *A coherence theorem for Martin-Löf's type theory*, J. Functional Programming, (1998)

* [[Nicolai Kraus]], *A direct proof of Hedberg's theorem*, [blog post](http://homotopytypetheory.org/2012/03/30/a-direct-proof-of-hedbergs-theorem/)

* {#KECA} [[Nicolai Kraus]], [[Martin Escardo]], [[Thierry Coquand]] , [[Thorsten Altenkirch]], _Generalizations of Hedberg's theorem_, in M. Hasegawa (Ed.): TLCA 2013, LNCS 7941, pp. 173-188. Springer, Heidelberg 2013. ([pdf](http://www.cs.bham.ac.uk/~mhe/papers/hedberg.pdf))

Formalization of [[set theory]] via h-sets in [[homotopy type theory]] (see at [[Set in homotopy type theory]])

* {#RijkeSpitters13} [[Egbert Rijke]], [[Bas Spitters]], _Sets in homotopy type theory_, Mathematical Structures in Computer Science, Volume 25, Issue 5 (From type theory and homotopy theory to Univalent Foundations of Mathematics)  ([arXiv:1305.3835](http://arxiv.org/abs/1305.3835))

which became one chapter in

* [[Univalent Foundations Project]], _[[Homotopy Type Theory – Univalent Foundations of Mathematics]]_

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