game theory

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## Idea

Game theory is the study of strategic interaction between agents. It is often used for mathematical modelling of real-world situations in which agents interact, including in microeconomics, computer science and ecology.

Game theory should not be confused with [[game semantics]], which has some terminology and history in common but has little overlap in practice.

## Via coalgebras

Several authors have approached game theory through coalgebraic systems theory, defining games as elements of the [[terminal coalgebra for an endofunctor|terminal coalgebra]] of an appropriate functor. This puts the concepts of _state_ and _state transition_ in a game to centre stage. It is closely related to [extensive form games](https://en.wikipedia.org/wiki/Extensive-form_game), whereby a game is defined by its tree of plays.

Perhaps the first paper to use this approach is [Pavlovic 2009](#Pavlovic09). Others are [Abramsky and Winschel 2017](#AbramskyWinschel17) and [Ghani, Kupke, Lambert and Forsberg 2018](#Ghani18a), the latter of which connects coalgebraic game theory with compositional game theory.

##Compositional game theory##

Open games approach games as open systems, making them morphisms of a [[symmetric monoidal category]] which can be depicted as [[string diagram|string diagrams]], in the spirit of [[applied category theory]]. See [Ghani, Hedges, Winschel and Zahn 2018](#Ghani18b).

Open games have a close connection to [[lens (in computer science)|lenses]]. This can be used to situate open games inside a symmetric monoidal [[double category]] in a game-theoretically interesting way, see [Hedges 2018](#Hedges18).

##Topology and game theory##

Deformations between strategy profiles have been considered as a practical method to compute Nash equilibria (see [Herings and Peeters 2010](#Herings10)).

## References

### General

* $n$Cafe: [combinatorial game categories](http://golem.ph.utexas.edu/category/2009/11/combinatorialgame_categories.html), [zero determinant strategies](http://golem.ph.utexas.edu/category/2012/07/zerodeterminant_strategies_in.html) in the iterated prisoner's dilemma
* wikipedia: [game theory](https://secure.wikimedia.org/wikipedia/en/wiki/Game_theory), [combinatorial game theory](http://en.wikipedia.org/wiki/Combinatorial_game_theory), [dominance (game theory)](http://en.wikipedia.org/wiki/Dominance_%28game_theory%29), [Nash equilibrium](http://en.wikipedia.org/wiki/Nash_equilibrium), [wild type](http://en.wikipedia.org/wiki/Wild_type)
* William Press, Freeman Dyson, _Iterated prisoner's dilemma contains strategies that dominate any evolutionary opponent_, [PNAS open access](http://www.pnas.org/content/early/2012/05/16/1206569109.full.pdf+html), March 2012.
* [[J.R.B. Cockett]], G.S.H. Cruttwell and K. Saff, _Combinatorial game categories_, [pdf](https://www.mta.ca/uploadedFiles/Community/Bios/Geoff_Cruttwell/CGC.pdf)
* [[André Joyal]], _Remarques sur la théorie des jeux à deux personnes_, Gazette des
Sciences Mathematiques du Québec 1(4):46–52, 1977; Robin Houston's rough translation [can be found here](https://bosker.wordpress.com/2009/11/16/the-category-of-conway-games/)
* André Joyal, _Free lattices, communication and money games_, in: Logic and
scientific methods. Volume one of the proceedings of the tenth international
congress of logic, methodology and philosophy of science, Florence, Italy, Synth.
Libr. 259, pages 29–68. Dordrecht: Kluwer Academic Publishers, 1997.
* {#Pavlovic09} [[Dusko Pavlovic]], _A semantical approach to equilibria and rationality_, CALCO 2009. ([arXiv:0905.3548](https://arxiv.org/abs/0905.3548), [doi:10.1007/978-3-642-03741-2_22](http://dx.doi.org/10.1007/978-3-642-03741-2_22))
* {#AbramskyWinschel17} [[Samson Abramsky]] and [[Viktor Winschel]], _Coalgebraic analysis of subgame-perfect equilibria in infinite games without discounting_, Mathematical structures in computer science 2017. ([arXiv:1210.4537](https://arxiv.org/abs/1210.4537), [doi:10.1017/S0960129515000365](https://doi.org/10.1017/S0960129515000365))
* {#Ghani18a} [[Neil Ghani]], [[Clemens Kupke]], [[Alasdair Lambert]] and [[Fredrik Nordvall Forsberg]], _A compositional treatment of iterated open games_, Theoretical computer science 2018. ([arXiv:1711.07968](https://arxiv.org/abs/1711.07968), [doi:10.1016/j.tcs.2018.05.026](https://doi.org/10.1016/j.tcs.2018.05.026))
* {#Ghani18b} [[Neil Ghani]], [[Jules Hedges]], [[Viktor Winschel]] and [[Philipp Zahn]], _Compositional game theory_, LiCS 2018. ([arXiv:1603.04641](https://arxiv.org/abs/1603.04641), [doi:10.1145/3209108.3209165](https://doi.org/10.1145/3209108.3209165))
* {#Hedges18} [[Jules Hedges]], _Morphisms of open games_, MFPS 2018. ([arXiv:1711.07059](https://arxiv.org/abs/1711.07059), [doi:10.1016/j.entcs.2018.11.008](https://doi.org/10.1016/j.entcs.2018.11.008))
* {#Herings10} P. Jean-Jacques Herings and Ronald Peeters, _Homotopy methods to compute equilibria in game theory_, Economic Theory 2010. ([doi:10.1007/s00199-009-0441-5](https://doi.org/10.1007/s00199-009-0441-5))

### Via modal logic
 {#ReferencesViaModalLogic}

On [[epistemic logic|epistemic]] ([[S5 modal logic|S5]]) [[modal logic]] in [[game theory]] and [[information science]]:

* [[Robert J. Aumann]], *Interactive epistemology I: Knowledge*, International Journal of Game Theory **28** (1999) 263–300 $[$[doi:10.1007/s001820050111](https://doi.org/10.1007/s001820050111)$]$

* [[Wiebe van der Hoek]], [[Marc Pauly]], *Modal logic for games and information*, Ch. 20 in: *The Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 85-183 $[$doi:10.1016/S1570-2464(07)80023-1, [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/)$]$

* {#DitmarschHoekKooi08} [[Hans Ditmarsch]], [[Wiebe Hoek]], Barteld Kooi, *Dynamic Epistemic Logic*, Studies In Epistemology, Logic, Methodology, And Philosophy Of Science (SYLI) **337**, Springer (2008) $[$[doi:10.1007/978-1-4020-5839-4_2](https://doi.org/10.1007/978-1-4020-5839-4_2), [pdf](http://www.ivanociardelli.altervista.org/wp-content/uploads/2018/04/Epistemic-logic.pdf)$]$


Gamma function

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# The $\Gamma$ function
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## Idea and Definition

[[Leonhard Euler]] solved the problem of finding a [[function]] of a continuous [[variable]] $x$ which for [[integer]] values of $x=n$ agrees with the [[factorial]] function $n\mapsto n!$. The _gamma function_ is a shift by one of the solution to this problem.

For a complex variable $x\neq -1,-2,\ldots$, we define $\Gamma(x)$ by the formula

$$
  \Gamma(x) = \lim_{k\to \infty} \frac{k! \cdot k^{x-1}}{(x)_k}
$$

where $(x)_0 =1$ and for positive integer $k = 1,2,\ldots$,

$$
(x)_k = x (x+1) (x+2)\cdots (x+ k-1)
$$

is the "Pochhammer symbol" (or rising factorial). It easily follows that $\Gamma(n+1) = n!$ for natural numbers $n = 0, 1, 2, \ldots$.

## Properties

As a function of a complex variable, the Gamma function $\Gamma(x)$ is a [[meromorphic function]] with simple poles at $x = 0, -1, -2, \ldots$.

Extending the recursive definition of the ordinary factorial function, the Gamma function satisfies the following translation formula:

\[
  \label{TranslationFormula}
  \Gamma(x+1)
  \;=\;
  x\,\Gamma(x)
\]

away from $x = 0, -1, -2, \ldots$.

It also satisfies a reflection formula, due to Euler:

$$\Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin(\pi x)}.$$

\begin{prop}\label{GaussMultiplicationFormula}
**([[Gauss multiplication formula]])**

For

* any [[positive number|positive]] [[integer]] $N \in \mathbb{N}_+$,

* any $z \in \mathbb{R} \setminus  \{0, -1/N, -2/N, \cdots\} $,

the [[Gamma function]] $\Gamma(-)$ satisfies

$$
  \underoverset
    {j = 0}
    {N-1}
    {\prod}
  \Gamma
  \left(
    z + \tfrac{j}{N}
  \right)
  \;=\;
  (2 \pi)^{ \tfrac{1}{2}(N-1) }
  \cdot
  N^{ \tfrac{1}{2} - N z }
  \cdot
  \Gamma( N z )
  \,.
$$

\end{prop}

Quite remarkably, the Gamma function (this time as a function of a real variable) is uniquely characterized in the following theorem:

\begin{theorem}\label{BohrMollerupTheorem}
**(Bohr-Mollerup)**
\linebreak

The [[restriction]] of the Gamma function to the [[open interval]] $(0, \infty)$ is the unique function $f$ such that

1. $f(x+1) = x f(x)$,

1. $f(1) = 1$,

1. $\log f$ is [[convex function|convex]].

\end{theorem}

([Artin (1931), Thm. 2.1](#Artin31))

A number of other representations of the Gamma function are known and frequently utilized, e.g.,

* Product representation:

  $$
    \frac1{\Gamma(x)}
    \;=\;
    x e^{\gamma x}
    \prod_{n=1}^{\infty}
    \left(
      1 + \frac{x}{n}
    \right)
    e^{-x/n}
    \,,
  $$

  where $\gamma$ is [[Euler's constant]].

* Integral representation:

  $$
    \Gamma(x)
    \;=\;
    \int_{0}^{\infty} t^x e^{-t} \frac{d t}{t}
    \,.
  $$

## Related concepts

* [[factorial]]

* [[Euler beta function]]

* [[Riemann zeta function]]

* [[Barnes G-function]]

* [[Hölder's inequality]]

## References

* {#Artin31} [[Emil Artin]], *Einführung in die Theorie der Gammafunktion*, Hamburger Mathematische Einzelschriften
l. Heft, Verlag B. G. Teubner, Leipzig (1931)

  English translation by Michael Butler: *The Gamma Function*, Holt, Rinehart and Winston (1931) [[[Artin-TheGammaFunction.pdf:file]]]

* George Andrews, Richard Askey, Ranjan Roy, _Special Functions_. Encyclopedia of Mathematics and Its Applications 71, Cambridge University Press, 1999.

See also:

* Wikipedia, *[Gamma function](https://en.wikipedia.org/wiki/Gamma_function)*

* Wikipedia, *[Bohr-Mollerup theorem](https://en.wikipedia.org/wiki/Bohr%E2%80%93Mollerup_theorem)*

[[!redirects gamma function]]
[[!redirects Gamma function]]

[[!redirects Gamma-function]]
[[!redirects Gamma-functions]]

[[!redirects Bohr-Mollerup theorem]]


Gamma-set

Let $\Gamma^{op}$ (see [[Segal's category]]) be the [[skeleton]] of the category of [[finite set|finite]] [[pointed sets]].  We write $\underline{n}$ for the finite pointed set with $n$ non-basepoint elements.  Then a **$\Gamma$-set** is a functor $X\colon \Gamma^{op}\to Set$.

The [[topos]] $\Set^{\Gamma^{op}}$ of $\Gamma$-sets is the [[classifying topos]] for [[pointed objects]] ([MO question](http://mathoverflow.net/questions/85600/what-do-gamma-sets-classify)). For more on this see also at _[[classifying topos for the theory of objects]]_.

Related $n$Lab entries include [[Gamma-space]], [[Segal's category]].

* [[Graeme Segal]], _Categories and cohomology theories_, Topology 13 (1974).
* [[Alain Connes]], [[Caterina Consani]], _Absolute algebra and Segal's Gamma sets_, [arxiv/1502.05585](https://arxiv.org/abs/1502.05585)

[[!redirects Gamma-sets]]


Gamma-space

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## Idea

The concept of _$\Gamma$-spaces_ is a model for [[∞-groupoids]] equipped with a multiplication that is unital, associative, and commutative up to higher [[coherence|coherent]]  [[homotopies]]:  they are models for [[E-∞ spaces]] and hence, if grouplike ("very special" $\Gamma$-spaces), for [[infinite loop spaces]] / [[connective spectra]] / [[abelian ∞-groups]].

The notion of $\Gamma$-space is a close variant of that of [[Segal category]] for the case that the underlying [[(∞,1)-category]] happens to be an [[∞-groupoid]], happens to be [[n-connected object of an (∞,1)-category|connected]] and is equipped with extra structure.

$\Gamma$-spaces differ from [[operad|operadic]] models for $E_\infty$-spaces, such as in terms of  [[algebra over an operad|algebras]] over an [[E-∞ operad]], in that their multiplication is specified "[[geometric definition of higher categories|geometrically]]" rather than [[algebraic definition of higher categories|algebraically]].

## Definition

Let $\Gamma^{op}$ (see [[Segal's category]]) be the [[skeleton]] of the category of [[finite set|finite]] [[pointed sets]].  We write $\underline{n}$ for the finite pointed set with $n$ non-basepoint elements.  Then a **$\Gamma$-space** is a functor $X\colon \Gamma^{op}\to Top$ (or to [[simplicial sets]], or whatever other model one prefers).

We think of $X(\underline{1})$ as the "underlying space" of a $\Gamma$-space $X$, with $X(\underline{n})$ being a "model for the cartesian power $X^n$".  In order for this to be valid, and thus for $X$ to present an infinite loop space, a $\Gamma$-space must satisfy the further condition that all the [[Segal map]]s
$$ X(\underline{n}) \to X(\underline{1}) \times \dots \times X(\underline{1})$$
are [[weak equivalences]].  We include in this the $0$th Segal map $X(\underline{0}) \to *$, which therefore requires that $X(\underline{0})$ is contractible.  Sometimes the very definition of *$\Gamma$-space* includes this homotopical condition as well.

## Delooping

One of the main advantages of $\Gamma$-spaces (and,
more generally, $\Gamma$-objects) is that the [[delooping]] construction
is very easy to express in this language.

The delooping construction is a functor
$$B\colon Fun(\Gamma^{op},M) \to Fun(\Gamma^{op},M),$$
where $M$ is the [[relative category]] for which we are considering $\Gamma$-objects.
The most common choices are $M=sSet$, the [[model category of simplicial sets]],
and $M=Top$, the [[model category of topological spaces]].

We define
$$(B A)(S) := hocolim (T\mapsto A(S\times T)),$$
where $T\in\Delta^\op$ and the argument of the [[homotopy colimit]] functor
is a [[simplicial object]] in $M$.
Here $T\in\Delta$ is converted first to an object of $\Gamma$
via the functor $\Delta\to\Gamma$ described below.

## Properties

### Relation to simplicial sets

We have a functor $\Delta\to\Gamma$, where $\Delta$ is the [[simplex category]], which takes $[n]$ to $\underline{n}$.  Thus, every $\Gamma$-space has an underlying simplicial space. This simplicial space is in fact a [[special Delta-space]] which exhibits the 1-fold delooping of the corresponding $\Gamma$-space.

### Model category structure

A [[model category]] structure on $\Gamma$-spaces is due to ([Bousfield-Friedlander 77](#BousfieldFriedlander77)). See at _[[model structure for connective spectra]]_.

## Related notions

* [[Gamma-set]]

[[!include k-monoidal table]]

## References

The concept goes back to

* [[Graeme Segal]], _Categories and cohomology theories_, Topology 13 (1974).

Another early reference considers $\Gamma$-objects in [[simplicial groups]].
It is also the first reference that uses the terms “special Γ-spaces”
and “very special Γ-spaces”, which it attributes to Segal.

* [[Donald W. Anderson]], _Chain functors and homology theories_, Symposium on Algebraic Topology, Lecture Notes in Mathematics (1971), 1–12.  [doi](http://dx.doi.org/10.1007/bfb0060890).

The [[model category]] structure on $\Gamma$-spaces (a [[generalized Reedy model structure]]) was established in

* {#BousfieldFriedlander77} [[A. K. Bousfield]], [[E. M. Friedlander]], _Homotopy Theory of Γ-spaces, Spectra, and Bisimplicial Sets_, Geometric Applications of Homotopy Theory (1977) ([pdf](http://dodo.pdmi.ras.ru/~topology/books/bousfield-friedlander.pdf))

Discussion of the [[smash product of spectra]] on connective spectra via $\Gamma$-spaces is due to

* {#Lydakis98} Lydakis, _Smash products and $\Gamma$-spaces_, Math. Proc. Cam. Phil. Soc. 126 (1999), 311-328 ([pdf](http://hopf.math.purdue.edu/Lydakis/smash_gamma.pdf))

and of the corresponding [[monoid objects]], hence [[ring spectra]], in

* {#Schwede99} [[Stefan Schwede]], _Stable homotopical algebra and $\Gamma$-spaces_, Math. Proc. Camb. Phil. Soc. (1999), 126, 329 ([pdf](http://www.math.uni-bonn.de/people/schwede/Gamma.pdf))

* {#Lawson09} [[Tyler Lawson]], _Commutative Γ-rings do not model all commutative ring spectra_, Homology Homotopy Appl. Volume 11, Number 2 (2009), 189-194. ([Euclid](http://projecteuclid.org/euclid.hha/1296138517))

Discussion in relation to [[symmetric spectra]] includes

* {#Schwede12} [[Stefan Schwede]], chapter I, section 7.4 of _[[Symmetric spectra]]_ (2012)

Discussion of $\Gamma$-spaces in the broader context of [[higher algebra]] in [[(infinity,1)-operad]] theory is around remark 2.4.2.2 of

* {#Lurie} [[Jacob Lurie]], _[[Higher Algebra]]_

See also

*  C. Balteanu, Z. Fiedorowicz, [[R. Schwanzl]] and [[R. Vogt]], _Iterated
Monoidal Categories_, Advances in Mathematics (2003).

* B. Badzioch, _Algebraic Theories in Homotopy Theory_, Annals of Mathematics, 155, 895--913 (2002).

[[!redirects Gamma-spaces]]
[[!redirects Γ-space]]
[[!redirects Γ-spaces]]
[[!redirects ∞-space]]
[[!redirects ∞-spaces]]

[[!redirects Gamma space]]
[[!redirects Gamma spaces]]

[[!redirects Segal's category Gamma]]
[[!redirects Segal's category Γ]]

[[!redirects model structure for Gamma-spaces]]
[[!redirects model structure for ∞-spaces]]


Gandalf Lechner

* [webpage](http://gandalflechner.eu/)

## related $n$Lab entries

* [[AQFT]]

category: people


Gang Tian

__Gang Tian__ is a Chinese-American differential geometer. With [[Yongbin Ruan]], he is one of the pioneers of differential geometric formalization of [[quantum cohomology]].

## Selected writings

Introducing a mathematical definition of [[quantum cohomology rings]]:

* [[Yongbin Ruan]], [[Gang Tian]], _A mathematical theory of quantum cohomology_, Mathematical Research Letters **1** 2 (1994) 269-278 [[doi:10.4310/MRL.1994.v1.n2.a15](https://doi.org/10.4310/MRL.1994.v1.n2.a15), [pdf](https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1994/0001/0002/MRL-1994-0001-0002-a015.pdf)]

* [[Bernd Siebert]], [[Gang Tian]], *On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator*,  Asian Journal of Mathematics **1** 4 (1997) 679 – 695 [[arXiv:alg-geom/9403010](https://arxiv.org/abs/alg-geom/9403010), [doi:10.4310/AJM.1997.v1.n4.a2](https://dx.doi.org/10.4310/AJM.1997.v1.n4.a2)]

Related work on technique of virtual classes

* Jun Li, Gang Tian, _Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties_, J. Amer. Math. Soc. 11 (1998) 119-174 [doi](https://doi.org/10.1090/S0894-0347-98-00250-1)

* [[Yongbin Ruan]], [[Gang Tian]], _A mathematical theory of quantum cohomology_, J. Diff. Geometry __42__ 2 (1995) 259-367 [[doi:10.4310/jdg/1214457234](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-42/issue-2/A-mathematical-theory-of-quantum-cohomology/10.4310/jdg/1214457234.full)]

On [[special holonomy]]:

* [[Jeff Cheeger]], [[Gang Tian]], _Anti-self-duality of curvature and degeneration of metrics with special holonomy_, Commun. Math. Phys. __255__, 391--417 (2005) [doi](https://doi.org/10.1007/s00220-004-1279-0)
* Paolo de Bartolomeis, [[Gang Tian]], _Stability of complex vector bundles_, J. Diff. Geometry __43__:2 (1996) [pdf](https://www.intlpress.com/JDG/archive/1996/43-2-231.pdf)

A monograph on Perelman's solution of geometrization conjecture is

* John W. Morgan, Gang Tian, _Ricci Flow and the Poincare Conjecture_, Clay Mathematics Monographs __3__, 2007
By

category: people


GAP

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GAP is a software system for computational discrete [[algebra]], with particular emphasis on computational [[finite group|finite]] [[group theory]]. The acronym stands for _Groups, Algorithms, Programming_.

It is the basis on which the [[HAP]] homological algebra library are built.

## References

### General

* [GAP Homepage](http://www.gap-system.org/)

  [Reference Manual](https://www.gap-system.org/Manuals/doc/ref/chap0.html)

### Applications

classification of [[crystallographic groups]]:

* [[GAP]] package, _The Crystallographic Groups Catalog_ ([web](http://www.math.rwth-aachen.de/~Greg.Gamble/gap4r3/pkg/crystcat/htm/CHAP001.htm))

[[table of marks]]:

* {#GAP} [GAP](https://www.gap-system.org/) [Reference Manual](https://www.gap-system.org/Manuals/doc/ref/chap0.html), _[70 Tables of Marks](https://www.gap-system.org/Manuals/doc/ref/chap70.html)_


gapped Hamiltonian

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#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]] a material is said to have a _gapped Hamiltonian_ if in the [[thermodynamic limit]] of infinite [[volume]] it [[Hamiltonian]] has a [[mass gap]].

## Related concepts

* [[confinement]]

* [[topological phase of matter]]

* [[tensor network]]

* [[mass term]]

* [[mass gap problem]] in [[Yang-Mills theory]]/[[QCD]]

## References

### General

See also

* Wikipedia, _[Gapped Hamiltonian](https://en.m.wikipedia.org/wiki/Gapped_Hamiltonian)

### K-Theory classification of topological phases of matter

[[topological phases of matter via K-theory -- references]]

Classification of [[condensed matter physics|condensed matter]] with [[gapped Hamiltonians]] ([[topological insulators]], [[topological phases of matter]]) by [[twisted K-theory|twisted]] [[equivariant K-theory|equivariant]] [[topological K-theory]]:

* [[Alexei Kitaev]], _Periodic table for topological insulators and superconductors_, talk at: L.D.Landau Memorial Conference "Advances in Theoretical Physics", June 22-26, 2008, In:AIP Conference Proceedings 1134, 22 (2009) ([arXiv:0901.2686](https://arxiv.org/abs/0901.2686), [doi:10.1063/1.3149495](https://doi.org/10.1063/1.3149495))

* [[Daniel Freed]], [[Gregory Moore]], _Twisted equivariant matter_, Ann. Henri Poincaré (2013) 14: 1927 ([arXiv:1208.5055](https://arxiv.org/abs/1208.5055))

* [[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))

* Ralph M. Kaufmann, Dan Li, Birgit Wehefritz-Kaufmann, _Topological insulators and K-theory_ ([arXiv:1510.08001](https://arxiv.org/abs/1510.08001), [spire:1401095/](http://inspirehep.net/record/1401095/))

* [[Daniel Freed]], [[Michael Hopkins]], _Reflection positivity and invertible topological phases_ ([arXiv:1604.06527](https://arxiv.org/abs/1604.06527))

* [[Daniel Freed]], _Lectures on field theory and topology_ ([cds:2699265](https://cds.cern.ch/record/2699265))

* Charles Zhaoxi Xiong, _Classification and Construction of Topological Phases of Quantum Matter_ ([arXiv:1906.02892](https://arxiv.org/abs/1906.02892))

[[!redirects gapped Hamiltonians]]


Gareth Jones

[webpage](http://www.southampton.ac.uk/maths/about/staff/gaj2.page)

category: people

[[!redirects Gareth A. Jones]]


Garnet Kin-Lic Chan

* [institute page](https://cce.caltech.edu/people/garnet-k-chan)

* [research group page](https://www.chan-lab.caltech.edu/)

## Selected writings

On elusive [[quantum supremacy]] for [[quantum computation]] applied to [[quantum chemistry]]:

* [[Garnet Kin-Lic Chan]] et al., *Is there evidence for exponential quantum advantage in quantum chemistry?* [[arXiv:2208.02199](https://arxiv.org/abs/2208.02199), [talk video](https://youtu.be/DZPH7ENcRLU)]

category: people

[[!redirects Garnet K. Chan]]
[[!redirects Garnet Chan]]


Garrett Birkhoff

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

## Selected writings

On [[monoids]] (*not* on *[[groupoids]]* in the modern sense of that term):

* [[Garrett Birkhoff]], *Hausdorff Groupoids*, Annals of Mathematics, Second Series **35** 2 (1934) 351-360 [[jstor:1968437](https://www.jstor.org/stable/1968437), [doi:10.2307/1968437](https://doi.org/10.2307/1968437)]

On [[quantum logic]]:

* {#BirkhoffvonNeumann36} [[Garrett Birkhoff]], [[John von Neumann]]: *The logic of quantum mechanics*, Annals of Mathematics **37** 823-843 (1936) [[doi:10.2307/1968621](https://doi.org/10.2307/1968621), [pdf](http://www.fulviofrisone.com/attachments/article/451/the%20logic%20of%20quantum%20mechanics%201936.pdf)]

category: people


Gary Gibbons

Gray Gibbons is professor for theoretical [[physics]] at Cambridge.

* [webpage](http://www.damtp.cam.ac.uk/people/g.w.gibbons/)

## Selected writings

On [[D1-D3 brane intersections]] as spikes/[[BIons]] in the [[D3-brane]] [[DBI action|DBI-theory]]:

* [[Gary Gibbons]], _Born-Infeld particles and Dirichlet p-branes_, Nucl. Phys. B514: 603-639, 1998 ([arXiv:hep-th/9709027](https://arxiv.org/abs/hep-th/9709027))

On [[Yang-Mills monopoles]]:

* {#GibbonsManton86} [[Gary Gibbons]], [[Nicholas Manton]], _Classical and Quantum Dynamics of BPS Monopoles_, Nucl. Phys. B274 (1986) 183-224 ([spire:18322](http://inspirehep.net/record/18322), doi:10.1016/0550-3213(86)90624-3)

On [[G2-conifolds]]:

* [[Gary Gibbons]], [[Don Page]], [[Christopher Pope]], _Einstein metrics on $S^3$, $\mathbb{R}^3$ and $\mathbb{R}^4$ bundles_, Comm. Math. Phys. Volume 127, Number 3 (1990), 529-553 ([euclid:cmp/1104180218](https://projecteuclid.org/euclid.cmp/1104180218))

On open [[M5-branes]] [[intersecting brane|intersecting]] [[M9-branes]] in a [[Yang monopole]]:

* [[Eric Bergshoeff]], [[Gary Gibbons]], [[Paul Townsend]], _Open M5-branes_, Phys. Rev. Lett. 97:231601 2006 ([arXiv:hep-th/0607193](http://arxiv.org/abs/hep-th/0607193), [doi:10.1103/PhysRevLett.97.231601](https://doi.org/10.1103/PhysRevLett.97.231601))

On [[gravitational lensing]] via the [[Gauss-Bonnet theorem]]:

*  [[Gary Gibbons]] and M. C. Werner, _Applications of the Gauss-Bonnet theorem to gravitational lensing_, Class. Quant. Grav. 25, 235009 (2008) ([arXiv:0807.0854](https://arxiv.org/abs/0807.0854))

## Related $n$Lab entries

* [[gravity]]

* [[M5-brane]]

* [[Yang monopole]]

category: people

[[!redirects Gary W. Gibbons]]


Gary Horowitz

* [webpage](http://web.physics.ucsb.edu/~gary/)

## Selected writings

On [[Lorentzian orbifolds]] as [[target spacetimes]] in [[string theory]]:

* [[Gary Horowitz]], Alan R. Steif, _Singular string solutions with nonsingular initial data_, Physics Letters B Volume 258, Issues 1–2, 4 April 1991, Pages 91-96 (doi:10.1016/0370-2693(91)91214-G)

On [[AdS/CFT duality]]:

* [[Tom Banks]], [[Michael Douglas]], [[Gary Horowitz]], [[Emil Martinec]], _AdS Dynamics from Conformal Field Theory_ ([arXiv:hep-th/9808016](https://arxiv.org/abs/hep-th/9808016), [spire:474214](http://inspirehep.net/record/474214))

* [[Gary Horowitz]], [[Joseph Polchinski]], _Gauge/gravity duality_ ([gr-qc/0602037](http://arxiv.org/abs/gr-qc/0602037))

Discussion of [[superconductivity]] via [[AdS/CFT in condensed matter physics]]:

* [[Sean Hartnoll]], [[Christopher Herzog]], [[Gary
 Horowitz]], _Building an AdS/CFT superconductor_, Phys. Rev. Lett. 101:031601, 2008 ([arXiv:0803.3295](https://arxiv.org/abs/0803.3295))

On the [[cosmic censorship hypothesis]] and the [[weak gravity conjecture]]:

* {#HorowitzSantos19} [[Gary Horowitz]], [[Jorge Santos]], _Further evidence for the weak gravity - cosmic censorship connection_ ([arXiv:1901.11096](https://arxiv.org/abs/1901.11096))

## Related $n$Lab entries

* [[universal spacetime]]

* [[bottom-up and top-down model building]]

* [[black holes in string theory]]

* [[bosonic M-theory]]

* [[BF-theory]]

* [[black brane]]

* [[cosmic censorship hypothesis]]

* [[weak gravity conjecture]]

category: people


Gary McCartor

* [InSpire page](https://inspirehep.net/authors/998131)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=163290)

## Selected writings

On [[light-cone gauge quantization]]:

* [[Stanley J. Brodsky]], [[Gary McCartor]], [[Hans-Christian Pauli]], [[Stephen S. Pinsky]], *The challenge of light-cone quantization of gauge field theory*, Particle World **3** 3 (1993) 109-124  [[cds:240388](http://cds.cern.ch/record/240388), [inspire:335247](https://inspirehep.net/literature/335247)]

* {#McCartorRobertson94} [[Gary McCartor]], David G. Robertson, *Light-cone quantization of gauge fields*, Zeitschrift für Physik C Particles and Fields **62** (1994) 349–355 [[doi:10.1007/BF01560250](https://doi.org/10.1007/BF01560250)]

category: people


Gary Shiu

* [webpage](http://uw.physics.wisc.edu/~strings/shiu/shiu.html)

## Selected writings

On [[orientifolds]] and [[RR-field tadpole cancellation]]

* Zurab Kakushadze, [[Gary Shiu]], S.-H. Henry Tye, _Type IIB Orientifolds, F-theory, Type I Strings on Orbifolds and Type I - Heterotic Duality_, Nucl.Phys. B533 (1998) 25-87 ([arXiv:hep-th/9804092](http://arxiv.org/abs/hep-th/9804092))

* {#BuchelShiuTye99} Alex Buchel, [[Gary Shiu]], S.-H. Henry Tye, _Anomaly Cancelations in Orientifolds with Quantized B Flux_, Nucl.Phys. B569 (2000) 329-361 ([arXiv:hep-th/9907203](https://arxiv.org/abs/hep-th/9907203))

* {#CveticShiuUranga01} [[Mirjam Cvetic]], [[Gary Shiu]], [[Angel Uranga]], _Chiral Four-Dimensional $N=1$ Supersymmetric Type IIA Orientifolds from Intersecting D6-Branes_, Nucl. Phys. B615:3-32,2001 ([arXiv:hep-th/0107166](http://arxiv.org/abs/hep-th/0107166))

and with [[KO-theory]]-classification:

* {#MaidenShiuStefanski06} John Maiden, [[Gary Shiu]], [[Bogdan Stefanski]], _D-brane Spectrum and K-theory Constraints of D=4, N=1 Orientifolds_, JHEP0604:052,2006 ([arXiv:hep-th/0602038](https://arxiv.org/abs/hep-th/0602038))

On [[intersecting D-brane models]] in [[string phenomenology]]:

* [[Ralph Blumenhagen]], [[Mirjam Cvetic]], [[Paul Langacker]], [[Gary Shiu]], *Towards Realistic Intersecting D-Brane Models*, Ann. Rev. Nucl. Part. Sci. **55** (2005) 71-139 [[arXiv:hep-th/0502005](http://arxiv.org/abs/hep-th/0502005), [doi:10.1146/annurev.nucl.55.090704.151541](https://doi.org/10.1146/annurev.nucl.55.090704.151541)]

and with [[flux compactification]]:

* {#MarchesanoShiu04} [[Fernando Marchesano]], [[Gary Shiu]], _Building MSSM Flux Vacua_, JHEP0411:041, 2004 ([arXiv:hep-th/0409132](https://arxiv.org/abs/hep-th/0409132))

On [[MSSM]] model building:

* {#MarchesanoShiu04} [[Fernando Marchesano]], [[Gary Shiu]], _Building MSSM Flux Vacua_, JHEP0411:041, 2004 ([arXiv:hep-th/0409132](https://arxiv.org/abs/hep-th/0409132))

On the [[Higgs mechanism]]:

* {#HINSS18} Manami Noumi Hashi, Hiroshi Isono, Toshifumi Noumi, [[Gary Shiu]], Pablo Soler, _Higgs Mechanism in Nonlocal Field Theories_, JHEP08 (2018) 064 ([arXiv:1805.02676](https://arxiv.org/abs/1805.02676))

Application of [[topological data analysis]] ([[persistent homology]]) to analysis of [[cosmological structure formation]]:

* Matteo Biagetti, [[Alex Cole]], [[Gary Shiu]], _The Persistence of Large Scale Structures I: Primordial non-Gaussianity_ ([arXiv:2009.04819](https://arxiv.org/abs/2009.04819))

and to analysis of [[phase transitions]]:

* [[Alex Cole]], Gregory J. Loges, [[Gary Shiu]], _Quantitative and Interpretable Order Parameters for Phase Transitions from Persistent Homology_ ([arXiv:2009.14231](https://arxiv.org/abs/2009.14231))

On the [[weak gravity conjecture]]:

* Lars Aalsma, [[Alex Cole]], Gregory J. Loges, [[Gary Shiu]], _A New Spin on the Weak Gravity Conjecture_ ([arXiv:2011.05337](https://arxiv.org/abs/2011.05337))

Survey of [[string phenomenology]]:

* [[Fernando Marchesano]], [[Gary Shiu]], [[Timo Weigand]], *The Standard Model from String Theory: What Have We Learned?* [[arXiv:2401.01939](https://arxiv.org/abs/2401.01939)]

category: people


Gaston Giribet

__Gaston Giribet__ is a physicist from Buenos Aires.

* [webpage](http://www.df.uba.ar/users/gaston)

category: people


Gaudin integrable model

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## Idea/Definition

A **Gaudin integrable model** is a special case of [[Hitchin integrable system]] where the [[spectral curve]] is [[elliptic curve|elliptic]].

## References

* O. Babelon, D. Bernard, M. Talon, _Introduction to classical integrable systems_, Cambridge Univ. Press 2003.
* [[Boris Feigin]], [[Edward Frenkel]], [[Valerio Toledano Laredo]], _Gaudin models with irregular singularities_, [math.QA/0612798](http://arxiv.org/abs/math/0612798)
* L. Aguirre, G. Felder, A. P. Veselov, _Gaudin subalgebras and stable rational curves_, [arxiv/1004.3253](http://arxiv.org/abs/1004.3253)
* E. K. Sklyanin, _Separation of variables in the Gaudin model_, J. Math. Sci. __47__, 2 (1989) 2473-2488, [doi](http://dx.doi.org/10.1007/BF01840429)
* [[Branislav Jurčo]], _Classical Yang-Baxter equations and quantum integrable systems (Gaudin models)_, pp. 1616-6361 in Springer Lect. Notes in Physics __370__ (1990) [doi](http://dx.doi.org/10.1007/3-540-53503-9)
* [[Boris Feigin|B. Feigin]], E. Frenkel, N. Reshetikhin, _Gaudin model, Bethe Ansatz and critical level_, Commun. Math. Phys __166__ (1994), pp. 27–62, [euclid](http://projecteuclid.org/euclid.cmp/1104271502), [MR1309540](http://www.ams.org/mathscinet-getitem?mr=1309540)
* L.G. Rybnikov, _Uniqueness of higher Gaudin Hamiltonians_, doi, Reports on Math. Phys. __61__, 2 (2008) 247-252
* B. Enriquez, V. Rubtsov, _Hitchin systems, higher Gaudin Hamiltonians and r-matrices_, Math. Res. Letters __3__ (1996) n° 3, 343-357,  [alg-geom/9503010](http://arxiv.org/abs/alg-geom/9503010)
* K. Takesaki, _Gaudin model, KZ equation and an isomonodromic problem on the torus_, 	__44__, 2 (1998) 143-156, [doi](http://dx.doi.org/10.1023/A:1007417518021), [hep-th/9711058](http://arxiv.org/abs/hep-th/9711058)
* M. Gaudin, _La fonction d'onde de Bethe_, Paris: Masson 1983, xvi+331.
* M. Gaudin, _Diagonalisation d'une classe d'hamiltoniens de spin_, J. Physique __37__, 1087–1098 (1976).
* (physics applications:) G. Ortiz, R. Somma, J. Dukelsky, S. Rombouts, _Exactly-solvable models derived from a generalized Gaudin algebra_, [cond-mat/0407429](http://arxiv.org/abs/cond-mat/0407429)

Quantization of the Gaudin model:

* V. Talalaev, _The quantum Gaudin system_, Funct. Anal. Appl. __40__, 73–77 (2006) ( _Квантовая система Годена_, Функц. анализ и его прил. __40__, вып. 1, 2006, с. 86–91, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=25&what=fullt&option_lang=rus))
[[!redirects Gaudin model]]
[[!redirects Gaudin models]]
[[!redirects Gaudin integrable models]]


gauge

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## Idea

According to ([Hegel, Science of Logic, §714 ](Science+of+Logic#714)):

> Ein Maaß, als Maaßstab im gewöhnlichen Sinne, ist ein Quantum, das als die an sich bestimmte Einheit gegen äußerliche Anzahl willkürlich angenommen wird. Eine solche Einheit kann zwar auch in der That an sich bestimmte Einheit seyn, wie Fuß und dergleichen ursprüngliche Maaße; insofern sie aber als Maaßstab zugleich für andere Dinge gebraucht wird, ist sie für diese nur äußerliches, nicht ihr ursprüngliches Maaß.—So mag der Erddurchmesser, oder die Pendellänge, als specifisches Quantum für sich genommen werden. Aber es ist willkürlich, den wievielsten Theil des Erddurchmessers oder der Pendellänge und unter welchem Breitengrade man diese nehmen wolle, um sie als Maaßstab zu gebrauchen. Noch mehr aber ist für andere Dinge ein solcher Maaßstab etwas Äußerliches. Diese haben das allgemeine specifische Quantum wieder auf besondere Art specificirt, und sind dadurch zu besondern Dingen gemacht. Es ist daher thöricht, von einem natürlichen Maaßstab der Dinge zu sprechen. Ohnehin soll ein allgemeiner Maaßstab nur für die äußerliche Vergleichung dienen; in diesem oberflächlichsten Sinne, in welchem er als allgemeines Maaß genommen wird, ist es völlig gleichgültig, was dafür gebraucht wird. Es soll nicht ein Grundmaaß in dem Sinne seyn, daß die Naturmaaße der besondern Dinge daran dargestellt und daraus nach einer Regel, als Specifikationen Eines allgemeinen Maaßes, des Maaßes ihres allgemeinen Körpers, erkannt würden. Ohne diesen Sinn aber hat ein absoluter Maaßstab nur das Interesse und die Bedeutung eines Gemeinschaftlichen, und ein solches ist nicht an sich, sondern durch Übereinkommen ein Allgemeines.

> A measure taken as a gauge in the usual meaning of the word is a quantum which is arbitrarily assumed as the intrinsically determinate unit relatively to an external amount. Such a unit can, it is true, also be in fact an intrinsically determinate unit, like a foot and suchlike original measures; but in so far as it is also used as a standard for other things it is in regard to them only an external measure, not their original measure. Thus the diameter of the earth or the length of a pendulum may be taken, each on its own account, as a specific quantum; but the selection of a particular fraction of the earth's diameter or of the length of the pendulum, as well as the degree of latitude under which the latter is to be taken for use as a standard, is a matter of choice. But for other things such a standard is still more something external. These have further specified the general specific quantum in a particular way and have thereby become particular things. It is therefore foolish to speak of a natural standard of things. Moreover, a universal standard ought only to serve for external comparison; in this most superficial sense in which it is taken as a universal measure it is a matter of complete indifference what is used for this purpose. It ought not to be a fundamental measure in the sense that it forms a scale on which the natural measures of particular things could be represented and from which, by means of a rule, they could be grasped as specifications of a universal measure, i.e. of the measure of their universal body. Without this meaning, however, an absolute measure is interesting and significant only as a common element, and as such is a universal not in itself but only by agreement.

## Related entries

* [[physical unit]]

* [[gauge symmetry]]

* [[gauge theory]], [[gauge field]]

* [[gauge transformation]]

* [[gauge group]]

* [[gauge fixing]]

## References

A discussion of "gauge" and [[gauge transformation]] in [[metaphysics]] is in

* [[Georg Hegel]], [§714](Science+of+Logic#714) of _[[Science of Logic]]_

[[Hermann Weyl]]'s historical argument motivating [[gauge theory]] in [[physics]] from rescaling of units of length was given in 1918 in

* {#Weyl23} [[Hermann Weyl]], _Raum, Zeit, Materie: Voerlesungen über die Allgemeine Relativitätstheorie_, Springer Berlin Heidelberg 1923

Quick reviews include

* Quigley, _On the origins of gauge theory_ ([pdf](http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf))

* Afriat, _Weyl's gauge argument_ ([pdf](http://hal.inria.fr/docs/00/76/96/90/PDF/WGA.pdf))

[[!redirects gauges]]


gauge boson

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## Idea

In [[quantum field theory]] the term _gauge boson_ denotes the [[quantum]] of a [[gauge field]].

In terms of [[Chern-Weil theory]]/[[differential cohomology]] we have: gauge bosons are the quanta of [[connections on bundles]].

For instance

* a [[photon]] is the quantum of the [[electromagnetic field]]: a connection on a [[line bundle]];

* a [[gluon]] is the quantum of the [[Yang-Mills field]]: a [[connection on a bundle|connection]] on a $SU(3)$-[[principal
bundle]].

* if one regards [[first-order formulation of gravity|gravity as a gauge theory]], then the [[graviton]] may be regaded as the gauge boson of a connection on a [[Poincare group|ISO]]-[[principal bundle]].

## Related concepts

* [[boson]],

  * [[gauge boson]]

    * [[photon]], [[gluon]], [[graviton]]

* [[fermion]]

  * [[electron]], [[quark]],

  * [[gravitino]],[[gaugino]] , [[dilatino]]

[[!redirects gauge bosons]]


gauge coupling unification

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## Idea

The [[renormalization group flow]] of the [[coupling constants]] ("[[running coupling constants]]") of the three [[gauge field|gauge]] [[forces]] of the [[standard model of particle physics]] come close to all intersecting at a single point at some high energy. This observation has led to the speculation that there ought to be a refinement of the standard model such as [[GUT]]-theories and/or [[supersymmetry]]-extensions such as the [[MSSM]] in which the gauge coupling constants unify exactly at high energy. This idea is called _gauge coupling unification_.

Surveys include [Uzan, section 5.3.1](#Uzan)

From [Ellis-Wells 15, p. 2](#EllisWells15):

> Famously, the [[standard model of particle physics|Standard Model]] (SM) does not unify under that rubric but the [[MSSM|Minimal Supersymmetric Standard Model]] (MSSM) does according to many [1–7].  However, neither statement is true.  The SM cannot be ruled out as the IR manifestation of a [[GUT|Grand Unified Theory]] (GUT), nor do the three gauge couplings meet at precisely one point in the MSSM. The key to understanding both claims is that high-scale thresholds are generically expected, which are the multiplets at the high scale that get masses by the same mechanism that breaks the GUT symmetry. High-scale threshold corrections kick the couplings further into exact unification if the underlying theory is gauge-coupling unified.

and further from [Ellis-Wells 17, p. 23 to 26](#EllisWells17):

> Thus, even the Standard Model up to the high scale is compatible with gauge coupling unification from this perspective, although the corrections becomes quite large in that case, and one has to ask whether nature would rather have large corrections at the GUT scale for a SM GUT or very small corrections
for a low-scale SUSY GUT.

> $[...]$ the  introduction  of  supersymmetry  both  reduces  the  needed  threshold corrections at the high-scale and increases the GUT scale $[...]$.  This  latter  element  is  helpful  since  one generally  requires  that  the  GUT  scale  be  above  about  $10^{15}$ [[GeV]] so that the $X$,$Y$ GUT gauge bosons do not induce too large dimension-six operators that cause the [[proton decay|proton to decay]] faster than current limits allow.

> Thus, exact gauge coupling unification is viable for intermediate values of supersymmetry breaking, which are also compatible with the Higgs boson mass constraint.

> $[...]$

> One important experimental prediction of minimal supersymmetry, even for superpartners at very high scales,  is the existence of a relatively light [[Higgs boson]].  This has been seen  by  the  LHC.  We  have  shown  in  this  paper  that  even  arbitrary  high  scales  of  super-partners allow the light Higgs boson, due to the required matching of the SM effective theory Higgs self-interaction coupling to gauge couplings in the supersymmetric theory.

> We have discussed how gauge coupling unification generally only improves in a supersymmetric theory, even at very high scales, with respect to the SM. Thus, we find that PeV scale [23–27] or intermediate scale supersymmetry [3, 5], two ideas that are prevalent in the literature for other reasons involving dark matter and neutrino physics, are compatible with the Higgs boson constraint and gauge coupling unification under the conditions described above.

## Related concepts

* [[GUT]], [[supersymmetry]]

* [[theory of everything]]

* [[string theory FAQ]]

## References

Reviews include (roughly in increasing order of technical detail)

* [[Savas Dimopoulos]], [[Stuart Raby]], [[Frank Wilczek]], _Unification of couplings_, Physics Today, October 1991 [[WilczekGaugeCouplingUnification.pdf:file]]

* Sander Mooij, section 1.3.4 of _Supersymmetric Grand unification_, 2008 ([[MooijGUT.pdf:file]])

* {#Uzan} Jean-Philippe Uzan, section 5.3.1 of _Varying Constants, Gravitation and Cosmology_, Living Reviews in Relativity ([arXiv:1009.5514](https://arxiv.org/abs/1009.5514))

* {#EllisWells15} Sebastian A. R. Ellis, [[James Wells]], _Visualizing gauge unification with high-scale thresholds_, Phys. Rev. D 91, 075016 (2015) ([arXiv:1502.01362](https://arxiv.org/abs/1502.01362))

* {#EllisWells17} Sebastian A. R. Ellis, [[James Wells]], _High-scale Supersymmetry, the Higgs Mass and Gauge Unification_, Phys. Rev. D 96, 055024 (2017) ([arXiv:1706.00013](https://arxiv.org/abs/1706.00013))

Original articles:

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

* P.H. Chankowski, Z. Pluciennik, S. Pokorski, C.E. Vayonakis, _Gauge Coupling Unification in GUT and String Models_ ([arXiv:hep-ph/9506393](http://arxiv.org/abs/hep-ph/9506393))

* Dumitru M. Ghilencea, Graham G. Ross, _Precision prediction of gauge couplings and the profile of a string theory_ ([arXiv:hep-ph/0102306](http://arxiv.org/abs/hep-ph/0102306))

Potential 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)]


gauge field - table

**[[gauge field]]**: models and components

| **[[physics]]** | **[[differential geometry]]** | **[[differential cohomology]] |
|--|--|--|
| [[gauge field]] | [[connection on a bundle]] | [[cocycle]] in [[differential cohomology]] |
| [[instanton sector|instanton]]/[[charge]] sector | [[principal bundle]] | [[cocycle]] in underlying [[cohomology]] |
| [[gauge potential]] | local connection [[differential form]] | local connection [[differential form]] |
| [[field strength]] | [[curvature]] | underlying [[cocycle]] in [[de Rham cohomology]] |
| [[gauge transformation]] | [[equivalence]] | [[coboundary]] |
| [[minimal coupling]] | [[covariant derivative]] | [[twisted cohomology]] |
| [[BRST complex]] | [[Lie algebroid]] of [[moduli stack]] | [[Lie algebroid]] of [[moduli stack]] |
| [[extended Lagrangian]] | [[universal Chern-Simons n-bundle]] | [[universal characteristic map]] |


Gauge Field Theory and Complex Geometry

This page collects material related to the book

* [[Yuri Manin]],

  _Gauge Field Theory and Complex Geometry_,

  Grundlehren der Mathematischen Wissenschaften 289,

  Springer 1988

on the [[Penrose-Ward transform]] relating [[twistor space]] to [[Minkowski spacetime]], on its generalization to [[superalgebra]] and [[supergeometry]], and on applications to [[super Yang-Mills theory]] and [[supergravity]].

Similar material is also for instance in

* R. Fioresi, M. A. Lledo, [[Veeravalli Varadarajan]], _The Minkowski and conformal superspaces_, J.Math.Phys.48:113505,2007 ([arXiv:0609813](https://arxiv.org/abs/math/0609813))

category: reference


gauge fixing

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## Idea

See _[[A first idea of quantum field theory]]_ the chapter _[Gauge fixing](A+first+idea+of+quantum+field+theory#GaugeFixing)_.

## Examples

* [[temporal gauge]]

* [[Lorenz gauge]]

* [[Coulomb gauge]]

* Feynman gauge

* light cone gauge

## Related concepts

* [[gauge]]

* [[gauge group]], [[gauge transformation]]

* [[spontaneously broken symmetry]]

* [[Gribov ambiguity]]

* [[gauge fixing fermion]]

* [[Nakanishi-Lautrup field]], [[antighost field]]

* [[slice theorem]]

## References

Discussion in [[BV-BRST formalism]] via [[gauge fixing Lagrangian densities]] includes

* {#Henneaux90} [[Marc Henneaux]], from section 8.2 on in _Lectures on the Antifield-BRST formalism for gauge theories_, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 ([pdf](http://www.math.uni-hamburg.de/home/schweigert/ws07/henneaux2.pdf))

Discussion of the [[Peierls brackets]] ([[covariant phase space]]) after gauge fixing is discussed in

* {#Khavkine14} [[Igor Khavkine]], section 3.3 of _Covariant phase space, constraints, gauge and the Peierls formula_, Int. J. Mod. Phys. A, 29, 1430009 (2014) ([arXiv:1402.1282](https://arxiv.org/abs/1402.1282))

[[!redirects gauge fixings]]

[[!redirects gauge-fixed]]
[[!redirects gauge fixed]]


gauge fixing Lagrangian density

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## Idea

In [[BV-BRST formalism]] a _gauge fixing Lagrangian density_ (traditionally called a "gauge fixing fermion") is a [[Lagrangian density]] $\mathbf{L}_{gf}$ whose [[Lagrangian function]] $L_{gf}$ is in degree -1 in a [[resolution|resolved]] [[BV-BRST complex]] whose [[Hamiltonian flow]] under the [[local antibracket]] is used to [[isomorphism|isomorph]] the complex into one that admits a degreewise [[covariant phase space]]. This is part of the process of "[[gauge fixing]]" in [[BV-BRST formalism]].

This is called a "fermion" only because it has to sit in BV-BRST degree (-1). Besides this odd grading, it has nothing to do with the concept of [[fermions]]. Of course, if the [[field theory]] in question does contain [[fermion fields]], then the "gauge fixing fermion" may depend on these.

## Details

For details see at _[[A first idea of quantum field theory]]_ the chapter _[Gauge fixing](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#GaugeFixing)_.

## Related concepts

* [[gauge fixing]]

* [[BV-Lagrangian density]]

[[!redirects gauge fixing Lagrangian densities]]

[[!redirects gauge fixing fermion]]
[[!redirects gauge fixing fermions]]

[[!redirects gauge fixing Lagrangian]]
[[!redirects gauge fixing Lagrangians]]


gauge fixing operator

In the context of [[BV-quantization]] for classical BV-operator $Q$, a _gauge fixing operator_ is a $Q^{GF}$ such that $H \coloneqq [Q,G^{GF}]$ is a [[generalized Laplace operator]].

See at _[[renormalization]]_ in the section _[Of theories in BV-CS form](renormalization#OfTheoriesInBVForm)_ for more details.

[[!redirects gauge fixing operators]]


gauge group

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# Contents
* table of contents
{:toc}

## Idea

In [[physics]] there are (at least) two different concepts that go by the name **gauge group**:

* a _local gauge group_ $G$ is a structure group of $G$-[[principal bundle]]s in the configuration space of a classical [[gauge theory]]: it acts by [[gauge transformation]]s on the space of field configurations.

* a _global gauge group_ is a group of [[automorphism]]s that acts on the ([[local net]] of) [[observable]]s of a [[quantum field theory]].

Notably after [[quantization]] the gauge group in the first sense does _not_ become a gauge group in the second sense. On the contrary, observables in quantum field theory are required _not_ to depend on gauge transformations in the first sense, and part of what makes quantization of [[gauge theory]] nontrivial is to find among all potential candidate observables those that actually are invariant under gauge transformations, i.e. under isomorphisms of [[principal bundle]]s with [[connection on a bundle|connection]] in the configuration space of the gauge theory.

### Global gauge group

The concept of gauge groups is most prominent in [[quantum field theory]], where the gauge group of a [[physical system]] is the [[group]] of transformations of the mathematical model of the system that do not correspond to any measurable [[physics|physical]] effects. In this sense, nontrivial gauge groups arise from redundancies of the mathematical description. Gauge groups are a central ingredient of [[gauge theories]].

In  [[AQFT]] gauge groups are introduced via a [[net of C-star-systems]].

### Local gauge groups

In [[Yang-Mills theory]] and other [[gauge theories]] the _gauge groups_ is the structure group $G$ of the $G$-[[principal bundle]] on which the [[Yang-Mills field]] is a [[connection on a bundle|connection]].

Local gauge groups are visible in the [[Lagrangian mechanics|Lagrangian]] approach to [[quantum field theory]], where they act on the configuration space on which the [[action functional]] is a function by [[gauge transformation]]s. A large machinery has been developed to handle the ([[path integral]]) [[quantization]] of [[action functional]]s on such configuration spaces. See for instance [[BV-BRST formalism]].

## Properties
 {#Properties}

### Not a redundancy
 {#NotARedundancy}

Sometimes one sees the statement being made that gauge symmetry in theories of [[physics]] is just a sign of a _redundancy_ in the theory, since, after all, gauge equivalent configurations are equivalent.

But, instead, there is genuine information contained in the gauge group of a physical theory: it encodes the [[homotopy type]] of the [[moduli space]] of configurations; or in other words: the higher [[homotopy group]]s in the [[∞-groupoid]] of [[configuration space|configurations]] of the system.  [[infinitesimal object|Infinitesimally]] this is given by the [[BRST complex]] of the system, and the nature of the gauge group controls its higher [[cochain cohomology|cohomology groups]]. For instance the degree-1 cohomology of the BRST complex (meaning: "ghost degree-1") of a system contains the possible gauge [[quantum anomalies]] (as discussed there) of the system. This is clearly not redundant information.

Abstractly speaking, the idea that symmetry is _just_ a redundancy is a mistake of [[decategorification]]: passing from a [[groupoid]] of [[configuration space|configurations]] -- where different configurations are related by [[morphism]]s called [[gauge transformation]]s -- to the [[quotient]] space of configurations modulo gauge transformations is the [[decategorification]] of the groupoid. More technically speaking, it is the _[[0-truncated|0-truncation]]_ . It computes the 0-th [[homotopy group]] and forgets all the higher homotopy groups.

While there is indeed some information extracted by this process, that about the gauge equivalence classes of configurations, other information is lost.

Physically speaking, notably the _locality_ of field theory would break down if one insisted on always passing to gauge equivalence classes of configurations.

For instance in a finite [[QFT]] such as [[Dijkgraaf-Witten theory]] with gauge group $G$, the [[moduli space]] is  the [[delooping]]/[[classifying space]] $\mathbf{B}G$. This space is [[connected]], reflecting the fact that _locally_ every configuration of DW-theory is gauge equivalent to every other: namely the field configurations are $G$-[[principal bundle]]s and locally on a piece of space $U$ these are all equivalent to the trivial principal bundle $U \times G$. If one insisted that only gauge equivalence classes of configuration contain non-redundant information, then one would find that either DW theory is not a local QFT or else that it contains the trivial configuration.

Entirely analogous comments apply to more interesting theories such as [[Yang-Mills theory]], only that there the moduli space is richer: it is the [[differential cohomology|differential refinement]] $\mathbf{B}G_{conn} := [P_1(-),\mathbf{B}G]$ of $\mathbf{B}G$ (see _[[connection on a bundle]]_ for details). Again, this cannot be reconstructed from its decategorification and if one insisted on passing to gauge equivalence classes of configurations then either Yang-Mills theory would no longer appear as a local QFT, or else it would have just the the globally trivial configurations.

To say this more precisely, here instead of "[[moduli space]]" we should be saying "[[moduli stack]]". The mathematical theory of [[stacks]] is the theory that deals with systems that are both _local_ (in that global configurations are glued together from local data) as well as equipped with (gauge) symmetries. In that more refined language of [[higher topos theory]] one would find and say that the [[0-truncated|0-truncation]] of $\mathbf{B}G_{conn}$ is the [[sheaf]] of gauge equivalence classes of [[Lie algebra valued differential form]]s. Insisting that this is the only relevant information amounts to insisting that either Yang-Mills theory is not a local theory, or else that all configurations of Yang-Mills theory are given by globally defined differential form data. This would exclude from the theory all configurations with nontrivial [[Chern class]]es, hence in particular it would exclude the [[instanton]] solutions from the theory, which are known to crucuially encode information about the [[quantum field theory|quantum theory]].

That all said, one should note that in _higher_ [[gauge theory]] redundancies may appear after all. This is related to the fact that in [[higher category theory]]/[[homotopy theory]] the notion of _[[resolution]]s_ becomes relevant: there are higher [[moduli stack]]s that look very rich on first sight but turn out to be [[equivalence in an (infinity,1)-category|equivalent]] to simpler moduli stacks.

For instance if $U(1) \to \hat G \to G$ is a [[central extension]] of the gauge group $G$, then there is a [[2-group]] denoted $(U(1) \to \hat G)$ (see _[[crossed module]]_) which is equivalent to $G$. Therefore 2-gauge theory (such as, say, the [[Courant sigma-model]]) with gauge 2-group $(U(1) \to \hat G)$ is in fact equivalent to ordinary $G$-gauge theory: there is indeed a lot of redundancy in $(U(1) \to \hat G)$. One can detect this by looking at the invariant information: the higher [[homotopy group]]s of $(U(1) \to \hat G)$ or else of its [[moduli stack]] (now a [[infinity-stack|2-stack]]) $\mathbf{B}(U(1) \to \hat G)$ are the same as that of $\mathbf{B}G$.

But even in such a situation, these redundant [[resolution]]s have a good use, in general as well as in applications to physics. In the above example the resolution serves to support an evident morphism $(U(1) \to \hat G) \to (U(1) \to 1) = \mathbf{B}U(1)$. Together with the equivalence to $G$ this constitutes an [[infinity-anafunctor|2-anafunctor]]

$$
  \array{
     \mathbf{B}(U(1) \to \hat G)
     &\to&
     \mathbf{B}^2 U(1)
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     \mathbf{B}G
  }
  \,.
$$

This structure exhibits a [[characteristic class]] of $G$-gauge theory (namely the class that classifies the [[group extension]]). The nontriviality of this class measures the failure of $G$-gauge theory to lift to $\hat G$-gauge theory.

A famous example of this in [[string theory]] comes from the case where $\hat G \to G$ is the projection $U(n) \to PU(n)$ from the [[unitary group]] to the [[projective unitary group]]. In this case a configuration of $PU(n)$-gauge theory is a [[twisted bundle]] representing a class in [[twisted K-theory]] as it appears on the worldvolume of [[D-brane]]s. The corresponding higher $\mathbf{B}U(1)$-gauge theory configuration is the [[Kalb-Ramond field]]([[B-field]]) restricted to the brane, which is the _twist_ that twists the bundle and prevents it from being a configuration in genuine $U(n)$-gauge theory.

## Examples
 {#Examples}

We list examples of local gauge groups and [[∞-groups]] for various higher [[gauge theories]].

* the gauge group of $G$-[[Yang-Mills theory]] is the given [[Lie group]] $G$; for the Yang-Mills theory appearing in the [[standard model of particle physics]] this is the [[unitary group]] $U(3) \times SU(2) \times U(1)$;

* the local gauge group of [[gravity]] on a [[manifold]] $X$ is the [[Poincare group]];

* the gauge [[2-group]] of the [[Kalb-Ramond field]] is the [[circle n-group|circle 2-group]] $\mathbf{B} U(1) = (U(1) \to 1)$;

* the gauge [[3-group]] of the [[supergravity C-field]] is the [[circle n-group|circle 3-group]] $\mathbf{B}^2 U(1) = (U(1) \to 1 \to 1)$;

* the gauge group of abelian [[higher dimensional Chern-Simons theory]] in dimension $4 k+3$ is the [[circle n-group|circle (2k+1)-group]] $\mathbf{B}^{2k} U(1)$;

* the 7-dimensional "[[differential fivebrane structure|fivebrane Chern-Simons theory]]" has as gauge 2-group the [[string 2-group]];

* the symmetry group of [[string field theory]] is some [[∞-group]] that is not an $n$-group for any finite $n$;

* an [[schreiber:∞-Chern-Simons theory]] has in general not only a gauge [[∞-group]] but an [[∞-groupoid]] of symmetries:

  * the [[Poisson sigma-model]] has as gauge groupoid the [[symplectic groupoid]] that is the [[Lie integration]] of the given [[Poisson Lie algebroid]];

  * the [[Courant sigma-model]] has as gauge [[2-groupoid]] the [[symplectic infinity-groupoid|symplectic 2-groupoid]] that integrates the given [[Courant Lie 2-algebroid]];

  * generally, the [[AKSZ sigma-model]] in grade $n$ has as gauge $\infty$-groupoid a [[symplectic infinity-groupoid|symplectic Lie n-groupoid]].

## Related concepts

* [[gauge]]

* [[structure group]], [[equivariance group]]

* [[gauge symmetry]]

* [[gauge fixing]]

* [[spontaneously broken symmetry]]

* [[enhanced gauge symmetry]]

## References

* Wikipedia: [gauge group](http://en.wikipedia.org/wiki/Gauge_group)

### Global gauge group

Chapter IV of

* [[Rudolf Haag]], _[[Local Quantum Physics]]_

### Local gauge group

(...)

### Local gauging of global symmetries in quantum gravity

It is being argued that after embedding into consistent [[quantum gravity]], all [[global symmetries]] must become [[local symmetries]].

A substantiation of this argument via [[AdS/CFT]] is given in:

* [[Daniel Harlow]], [[Hirosi Ooguri]], _Symmetries in quantum field theory and quantum gravity_ ([arXiv:1810.05338](https://arxiv.org/abs/1810.05338))

* [[Daniel Harlow]], [[Hirosi Ooguri]], _Constraints on symmetry from holography_, Phys. Rev. Lett. 122, 191601 (2019) ([arXiv:1810.05337](https://arxiv.org/abs/1810.05337))

See also:

* Sylvain Fichet, Prashant Saraswat, _Approximate Symmetries and Gravity_ ([arXiv:1909.02002](https://arxiv.org/abs/1909.02002))

[[!redirects gauge group]]
[[!redirects gauge groups]]

[[!redirects global gauge group]]
[[!redirects local gauge group]]
[[!redirects global gauge groups]]
[[!redirects local gauge groups]]


gauge invariance

#Contents#
* table of contents
{:toc}

## Idea

[[invariant|Invariance]] under [[gauge transformations]]. See there for details.

## Related concepts

* [[gauge theory]]

* [[group averaging]]

[[!redirects gauge invariances]]

[[!redirects gauge invariant]]


gauge invariant observables

This [[observables]] which are [[gauge invariance|gauge invariant]].

(...)


gauge orbit

The [[orbit]] of a [[gauge group]] [[action]].


gauge parameter

> under construction

+-- {: .num_defn #GaugeParametrizedInfinitesimalGaugeTransformation}
###### Definition
**([[gauge parameter|gauge parameterized]] implicit [[infinitesimal gauge transformations]])

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
Then a collection of _[[gauge parameters]]_ for $(E,\mathbf{L})$ is

1. a [[vector bundle]] $\mathcal{G} \overset{gb}{\longrightarrow} \Sigma$ over [[spacetime]] $\Sigma$;
   the [[sections]] $\epsilon$ of which are to be called the _[[gauge parameters]]_;

1. a [[bundle morphism]] $R$ from the [[fiber product]] of its [[jet bundle]] with that of the [[field bundle]]
   to the [[vertical tangent bundle]] of $E$:

   $$
     \array{
        J^\infty_\Sigma(\mathcal{G}) \times_\Sigma J^\infty_\Sigma(E)
         &\overset{R}{\longrightarrow}&
        V_\Sigma E
        \\
        \downarrow & \swarrow_{}
        \\
        E
     }
   $$

   such that

   1. $R$ is linear in the first argument.

   1. $R$ takes values in the sub-bundle of those [[evolutionary vector field|evolutionary vectors]] which are [[infinitesimal symmetries of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents});

   For every [[gauge parameter]] $\epsilon$ of [[compact support]] the composite of $R$ with the
   [[jet prolongation]] $j^\infty_\Sigma(\epsilon)$ (def. \ref{JetProlongation})

   $$
     v_\epsilon
      \;\colon\;
     J^\infty_\Sigma(E)
      =
     \Sigma \times_\Sigma J^\infty_\Sigma(E)
      \overset{(j^\infty_\Sigma(\epsilon), id)}{\longrightarrow}
     J^\infty_\Sigma(\mathcal{G}) \times_\Sigma J^\infty_\Sigma(E)
      \overset{R}{\longrightarrow}
     V_\Sigma E
   $$

   is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}).

If the [[field bundle]] $E$ is a [[trivial vector bundle]] with [[field (physics)|field]] coordinates
$(\phi^a)$ (example \ref{TrivialVectorBundleAsAFieldBundle})
and also $\mathcal{G}$ happens to be a [[trivial vector bundle]] equipped with [[fiber]] [[coordinates]] $(e^\alpha)$
then this mean that $v_\epsilon$ is of the form

$$
  v_\epsilon
    \;=\;
  \left(
    \epsilon^\alpha R^a_\alpha
    +
    \frac{d \epsilon^\alpha}{d x^\mu} R^{a \mu}_\alpha
    +
    \frac{d^2 \epsilon^\alpha}{d x^{\mu_1} d x^{\mu_2}} R^{a \mu_1 \mu_2}_\alpha
    +
    \cdots
  \right)
  \partial_{\phi^a}
  \,,
$$

where the $R^{a \mu_1 \cdots \mu_k}_\alpha$ are smooth functions on the jet bundle of $E$ (prop. \ref{JetBundleIsLocallyProManifold}).

=--

[[!redirects gauge parameters]]

[[!redirects gauge parameter bundle]]
[[!redirects gauge parameter bundles]]


gauge potential

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> see also at *[[electromagnetic potential]]*

#Contents#
* table of contents
{:toc}

## Idea

In [[gauge theory]] in [[physics]] a [[field (physics)|field configuration]] is modeled, in [[mathematics]]-terminology by a [[connection on a bundle|connection]] $\nabla$ on a [[principal bundle]] $P$ (or some [[associated bundle]]). In [[physics]]-terminology the connection $\nabla$ is sometimes called the **gauge potential** (while the [[equivalence class]] of $P$ is called the [[instanton sector]] and the [[curvature]] $F_\nabla$ is called the [[field strength]]).

Specfically in _[[electromagnetism]]_ one also speaks of [[electromagnetic potential]] or _[[vector potential]]_.

## Related concepts

* [[electromagnetic potential]]

* [[Aharonov-Bohm effect]]

[[!include gauge field - table]]

## References

The understanding of gauge potentials as [[connection on a bundle|connections]] on [[fiber bundles]] is due to

The identification of [[gauge potentials]] with [[connections]] on [[fiber bundles]] is due to:

* {#WuYang75} [[Tai Tsun Wu]], [[Chen Ning Yang]], *Concept of nonintegrable phase factors and global formulation of gauge fields*, Phys. Rev. D **12** (1975) 3845 [[doi:10.1103/PhysRevD.12.3845](https://doi.org/10.1103/PhysRevD.12.3845)]

See also at *[[fiber bundles in physics]]*.

History:

* {#WuYang06} A. C. T. Wu, [[Chen Ning Yang]], *Evolution of the concept of vector potential in the description of the fundamental interactions*,  International Journal of Modern Physics A **21** 16 (2006) 3235-3277 [[doi:10.1142/S0217751X06033143](https://doi.org/10.1142/S0217751X06033143)]

* {#Yang14} [[Chen Ning Yang]], *The conceptual origins of Maxwell’s equations and gauge theory*, Phyics Today **67** 11 (2014) [[doi:10.1063/PT.3.2585](https://doi.org/10.1063/PT.3.2585), [pdf](http://home.ustc.edu.cn/~lxsphys/2021-3-18/The%20conceptual%20origins%20of%20Maxwell%27s%20equations%20and%20gauge%20theory.pdf)]

Review:

* [[Theodore Frankel]], Section 5 of: _[[The Geometry of Physics - An Introduction]]_

* [[geometry of physics]], _[The electromagnetic field strength](geometry+of+physics#ElectromagneticFieldStrength)_

* Wikipedia, *[Magnetic vector potential](https://en.wikipedia.org/wiki/Magnetic_vector_potential)*

[[!redirects gauge potentials]]


gauge reduction

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

## Idea

Given a [[presymplectic manifold]] $(X,\omega)$, the [[quotient]] (if it exists suitably) by the [[flow]] of the [[kernel]] of the [[presymplectic form]] $\omega : T X \to T^* X$ is the [[symplectic manifold]] obtained by _gauge reduction_. Also called _[[presymplectic reduction]]_

In the interpretation in [[physics]] this takes one from a [[covariant phase space]] to a _[[reduced phase space]]_.

In constrast, in _[[symplectic reduction]]_ one takes the quotient of a pre-specified [[Hamiltonian action]] _and_ restricts to the 0-locus of the corresponding [[Hamiltonians]] (the [[momentum map]]).

## Related concepts

[[!include symplectic reduction - table]]

* [[quantization commutes with reduction]]

## References

Around page 11 of

* A. Echeverría-Enríquez, M.C. Muñoz-Lecanda, N. Román-Roy, _Reduction of Presymplectic Manifolds with Symmetry_ ([arXiv:math-ph/9911008](http://arxiv.org/abs/math-ph/9911008))
 {#EMR}


gauge space

# Gauge spaces
* table of contents
{: toc}

## Idea

A gauge space is a [[topological space]] (necessary completely regular) whose topology is given by a family of [[metric space|pseudometrics]].  More generally, a quasigauge space is a space (not necessarily completely regular) whose topology is given by a family of quasipseudometrics.

Actually, a gauge space has additional structure, so that it can be seen as giving a (completely regular) [[Cauchy space]], a [[uniform space]], or even a generalisation of a [[metric space]] in which the category $Met$ of metric spaces and short maps is a [[full subcategory]].

Please note that, while this is based on the presentation in _[[HAF]]_, the precise definitions of the objects and morphisms of the [[category]] of gauge spaces below constitute original research.  (In particular, _HAF_ really considers the category of pregauge spaces and uniformly continuous maps, which is equivalent to the category of [[uniform spaces]], since it uses them only to study that category.)

## Definitions

Given a [[set]] $X$, a __pregauge__ on $X$ is simply a family of [[metric space|pseudometrics]] on $X$.  A __gauge__ is a $\geq$-[[filter]] of pseudometrics on $X$, that is a collection $G$ of __gauging distances__ such that

1.  There is an element of $G$; in the light of (3), the zero pseudometric $( x, y \mapsto 0 )$ is a gauging distance.
2.  Given $d, e \in G$, some $f \in G$ satisfies
    $$ d(x,y), e(x,y) \leq f(x,y) $$
    for all $x, y$ in $X$; in the light of (3), the pseudometric $( x, y \mapsto max(d(x,y), e(x,y)) )$ is a gauging distance.
3.  Given $d \in G$ and any pseudometric $e$ on $X$, if
    $$ e(x,y) \leq d(x,y) $$
    for all $x, y$ in $X$, then $e \in G$.

A pregauge satisfying axioms (1&2) is a __[[base]]__ for a gauge; a base is precisely what generates a gauge by taking the downward closure.  Any pregauge whatsoever is a __[[subbase]]__ for a gauge; a subbase is precisely what generates a base by closing under finitary [[join]]s.

A __gauge space__ is a set equipped with a gauge.  A __quasigauge__ is a collection of quasipseudometrics satisfying (1--3); a __quasigauge space__ is a set equipped with a quasigauge.

Given (quasi)gauge spaces $X$ and $X'$, a __[[short map]]__ from $X$ to $X'$ is a [[function]] $F$ (on their underlying sets) such that the [[composite]] with $F$ (or with $F \times F$, to be precise) of any gauging distance on $X'$ is a gauging distance on $X$.  That is,

*  for every $e \in G'$, there is a $d \in G$ such that
   $$ e(F(x),F(y)) \leq d(x,y) $$
   for all $x, y$ in $X$; in the light of (3), the pseudometric $( x, y \mapsto e(F(x),F(y)) )$ (this is the composite) is a gauging distance.

(Warning: this definition of short map is probably the most significant original research on this page.)

Gauge spaces and short maps between them form the [[category]] $Gau$ of gauge spaces; quasigauge spaces and short maps between them form the category $QGau$ of quasigauge spaces.  Note that any gauge is a base for a quasigauge; in this way, $Gau$ is ([[equivalence of categories|equivalent to]]) a [[full subcategory]] of $QGau$.

## Examples

The categories $Gau$ and $QGau$ are not well known, but some of their subcategories are.

*  A [[metric space]] (or pseudometric space) defines a gauge space, taking its single (pseudo)metric as a base; similarly, a quasi(psuedo)metric space defines a quasigauge space.  Taking short maps (in the usual sense, that is distance-nonincreasing functions) as morphisms, the categories $Met$ and $PsMet$ are full subcategories of $Gau$; similarly, $QMet$ and $QPsMet$ are full subcategories of $QGau$.

*  A [[uniform space]] $X$ defines a gauge space, consisting of all of the pseudometrics on $X$ that are uniformly continuous as maps from $X \times X$ to the real line; similarly, a quasiuniform space defines a quasigauge space.  Taking uniformly continuous maps as morphisms, the category $Unif$ is a full subcategory of $Gau$; similarly, $QUnif$ is a full subcategory of $QGau$.  Thus, uniform spaces can be viewed as gauge spaces whose collection of gauging distances is "saturated" under uniform continuity.

*  A completely regular [[topological space]] $X$ defines a gauge space, consisting of all the pseudometrics on $X$ that are continuous as maps from $X \times X$ to the real line.  In this way the category $CReg Top$ of completely regular spaces and continuous maps is a full subcategory of $Gau$; it is in fact contained in $Unif$.  (In general, a completely regular space can be uniformized in many ways; this inclusion corresponds to the "initial" uniformity.)

*  An arbitrary topological space defines a quasigauge space in a more complicated way.  Given any open set $U$ in a space $X$, define the pseudometric (in fact a pseudoultrametric) $d_U$ as follows:
   $$ d_U(x,y) = \begin{cases} 0 & if\; x \in U \;\Rightarrow\; y \in U ,\\ 1 & if\; x \in U \;\wedge\; y \notin U ;\end{cases} $$
   then the $d_U$ form a base for a quasigauge, which induces the original topology on $X$.  In other words, every space is "quasigaugeable."  In this way $Top$ also becomes a full subcategory of $QGau$.  Note, though that replacing $1$ by any other positive real number gives a _different_ embedding of $Top$ in $QGau$.

*  Another way to get a quasigauge space from a topological space is to take as a base the set of all quasi-pseudometrics $d$ on $X$ such that for each $x\in X$, the function $d(x,-):X\to \mathbb{R}$ is [[semicontinuous function|upper semicontinuous]].  This is also a full embedding of $Top$ in $QGau$, which is perhaps more "canonical."

## Reflections

Many of these full subcategories of $Gau$ and $QGau$ are [[reflective subcategory|reflective]].

>Details to come

## References

*  Eric Schechter; 1997; [[HAF|Handbook of Analysis and its Foundations]]; Academic Press, 1st ed (1996).

* {#ShulmanSOI} [[Mike Shulman]], *The shape of infinity*, [arxiv](https://arxiv.org/abs/1209.2735)

[[!redirects gauge space]]
[[!redirects gauge spaces]]

[[!redirects quasigauge space]]
[[!redirects quasigauge spaces]]

[[!redirects Gau]]

[[!redirects QGau]]


gauge theory

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#Contents#
* table of contents
{:toc}

## Idea

A _gauge theory_ may denote either a [[classical field theory]] or a [[quantum field theory]] whose [[space of trajectories|field configurations]] are [[cocycles]] in [[differential cohomology]] ([[ordinary differential cohomology|abelian]] or [[Chern-Weil theory|nonabelian]]).

### Ordinary gauge theories

An _ordinary gauge theory_ is a [[quantum field theory]] whose field configurations are [[vector bundles]]
[[connection on a bundle|with connection]].

This includes notably the [[field (physics)|fields]] that carry the three fundamental forces of the [[standard model of particle physics]]:

* Ordinary [[electromagnetism]] in the absence of magnetic charges is a gauge theory of $U(1)$-[[principal bundle|principal bundles]] with [[connection on a bundle|connection]].

* Fields in [[Yang-Mills theory]] (such as appearing in the [[standard model of particle physics]] and in [[GUT|GUTs]]) are [[vector bundle|vector bundles]] [[connection on a bundle|with connection]].

Other examples include formal physical models.

* [[Dijkgraaf-Witten theory]] is a gauge theory whose field configurations are $G$-[[principal bundle|principal bundles]] for $G$ a finite group (these come with a unique [[connection on a bundle|connection]], so that in this simple case the connection is no extra datum).

The group $G$ in these examples is called the [[gauge group]] of the theory.

### Higher and generalized gauge theories

The above examples of gauge fields consisted of cocycles in degree-$1$ [[differential cohomology]].

More generally, a _[[higher gauge theory]]_ is a [[quantum field
theory]] whose field configurations are cocycles in more general [[differential cohomology]], for instance higher degree [[Deligne cohomology|Deligne cocycles]] or more generally cocycles in other differential refinements, such as in differential [[K-theory]].

This generalization does contain experimentally visible physics such as

* The [[magnetic current]] in [[electromagnetism]] is a [[bundle gerbe|bundle gerbe with connection]], a [[Deligne cohomology|Deligne cocycle]] refining a cocycle in degree-$3$ [[Eilenberg-MacLane spectrum|Eilenberg-MacLane cohomology]]: the _magnetic charge_ .

But a whole tower of higher and generalized gauge theories became visible with the study of higher [[supergravity]] theories,

* The [[Kalb-Ramond field]] is a [[bundle gerbe|bundle gerbe with connection]], a [[Deligne cohomology|Deligne cocycle]] with curvature 3-form.

* The [[supergravity C-field]] is a [[Deligne cohomology|Deligne cocycle]] with curvature 4-form.

* The [[RR-field]] is a cocycle in [[differential K-theory]].

### Gravity as a (non-)gauge theory

In the [[first order formulation of gravity]] also the theory of [[gravity]] looks a little like a gauge theory. However, there is a crucial difference. What really happens here is _[[Cartan geometry]]_: the field of gravity may be encoded in a [[vielbein]] field, namelely an [[orthogonal structure]] on the [[tangent bundle]], hence as an example of a [[G-structure]], and the [[torsion of a G-structure|torsion freedom]] of this G-structure may be encoded by an auxiliary [[connection on a bundle|connection]], namely a [[Cartan connection]], often called the "[[spin connection]]" in this context. Hence while in the formulation of [[Cartan geometry]] gravity is described by many of the ingredients from [[differential geometry]] that also govern pure gauge theory, it's not quite the same. In particular there is a constraint on a [[Cartan connection]], which in terms of [[vielbein]] fields is the constraint that the vielbein (which is part of the Cartan connection) is non-degenerate, and hence really a "[[soldering form]]". Such a constraint is absent in a "genuine" gauge theory such as [[Yang-Mills theory]] or [[Chern-Simons theory]].

## Properties

### Non-redundancy and locality

Sometimes one see the view expressed that [[gauge symmetry]] is "just a redundancy" in the description of a [[theory (physics)|theory of physics]], for instance in that among [[observables]] it is only the [[gauge invariance|gauge invariant]] ones which are physically meaningful.

This statement however

### Anomalies

In the presence of [[magnetic charge]] (and then even in the absence of [[chiral fermion anomaly|chiral fermion anomalies]]) the standard would-be action functional for higher gauge theories may be ill-defined. The [[Green-Schwarz mechanism]] is a famous phenomenon in [[differential cohomology]] by which such a [[quantum anomaly]] cancels against that given by chiral fermions.

## List of gauge fields and their models

The following tries to give an overview of some collection of gauge fields in physics, their models by [[differential cohomology]] and further details.

* [[Yang-Mills field]]

  * cocycle in lowest degree [[schreiber:Differential Nonabelian Cohomology|nonabelian differential cohomology]]

    * originally realized in terms of differential [[Čech cohomology|Čech cocycles]]
      $$
        \hat F \in \mathbf{H}(X, \bar \mathbf{B} G)
      $$
      with coefficients in the [[groupoid of Lie-algebra valued forms]],

      then traditionally in terms of [[vector bundle|vector bundles]] [[connection on a bundle|with connection]]

  * [[field strength]] depending on the [[group]] $G$ we have

    * $G = U(1)$ - [[electromagnetism]] (see below)

    * $G = SU(2)\times U(1)$ - electroweak force field strength

    * $G = SU(3)$ - strong nuclear force field

  * [[parallel transport]]: [[Wilson line|Wilson lines]]

* [[electromagnetic field]]

  * cocycle in degree-$2$ [[Eilenberg-MacLane spectrum|ordinary]] [[differential cohomology]]

    * naturally/historically realized in terms of Maxwell-Dirac presentation as a cocycle in [[Čech cohomology|Čech]]--[[Deligne cohomology|Deligne]] cocycle
      $$
        \hat F \in \mathbf{H}(X,\bar \mathbf{B} U(1))
      $$

  * [[field strength]]: the electric field $E$ and magnetic field $B$, locally at a point $x \in X$
    $$
      F = E \wedge d t + \star_3 B
    $$

  * on $X = \mathbb{R}^3\backslash \{0\}$: underlying class in integral cohomology $cl(\hat F) \in H(X,\mathbf{B} U(1)) \simeq H^2(X,\mathbb{Z})$ is the [[magnetic charge]]

  * [[parallel transport]]: gauge interaction piece of [[path integral|action functional]] of the electrically charged quantum 1-particle

* [[Kalb-Ramond field]]

  * cocycle in degree-$3$ [[Eilenberg-MacLane spectrum|ordinary]] [[differential cohomology]]

    * naturally/historically realized in terms of

      * a cocycle in [[Čech cohomology|Čech]]--[[Deligne cohomology|Deligne]] cocycle

        $$
          \hat H \in \mathbf{H}(X,\bar \mathbf{B}^2 U(1))
        $$

      * a [[bundle gerbe]] with connection

  * [[field strength]]: $H \in \Omega^3(X)$ the "$H$-field" -- on a D-[[brane]] this is the [[magnetic current]] for the  [[Yang-Mills field]] on the [[brane]]

  * [[parallel transport]]: gauge interaction piece of [[path integral|action functional]] of the electrically charged quantum 2-particle (the [[string theory|string]]).

* [[supergravity C-field]]

  * cocycle in degree-$4$ [[Eilenberg-MacLane spectrum|ordinary]] [[differential cohomology]]

    * naturally/historically realized in terms of as a cocycle in [[Čech cohomology|Čech]]--[[Deligne cohomology|Deligne]] cocycle
      $$
        \hat H \in \mathbf{H}(X,\bar \mathbf{B}^3 U(1))
      $$

      using the [[D'Auria-Fre formulation of supergravity]] it may also be thought of as a [[schreiber:theory of differential nonabelian cohomology|nonabelian differential cocycle]] given by a [[schreiber:Cartan-Ehresmann ∞-connection|Cartan-Ehresmann ∞-connection]]

  * [[field strength]]: $H \in \Omega^4(X)$ the "$G$-field" -- in heterotic [[supergravity]] this is the 5-brane [[magnetic current]] for the  [[twisted cohomology|twisted]] [[Kalb-Ramond field]]

  * [[parallel transport]]: gauge interaction piece of [[path integral|action functional]] of the electrically charged quantum 3-particle (the [[brane|membrane]]).

* [[RR field]]

  * cocycle in [[differential K-theory]]

    * in presence of nontrivial [[Kalb-Ramond field]]: cocycle in differential [[twisted K-theory]]

  * [[field strength]]: RR-forms

## Related concepts

* [[fiber bundles in physics]]

* [[gauge]]

* [[gauge group]]

* [[gauge transformation]], [[higher gauge transformation]]

* [[BRST complex]], [[BV-BRST formalism]]

* [[ghost field]], [[ghost-of-ghost field]]

* [[gauge fixing]], [[gauge fixing fermion]], [[gauge invariant]]

* [[lattice gauge theory]]

* [[Gribov ambiguity]]

* [[quiver gauge theory]]

* [[arithmetic gauge theory]]

[[!include gauge field - table]]

* [[higher U(1)-gauge theory]]

  * [[higher electric background charge coupling]]

* [[self-dual higher gauge theory]]

* [[higher spin gauge theory]]

* [[quantum anomaly]]

  * [[Green-Schwarz mechanism]]

* [[schreiber:infinity-Chern-Simons theory]]

* [[free field theory]]

## References

### General

Historical origins:

* A. C. T. Wu, [[Chen Ning Yang]], *Evolution of the concept of vector potential in the description of the fundamental interactions*,  International Journal of Modern Physics A **21** 16 (2006) 3235-3277 [[doi:10.1142/S0217751X06033143](https://doi.org/10.1142/S0217751X06033143)]

* [[Chen Ning Yang]], *The conceptual origins of Maxwell’s equations and gauge theory*, Phyics Today **67** 11 (2014) [[doi:10.1063/PT.3.2585](https://doi.org/10.1063/PT.3.2585), [pdf](http://home.ustc.edu.cn/~lxsphys/2021-3-18/The%20conceptual%20origins%20of%20Maxwell%27s%20equations%20and%20gauge%20theory.pdf)]

Textbook accounts:

* [[Yuri Makeenko]], *Methods of contemporary gauge theory*, Cambridge Monographs on Math. Physics, Cambridge University Press (2002) [[doi:10.1017/CBO9780511535147]( https://doi.org/10.1017/CBO9780511535147), [gBooks](http://books.google.com/books?id=9W-W2w75ulAC)]

* [[Mikio Nakahara]], Section 10.5 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

* Mike Guidry, _Gauge Field Theories: An Introduction with Applications_, Wiley 2008 ([ISBN:978-3-527-61736-4](https://www.wiley.com/en-us/Gauge+Field+Theories%3A+An+Introduction+with+Applications-p-9783527617364))

Basics on [[fiber bundles in physics]] are recalled in

* Adam Marsh, _Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results_ ([arXiv:1607.03089](https://arxiv.org/abs/1607.03089))

An introduction to concepts in the [[quantization]] of gauge theories is in

* [[Nicolai Reshitikhin]], _Lectures on quantization of gauge systems_ ([pdf](http://staff.science.uva.nl/~nresheti/Holb-Quant-Gauge.pdf))

A standard textbook on the [[BV-BRST formalism]] for the quantization of gauge systems is in

* [[Marc Henneaux]], [[Claudio Teitelboim]], *[[Quantization of Gauge Systems]]*, Princeton University Press (1992) [[doi:10.2307/j.ctv10crg0r](https://doi.org/10.2307/j.ctv10crg0r)]

Comprehensive lecture notes on this are at

* _[[geometry of physics -- perturbative quantum field theory]]_.

Discussion of abelian higher gauge theory in terms of [[differential cohomology]] is in

* [[Dan Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_

* [[Alessandro Valentino]], _Differential cohomology and quantum gauge fields_ ([pdf](http://www.uni-math.gwdg.de/sandro/seminarGoett.pdf))

* [[José Figueroa-O'Farrill]], _Gauge theory_ ([web page](http://empg.maths.ed.ac.uk/Activities/GT/index.html))

* [[Tohru Eguchi]], Peter Gilkey, Andrew Hanson, _Gravitation, gauge theories and differential geometry_, Physics Reports __66__:6 (1980) 213—393 ([pdf](http://empg.maths.ed.ac.uk/Activities/GT/EGH.pdf))

For discussion in the context of [[gravity]] see also

* {#Witten17} [[Edward Witten]], _Symmetry and Emergence_ ([arXiv:1710.01791](https://arxiv.org/abs/1710.01791))

### In AQFT
 {#ReferencesInAQFT}

Standard discussion of gauge theory in the context of [[algebraic quantum field theory]] ([[AQFT]]) includes

* [[Franco Strocchi]], section 4 of _Relativistic Quantum Mechanics and Field Theory_, Found.Phys. 34 (2004) 501-527 ([arXiv:hep-th/0401143](http://arxiv.org/abs/hep-th/0401143))

For [[AQFT on curved spacetimes]] the axioms of AQFT need to be promoted to a context of [[higher geometry]] unless [locality](field+%28physics%29#IdeaLocality) is broken, see the expositions at

* _[[schreiber:Higher field bundles for gauge fields]]_

* {#Schenkel14} [[Alexander Schenkel]], _On the problem of gauge theories
in locally covariant QFT_, talk at _[Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html)_ Trento, 2014 ([web](field+bundle#Schenkel14))

This was established in

* {#BDS} Marco Benini, [[Claudio Dappiaggi]], [[Alexander Schenkel]], _Quantized Abelian principal connections on Lorentzian manifolds_, Communications in Mathematical Physics 2013 ([arXiv:1303.2515](http://arxiv.org/abs/1303.2515))

and the program of improving the axioms of [[AQFT on curved spacetimes]] to the [[stack|stacky]] context in order to accomodate gauge theory includes the following articles:

* {#BeniniSchenkelSzabo15} [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Homotopy colimits and global observables in Abelian gauge theory_ ([arXiv:1503.08839](http://arxiv.org/abs/1503.08839))

* [[Marco Benini]], [[Alexander Schenkel]], _Quantum field theories on categories fibered in groupoids_ ([arXiv:1610.06071](https://arxiv.org/abs/1610.06071))

* [[Marco Benini]], [[Alexander Schenkel]], [[Urs Schreiber]], _The stack of Yang-Mills fields on Lorentzian manifolds_ ([arXiv:1704.01378](https://arxiv.org/abs/1704.01378))

### Dualities

An exposition of the relation to [[geometric Langlands duality]] is in

* [[Edward Frenkel]], _Gauge theory and Langlands duality_ ([pdf](http://math.berkeley.edu/~frenkel/bourbaki-gauge.pdf))

### History
 {#History}

A discussion of "[[gauge]]" and [[gauge transformation]] in [[metaphysics]] is in

* [[Georg Hegel]], [§714](Science+of+Logic#714) of _[[Science of Logic]]_, 1812

[[Hermann Weyl]]'s historical argument motivating gauge theory in [[physics]] from rescaling of units of length was given in 1918 in

* [[Hermann Weyl]], _Raum, Zeit, Materie: Vorlesungen über die Allgemeine Relativitätstheorie_, Springer Berlin Heidelberg 1923

  > The manuscript of Weyl's first book on mathematical physics, Space – Time – Matter (STM) (Raum – Zeit – Materie), delivered to the publishing house (Springer) Easter 1918, did not contain Weyl's new geometry and proposal for a UFT. It was prepared from the lecture notes of a course given in the Summer semester of 1917 at the Polytechnical Institute (ETH) Zürich. Weyl included his recent findings only in the 3rd edition (1919) of the book. The English and French versions (Weyl 1922b, Weyl 1922a), translated from the fourth revised edition (1921), contained a short exposition of Weyl's generalized metric and the idea for a scale gauge theory of electromagnetism. ([Scholz](#Scholz))

See

* {#Scholz} Erhard Scholz, _H. Weyl's and E. Cartan's proposals for infinitesimal geometry in the early 1920s_ ([pdf](http://www2.math.uni-wuppertal.de/~scholz/preprints/weyl_cartan.pdf))

Early surveys include

* {#Iliopoulos75} [[John Iliopoulos]], _Progress in Gauge Theories_, 1975 ([spire:3000](http://inspirehep.net/record/3000))

Quick reviews include

* Quigley, _On the origins of gauge theory_ ([pdf](http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf))

* Afriat, _Weyl's gauge argument_ ([pdf](http://hal.inria.fr/docs/00/76/96/90/PDF/WGA.pdf))

More comprehensive historical accounts include

* [[Lochlainn O'Raifeartaigh]], _The Dawning of Gauge Theory_, Princeton University Press (1997)

* [[Lochlainn O'Raifeartaigh]], Norbert Straumann, _Gauge Theory: Historical Origins and Some Modern Developments_ Rev. Mod. Phys. 72, 1-23 (2000).

* [[Norbert Straumann]], _Early History of Gauge Theories and Weak Interactions_ ([arXiv:hep-ph/9609230](http://arxiv.org/abs/hep-ph/9609230))

* [[Norbert Straumann]], _Gauge principle and QED_, talk at PHOTON2005, Warsaw (2005) ([arXiv:hep-ph/0509116](http://arxiv.org/abs/hep-ph/0509116))

[[!redirects gauge theories]]

[[!redirects gauge field theory]]
[[!redirects gauge field theories]]

[[!redirects gauge principle]]

[[!redirects gauge field]]
[[!redirects gauge fields]]


gauge theory from AdS-CFT -- table

**[[gauge theory]] induced via [[AdS-CFT correspondence]]**

| **[[M-theory]] perspective via [[AdS7-CFT6]]** | [[F-theory]] perspective |
|--|--|
| [[11d supergravity]]/[[M-theory]] |  |
| $\;\;\;\;\downarrow$ [[Kaluza-Klein compactification]] on $S^4$ |  compactificationon [[elliptic fibration]] followed by [[T-duality]] |
| [[7-dimensional supergravity]] |  |
| $\;\;\;\;\downarrow$ topological sector |  |
| [[7-dimensional Chern-Simons theory]] |  |
| $\;\;\;\;\downarrow $ [[AdS-CFT|AdS7-CFT6 holographic duality]] |  |
| [[6d (2,0)-superconformal QFT]] on the [[M5-brane]] with [[CFT|conformal invariance]] | [[M5-brane]] worldvolume theory |
| $\;\;\;\; \downarrow$ [[KK-compactification]] on [[Riemann surface]] | [[double dimensional reduction]] on [[M-theory]]/[[F-theory]] [[elliptic fibration]]  |
| [[N=2 D=4 super Yang-Mills theory]] with [[Montonen-Olive duality|Montonen-Olive]] [[S-duality]] invariance; [[AGT correspondence]] | [[D3-brane]] worldvolume theory with type IIB [[S-duality]]  |
| $\;\;\;\;\; \downarrow$ [[topological twist]] |  |
| [[topologically twisted N=2 D=4 super Yang-Mills theory]] |  |
| $\;\;\;\; \downarrow$ [[KK-compactification]] on [[Riemann surface]] |  |
| [[A-model]] on $Bun_G$, [[Donaldson theory]] |  |

$\,$

| **[[gauge theory]] induced via [[AdS5-CFT4]]** |
|--|
| [[type II string theory]] |
| $\;\;\;\;\downarrow$ [[Kaluza-Klein compactification]] on $S^5$ |
| $\;\;\;\; \downarrow$ topological sector |
| [[5-dimensional Chern-Simons theory]] |
| $\;\;\;\;\downarrow $ [[AdS-CFT|AdS5-CFT4 holographic duality]] |
| [[N=4 D=4 super Yang-Mills theory]] |
| $\;\;\;\;\; \downarrow$ [[topological twist]] |
| [[topologically twisted N=4 D=4 super Yang-Mills theory]] |
| $\;\;\;\; \downarrow$ [[KK-compactification]] on [[Riemann surface]] |
| [[A-model]] on $Bun_G$ and [[B-model]] on $Loc_G$, [[geometric Langlands correspondence]] |

[[!redirects gauge theory from AdS7-CFT6 -- table]]


gauge theory from AdS5-CFT4 -- table

| **[[gauge theory]] induced via [[AdS5-CFT4]]** |
|--|
| (see also _[[gauge theory from AdS7-CFT6 -- table]]_) |
| [[type II string theory]] |
| $\;\;\;\;\downarrow$ [[Kaluza-Klein compactification]] on $S^5$ |
| $\;\;\;\; \downarrow$ topological sector |
| [[5-dimensional Chern-Simons theory]] |
| $\;\;\;\;\downarrow $ [[AdS-CFT|AdS5-CFT4 holographic duality]] |
| [[N=4 D=4 super Yang-Mills theory]] |
| $\;\;\;\;\; \downarrow$ [[topological twist]] |
| [[topologically twisted N=4 D=4 super Yang-Mills theory]] |
| $\;\;\;\; \downarrow$ [[KK-compactification]] on [[Riemann surface]] |
| [[A-model]] on $Bun_G$ and [[B-model]] on $Log_G$, [[geometric Langlands correspondence]] |