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\section*{confinement}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{OpenProblem}{Open problem}\dotfill \pageref*{OpenProblem} \linebreak
\noindent\hyperlink{potential_solutions}{Potential solutions}\dotfill \pageref*{potential_solutions} \linebreak
\noindent\hyperlink{ViaCalorons}{Via Calorons}\dotfill \pageref*{ViaCalorons} \linebreak
\noindent\hyperlink{via_skyrmions_and_d4brane_models}{Via Skyrmions and D4-brane models}\dotfill \pageref*{via_skyrmions_and_d4brane_models} \linebreak
\noindent\hyperlink{InN2SYM}{In $\mathcal{N}=2$ super Yang-Mills theory via Seiberg-Witten theory}\dotfill \pageref*{InN2SYM} \linebreak
\noindent\hyperlink{ForN1SYMViaMTheoryOnG2Manifolds}{In $\mathcal{N} = 1$ super Yang-Mills theory via M-theory on $G_2$-manifolds}\dotfill \pageref*{ForN1SYMViaMTheoryOnG2Manifolds} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{ReferencesInYangMillsTheory}{In Yang-Mills theory}\dotfill \pageref*{ReferencesInYangMillsTheory} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ViaMonopoleCondensation}{Via monopole condensation}\dotfill \pageref*{ViaMonopoleCondensation} \linebreak
\noindent\hyperlink{interacting_field_vacuum}{Interacting field vacuum}\dotfill \pageref*{interacting_field_vacuum} \linebreak
\noindent\hyperlink{ReferencesInSuperYangMillsTheory}{In super-Yang-Mills theory}\dotfill \pageref*{ReferencesInSuperYangMillsTheory} \linebreak
\noindent\hyperlink{under_the_adscft_correspondence}{Under the AdS/CFT correspondence}\dotfill \pageref*{under_the_adscft_correspondence} \linebreak
\noindent\hyperlink{in_mtheory_on_manifolds}{In M-theory on $G_2$-manifolds}\dotfill \pageref*{in_mtheory_on_manifolds} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Confinement (e.g \hyperlink{Espiru94}{Espiru 94}) is the (expected) phenomenon in [[Yang-Mills theory]] generally and especially in [[quantum chromodynamics]] that the fundamental [[quarks]], which the [[Yang-Mills theory|YM]]/[[QCD]]-[[Lagrangian density]] actually describes, must form [[baryon|baryonic]] [[bound state|bound states]] which are neutral under the color charge -- the [[mesons]] and [[hadron|hadrons]] ([[proton|protons]], [[neutron|neutrons]]). Hence confinement in particular concerns the emergence and existence of [[atomic nuclei]], hence of ordinary [[matter]], which is not manifest at all in the [[quark]]-model.

Part of the issue is that confinement is a [[non-perturbative effect]] (e.g \hyperlink{Espiru94}{Espiru 94}) outside the range of validity of [[perturbative quantum field theory]].

\hypertarget{OpenProblem}{}\subsection*{{Open problem}}\label{OpenProblem}

While [[experiment]] as well as [[lattice gauge theory]]-computer simulation clearly show that confinement takes place, a real theoretical understanding is missing (see also at \emph{[[mass gap problem]]}).

Here are quotes from some references highlighting the open problem:

\begin{itemize}%
\item Robert Kutschke, section 3.1 \emph{Heavy flavour spectroscopy}, in D. Bugg (ed.), \emph{Hadron Spectroscopy and the Confinement Problem}, Proceedings of a NATO Advanced Study Institute, Plenum Press 1996 (\href{https://www.springer.com/cn/book/9780306453038}{doi:10.1007/978-1-4613-0375-6})

\end{itemize}
\begin{quote}%
While it is generally believed that QCD is the correct fundamental theory of the strong interactions there are, as yet, no practical means to produce full QCD calculations of hadron masses and their decay widths.


\end{quote}
\begin{itemize}%
\item Jeff Greensite, \emph{An Introduction to the Confinement Problem}, Lecture Notes in Physics, Volume 821, 2011 (\href{https://link.springer.com/book/10.1007%2F978-3-642-14382-3}{doi:10.1007/978-3-642-14382-3})

\end{itemize}
\begin{quote}%
Because of the great importance of the standard model, and the central role it plays in our understanding of particle physics, it is unfortunate that, in one very important respect, we don’t really understand how it works. The problem lies in the sector dealing with the interactions of quarks and gluons, the sector known as Quantum Chromodynamics or QCD. We simply do not know for sure why quarks and gluons, which are the fundamental fields of the theory, don’t show up in the actual spectrum of the theory, as asymptotic particle states. There is wide agreement about what must be happening in high energy particle collisions: the formation of color electric flux tubes among quarks and antiquarks, and the eventual fragmentation of those flux tubes into mesons and baryons, rather than free quarks and gluons. But there is no general agreement about why this is happening, and that limitation exposes our general ignorance about the workings of non-abelian gauge theories in general, and QCD in particular, at large distance scales.


\end{quote}
\begin{itemize}%
\item Csaba Csaki, [[Matthew Reece]], \emph{Toward a Systematic Holographic QCD: A Braneless Approach}, JHEP 0705:062, 2007 (\href{https://arxiv.org/abs/hep-ph/0608266}{arxiv:hep-ph/0608266})

(in motivation of [[Ads/QCD]])



\end{itemize}
\begin{quote}%
QCD is a perennially problematic theory. Despite its decades of experimental support, the detailed low-energy physics remains beyond our calculational reach. The lattice provides a technique for answering nonperturbative questions, but to date there remain open questions that have not been answered.


\end{quote}
\begin{itemize}%
\item Brambilla et al., \emph{QCD and strongly coupled gauge theories: challenges and perspectives}, Eur Phys J C Part Fields. 2014; 74(10): 2981 (\href{https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5}{doi:10.1140/epjc/s10052-014-2981-5})

\end{itemize}
\begin{quote}%
The success of the technique does not remove the challenge of understanding the non-perturbative aspects of the theory. The two aspects are deeply intertwined. The Lagrangian of QCD is written in terms of quark and gluon degrees of freedom which become apparent at large energy but remain hidden inside hadrons in the low-energy regime. This confinement property is related to the increase of $\alpha_s$ at low energy, but it has never been demonstrated analytically.

We have clear indications of the confinement of quarks into hadrons from both experiments and lattice QCD. Computations of the heavy quark–antiquark potential, for example, display a linear behavior in the quark–antiquark distance, which cannot be obtained in pure perturbation theory. Indeed the two main characteristics of QCD: confinement and the appearance of nearly massless pseudoscalar mesons, emergent from the spontaneous breaking of chiral symmetry, are non-perturbative phenomena whose precise understanding continues to be a target of research.

Even in the simpler case of gluodynamics in the absence of quarks, we do not have a precise understanding of how a gap in the spectrum is formed and the glueball spectrum is generated.


\end{quote}
\begin{itemize}%
\item J J Cobos-Martínez, \emph{Non-perturbative QCD and hadron physics} 2016 J. Phys.: Conf. Ser. 761 012036 (\href{http://iopscience.iop.org/article/10.1088/1742-6596/761/1/012036}{doi:10.1088/1742-6596/761/1/012036})

\end{itemize}
\begin{quote}%
$[\cdots]$ the QCD Lagrangian does not by itself explain the data on strongly interacting matter, and it is not clear how the observed bound states, the hadrons, and their properties arise from QCD. Neither confinement nor dynamical chiral symmetry breaking (DCSB) is apparent in QCD’s lagrangian, yet they play a dominant role in determining the observable characteristics of QCD. The physics of strongly interacting matter is governed by emergent phenomena such as these, which can only be elucidated through the use of non-perturbative methods in QCD 4, 5, 6, 7


\end{quote}
\begin{itemize}%
\item \href{https://web.infn.it/csn1/index.php/en/}{Istituto Nazionale di Fisica Nucleare}, \emph{What Next: White Paper of the INFN-CSN1} \emph{Proposal for a long term strategy for accelerator based experiments}, Frascati Phys.Ser. 60 (2015) 1-302 (2015-05-29) (\href{http://inspirehep.net/record/1374543/}{spire:1374543}, \href{http://www.lnf.infn.it/sis/frascatiseries/Volume60/Volume60.pdf}{pdf} ) chapter 7: \emph{Hadron Physics and non-perturbative QCD} (\href{http://www.infn.it/csn1/White_paper_documents/NPQCD.pdf}{pdf})

\end{itemize}
\begin{quote}%
Experimentally, there is a large number of facts that lack a detailed qualitative and quantitative explanation. The most spectacular manifestation of our lack of theoretical understanding of QCD is the failure to observe the elementary degrees of freedom, quarks and gluons, as free asymptotic states (color con- finement) and the occurrence, instead, of families of massive mesons and baryons (hadrons) that form approximately linear Regge trajectories in the mass squared. The internal, partonic structure of hadrons, and nucleons in particular, is still largely mysterious. Since protons and neutrons form almost all the visible matter of the Universe, it is of basic importance to explore their static and dynamical properties in terms of quarks and gluons interacting according to QCD dynamics.


\end{quote}
\begin{itemize}%
\item Hideo Suganuma, Yuya Nakagawa, Kohei Matsumoto, \emph{1+1 Large $N_c$ QCD and its Holographic Dual -Soliton Picture of Baryons in Single-Flavor World}, Proceedings of the 14th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU2016) (\href{https://arxiv.org/abs/1610.02074}{arxiv:1610.02074})

\end{itemize}
\begin{quote}%
Since 1973, quantum chromodynamics (QCD) has been established as the fundamental theory of the strong interaction. Nevertheless, it is very difficult to solve QCD directly in an analytical manner, and many effective models of QCD have been used instead of QCD, but most models cannot be derived from QCD and its connection to QCD is unclear. To analyze nonperturbative QCD, the lattice QCD Monte Carlo simulation has been also used as a first-principle calculation of the strong interaction. However, it has several weak points. For example, the state information (e.g. the wave function) is severely limited, because lattice QCD is based on the path-integral formalism. Also, it is difficult to take the chiral limit, because zero-mass pions require infinite volume lattices. There appears a notorious ``sign problem'' at finite density.

On the other hand, [[holographic QCD]] has a direct connection to QCD, and can be derived from QCD in some limit. In fact, [[holographic QCD]] is equivalent to infrared QCD in large $N_c$ and strong `t Hooft coupling $\lambda$, via gauge/gravity correspondence. Remarkably, holographic QCD is successful to reproduce many hadron phenomenology such as vector meson dominance, the KSRF relation, hidden local symmetry, the GSW model and the Skyrme soliton picture. Unlike lattice QCD simulations, holographic QCD is usually formulated in the chiral limit, and does not have the sign problem at finite density.


\end{quote}
\begin{itemize}%
\item V. A. Petrov, \emph{Asymptotic Regimes of Hadron Scattering in QCD} (\href{https://arxiv.org/abs/1901.02628}{arXiv:1901.02628})

\end{itemize}
\begin{quote}%
This is a commonplace that so far we do not have a full-fledged theory of interaction of [[hadrons]], derived from the first principles of [[QCD]] and having a regular way of calculating of hadronic [[scattering amplitude|amplitudes]], especially at high energies and small momentum transfers. The problem is related to a more general problem that QCD Lagrangian would yield [[confinement|colour confinement]] with massive colourless physical states (hadrons).


\end{quote}
\begin{itemize}%
\item Christian Drischler, Wick Haxton, Kenneth McElvain, Emanuele Mereghetti, Amy Nicholson, Pavlos Vranas, André Walker-Loud, \emph{Towards grounding nuclear physics in QCD} (\href{https://arxiv.org/abs/1910.07961}{arxiv:1910.07961})

\end{itemize}
\begin{quote}%
the entirety of the rich field of [[nuclear physics]] emerges from [[QCD]]: from the [[forces]] binding [[protons]] and [[neutrons]] into the nuclear landscape, to the fusion and fission reactions between [[atomic nucleus|nuclei]], to the prospective interactions of nuclei with BSM physics, and to the unknown [[phase of matter|state of matter]] at the cores of [[neutron stars]].

How does this emergence take place exactly? How is the clustering of [[quarks]] into [[nucleons]] and [[alpha particles]] realized? What are the mechanisms behind collective phenomena in nuclei as strongly correlated many-body systems? How does the extreme fine-tuning required to reproduce nuclear binding energies proceed? – are big open questions in [[nuclear physics]].


\end{quote}
\hypertarget{potential_solutions}{}\subsection*{{Potential solutions}}\label{potential_solutions}

\hypertarget{ViaCalorons}{}\subsubsection*{{Via Calorons}}\label{ViaCalorons}

It has been argued that, after [[Wick rotation]], confinement may be derived from the behaviour of [[instantons]] (\hyperlink{SchaeferShuryak96}{Schaefer-Shuryak 96, section III D}), or rather their [[thermal field theory|positive temperature]]-incarnations as \emph{[[calorons]]}, \hyperlink{Greensite11}{Greensite 11, section 8.5}:

\begin{quote}%
it is natural to wonder if [[confinement]] could be derived from some [[semiclassical approximation|semiclassical]] treatment of [[Yang–Mills theory]] based on the [[instanton]] solutions of [[nonabelian group|non-abelian]] [[gauge theories]]. The [[BPST instanton|standard]] [[instantons]], introduced by Belavin et al. (\href{BPTS-instanton#BelavinPolyakovSchwartzTyupkin75}{40}), do not seem to work; their [[field strengths]] fall off too rapidly to produce the desired magnetic disorder in the vacuum.

In recent years, however, it has been realized that instanton solutions at [[thermal field theory|finite temperature]], known as \emph{[[calorons]]}, might do the job. These caloron solutions were introduced independently by Kraan and van Baal (\href{caloron#KraanVanBaal98}{41}, \href{caloron#KraanVanBaal98b}{42}) and Lee and Lu (\href{caloron#LeeLu98}{43}) (KvBLL), and they have the remarkable property of containing [[monopole]] constituents which may, depending on the type of [[caloron]], be widely separated.

$[...]$

The [[caloron]] idea is probably the most promising current version of [[monopole]] [[confinement]] in pure non-abelian gauge theories, but it is basically (in certain [[gauge fixing|gauges]]) a superposition of [[monopoles]] with spherically symmetric abelian fields, and this leads to the same questions raised in connection with monopole Coulomb gases.


\end{quote}
See also at \emph{[[glueball]]}.

\hypertarget{via_skyrmions_and_d4brane_models}{}\subsubsection*{{Via Skyrmions and D4-brane models}}\label{via_skyrmions_and_d4brane_models}

A good qualitative and moderate quantitative explanation of confinement in [[quantum chromodynamics]] is found in [[intersecting D-brane models]], specifically in the [[Witten-Sakai-Sugimoto model]] which [[geometric engineering of QFT|geometrically engineers]] [[QCD]] on [[D4-D8 brane bound states]].

(\hyperlink{Witten98}{Witten 98}, followed up on in \href{Ads/QCD#SakaiSugimoto04}{Sakai-Sugimoto 04}, \href{Ads/QCD#SakaiSugimoto05}{Sakai-Sugimoto 05})

\begin{quote}%
graphics grabbed from \href{Ads/QCD#Erlich09}{Erlich 09, section 1.1}


\end{quote}
\begin{quote}%
graphics grabbed from \href{Ads/QCD#Rebhan14}{Rebhan 14}


\end{quote}
In this [[Witten-Sakai-Sugimoto model]] for [[non-perturbative effect|strongly coupled]] [[QCD]] the [[hadrons]] in [[QCD]] correspond to [[string theory|string-theoretic]]-phenomena in the [[bulk field theory]]:

\begin{enumerate}%
\item the [[mesons]] ([[bound states]] of 2 [[quarks]]) correspond to [[open strings]] in the bulk, whose two endpoints on the [[asymptotic boundary]] correspond to the two quarks


\item [[baryons]] ([[bound states]] of $N_c$ [[quarks]]) appear in two different but equivalent (\href{Ads/QCD#Sugimoto16}{Sugimoto 16, 15.4.1}) guises:

\begin{enumerate}%
\item as [[wrapped brane|wrapped]] [[D4-branes]] with $N_c$ [[open strings]] connecting them to the [[D8-brane]]

(\href{Ads/QCD#Witten98b}{Witten 98b}, \href{Ads/QCD#GrossOoguri98}{Gross-Ooguri 98, Sec. 5}, \href{Ads/QCD#BISY98}{BISY 98}, \href{Ads/QCD#CGS98}{CGS98})


\item as [[skyrmions]]

(\href{Ads/QCD#SakaiSugimoto04}{Sakai-Sugimoto 04, section 5.2}, \href{Ads/QCD#SakaiSugimoto05}{Sakai-Sugimoto 05, section 3.3}, see \href{Ads/QCD#Bartolini17}{Bartolini 17}).



\end{enumerate}


\end{enumerate}
For review see \href{Ads/QCD#Sugimoto16}{Sugimoto 16}, also \href{Ads/QCD#Rebhan14}{Rebhan 14, around (18)}.

\begin{quote}%
graphics grabbed from \href{Ads/QCD#Sugimoto16}{Sugimoto 16}


\end{quote}
Equivalently, these baryon states are the [[Yang-Mills instantons]] on the [[D8-brane]] giving the [[D4-D8 brane bound state]] (\href{Ads/QCD#SakaiSugimoto04}{Sakai-Sugimoto 04, 5.7}) as a special case of the general situation for [[Dp-D(p+4)-brane bound states]] (e.g. \href{Dp-D%28p%2B4%29-brane+bound+state#Tong05}{Tong 05, 1.4}).

For more on this see at

\begin{itemize}%
\item \emph{[[AdS/QCD correspondence]]}

\end{itemize}
and at

\begin{itemize}%
\item \emph{[[skyrmion]]} .

\end{itemize}
\hypertarget{InN2SYM}{}\subsubsection*{{In $\mathcal{N}=2$ super Yang-Mills theory via Seiberg-Witten theory}}\label{InN2SYM}

While confinement in plain [[Yang-Mills theory]] is still waiting for mathematical formalization and [[proof]] (see \hyperlink{JaffeWitten}{Jaffe-Witten}), there is a variant of [[Yang-Mills theory]] with more [[symmetry]], namely [[supersymmetry]], where the phenomenon has been giving a decent argument, namely in [[N=2 D=4 super Yang-Mills theory]] (\hyperlink{SeibergWitten94}{Seiberg-Witten 94}).

Also a strategy for a proof for [[N=1 D=4 super Yang-Mills theory]] has been proposed, see \hyperlink{ForN1SYMViaMTheoryOnG2Manifolds}{below}.

\hypertarget{ForN1SYMViaMTheoryOnG2Manifolds}{}\subsubsection*{{In $\mathcal{N} = 1$ super Yang-Mills theory via M-theory on $G_2$-manifolds}}\label{ForN1SYMViaMTheoryOnG2Manifolds}

An idea for a strategy towards a proof of confinement in [[N=1 D=4 super Yang-Mills theory]] via two different but conjecturally equivalent realizations as [[M-theory on G2-manifolds]] has been given in \hyperlink{AtiyahWitten01}{Atiyah-Witten 01, section 6}, review is in \hyperlink{AcharyaGukov04}{Acharya-Gukov 04, section 5.3}.

The idea here is to consider a [[KK-compactification]] of [[M-theory]] on [[fibers]] which are [[G2-manifolds]] that locally around a special point are of the form

\begin{displaymath}
X_{1,\Gamma} 
  \;\coloneqq\;
  \big( S^3 / \Gamma \big) \times Cone\big(S^3\big)
  \phantom{AA}
  \text{or}
  \phantom{AA}
  X_{2,\Gamma} 
  \;\coloneqq\;
  S^3 \times Cone\big(S^3/\Gamma\big)
\end{displaymath}
where

\begin{itemize}%
\item $\Gamma$ is a [[finite subgroup of SU(2)]] that [[action|acts]] canonically by left-multiplication on $S^3 \simeq$ [[SU(2)]];


\item $Cone(\cdots)$ denotes the [[metric cone]] construction.



\end{itemize}
This means that $X_{1,\Gamma}$ is a [[smooth manifold]], but $X_{2,\Gamma}$, as soon as $\Gamma$ is not the [[trivial group]], $\Gamma \neq 1$, is an [[orbifold]] with an [[ADE singularity]].

Now the lore of [[M-theory on G2-manifolds]] predicts that [[KK-compactification]]

\begin{enumerate}%
\item on $X_{1,\Gamma}$ yields a 4d theory without massless fields (since there are no massless modes on the [[covering space]] $S^3$ of $X_{1,\Gamma}$)


\item on the [[ADE-singularity]] $X_{2,\Gamma}$ yields [[non-abelian group|non-abelian]] [[Yang-Mills theory]] in 4d coupled to [[chiral fermions]].



\end{enumerate}
So in the first case a [[mass gap]] is manifest, while non-abelian gauge theory is not visible, while in the second case it is the other way around.

But if there were an argument that [[M-theory on G2-manifolds]] is in fact equivalent for compactification both on $X_{1,\Gamma}$ and on $X_{2,\Gamma}$. To the extent that this is true, it looks like an argument that could demonstrate confinement in non-abelian 4d gauge theory.

This approach is suggested in \hyperlink{AtiyahWitten01}{Atiyah-Witten 01, pages 84-85}. An argument that this equivalence is indeed the case is then provided in sections 6.1-6.4, based on an argument in \hyperlink{AtiyahMaldacenaVafa00}{Atiyah-Maldacena-Vafa 00}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[asymptotic freedom]]


\item [[quantization of Yang-Mills theory]]


\item [[proton spin crisis]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{ReferencesInYangMillsTheory}{}\subsubsection*{{In Yang-Mills theory}}\label{ReferencesInYangMillsTheory}

\hypertarget{general}{}\paragraph*{{General}}\label{general}

\begin{itemize}%
\item [[Kenneth Wilson]], \emph{Confinement of quarks, Phys. Rev. D10, 2445, 1974 (\href{https://doi.org/10.1103/PhysRevD.10.2445}{doi:10.1103/PhysRevD.10.2445})}

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item Jeff Greensite, \emph{An Introduction to the Confinement Problem}, Lecture Notes in Physics, Volume 821, 2011 (\href{https://link.springer.com/book/10.1007%2F978-3-642-14382-3}{doi:10.1007/978-3-642-14382-3})


\item Robert Iengo, section 9.1 of \emph{Quantum Field Theory} (\href{http://iopscience.iop.org/chapter/978-1-6432-7053-1/bk978-1-6432-7053-1ch9.pdf}{pdf})


\item D. Bugg (ed.), \emph{Hadron Spectroscopy and the Confinement Problem}, Proceedings of a NATO Advanced Study Institute, Plenum Press 1996 (\href{https://www.springer.com/cn/book/9780306453038}{doi:10.1007/978-1-4613-0375-6})



\end{itemize}
Introductions and surveys include

\begin{itemize}%
\item Yuri L. Dokshitzer, around section 1.2 of \emph{QCD Phenomenology} (\href{https://arxiv.org/abs/hep-ph/0306287}{arXiv:hep-ph/0306287})


\item D. Espriu, section 7 of \emph{Perturbative QCD} (\href{http://arxiv.org/abs/hep-ph/9410287}{arXiv:hep-ph/9410287})


\item Erhard Seiler, \emph{The Confinement Problem} (\href{http://physik.uni-graz.at/~dk-user/talks/Seiler20081113-14.pdf}{pdf})


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Color_confinement}{Color confinement}}



\end{itemize}
Discussion includes

\begin{itemize}%
\item \emph{\href{https://ensnaredinvacuum.wordpress.com/2015/02/11/what-is-strongly-coupled-quantum-chromodynamics/}{What is Strongly-Coupled Quantum Chromodynamics?}}

\end{itemize}
A formulation of confinement as an open problem of mathematical physics, together with many references, is in

\begin{itemize}%
\item [[Arthur Jaffe]], [[Edward Witten]], \emph{Quantum Yang-Mills theory} (\href{http://www.claymath.org/millennium/Yang-Mills_Theory/yangmills.pdf}{pdf})

\end{itemize}
Other technical reviews include

\begin{itemize}%
\item G. M. Prosperi, \emph{Confinement and bound states in QCD} (\href{http://arxiv.org/abs/hep-ph/0202186}{arXiv:hep-ph/0202186})


\item Christian Drischler, Wick Haxton, Kenneth McElvain, Emanuele Mereghetti, Amy Nicholson, Pavlos Vranas, André Walker-Loud, \emph{Towards grounding nuclear physics in QCD} (\href{https://arxiv.org/abs/1910.07961}{arxiv:1910.07961})



\end{itemize}
\hypertarget{ViaMonopoleCondensation}{}\paragraph*{{Via monopole condensation}}\label{ViaMonopoleCondensation}

An original suggestion that confinement in [[Yang-Mills theory]] may be understood via [[monopole]] [[condensate|condensation]] as a dual [[Meissner effect]] is due to

\begin{itemize}%
\item [[Gerard `t Hooft]], in Proceed.of the Europ.Phys.Soc. 1975, ed.by A.Zichichi (Editrice Compositori, Bologna, 1976), p.1225.


\item S. Mandelstam, Phys.Rep. 23C (1976) 145;



\end{itemize}
(That this is indeed the case has not yet been demonstarted for plain Yang-Mills theory, but it was later shown for [[N=2 D=4 super Yang-Mills theory]] in (\hyperlink{SeibergWitten94}{Seiberg-Witten 94}). What this does or does not imply for the case of [[QCD]] is discussed in (\hyperlink{Yung00}{Yung 00}) ).

The relation to [[QCD instantons]]/[[monopole|monopoles]] in the [[QCD vacuum]] is discussed in

\begin{itemize}%
\item T. Schaefer, [[Edward Shuryak]], section III D of \emph{Instantons in QCD}, Rev. Mod. Phys.70:323-426,1998 (\href{http://arxiv.org/abs/hep-ph/9610451}{arXiv:hep-ph/9610451})

\end{itemize}
and analogously (at [[thermal field theory|positive temperature]]) relation to \emph{[[calorons]]}:

\begin{itemize}%
\item \hyperlink{Greensite11}{Greensite 11, section 8.5}


\item P. Gerhold, E.-M. Ilgenfritz, M. Müller-Preussker, \emph{An $SU(2)$ KvBLL caloron gas model and confinement}, Nucl.Phys.B760:1-37, 2007 (\href{https://arxiv.org/abs/hep-ph/0607315}{arXiv:hep-ph/0607315})


\item Rasmus Larsen, [[Edward Shuryak]], \emph{Classical interactions of the instanton-dyons with antidyons}, Nucl. Phys. A \textbf{950}, 110 (2016) (\href{https://arxiv.org/abs/1408.6563}{arXiv:1408.6563})


\item Rasmus Larsen, [[Edward Shuryak]], \emph{Interacting Ensemble of the Instanton-dyons and Deconfinement Phase Transition in the SU(2) Gauge Theory}, Phys. Rev. D 92, 094022, 2015 (\href{https://arxiv.org/abs/1504.03341}{arXiv:1504.03341})



\end{itemize}
For further developments see

\begin{itemize}%
\item \emph{Dimensional Transmutation by Monopole Condensation in QCD} (\href{http://arxiv.org/abs/1206.6936}{arXiv:1206.6936})

\end{itemize}
\hypertarget{interacting_field_vacuum}{}\subsubsection*{{Interacting field vacuum}}\label{interacting_field_vacuum}

Discussion of confinement as a result of the [[interacting vacuum]] includes

\begin{itemize}%
\item [[Johann Rafelski]], \emph{Vacuum structure -- An Essay}, in pages 1-29 of H. Fried, Berndt Müller (eds.) \emph{Vacuum Structure in Intense Fields}, Plenum Press 1990 (\href{https://books.google.de/books?id=5uXcBwAAQBAJ&pg=PA14&lpg=PA14&dq=confinement+%22interacting+vacuum%22&source=bl&ots=xPGYJ-JOc-&sig=AoYbqWQeNRMg6hMSRZjJ3nzq8B0&hl=en&sa=X&ved=0ahUKEwjIgK68-tnYAhVCESwKHThMDykQ6AEIKTAA#v=onepage&q=confinement%20%22interacting%20vacuum%22&f=false}{GBooks})

\end{itemize}
See also

\begin{itemize}%
\item Yu. A. Simonov, \emph{What is the confinement mechanism in QCD?} (\href{https://arxiv.org/abs/1804.08946}{arXiv:1804.08946})

\end{itemize}
\hypertarget{ReferencesInSuperYangMillsTheory}{}\subsubsection*{{In super-Yang-Mills theory}}\label{ReferencesInSuperYangMillsTheory}

Confinement in [[N=2 D=4 super Yang-Mills theory]] by a version of the monopole condensation of (\hyperlink{Hooft75}{t Hooft 75}, \hyperlink{Mandelstam76}{Mandelstam 76}) was demonstrated in

\begin{itemize}%
\item [[Nathan Seiberg]], [[Edward Witten]], \emph{Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory}, Nucl.Phys.B426:19-52,1994; Erratum-ibid.B430:485-486,1994 (\href{http://arxiv.org/abs/hep-th/9407087}{arXiv:hep-th/9407087})


\item [[Nathan Seiberg]], [[Edward Witten]], \emph{Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD}, Nucl.Phys.B431:484-550,1994 (\href{http://arxiv.org/abs/hep-th/9408099}{arXiv:hep-th/9408099})



\end{itemize}
Reviews with discussion of the impact on confinement in plain YM include

\begin{itemize}%
\item Alexei Yung, \emph{What Do We Learn about Confinement from the Seiberg-Witten Theory} (\href{http://arxiv.org/abs/hep-th/0005088}{arXiv:hep-th/0005088})

\end{itemize}
\hypertarget{under_the_adscft_correspondence}{}\subsubsection*{{Under the AdS/CFT correspondence}}\label{under_the_adscft_correspondence}

Discussion in the context of the [[AdS-CFT correspondence]] is in

\begin{itemize}%
\item [[Edward Witten]], \emph{Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories}, Adv. Theor. Math. Phys.2:505-532, 1998 (\href{https://arxiv.org/abs/hep-th/9803131}{arXiv:hep-th/9803131})


\item [[David Berman]], Maulik K. Parikh, \emph{Confinement and the AdS/CFT Correspondence}, Phys.Lett. B483 (2000) 271-276 (\href{https://arxiv.org/abs/hep-th/0002031}{arXiv:hep-th/0002031})


\item Henrique Boschi Filho, \emph{AdS/QCD and confinement}, Seminar at the \emph{Workshop on Strongly Coupled QCD: The confinement problem}, November 2011 (\href{http://www.if.ufrj.br/~boschi/pesquisa/seminarios/AdS_QCD_Confinement_UERJ_2011.pdf}{pdf})



\end{itemize}
\hypertarget{in_mtheory_on_manifolds}{}\subsubsection*{{In M-theory on $G_2$-manifolds}}\label{in_mtheory_on_manifolds}

An idea for how to demonstrate confinement in models of [[M-theory on G2-manifolds]] is given in

\begin{itemize}%
\item [[Michael Atiyah]], [[Edward Witten]], section 6 of \emph{$M$-Theory dynamics on a manifold of $G_2$-holonomy}, Adv. Theor. Math. Phys. 6 (2001) (\href{http://arxiv.org/abs/hep-th/0107177}{arXiv:hep-th/0107177})

\end{itemize}
based on

\begin{itemize}%
\item [[Michael Atiyah]], [[Juan Maldacena]], [[Cumrun Vafa]], \emph{An M-theory Flop as a Large N Duality}, J.Math.Phys.42:3209-3220, 2001 (\href{https://arxiv.org/abs/hep-th/0011256}{arXiv:hep-th/0011256})

\end{itemize}
See also

\begin{itemize}%
\item [[Bobby Acharya]], \emph{Confining Strings from $G_2$-holonomy spacetimes} (\href{https://arxiv.org/abs/hep-th/0101206}{arXiv:hep-th/0101206})

\end{itemize}
For review see

\begin{itemize}%
\item [[Bobby Acharya]], [[Sergei Gukov]], section 5.3 of \emph{M theory and Singularities of Exceptional Holonomy Manifolds}, Phys.Rept.392:121-189,2004 (\href{https://arxiv.org/abs/hep-th/0409191}{arXiv:hep-th/0409191})

\end{itemize}
[[!redirects confinement]] [[!redirects color confinement]] [[!redirects colour confinement]]



\end{document}