# nLab confinement

Contents

## Surveys, textbooks and lecture notes

#### Differential cohomology

differential cohomology

# Contents

## Idea

Confinement (e.g Espiru 94) is the (expected) phenomenon in Yang-Mills theory generally and especially in quantum chromodynamics that the fundamental quarks, which the YM/QCD-Lagrangian density actually describes, must form baryonic bound states which are neutral under the color charge – the mesons and hadrons (protons, 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. In order to prove, via QCD itself, that the potential energy of a colorless package of quarks tends to infinity as the distance between them grows, one must probe the long-range regime of the quark-quark interaction, in which the methods of Feynman perturbation theory fail.

Part of the issue is that confinement is a non-perturbative effect (e.g Espiru 94) outside the range of validity of perturbative quantum field theory.

### The open problem of confinement

While experiment as well as lattice gauge theory-computer simulation clearly show that confinement takes place, a real theoretical understanding has been missing (though AdS-QCD is now on a good track). This is the confinement problem. The same problem from the point of view of mathematics is called the mass gap Millennium Problem. A related problem is the flavor problem.

The following is a list of quotes highlighting the open problem of confinement:

many of the essential properties that the theory $[$QCD$]$ is presumed to have, including confinement, dynamical mass generation, and chiral symmetry breaking, are only poorly understood. And apart from the low-lying bound states of heavy quarks, which we believe can be described by a nonrelativistic Schroedinger equation, we are unable to derive from the basic theory even the grossest features of the partticle spectrum, or of traditional strong interaction phenomenology

There are theoretical attempts to connect the fundamental theory of QCD with the very successful meson picture at low energy. The Skyrme model is an example. In other attempts, one tries to derive the NN interaction more or less directly from QCD. At present, the predictions are more of a qualitative kind. For quantitative results, the one-pion and two-pion contributions have to be added by hand, as they do not emerge naturally out of QCD-inspired models. Knowing that $\pi$ and $2\pi$ are the most important parts of the nuclear force, this defect of present quark model calculations is serious.

• Robert Kutschke, section 3.1 Heavy flavour spectroscopy, in D. Bugg (ed.), Hadron Spectroscopy and the Confinement Problem, Proceedings of a NATO Advanced Study Institute, Plenum Press 1996 (doi:10.1007/978-1-4613-0375-6)

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.

• Barry R. Holstein, A Brief Introduction to Chiral Perturbation Theory, Czech. J. Phys. 50S4:9-23, 2000 (arXiv:hep-ph/9911449)

the holy grail sought by particle/nuclear knights has been to verify the correctness of the “ultimate” theory of strong interactions – quantum chromodynamics (QCD).

The theory is, of course, deceptively simple on the surface. $[...]$ So why are we still not satisfied? While at the very largest energies, asymptotic freedom allows the use of perturbative techniques, for those who are interested in making contact with low energy experimental findings there exist at least three fundamental difficulties:

i) QCD is written in terms of the “wrong” degrees of freedom – quarks and gluons – while low energy experiments are performed with hadronic bound states;

ii) the theory is non-linear due to gluon self-interactions;

iii) the theory is one of strong coupling so that perturbative methods are not practical

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.

• Mike Guidry, Gauge Field Theories: An Introduction with Applications, Wiley 2008 (ISBN:978-3-527-61736-4)

Section 13.1.9:

The holy grail of QCD is the proof that a color SU(3) gauge theory confines in the non-perturbative regime.

This is not difficult to show for lattices with large spacing; unfortunately, such a demonstration does not constitute a proof of QCD confinement: to do that we must also demonstrate that the same theory that confines at large lattice spacing (strong coupling) has a continuum limit (weak coupling) that is consistent with the asymptotically free short distance behavior of QCD.

QCD is a challenging theory. Its most interesting aspects, namely the confinement of color and the chiral symmetry breaking, have defied all analytical approaches. While there are now many data accumulated from the lattice gauge theory, the methodology falls well short of giving us insights on how one may understand these phenomena analytically, nor does it give us a systematic way of obtaining a low energy theory of QCD below the confinement scale.

Nuclear physics is one of the oldest branches of high energy physics, yet remains one of more difficult. Despite the fact that we know the underlying fundamental theory, i.e. QCD, we are still unable to predict, reliably and analytically, behavior of nuclei or even a single proton. The problem is of course that one must understand the strong-coupling regime of QCD, which by and large remains inaccessible except by large-scale lattice simulations. Traditionally, this sets nuclear physics apart from the rest of high energy physics in many aspects. However, recent developments in the so-called gauge/gravity duality began to solve certain strongly coupled field theories,possibly including QCD or its close relatives, allowing the two communities to merge with each other.

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.

The problem with a derivation of nuclear forces from QCD is that this theory is non-perturbative in the low-energy regime characteristic of nuclear physics, which makes direct solutions very difficult. Therefore, during the first round of new attempts, QCD-inspired quark models became popular. The positive aspect of these models is that they try to explain hadron structure and hadron-hadron interactions on an equal footing and, indeed, some of the gross features of the nucleon-nucleon interaction are explained successfully.

However, on a critical note, it must be pointed out that these quark-based approaches are nothing but another set of models and, thus, do not represent fundamental progress. For the purpose of describing hadron-hadron interactions, one may equally well stay with the simpler and much more quantitative meson models.

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.

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.

$[\cdots]$ 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]

• Hideo Suganuma, Yuya Nakagawa, Kohei Matsumoto, 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) (arxiv:1610.02074)

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.

• V. A. Petrov, Asymptotic Regimes of Hadron Scattering in QCD (arXiv:1901.02628)

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 amplitudes, especially at high energies and small momentum transfers. The problem is related to a more general problem that QCD Lagrangian would yield colour confinement with massive colourless physical states (hadrons).

One of the long-standing problems in QCD is to reproduce profound nuclear physics. The strong coupling nature of QCD prevents us from solving it analytically, and even numerical simulations have a limitation such as the volume of the atomic nucleus versus the lattice size. It is quite important to bridge the particle physics and the nuclear physics, by solving QCD to derive typical fundamental notions of the nuclear physics, such as the magic numbers, the nuclear binding energy and the nuclear shell model.

Holographic QCD is an analytic method to approach these problems in the strong coupling limit and at a large $N_c$. The nuclear matrix model is a many-body quantum effective mechanics for multiple baryons, derived by the AdS/CFT correspondence applied to QCD.

• Gergely Gábor Barnaföldi, Vakhtang Gogokhia, The Mass Gap Approach to QCD (arXiv:1904.07748)

QCD as a fundamental quantum field gauge theory still suffers from a few important conceptual problems. We focus on some of them as follows: (A) The dynamical generation of a mass squared at the fundamental quark-gluon level, since the QCD Lagrangianforbids such kind of terms apart from the current quark mass. $[\ldots]$ (D) The non-observation of the colored objects as physical states which does not follow from the QCD Lagrangian,i.e., it cannot explain confinement of gluons and quarks.

• Yonggoo Heo, C. Kobdaj, M.F.M. Lutz, Constraints from a large-$N_c$ analysis on meson-baryon interactions at chiral order $Q^3$, Phys. Rev. D 100, 094035 (2019) (arXiv:1908.11816)

Still after many decades of vigorous studies the outstanding challenge of modern physics is to establish a rigorous link of QCD to low-energy hadron physics as it is observed in the many experimental cross section measurements.

• Christian Drischler, Wick Haxton, Kenneth McElvain, Emanuele Mereghetti, Amy Nicholson, Pavlos Vranas, André Walker-Loud, Towards grounding nuclear physics in QCD (arxiv:1910.07961)

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 nuclei, to the prospective interactions of nuclei with BSM physics, and to the unknown 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.

More than 98% of visible mass is contained within nuclei. In first approximation, their atomic weights are simply the sum of the masses of all the neutrons and protons (nucleons) they contain. Each nucleon has a mass $m_N \sim 1$ GeV, i.e. approximately 2000-times the electron mass. The Higgs boson produces the latter, but what produces the masses of the neutron and proton? This is the question posed above, which is pivotal to the development of modern physics: how can science explain the emergence of hadronic mass (EHM)? $[\cdots]$

Modern science is thus encumbered with the fundamental problem of gluon and quark confinement; and confinement is crucial because it ensures absolute stability of the proton. $[\cdots]$ Without confinement,our Universe cannot exist.

As the 21st Century began, the Clay Mathematics Institute established seven Millennium Prize Problems. Each represents one of the toughest challenges in mathematics. The set contains the problem of confinement; and presenting a sound solution will win its discoverer 1,000,000 bucks. Even with such motivation, today, almost fifty years after the discovery of quarks, no rigorous solution has been found. Confinement and EHM are inextricably linked. Consequently, as science plans for the next thirty years, solving the problem of EHM has become a grand challenge. $[\cdots]$

In trying to match QCD with Nature, one confronts the many complexities of strong, nonlinear dynamics in relativistic quantum field theory, e.g. the loss of particle number conservation, the frame and scale dependence of the explanations and interpretations of observable processes, and the evolving character of the relevant degrees-of-freedom. Electroweak theory and phenomena are essentially perturbative; hence, possess little of this complexity. Science has never before encountered an interaction such as that at work in QCD. Understanding this interaction, explaining everything of which it is capable, can potentially change the way we look at the Universe.

The confinement of quarks is one of the enduring mysteries of modern physics. $[\ldots]$ In spite of many decades of research, physically relevant quantum gauge theories have not yet been constructed in a rigorous mathematical sense. $[$ non-perturbatively, that is $]$ $[\ldots]$ Taking this program to its completion is one of the Clay millennium problems. $[\ldots]$ Perhaps the most important example is four-dimensional SU(3)-lattice gauge theory. If one can show that this theory has a mass gap at all values of the coupling strength, that would explain why particles known as glue-balls in the theory of strong interactions have mass. All such questions remain open.

The second big open question is the problem of quark confinement. Quarks are the constituents of various elementary particles, such as protons and neutrons. It is an enduring mystery why quarks are never observed freely in nature. The problem of quark confinement has received enormous attention in the physics literature, but the current consensus seems to be that a satisfactory theoretical explanation does not exist.

QCD, the theory of the strong interactions, involves quarks interacting with non-Abelian gluon fields. This theory has many features that are difficult to impossible to see in conventional diagrammatic perturbation theory. This includes quark confinement, mass generation, and chiral symmetry breaking.

The origin of the proton mass, and with it the basic mass-scale for all nuclear physics, is one of the most profound puzzles in Nature.

Although QCD is defined by a seemingly simple Lagrangian, it specifies a problem that has defied solution for more than forty years. The key challenges in modern nuclear and high-energy physics are to reveal the observable content of strong QCD and, ultimately, therefrom derive the properties of nuclei.

In spite of the important progress of Euclidean lattice gauge theory, a basic understanding of the mechanism of color confinement and its relation to chiral symmetry breaking in QCD has remained an unsolved problem.

Recent developments based on superconformal quantum mechanics in light-front quantization and its holographic embedding on a higher dimensional gravity theory (gauge/gravity correspondence) have led to new analytic insights into the structure of hadrons and their dynamics.

## Potential solutions

### Via Skyrmions and D4-brane 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 geometrically engineers QCD on D4-D8 brane bound states.

(Witten 98, followed up on in Sakai-Sugimoto 04, Sakai-Sugimoto 05)

graphics grabbed from Erlich 09, section 1.1

graphics grabbed from Rebhan 14

In this Witten-Sakai-Sugimoto model for strongly coupled QCD the hadrons in QCD correspond to string-theoretic-phenomena in the bulk field theory:

1. 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

2. baryons (bound states of $N_c$ quarks) appear in two different but equivalent (Sugimoto 16, 15.4.1) guises:

1. as wrapped D4-branes with $N_c$ open strings connecting them to the D8-brane

For review see Sugimoto 16, also Rebhan 14, around (18).

graphics grabbed from Sugimoto 16

Equivalently, these baryon states are the Yang-Mills instantons on the D8-brane giving the D4-D8 brane bound state (Sakai-Sugimoto 04, 5.7) as a special case of the general situation for Dp-D(p+4)-brane bound states (e.g. Tong 05, 1.4).

Strictly speaking, since the number of colors in quantum chromodynamics is not large ($N_c = 3$), an accurate formulation of such holographic QCD requires understanding small N corrections:

For more on this see at

and at

### In $\mathcal{N}=2$ super Yang-Mills theory via Seiberg-Witten theory

While confinement in plain Yang-Mills theory is still waiting for mathematical formalization and proof (see 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 (Seiberg-Witten 94).

Also a strategy for a proof for N=1 D=4 super Yang-Mills theory has been proposed, see below.

### In $\mathcal{N} = 1$ super Yang-Mills theory via M-theory on $G_2$-manifolds

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 Atiyah-Witten 01, section 6, review is in 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

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

where

• $\Gamma$ is a finite subgroup of SU(2) that acts canonically by left-multiplication on $S^3 \simeq$ SU(2);

• $Cone(\cdots)$ denotes the metric cone construction.

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

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

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

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 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 Atiyah-Maldacena-Vafa 00

### Via Calorons

It has been argued that, after Wick rotation, confinement may be derived from the behaviour of instantons (Schaefer-Shuryak 96, section III D), or rather their positive temperature-incarnations as calorons, Greensite 11, section 8.5:

it is natural to wonder if confinement could be derived from some semiclassical treatment of Yang–Mills theory based on the instanton solutions of non-abelian gauge theories. The standard instantons, introduced by Belavin et al. (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 finite temperature, known as calorons, might do the job. These caloron solutions were introduced independently by Kraan and van Baal (41, 42) and Lee and Lu (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 gauges) a superposition of monopoles with spherically symmetric abelian fields, and this leads to the same questions raised in connection with monopole Coulomb gases.

effective field theories of nuclear physics, hence for confined-phase quantum chromodynamics:

## References

### In Yang-Mills theory

#### General

Textbook accounts:

• David Griffiths, chapter 8 of Introduction to Elementary Particles 2nd ed., 2008. (pdf)

• Jeff Greensite, An Introduction to the Confinement Problem, Lecture Notes in Physics, Volume 821, 2011 (doi:10.1007/978-3-642-14382-3)

• Robert Iengo, section 9.1 of Quantum Field Theory (pdf)

• D. Bugg (ed.), Hadron Spectroscopy and the Confinement Problem, Proceedings of a NATO Advanced Study Institute, Plenum Press 1996 (doi:10.1007/978-1-4613-0375-6)

Introductions and surveys include

Discussion includes

A formulation of confinement as an open problem of mathematical physics, together with many references, is in

Other technical reviews include

• G. M. Prosperi, Confinement and bound states in QCD (arXiv:hep-ph/0202186)

• Christian Drischler, Wick Haxton, Kenneth McElvain, Emanuele Mereghetti, Amy Nicholson, Pavlos Vranas, André Walker-Loud, Towards grounding nuclear physics in QCD (arxiv:1910.07961)

• Yuri A. Simonov, The fundamental scale of QCD (arXiv:2103.08223)

• Duifje Maria van Egmond, Urko Reinosa, Julien Serreau, Matthieu Tissier, A novel background field approach to the confinement-deconfinement transition (arXiv:2104.08974)

#### Via monopole condensation

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

• Gerard 't Hooft, in Proceed.of the Europ.Phys.Soc. 1975, ed.by A.Zichichi (Editrice Compositori, Bologna, 1976), p.1225.

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

(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 (Seiberg-Witten 94). What this does or does not imply for the case of QCD is discussed in (Yung 00) ).

The relation to QCD instantons/monopoles in the QCD vacuum is discussed in

and analogously (at positive temperature) relation to calorons:

• Greensite 11, section 8.5

• P. Gerhold, E.-M. Ilgenfritz, M. Müller-Preussker, An $SU(2)$ KvBLL caloron gas model and confinement, Nucl.Phys.B760:1-37, 2007 (arXiv:hep-ph/0607315)

• Rasmus Larsen, Edward Shuryak, Classical interactions of the instanton-dyons with antidyons, Nucl. Phys. A 950, 110 (2016) (arXiv:1408.6563)

• Rasmus Larsen, Edward Shuryak, Interacting Ensemble of the Instanton-dyons and Deconfinement Phase Transition in the SU(2) Gauge Theory, Phys. Rev. D 92, 094022, 2015 (arXiv:1504.03341)

For further developments see

• Dimensional Transmutation by Monopole Condensation in QCD (arXiv:1206.6936)

### Interacting field vacuum

Discussion of confinement as a result of the interacting vacuum includes

• Johann Rafelski, Vacuum structure – An Essay, in pages 1-29 of H. Fried, Berndt Müller (eds.) Vacuum Structure in Intense Fields, Plenum Press 1990 (GBooks)

### In super-Yang-Mills theory

Confinement in N=2 D=4 super Yang-Mills theory by a version of the monopole condensation of (t Hooft 75, Mandelstam 76) was demonstrated in

Reviews with discussion of the impact on confinement in plain YM include

• Alexei Yung, What Do We Learn about Confinement from the Seiberg-Witten Theory (arXiv:hep-th/0005088)

Discussion in the context of the AdS-QCD correspondence is in

• Edward Witten, Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories, Adv. Theor. Math. Phys.2:505-532, 1998 (arXiv:hep-th/9803131)

• David Berman, Maulik K. Parikh, Confinement and the AdS/CFT Correspondence, Phys.Lett. B483 (2000) 271-276 (arXiv:hep-th/0002031)

• Henrique Boschi Filho, AdS/QCD and confinement, Seminar at the Workshop on Strongly Coupled QCD: The confinement problem, November 2011 (pdf)

### In M-theory on $G_2$-manifolds

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

based on