nLab Yang-Mills theory

Idea

Yang–Mills theory is a gauge theory on a given 4-dimensional (pseudo-)Riemannian manifold $X$ whose field is the Yang–Mills field – a cocycle $\nabla \in \mathbf{H}(X,\bar \mathbf{B}U(n))$ in differential nonabelian cohomology represented by a vector bundle with connection – and whose action functional is

$$\nabla \mapsto \frac{1}{g^2 }\int_X tr(F_\nabla \wedge \star F_\nabla) \;+\; i \theta \int_X tr(F_\nabla \wedge F_\nabla)$$

for

• $F_\nabla$ the field strength, locally the curvature $\mathfrak{u}(n)$-Lie algebra valued differential form on $X$ ( with $\mathfrak{u}(n)$ the Lie algebra of the unitary group $U(n)$);

• $\star$ the Hodge star operator of the metric $g$;

• $\frac{1}{g^2}$ the Yang-Mills coupling constant and $\theta$ the theta angle, some real numbers (see at S-duality).

(See this example at A first idea of quantum field theory.)

Properties

Classification of solutions

• Narasimhan-Seshadri theorem

• Donaldson-Uhlenbeck-Yau theorem

Quantization

Despite its fundamental role in the standard model of particle physics, various details of the quantization of Yang-Mills theory are still open. See at quantization of Yang-Mills theory.

Applications

All gauge fields in the standard model of particle physics as well as in GUT models are Yang–Mills fields.

The matter fields in the standard model are spinors charged under the Yang-Mills field. See

• spinors in Yang-Mills theory

History

From Jaffe-Witten:

By the 1950s, when Yang–Mills theory was discovered, it was already known that the quantum version of Maxwell theory – known as Quantum Electrodynamics or QED – gives an extremely accurate account of electromagnetic fields and forces. In fact, QED improved the accuracy for certain earlier quantum theory predictions by several orders of magnitude, as well as predicting new splittings of energy levels.

So it was natural to inquire whether non-abelian gauge theory described other forces in nature, notably the weak force (responsible among other things for certain forms of radioactivity) and the strong or nuclear force (responsible among other things for the binding of protons and neutrons into nuclei). The massless nature of classical Yang–Mills waves was a serious obstacle to applying Yang–Mills theory to the other forces, for the weak and nuclear forces are short range and many of the particles are massive. Hence these phenomena did not appear to be associated with long-range fields describing massless particles.

In the 1960s and 1970s, physicists overcame these obstacles to the physical interpretation of non-abelian gauge theory. In the case of the weak force, this was accomplished by the Glashow–Salam–Weinberg electroweak theory with gauge group $H =$ SU(2) $\times$ U(1). By elaborating the theory with an additional “Higgs field”, one avoided the massless nature of classical Yang–Mills waves. The Higgs field transforms in a two-dimensional representation of $H$; its non-zero and approximately constant value in the vacuum state reduces the structure group from $H$ to a $U(1)$ subgroup (diagonally embedded in $SU(2) \times U(1)$. This theory describes both the electromagnetic and weak forces, in a more or less unified way; because of the reduction of the structure group to $U(1)$, the long-range fields are those of electromagnetism only, in accord with what we see in nature.

The solution to the problem of massless Yang–Mills fields for the strong interactions has a completely different nature. That solution did not come from adding fields to Yang–Mills theory, but by discovering a remarkable property of the quantum Yang–Mills theory itself, that is, of the quantum theory whose classical Lagrangian has been given $[$above$]$. This property is called “asymptotic freedom”. Roughly this means that at short distances the field displays quantum behavior very similar to its classical behavior; yet at long distances the classical theory is no longer a good guide to the quantum behavior of the field.

Asymptotic freedom, together with other experimental and theoretical discoveries made in the 1960s and 1970s, made it possible to describe the nuclear force by a non-abelian gauge theory in which the gauge group is $G =$ SU(3). The additional fields describe, at the classical level, “quarks,” which are spin 1/2 objects somewhat analogous to the electron, but transforming in the fundamental representation of $SU(3)$. The non-abelian gauge theory of the strong force is called Quantum Chromodynamics (QCD).

The use of QCD to describe the strong force was motivated by a whole series of experimental and theoretical discoveries made in the 1960s and 1970s, involving the symmetries and high-energy behavior of the strong interactions. But classical non-abelian gauge theory is very different from the observed world of strong interactions; for QCD to describe the strong force successfully, it must have at the quantum level the following three properties, each of which is dramatically different from the behavior of the classical theory:

(1) It must have a “mass gap;” namely there must be some constant $\Delta \gt 0$ such that every excitation of the vacuum has energy at least $\Delta$.

(2) It must have “quark confinement,” that is, even though the theory is described in terms of elementary fields, such as the quark fields, that transform non-trivially under SU(3), the physical particle states – such as the proton, neutron, and pion –are SU(3)-invariant.

(3) It must have “chiral symmetry breaking,” which means that the vacuum is potentially invariant (in the limit, that the quark-bare masses vanish) only under a certain subgroup of the full symmetry group that acts on the quark fields.

The first point is necessary to explain why the nuclear force is strong but short-ranged; the second is needed to explain why we never see individual quarks; and the third is needed to account for the “current algebra” theory of soft pions that was developed in the 1960s.

Both experiment – since QCD has numerous successes in confrontation with experiment – and computer simulations, carried out since the late 1970s, have given strong encouragement that QCD does have the properties cited above. These properties can be seen, to some extent, in theoretical calculations carried out in a variety of highly oversimplified models (like strongly coupled lattice gauge theory). But they are not fully understood theoretically; there does not exist a convincing, whether or not mathematically complete, theoretical computation demonstrating any of the three properties in QCD, as opposed to a severely simplified truncation of it.

This is the problem of non-perturbative quantization of Yang-Mills theory. See there for more.

Related concepts

• Gauss law

• D=5 Yang-Mills theory

• D=2 Yang-Mills theory

• massive Yang-Mills theory

• self-dual Yang-Mills theory

• super Yang-Mills theory

• minimal coupling

• 't Hooft double line notation

• Einstein-Yang-Mills theory

  • Einstein-Maxwell theory

  • Einstein-Yang-Mills-Dirac theory

  • Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory

• Yang-Mills equation

• standard model of particle physics

  • electromagnetism

  • spinors in Yang-Mills theory

  • QED, QCD,

  • electroweak field

• Yang monopole, 't Hooft-Polyakov monopole

• S-duality, Montonen-Olive duality

  • electric-magnetic duality

  • geometric Langlands duality

• Proca theory

• Chern-Simons theory

• Yang-Mills instanton

• confinement

• asymptotic freedom

• uncertainty of fluxes

References

General

Yang-Mills theory is named after the article

• Chen Ning Yang, Robert Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance. Physical Review 96 (1): 191– 195. (1954) (web)

which was the first to generalize the principle of electromagnetism to a non-abalian gauge group. This became accepted as formulation of QCD and weak interactions (only) after spontaneous symmetry breaking (the Higgs mechanism) was understood in the 1960s.

The identification of Yang-Mills gauge potentials with connections on fiber bundles is due to:

• 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]

On the historical origins:

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

• Chen Ning Yang, The conceptual origins of Maxwell’s equations and gauge theory, Phyics Today 67 11 (2014) [doi:10.1063/PT.3.2585, pdf]

Review of the basics:

• Arthur Jaffe, Edward Witten, Quantum Yang-Mills theory (2000) [pdf, pdf, pdf]

  (in the context of the mass gap problem)

• Mikio Nakahara, Section 10.5.4 of: Geometry, Topology and Physics, IOP (2003) [doi:10.1201/9781315275826, pdf]

• Simon Donaldson, Yang-Mills theory and geometry (2005) [pdf, pdf]

• Simon Donaldson, Gauge Theory: Mathematical Applications, Encyclopedia of Mathematical Physics, Academic Press (2006) 468-481 [doi:10.1016/B0-12-512666-2/00075-4, author pdf, pdf]

• José Figueroa-O'Farrill, Gauge theory, lecture notes (2006) [web]

• Karen Uhlenbeck, notes by Laura Fredrickson, Equations of Gauge Theory, lecture at Temple University, 2012 (pdf, pdf)

• Gerd Rudolph, Matthias Schmidt, Chapters 7-9 of: Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields, Springer (2017) [doi:10.1007/978-94-024-0959-8]

See also the references at QCD, gauge theory, Yang-Mills monopole, Yang-Mills instanton and at super Yang-Mills theory.

Classical discussion of YM-theory over Riemann surfaces (which is closely related to Chern-Simons theory, see also at moduli space of flat connections) is in

• Michael Atiyah, Raoul Bott, The Yang-Mills equations over Riemann surfaces, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences

  Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (jstor, lighning summary)

which is reviewed in the lecture notes

• Jonathan Evans, Aspects of Yang-Mills theory, (web)

Relation to instanton Floer homology:

• Simon Donaldson, Floer homology groups in Yang-Mills theory Cambridge University Press (2002) (pdf)

Relation to Tamagawa numbers:

• Aravind Asok, Brent Doran, Frances Kirwan, Yang-Mills theory and Tamagawa numbers (arXiv:0801.4733)

On combinatorics of YM scattering amplitudes:

• Nima Arkani-Hamed, Qu Cao, Jin Dong, Carolina Figueiredo, Song He: Scalar-Scaffolded Gluons and the Combinatorial Origins of Yang-Mills Theory [arXiv:2401.00041]

On the nonperturbative effect of “effective locality”:

• Yves Gabellini, Thierry Grandou, Ralf Hofmann: On the Yang-Mills propagator at strong coupling [arXiv:2412.12124]

Classical solutions

Wu and Yang (1968) found a static solution to the sourceless $SU(2)$ Yang-Mills equations. Recent references include

• J. A. O. Marinho, O. Oliveira, B. V. Carlson, T. Frederico, Revisiting the Wu-Yang Monopole: classical solutions and conformal invariance

There is an old review,

• Alfred Actor, Classical solutions of $SU(2)$ Yang—Mills theories, Rev. Mod. Phys. 51, 461–525 (1979),

that provides some of the known solutions of $SU(2)$ gauge theory in Minkowski (monopoles, plane waves, etc) and Euclidean space (instantons and their cousins). For general gauge groups one can get solutions by embedding $SU(2)$‘s.

For Yang-Mills instantons the most general solution is known, first worked out by

• Michael Atiyah, Nigel Hitchin, Vladimir Drinfeld, Yuri Manin, Construction of instantons, Physics Letters 65 A, 3, 185–187 (1978) pdf

for the classical groups SU, SO , Sp, and then by

• C. Bernard, N. Christ, A. Guth, E. Weinberg, Pseudoparticle Parameters for Arbitrary Gauge Groups, Phys. Rev. D16, 2977 (1977)

for exceptional Lie groups. The latest twist on the Yang-Mills instanton story is the construction of solutions with non-trivial holonomy:

• Thomas C. Kraan, Pierre van Baal, Periodic instantons with nontrivial holonomy, Nucl.Phys. B533 (1998) 627-659, hep-th/9805168

There is a nice set of lecture notes

• David Tong, TASI Lectures on Solitons (hep-th/0509216),

on topological solutions with different co-dimension (instantons, monopoles, vortices, domain walls). Note, however, that except for instantons these solutions typically require extra scalars and broken U(1)‘s, as one may find in super Yang-Mills theories.

Some of the material used here has been taken from

• P.SE, Which exact solutions of the classical Yang-Mills equations are known?

Another model featuring Yang-Mills fields has been proposed by Curci and Ferrari, see Curci-Ferrari model.

See also

• DispersiveWiki, Yang-Mills equations

Phase space and canonical quantization

On the phase space, Poisson brackets and their quantization in Yang-Mills theory:

• Taichiro Kugo, Izumi Ojima, Manifestly Covariant Canonical Formulation of the Yang-Mills Field Theories. I: General Formalism, Progress of Theoretical Physics 60 6 (1978) 1869–1889 [doi:10.1143/PTP.60.1869]

• John L. Friedman, Nicholas J. Papastamatiou, On the canonical quantization of Yang-Mills theories, Nuclear Physics B 219 1 (1983) 125-142 [doi:10.1016/0550-3213(83)90431-5]

• A. Bassetto, I. Lazzizzera, R. Soldati, Yang-Mills theories in space-like axial and planar gauges, Nuclear Physics B 236 2 (1984) 319-335 [doi:10.1016/0550-3213(84)90538-8]

• D. M. Gitman, S. L. Lyakhovich & I. V. Tyutin, Canonical quantization of the Yang-Mills Lagrangian with higher derivatives, Soviet Physics Journal 28 (1985) 554–556 [doi:10.1007/BF00896182]

• Kurt Haller, Yang-Mills theory and quantum chromodynamics in the temporal gauge, Phys. Rev. D 36 (1987) 1839 [doi:10.1103/PhysRevD.36.1839]

• P. E. Haagensen, On The Exact Implementation Of Gauss’ Law In Yang-Mills Theory [arXiv:hep-ph/9307319]

• Sarada G. Rajeev, O. T. Turgut, Poisson Algebra of Wilson Loops in Four-Dimensional Yang-Mills Theory, Int. J. Mod. Phys. A 10 (1995) 2479 [arXiv:hep-th/9410053, doi:10.1142/S0217751X95001194]

  (in light-front formalism)

• Jonathan Dimock, Canonical Quantization of Yang-Millson a circle, Reviews in Mathematical Physics 08 01 (1996) 85-102 [doi:10.1142/S0129055X96000044]

• Daniel N. Blaschke, François Gieres, On the canonical formulation of gauge field theories and Poincaré transformations, Nuclear Physics B 965 (2021) 115366 [doi:10.1016/j.nuclphysb.2021.115366, arXiv:2004.14406]

• Aldo Riello, Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back, SciPost Phys. 10 125 (2021) [doi:10.21468/SciPostPhys.10.6.125]