---
title: Yang-Mills Theories
url: https://www.emergentmind.com/topics/yang-mills-theories
type: topic
---

# Yang-Mills Theories

Yang–Mills theories are gauge field theories based on local, generally non-Abelian symmetries. Their fundamental variables are connections on principal bundles or Lie-algebra-valued one-forms, with curvature
\[
F=dA+\frac12[A\wedge A],
\]
and action constructed from an invariant quadratic form, conventionally
\[
S_{\mathrm{YM}}=-\frac14\int \operatorname{Tr}(F\wedge \star F).
\]
The commutator term in \(F\) distinguishes non-Abelian Yang–Mills theory from electromagnetism and produces gauge-boson self-interactions. The framework includes ordinary Lie-algebra gauge theories, supersymmetric and curved-space extensions, lattice and functional formulations, generalized gauge algebras, Lie algebroid theories, and geometric relations to gravity.

## 1. Gauge structure, curvature, and classical formulation

Let \(G\) be a compact Lie group with Lie algebra \(\mathfrak g\). For a trivial principal bundle, a connection is a \(\mathfrak g\)-valued one-form
\[
A=A_\mu^a\,dx^\mu\otimes \xi_a.
\]
The curvature is
\[
F_A=dA+\frac12[A\wedge A],
\]
or, in components,
\[
F_{\mu\nu}^a
=
\partial_\mu A_\nu^a-\partial_\nu A_\mu^a
+\epsilon_{bc}{}^a A_\mu^bA_\nu^c.
\]
For matrix-valued fields, the non-Abelian contribution is the commutator \([A_\mu,A_\nu]\). It vanishes for an Abelian group such as \(U(1)\), but produces three-gauge-boson and four-gauge-boson interactions in non-Abelian theories [1803.01950].

A gauge transformation \(g\colon M\to G\) acts on the connection in a convention-dependent manner, for example
\[
A\longmapsto A^g=gAg^{-1}-(dg)g^{-1},
\]
while the curvature transforms covariantly:
\[
F_{A^g}=gF_Ag^{-1}.
\]
Consequently, \(A\) is not itself gauge invariant, whereas traces of holonomies and the Yang–Mills action are gauge invariant.

For a spacetime metric \(h\), the classical action is
\[
S_{\mathrm{YM}}
=
-\frac14\int_M \operatorname{Tr}(F\wedge\star F).
\]
The invariant inner product on \(\mathfrak g\), often given by the Killing–Cartan form, contracts Lie-algebra indices. In ordinary Yang–Mills theory, gauge invariance of this quadratic form is equivalent to ad-invariance:
\[
g([u,v],w)+g(v,[u,w])=0.
\]

The Standard Model contains the gauge structure
\[
SU(3)_c\times SU(2)_L\times U(1)_Y.
\]
The \(SU(3)_c\) sector is QCD, while the electroweak \(SU(2)_L\times U(1)_Y\) theory is spontaneously broken. The non-Abelian gauge fields carry the corresponding gauge charge and therefore interact among themselves. In QCD, these interactions are associated with gluon self-couplings; in the electroweak sector they include interactions among \(W\) and \(Z\) bosons [1602.02307].

### Gauge fixing and physical observables

Gauge-related configurations represent the same physical field. Gauge-fixed correlation functions such as
\[
\langle A_\mu^a(x)A_\nu^b(y)\rangle
\]
are therefore defined only after a gauge choice. The Faddeev–Popov construction introduces ghost fields to represent the gauge-fixing Jacobian. In linear covariant gauges,
\[
\mathcal L_{\mathrm{gf+gh}}
=
\frac{1}{2\xi}(\partial_\mu A_\mu^a)^2
-\bar c^a\,\partial_\mu D_\mu^{ab}c^b,
\]
and Landau gauge is obtained in the limit \(\xi\to0\):
\[
\partial_\mu A_\mu^a=0.
\]

The Gribov problem arises because the Faddeev–Popov prescription does not select a unique representative of every gauge orbit. In Landau gauge, the first Gribov region is
\[
\Omega=\{A\mid \partial_\mu A_\mu=0,\;M[A]>0\},
\]
but even this region contains gauge copies. This complicates the relation between continuum functional equations and lattice gauge-fixing prescriptions [1808.05227].

## 2. Renormalization, asymptotic freedom, and nonperturbative dynamics

Yang–Mills theories in four dimensions are asymptotically free: the running coupling decreases at large momentum, making perturbation theory reliable in the ultraviolet. Schematically,
\[
g(\mu)\sim\frac{1}{\ln\mu}.
\]
For QCD,
\[
\alpha_s(\mu)=\frac{g_s^2(\mu)}{4\pi}
\approx
\frac{4\pi}{b_0\ln(\mu^2/\Lambda_{\mathrm{QCD}}^2)}.
\]
At low energies the coupling becomes large, and perturbative methods cease to determine confinement, bound states, and the mass spectrum.

Renormalization absorbs ultraviolet divergences into scale-dependent couplings, masses, and fields. The physical predictions of accelerator processes are obtained through perturbative expansions in the running coupling, together with infrared-safe definitions of observables, factorization, parton evolution, and Wilson-line methods [1602.02307].

### Nonperturbative scales and mass gaps

A dimensionless coupling can generate a physical scale through dimensional transmutation. One proposed mechanism for pure \(SU(3)\) Yang–Mills theory uses the conjectured beta function
\[
\beta_{\mathrm{YM}}(g)
=
-\frac{ag^3}{1-bg^2},
\qquad
a=\frac{11}{3}\frac{N}{(4\pi)^2},
\qquad
b=\frac{17}{11}\frac{N}{8\pi^2}.
\]
This expression reproduces the known perturbative coefficients through two loops but is not established as the exact nonsupersymmetric beta function. It has a pole at
\[
g_{\mathrm p}=\frac1{\sqrt b}.
\]
Within the model, the integrated renormalization-group trajectory has a minimum scale at this pole. That scale is interpreted as a dynamically generated infrared scale and, approximately, twice the lightest glueball mass. For \(SU(3)\), using the stated high-energy input, the resulting estimate is
\[
m_{\mathrm{lightest\ glueball}}\simeq 1.67~\mathrm{GeV},
\]
compared with a quenched lattice value
\[
m(0^+)=1648\pm58~\mathrm{MeV}.
\]
The agreement is model-dependent evidence rather than a proof of the beta function or of the Yang–Mills mass gap [1009.0265].

### Functional methods

Dyson–Schwinger equations (DSEs) arise from functional integration by parts and generate an infinite hierarchy of equations for propagators and vertices. In Landau gauge, the gluon propagator is
\[
D_{\mu\nu}^{ab}(p)
=
\delta^{ab}P_{\mu\nu}(p)\frac{Z(p^2)}{p^2},
\]
with
\[
P_{\mu\nu}(p)=\delta_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},
\]
and the ghost propagator is
\[
D_G^{ab}(p)=-\delta^{ab}\frac{G(p^2)}{p^2}.
\]

Two broad classes of infrared solutions occur:

- **Scaling**:
  \[
  Z(p^2)\propto(p^2)^{2\kappa},
  \qquad
  G(p^2)\propto(p^2)^{-\kappa}.
  \]
  The gluon propagator vanishes at zero momentum while the ghost dressing diverges.

- **Decoupling**:
  \[
  G(0)<\infty,
  \qquad
  D(0)<\infty.
  \]
  This gives an infrared-finite gluon propagator and finite ghost dressing.

Current lattice calculations predominantly find decoupling behavior, although the relation between lattice gauge copies and continuum boundary conditions remains unresolved. Both solution classes violate gluon reflection positivity, consistent with the absence of asymptotic physical gluons. The three-gluon vertex exhibits a robust infrared zero crossing in several dimensions and computational approaches [1808.05227].

Functional calculations require truncation. Relevant sources of systematic uncertainty include omitted tensor structures, unknown vertices, two-loop diagrams, spurious cutoff divergences, renormalization prescriptions, and the continuation to complex momenta. In four dimensions, correct anomalous logarithms require either explicit two-loop diagrams or carefully constructed renormalization-group improvement factors.

## 3. Lattice, probabilistic, and nonperturbative formulations

A lattice Yang–Mills configuration assigns a group element \(U(x,y)\in G\) to every oriented edge, with
\[
U(y,x)=U(x,y)^{-1}.
\]
For a plaquette \(p\),
\[
U_p=U(x_1,x_2)U(x_2,x_3)U(x_3,x_4)U(x_4,x_1).
\]
The Wilson action is
\[
S_\Lambda(U)
=
\sum_{p\in P(\Lambda)}
\operatorname{Re}\operatorname{Tr}(I-U_p),
\]
and the Gibbs measure is
\[
d\mu_{\Lambda,\beta}(U)
=
\frac1{Z_{\Lambda,\beta}}
e^{-\beta S_\Lambda(U)}\,d\sigma_\Lambda(U).
\]
Because \(G(\Lambda)\) is compact and finite-dimensional, this is a genuine probability measure, unlike the formal continuum expression involving an infinite-dimensional Lebesgue measure [1803.01950].

For a smooth connection and lattice spacing \(\varepsilon\), one may associate
\[
U(x,x+\varepsilon e_j)=e^{\varepsilon A_j(x)}.
\]
The plaquette expansion gives
\[
\operatorname{Re}\operatorname{Tr}(I-U_p)
=
-\frac{\varepsilon^4}{2}\operatorname{Tr}(F_{jk}^2)
+O(\varepsilon^5).
\]
The lattice action therefore approximates the continuum action after an appropriate relation between \(\beta\) and \(\varepsilon\). In four-dimensional non-Abelian theories, the expected asymptotic behavior is logarithmic,
\[
\beta\sim\log(1/\varepsilon),
\]
although establishing the correct scaling is part of the unresolved continuum problem.

### Wilson loops and confinement

For a closed curve \(\gamma\), the Wilson loop is
\[
W_\gamma
=
\operatorname{Tr}
\left(
P\exp\oint_\gamma A
\right).
\]
On the lattice it is the trace of the ordered product of link variables around the loop. For a rectangular loop of spatial width \(R\) and temporal length \(T\),
\[
V(R)
=
-\lim_{T\to\infty}
\frac1T\log\langle W_{\gamma_{R,T}}\rangle.
\]
An area law,
\[
\log\langle W_{\gamma_{R,T}}\rangle\sim-\sigma RT,
\]
defines a string tension \(\sigma\) and is a strong confinement criterion.

An area law is rigorously known at sufficiently small \(\beta\), but this does not establish physical confinement in the weak-coupling regime. For four-dimensional non-Abelian theories, proving the area law at arbitrarily large \(\beta\), constructing a nontrivial continuum probability measure, and establishing a positive mass gap remain open problems [1803.01950].

### Center-flux topology

For \(SU(2)\), the effective adjoint group is
\[
SU(2)/\mathbb Z_2\simeq SO(3),
\]
with
\[
\pi_1(SO(3))=\mathbb Z_2.
\]
A lattice center-flux order parameter \(z\) distinguishes:

- an **ordered phase**, with \(\langle z\rangle=1\), closed center vortices, and well-defined \(\mathbb Z_2\) topological sectors;

- a **disordered phase**, with \(\langle z\rangle=0\), open center vortices, and no consistent superselection decomposition into center-flux sectors.

The transition displays essential, Kosterlitz–Thouless-like scaling, but is argued to belong to a distinct universality class. Its behavior is controlled by the lattice action, representation content, dimensionality, and center symmetry rather than by temperature. In the fundamental-action continuum limit, the thermodynamic limit is generally disordered, whereas suitable adjoint or positive-plaquette formulations can support ordered continuum vacua [1412.1762].

## 4. Supersymmetric and curved-space Yang–Mills theories

Maximally supersymmetric Yang–Mills theories arise by dimensional reduction of ten-dimensional \(\mathcal N=1\) SYM. The ten-dimensional fields consist of a gauge field \(A_M\) and a sixteen-component Majorana–Weyl fermion. In \(d\) dimensions, the components of the ten-dimensional gauge field split into a \(d\)-dimensional gauge field and \(10-d\) adjoint scalars. Examples include four-dimensional \(\mathcal N=4\) SYM, three-dimensional \(\mathcal N=8\) SYM, and five-dimensional maximal SYM [1209.4320].

### Off-shell supersymmetry

Ordinary supersymmetry transformations close only after imposing equations of motion. The Berkovits construction introduces seven auxiliary bosonic fields \(K_m\) and realizes up to nine independent off-shell supercharges in the local covariant formalism. The full sixteen physical supercharges are not all manifest off shell.

On curved backgrounds, the supersymmetry parameter satisfies a generalized Killing-spinor equation,
\[
\nabla_\mu\epsilon
=
\alpha\,\widetilde\Gamma_\mu\Gamma\,\epsilon.
\]
Curvature-dependent scalar masses, fermion bilinears, cubic scalar couplings, and modified auxiliary-field transformations are required for invariance. In four dimensions, the scalar mass is the conformal mass,
\[
4\alpha^2(\Gamma)^2=\frac{R}{12}.
\]
A more general construction with \(\Gamma=\Gamma^{789}\) exists for \(d\leq7\), including spheres, anti-de Sitter spaces, lens spaces, Sasaki–Einstein manifolds, and nearly Kähler manifolds. The generic higher-dimensional curved-space action can be complex and need not be reflection positive.

A distinct \(\mathcal N=8\) theory on \(S^3\) follows by reducing four-dimensional \(\mathcal N=4\) SYM on \(\mathbb R\times S^3\). It has a real, reflection-positive action and retains the \(SO(6)_R\) symmetry. Curvature generates scalar mass and Myers-type terms in the corresponding reduced matrix model.

### Lattice supersymmetry

Naive lattice discretization destroys supersymmetry because lattice difference operators do not obey the ordinary Leibniz rule. Twisting and orbifolding instead preserve a nilpotent scalar supercharge
\[
Q^2=0
\]
at nonzero lattice spacing. Twisted fermions form a Kähler–Dirac multiplet of lattice \(p\)-forms, and gauge variables are placed on oriented lattice cells so that interactions form closed gauge-invariant loops.

The resulting lattice theories are local, gauge invariant, free of the usual fermion-doubling problem, and possess exact lattice supersymmetry. The examples include two-dimensional \(\mathcal N=(2,2)\) SYM, three-dimensional \(\mathcal N=4\) SYM, and four-dimensional \(\mathcal N=4\) SYM. Numerical simulations use rational hybrid Monte Carlo (RHMC), pseudofermions, rational approximations to fractional powers of \(M^\dagger M\), and multi-mass conjugate-gradient solvers [1108.1503].

## 5. Generalizations of Yang–Mills gauge structure

### Lie algebroid Yang–Mills theory

A Lie algebroid replaces the structural Lie algebra \(\mathfrak g\) by a vector bundle
\[
(E\to M,[\ ,\ ],\rho),
\]
with bracket on sections and anchor
\[
\rho:E\to TM.
\]
The anchor satisfies
\[
[\rho(\psi_1),\rho(\psi_2)]
=
\rho([\psi_1,\psi_2]).
\]
For a spacetime \(\Sigma\), the fields are a map \(X:\Sigma\to M\) and a one-form
\[
A\in\Omega^1(\Sigma,X^*E).
\]
The two field strengths are
\[
F^i=dX^i-\rho^i{}_aA^a,
\]
and
\[
F^a
=
dA^a+\frac12C^a{}_{bc}A^b\wedge A^c
+\Gamma^a{}_{ib}F^i\wedge A^b.
\]
The condition \(F^i=0\) is the anchor-compatibility condition.

A naive action that squares both field strengths is highly restrictive: gauge invariance forces
\[
R_{ij}{}^a{}_b=0,
\qquad
S^a{}_{bc\,i}=0,
\]
which implies that the algebroid is locally an action Lie algebroid. A genuinely algebroid theory instead introduces Lagrange multipliers:
\[
S_{\mathrm{LAYM}}
=
\int_\Sigma B_i\wedge F^i
-\frac12\int_\Sigma F^a\wedge\star F^b\,g_{ab}.
\]
Gauge invariance requires covariant constancy of the fiber metric with respect to a Lie-algebroid connection,
\[
\widetilde\nabla g=0.
\]
A weaker intrinsic condition involves only the anchor kernel:
\[
\nabla^{\mathrm{Bott}}\,g|_{\ker\rho}=0.
\]
For \(M\) a point, the construction reduces to ordinary Yang–Mills theory and \(\widetilde\nabla g=0\) becomes ad-invariance [0908.3161].

Matter fields can be sections of a vector bundle \(V\to M\), with gauge transformations governed by a flat \(E\)-connection:
\[
{}^ER=0.
\]
Gauge invariance of the matter kinetic term requires
\[
{}^E\nabla g_V=0.
\]
For nonlinear sigma-model matter, the Lie algebroid acts on a fiber bundle \(\widetilde M\to M\), and the vertical target metric must be invariant under the induced infinitesimal action.

### Generalized gauge algebras and gravity

A generalized Yang–Mills theory need not factorize into a finite-dimensional internal Lie algebra and an algebra of spacetime functions. Its structure constants may depend on momentum or spacetime data:
\[
[T_a(p),T_b(p')]
=
f_{ab}{}^c(p,p')T_c(p+p').
\]
The diffeomorphism algebra is an example. Its generators are differential operators, and the generalized gauge potential is
\[
A_\mu(x)=A_\mu{}^\nu(x)\partial_\nu.
\]
The covariant derivative
\[
D_\mu=\partial_\mu+A_\mu
\]
can be expressed in terms of a frame-like field
\[
\hat e_a{}^\mu=\delta_a{}^\mu+A_a{}^\mu.
\]
The generalized field strength
\[
F_{ab}=[D_a,D_b]
\]
is identified, after a change of basis, with Weitzenböck torsion. A suitable quadratic action becomes the teleparallel equivalent of the Einstein–Hilbert action. Generic quadratic choices contain a graviton, a dilaton-like scalar, and a rank-two antisymmetric tensor; the teleparallel parameter choice removes the additional fields and reproduces Einstein gravity [1501.05378].

### Covariant Hamiltonian and boundary formulations

In the covariant multisymplectic formulation, the Yang–Mills fields are a connection \(A\) and a covariant multimomentum \(P\). The first-order action is
\[
S_{\mathrm{YM}}(A,P)
=
-\int_M
\left[
P_a^{\mu\nu}F_{\mu\nu}^a
+\frac14P_a^{\mu\nu}P_{\mu\nu}^a
\right]\mathrm{vol}_M.
\]
The equations of motion give
\[
P_{\mu\nu}^a=-2F_{\mu\nu}^a,
\qquad
d_AP=0.
\]

On a boundary, the canonical variables are the boundary connection \(a\) and electric momentum \(p\), with symplectic form
\[
\omega_{\partial M}=\delta a\wedge\delta p.
\]
The boundary gauge group has moment map
\[
\mathcal J(a,p)=d_a^*p.
\]
The reduced Yang–Mills phase space is
\[
\mathcal R_{\mathrm{YM}}
=
\mathcal J^{-1}(0)/\mathcal G_{\partial M}.
\]
Thus Gauss’s law is the zero level set of the boundary moment map, and the physical boundary phase space is obtained by symplectic reduction [1506.00338].

A related field-space connection, particularly the Singer–DeWitt connection, separates perturbations into horizontal radiative directions and vertical gauge directions. In the canonical formulation, the electric field decomposes into a radiative part and a Coulombic part fixed by Gauss’s law. For topologically simple regions, regional radiative modes together with local charge and flux data suffice for gluing. Nontrivial topology can add global Wilson-loop or Aharonov–Bohm modes, but these are not new local boundary degrees of freedom [1906.00992].

## 6. Applications, dualities, and unresolved problems

### Accelerator physics

Yang–Mills theory underlies QCD and the electroweak theory tested at high-energy accelerators. Asymptotic freedom permits perturbative calculations at large momentum transfer, while infrared safety makes partonic predictions compatible with the confinement of quarks and gluons.

Factorization separates short-distance hard scattering from long-distance parton distributions:
\[
d\sigma_{H_1H_2}
=
\sum_{a,b}
\int dx_a\,dx_b\,
d\hat\sigma_{ab}
f_{a/H_1}(x_a,\mu)
f_{b/H_2}(x_b,\mu).
\]
DGLAP evolution transports parton distributions between scales. Wilson lines encode the coherent interaction of fast colored particles with soft and collinear gauge fields.

Jets, thrust distributions, three-jet events, vector-boson scattering, and Higgs production test the non-Abelian gauge structure. The 2012 LHC discovery of a Higgs-like scalar near \(125\ \mathrm{GeV}\) confirmed the essential mechanism of spontaneously broken gauge symmetry, while QCD production and electroweak decay connected the observation to the complete gauge-theory framework [1602.02307].

### Integrable deformations

An anisotropic \(2+1\)-dimensional \(SU(N)\) Yang–Mills theory can be reorganized as an array of coupled \(1+1\)-dimensional principal chiral sigma models. After a longitudinal rescaling and axial-gauge reduction,
\[
H=H_0+\lambda^2H_1,
\]
where
\[
H_0=\sum_{x^2}H_{\mathrm{PCSM}(x^2)}
\]
is an array of integrable principal chiral models and \(H_1\) is a current-current interaction generated by Gauss’s law. The principal chiral model is asymptotically free, massive, and integrable, with an exact S-matrix and bound-state masses
\[
m_r=m\frac{\sin(\pi r/N)}{\sin(\pi/N)}.
\]
Form factors of currents and stress tensors provide analytic input for glueball masses, string tensions, and correlation functions. The isotropic Yang–Mills theory is not solved; the construction is an expansion around an exactly solvable anisotropic limit [1409.8341].

### Double copy

Color–kinematics duality organizes Yang–Mills amplitudes in terms of kinematic numerators \(n_i\) satisfying Jacobi relations analogous to color factors:
\[
c_i+c_j+c_k=0
\quad\Longrightarrow\quad
n_i+n_j+n_k=0.
\]
Replacing color factors by a second set of kinematic numerators gives a gravity integrand:
\[
M_n^L
=
i^{L+1}
\left(\frac{\kappa}{2}\right)^{n-2+2L}
\sum_i
\int
\frac{n_i\widetilde n_i}{S_i\prod_a p_{a_i}^2}.
\]
At the level of states,
\[
\text{gluon}\otimes\text{gluon}
=
\text{graviton}\oplus B_{\mu\nu}\oplus\text{dilaton}.
\]
The most prominent supersymmetric example is
\[
\mathcal N=4\ \mathrm{SYM}\times\mathcal N=4\ \mathrm{SYM}
=
\mathcal N=8\ \mathrm{supergravity}.
\]
The amplitude double copy is strongly supported at tree level and by many loop-level calculations. A universal nonlinear off-shell field dictionary is not established, and ordinary Yang–Mills squared generally yields gravity coupled to additional fields rather than pure Einstein gravity [1602.08267].

### Open mathematical and physical problems

The principal unresolved problems include:

- construction of the four-dimensional non-Abelian continuum probability measure;
- rigorous determination of the lattice-spacing and coupling relation;
- proof of convergence and uniqueness of Wilson-loop limits;
- establishment of a positive Yang–Mills mass gap;
- proof of confinement in the weak-coupling continuum regime;
- control of Gribov copies and their relation to functional solutions;
- a complete four-dimensional functional treatment including all relevant vertices and tensor structures;
- quantum consistency, unitarity, and stability of higher-derivative tensor reformulations;
- a complete nonlinear off-shell formulation of double-copy field dictionaries;
- quantitative identification of effective scalar or defect degrees of freedom in the Yang–Mills infrared sector.

The status of these questions differs sharply across formulations. Lattice gauge theory provides finite-dimensional probability measures and nonperturbative numerical data; functional methods provide continuum equations but require truncation; supersymmetric constructions offer exact algebraic control in selected sectors; generalized geometric formulations broaden the notion of gauge symmetry; and amplitude methods expose hidden kinematic structures. Together, these approaches define the contemporary mathematical and physical scope of Yang–Mills theory without reducing its central nonperturbative problems to a single established framework.

Source: https://www.emergentmind.com/topics/yang-mills-theories