Lattice Gauge Theory

Lattice gauge theory preserves local gauge symmetry before any continuum limit is taken. Its fundamental variables are group elements on oriented links, not Lie-algebra fields at sites. This chapter develops that construction from parallel transport through actions and ensemble observables, then separates the logically different evidence supplied by strong-coupling series, Wilson and Polyakov loops, gauge-fixed correlators, topology, and gradient flow. The finite-group link construction and its loop observables originate in Wilson 1974, pp. 2445–2459.

Choose the claim before the observable

A gauge calculation is easiest to interpret when the intended claim is fixed first.

Intended claim	Start here	Essential control
Exact gauge covariance at nonzero spacing	Links, plaquettes, and gauge invariance	Link orientation, representation, trace normalization
Recovery of the Yang–Mills action	The Wilson gauge action and its continuum limit	Small-plaquette expansion and a tuned continuum trajectory
A gauge-dependent propagator	Gauge fixing, Gribov copies, and gauge-dependent correlators	Residual symmetry, stopping rule, and copy sensitivity
Static-source or thermal diagnostic	Wilson and Polyakov loops	Self energies, operator basis, center symmetry, finite volume
A renormalized quantity from configurations	Gauge ensembles and renormalized observables	Measure, projection, scale, operator map, covariance
Analytic control near strong coupling	Strong-coupling and character expansions	Convergence domain and minimal-surface counting
Topological sectors or susceptibility	Topology and lattice index diagnostics	Smoothness, dislocations, sector sampling, continuum agreement
A flow scale or flowed composite	Gradient flow and renormalized gauge observables	The window $a^2\ll t\ll L^2$ and ordered limits

These entries are connected, but they are not interchangeable. An area law at one strong bare coupling is not a continuum confinement theorem; a Polyakov loop is an exact order parameter only when the relevant center symmetry exists; a smooth, nearly integer charge is not proof of correct sector sampling; and gradient flow does not erase the need for a continuum extrapolation.

A common convention card

Unless a page states otherwise, take a hypercubic Euclidean lattice with spacing $a$ , periodic spatial boundaries, compact gauge group $G$ , and unit vector $\hat\mu$ . An oriented link from $x$ to $x+a\hat\mu$ is $U_\mu(x)\in G$ , while the reversed link is

U_{-\mu}(x+a\hat\mu)=U_\mu(x)^{-1}.

For $SU(N)$ in representation $R$ , generators are Hermitian and normalized by

\operatorname{tr}_R(T^aT^b)=T_R\delta^{ab};

in the fundamental representation, $T_F=1/2$ . A positively oriented $\mu\nu$ plaquette begins at $x$ and is

U_{\mu\nu}(x)=U_\mu(x)U_\nu(x+a\hat\mu) U_\mu(x+a\hat\nu)^{-1}U_\nu(x)^{-1}.

Changing generator normalization, representation, anisotropy, or plaquette orientation requires changing the accompanying coefficients. A safe comparison therefore records the gauge group and global form, representation, $T_R$ , link exponential, plaquette orientation, action normalization, boundary sector, observable normalization, smoothing or flow time, renormalization scheme, and order of limits.

The inference chain

A typical calculation has four layers:

choose a finite lattice measure $d\mu(U)e^{-S[U]}$ with exact gauge invariance;
construct an operator with its transformation law and boundary sector explicit;
extract a finite-volume, finite-spacing quantity with correlations and excited-state effects controlled;
attach a physical interpretation only after scale setting, renormalization where needed, and the required volume and continuum limits.

The chapter’s shared regime map, introduced on the Wilson-action page, makes the branches visible. Follow the solid arrows for definitions and transformations; treat each dashed test as a condition on the claim, not as optional presentation detail.

A compact gauge convention table

Item	Definition to record	A decisive check	Failure if omitted
Gauge data	$G$ , global form, representation, $T_R$	Translate one known Casimir or group integral	Couplings and loop normalizations disagree
Links	Orientation and $U_\mu(x)\simeq e^{iagA_\mu}$ convention	Reverse a link and obtain $U^{-1}$	Paths fail to compose consistently
Plaquettes and action	Ordered boundary and coefficient of $1-\operatorname{Re}\operatorname{tr}U_p/N$	Recover $\frac14F^a_{\mu\nu}F^a_{\mu\nu}$	Wrong sign or factor in the continuum action
Gauge fixing	Functional, residual group, copy selection, tolerance	Repeat from several gauge transforms	Algorithmic representative is mistaken for a unique gauge
Loop observable	Representation, contour, smearing, subtraction	Reverse orientation and complex conjugate	Static energy contains an unidentified self energy
Topology	Charge definition, admissibility or smoothing, volume	Compare independent definitions toward $a=0$	Dislocations are counted as physical sectors
Flow	Integrator, $t/a^2$ , smoothing radius, discretization	Step-size refinement and $a^2\ll t\ll L^2$	Smoothing dependence is mislabeled renormalization
Final quantity	Scale, scheme, covariance, limit order	Leave-one-spacing-out and volume stability	A bare finite-lattice number is labeled physical

Chapter standard

Every conclusion in this chapter should pass three separate questions: Is it exactly true of the regulated theory? Is it stable under the estimator and analysis choices? Does it survive the limits required by the intended continuum statement? Keeping those questions separate is the central discipline of lattice gauge theory.

References

Wilson, K. G. (1974). Confinement of quarks. Physical Review D, 10, 2445–2459. DOI.