Lattice Gauge Theory

Group-valued link variables allow a lattice gauge theory to retain local gauge symmetry at finite spacing. This chapter connects parallel transport to the Wilson action and to physical loop observables, then distinguishes gauge fixing, strong-coupling expansions, topology, and smoothing by gradient flow.

Enter this chapter

Begin with Links, Plaquettes, and Gauge Invariance to derive link and plaquette transformations before choosing an action.

Topics

1. Links, Plaquettes, and Gauge Invariance — Constructs parallel transport with consistent orientation.
2. The Wilson Gauge Action and Its Continuum Limit — Recovers the continuum gauge kinetic term.
3. Gauge Fixing, Gribov Copies, and Gauge-Dependent Correlators — Tracks the gauge-copy prescription of nonphysical correlators.
4. Wilson and Polyakov Loops, Static Energies, and Screening Diagnostics — Distinguishes static energies, thermal probes, and screening.
5. Gauge Ensembles and Renormalized Observables — Connects ensemble measurements to renormalized quantities.
6. Strong-Coupling and Character Expansions — Uses character expansions in their convergence regime.
7. Topology and Lattice Index Diagnostics — Compares lattice charge and index definitions.
8. Gradient Flow, Flow Scales, and Renormalized Gauge Observables — Separates smoothing scales from cutoff and physical scales.

How the ideas fit together

A link transforms at its two endpoints, so a traced closed product is gauge invariant. This gives both the plaquette action and extended probes of gauge dynamics. Gauge invariance, however, does not equate their physical meanings: a rectangular Wilson loop extracts a static energy only after a long-time projection, and screening can change its large-distance behavior. Positive flow time introduces another scale rather than taking the continuum limit. Wilson 1974, pp. 2445–2459. The table keeps these distinctions explicit.

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	Endpoint-to-base transport with $U_\mu(x)\simeq e^{-iagA_\mu}$	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

Where to go next

Continue to Lattice Fermions and Chirality to locate the extra zeros of the naive lattice Dirac operator.

References

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