Local (gauge) invariance of field theories under Lie-group transformations: U(1), SU(2), SU(3) and Yang–Mills. Covariant derivative, gauge fields, gauge fixing, BRST, instantons, anomalies.
Gauge Theory
Local gauge invariance
Invariance of an action under spacetime-dependent Lie-group transformations φ(x) → g(x) φ(x). Forces introduction of a connection (gauge…
Covariant derivative D_μ
D_μ = ∂_μ + ig A^a_μ T^a: minimal coupling of matter to the gauge connection. Transforms homogeneously under gauge transformations.
Yang–Mills theory
Non-Abelian gauge theory based on a compact Lie group G (usually SU(N)). Field strength F^a_{μν} = ∂_μ A^a_ν − ∂_ν A^a_μ + g f^{abc} A^b_μ…
Gauge fixing
Selection of a representative from each gauge orbit to make the path integral well-defined. Common choices: Lorenz (∂·A = 0), Coulomb,…
Faddeev–Popov procedure
Inserts 1 = Δ_{FP} ∫ Dα δ(F(A^α)) into the gauge-theory path integral to yield a well-defined finite measure; introduces ghost fields for…
Wilson loop
Gauge-invariant path-ordered holonomy W(C) = Tr P exp(ig ∮_C A_μ dx^μ). Order parameter for confinement (area law) vs Coulomb phase…
Instanton
Classical finite-action Euclidean solutions of Yang–Mills equations interpolating between topologically distinct vacua. Responsible for…
Anomaly (quantum)
Classical symmetry broken by quantum effects — typically appears as a non-conservation of an otherwise-Noether current after regularisation…
SU(2)_L × U(1)_Y electroweak theory
Glashow–Weinberg–Salam unification of EM and weak interactions as a chiral gauge theory broken to U(1)_em by the Higgs VEV.
Color confinement
Empirical/theoretical fact that colour-charged states (quarks, gluons) cannot appear as isolated asymptotic particles; only colour singlets…
Gauge symmetry
Invariance of a physical theory under spacetime-dependent (local) transformations of an internal symmetry group G. Demanding local…
BRST cohomology
Cohomological formulation of gauge theory quantization. Physical states = H⁰(Q_BRST). Handles constraints, gauge fixing, unitarity in…
Lattice QCD
Wilson action on Euclidean spacetime lattice; numerical computation of non-perturbative QCD quantities. Hadron masses, form factors,…
Yang–Mills action
S = -(1/4g²)∫ tr(F_μν F^μν) d⁴x with F_μν = ∂_μA_ν - ∂_νA_μ + [A_μ,A_ν]; non-linear self-interactions from non-Abelian structure.
Bianchi identity (gauge)
D_μ F_νρ + D_ν F_ρμ + D_ρ F_μν = 0; automatic from F = dA + A∧A; non-Abelian generalization of ∇·B = 0.
Wilson lines & parallel transport
U(x,y) = P exp(i∫_y^x A·dx'); gauge-covariant transport; closed loop = Wilson loop; basis of lattice gauge observables.
't Hooft–Polyakov monopole
Topological soliton in SU(2)→U(1) Higgs gauge theory; finite-energy magnetic monopole with g = 4π/e; BPS bound E ≥ |(vacuum charge)|.
BPS bound and states
Central charges of extended SUSY algebra bound masses M ≥ |Z|; saturation yields preserved supersymmetry; used in string dualities.
Yang–Mills instanton
Self-dual F = *F classical solution with topological charge k ∈ ℤ; exp(-8π²/g²) tunneling amplitude; generates θ-vacuum physics.
Yang–Mills θ-vacuum
True vacuum |θ⟩ = Σ e^(inθ)|n⟩ with CP-violating term θ tr(F F̃); measurement of neutron EDM constrains θ_QCD < 10⁻¹⁰.
Chern–Simons action
S_CS = (k/4π)∫ tr(A∧dA + (2/3)A∧A∧A); topological in 3D; level k quantized; describes FQH, knot invariants (Witten 1989).
Higgs, Coulomb, and confining phases
Fradkin–Shenker: gauge theories with matter have Higgs/confining regions; Wilson loop area vs perimeter law; order parameter for…
Color confinement & linear potential
V(r) ≈ σr at large r with string tension σ ≈ (420 MeV)²; lattice-QCD; quark-antiquark string breaks at energy 2 m_q.
Asymptotic freedom (QCD β-function)
β(g) = -(11N - 2N_f)g³/(48π²) + O(g⁵); αs runs weak at high Q²; Gross–Wilczek–Politzer (2004 Nobel).
Seiberg duality
N=1 SU(N_c) with N_f flavors dual to SU(N_f-N_c) with magnetic quarks + mesons; IR-equivalent despite different UV.
Seiberg–Witten theory
Exact low-energy effective action of N=2 SU(2); holomorphic prepotential F(a); dyon condensation → confinement on moduli space.
't Hooft operator & magnetic disorder
Disorder operator creating magnetic flux line; dual to Wilson electric line; 't Hooft algebra classifies gauge phases.
Wilson lattice gauge theory
Link variables U_μ(x) ∈ G on hypercubic lattice; plaquette action; nonperturbative definition; Monte Carlo simulations of QCD.
Gribov ambiguity & copies
Gauge-fixing conditions (e.g. Landau) have multiple solutions; Faddeev–Popov breaks down nonperturbatively; Gribov–Zwanziger confinement…
AdS/CFT gauge-gravity duality
Type IIB string on AdS_5×S⁵ ↔ N=4 SYM in 4D; strong coupling in CFT ↔ classical gravity in bulk; holographic principle in action.
Dirac magnetic-monopole quantization
Electric-magnetic duality requires eg = 2πnℏc; single monopole would explain electric-charge quantization; topological origin (bundle…
Chevalley–Eilenberg cohomology of a gauge Lie algebra
Gauge-theory application of L0 cochain complex cohomology and derived functors. The Chevalley–Eilenberg complex C^n(g, M) = Hom(Λⁿ g, M)…
Stora–Zumino descent equations (gauge anomalies)
Gauge-theory application of L0 long exact sequence of cohomology and chain complexes. Starting from the gauge-invariant Chern–Weil…