A Wilson loop is the gauge-invariant trace of the gauge field parallel-transported around a closed contour \(C\). It is the fundamental probe of confinement. A spatial Wilson loop is one whose contour lies entirely in a plane of the three spatial directions (no euclidean time edge); at finite temperature it measures the spatial string tension.

\[ W(C)=\mathrm{Tr}\,\mathcal{P}\exp\!\Big(i g\oint_C A_\mu\,dx^\mu\Big), \]

where \(\mathcal{P}\) denotes path ordering and \(A_\mu=A_\mu^aT^a\). Under a gauge transformation the trace is invariant. The comparator/parallel-transport idea is the same Wilson line \(U(y,x)\) used to build the covariant derivative.

Confinement is signalled by an area law: for a large loop enclosing area \(A\),

\[ \langle W(C)\rangle \sim \exp\!\big(-\sigma\,A\big), \]

with string tension \(\sigma\). A perimeter law (\(\langle W\rangle\sim e^{-\mu\,P}\)) signals deconfinement/screening.

• A temporal loop (\(R\times\beta\), one edge along \(\tau\)) gives the static quark–antiquark free energy; its tension \(\sigma(T)\to0\) above \(T_c\) (deconfinement).
• A spatial loop lies in a spatial plane and only feels the magnetic field \(A_i\). Its tension \(\sigma_s(T)\) stays finite and grows with \(T\). Because it involves purely magnetic, static gluons, it is exactly the observable computed in the reduced 3D theory MQCD.

Spatial Wilson loops are built from products of spatial link variables around rectangular contours; smearing (e.g. HYP/APE) and algorithms such as Bresenham path construction are used to improve the signal — see WilsonLoop.cpp and String Tension (Project 1). Fitting the area-law slope yields \(\sqrt{\sigma_s(T)}\), which is then compared with the dimensional reduction prediction of Why spatial string tension and dimensional reduction.