Lattice Gauge Theory

Definition

Lattice gauge theory is gauge theory formulated on a discrete spacetime lattice, with the gauge field represented by group-valued variables $U_{\ell }\in G$ on links rather than a Lie-algebra-valued connection $A_{\mu }$. It provides a gauge-invariant, nonperturbative regulator and is the natural home for the strong-coupling expansion, duality, and the modern operator view of phases.

The building blocks:

• Link variables $U_{\ell }=\mathcal{P}\exp (i{\int }_{\ell }A)$, one per oriented link, transforming as $U_{\ell }\to g_x{U}_{\ell }{g}_y^{†}$ under gauge transformations at the endpoints.
• Plaquette $U_p={\prod }_{\ell \in \partial p}{U}_{\ell }$, the smallest Wilson loop; the Wilson action $S=-\frac{\beta }{N}{\sum }_p\mathrm{Re}\mathrm{tr}{U}_p$ reduces to $\frac{1}{4}\int F^2$ in the continuum limit.
• Haar measure ${\prod }_{\ell }d{U}_{\ell }$ makes the path integral finite and gauge-invariant with no gauge fixing — the setting for Elitzur’s theorem (local symmetries do not break).

Two complementary formulations run through the course: the Euclidean (Wilson) action above, and the Hamiltonian (Kogut–Susskind) form $H=\frac{g^2}{2}{\sum }_{\ell }{E}_{\ell }^2-\frac{1}{g^2}{\sum }_p(U_p+U_p^{†})$ with the Gauss-law constraint selecting closed electric strings as physical states.

Role in Research

Lattice gauge theory is the technical spine of Semester I of the generalized-symmetries course and the concrete setting for the group’s confinement-duality revival. Its diagnostics — the wilson-loop area law, the string tension, the Villain dualities — are exactly the objects that the modern higher-form symmetry language reinterprets, and the lattice is where the group’s Julia–Toulouse/higher-gauging construction is most explicit (see condensation-defects).

• Strong-coupling expansion gives confinement (area law) for any compact group as a controlled calculation.
• The abelian cases dualize exactly (Villain form), producing the monopole pictures of confinement (dual-superconductor, julia-toulouse-mechanism).
• Adding matter yields the Fradkin–Shenker phase diagram and the Higgs/confinement complementarity, the backdrop for topological-order (the deconfined phase is the toric-code).

Relations

• wilson-loop — the central gauge-invariant order parameter; area vs perimeter law
• villain-action — the form in which abelian lattice gauge theories dualize exactly
• higher-form-symmetries — center/magnetic symmetries of lattice gauge theory are higher-form symmetries; confinement = unbroken 1-form symmetry
• toric-code — the ${\mathbb{Z}}_2$ Kogut–Susskind Hamiltonian in its deconfined phase
• julia-toulouse-mechanism, dual-superconductor — confinement via condensation of the dual (monopole) defects
• confinement — the phenomenon lattice gauge theory was built to capture nonperturbatively

Papers

Foundational: Wilson (1974); Wegner (1971); Kogut–Susskind (1975); Kogut’s review, Rev. Mod. Phys. 51 (1979) 659. See the course bibliography and confinement-duality for the full list.

Notes

• Elitzur’s theorem is why lattice gauge theory forced the field toward extended (loop) order parameters decades before “generalized symmetry” named the pattern.
• The continuum limit is taken along a line of increasing $\beta$ toward a critical point; asymptotic freedom fixes how $a\to 0$ relates to $\beta \to \infty$ for non-abelian groups.
• Monte Carlo (Creutz, 1980) turned the lattice into the primary quantitative tool for QCD; this course stays analytic, using the lattice for its exact dualities and its clean phase structure rather than for numbers.