Lattice gauge theories are based on the notion of parallel transport. They can be considered as non-perturbative regularizations of the continuum gauge theories in the sense of a low-temperature expansion. The chapter is mainly devoted on a study of matterless lattice gauge theories from the point of view of phase transitions. This means many properties of a realistic theory like quantum chromodynamics (QC) cannot be investigated, but the important question of confinement can still be studied: does the theory generate a force between charged particles increasing at large distances, so that heavy quarks in the fundamental representation cannot be separated? More generally, can one find charged asymptotic states like massless vector particles in the theory? Lattice gauge theories have properties quite different from the ferromagnetic systems. In particular the absence of a local order parameter requires a study of the behaviour of a non-local quantity, a functional of loops generally called Wilson's loop, to distinguish between the confined and deconfined phases, characterized by an area or perimeter law, respectively.
Eliminating fermionic matter fields in lattice gauge theories
