Invariance of a physical system (its action) under a continuous group of local transformations underlain by a global symmetry whose group parameters fixed in space-time have now been extended to vary in space-time instead. Use for buildup of the invariance, fixing the gauge, and accounting for the corresponding changes in the functional measure of the system.
Invariance of a physical system (often, formally, its Lagrangian) under a continuous group of local transformations underlain by a global symmetry whose group parameters fixed in space-time (transformation identically performed at every point) have now been extended to vary in space-time, instead, so the transformation may be different at each and every point. The term "gauge" refers to redundant degrees of freedom in the resulting Lagrangian.
Transformations between possible equivalent gauges, called gauge transformations, form a Lie group—referred to as the symmetry group (gauge group) of the theory. A Lie group is built up by exponentials of group generators in a corresponding Lie algebra. To each such group generator there corresponds a gauge field (usually a vector field). Gauge fields included in the Lagrangian ensure its gauge invariance. Upon quantization, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, the gauge theory is referred to as non-abelian, the usual example being the Yang–Mills theory.