Gauge group

The gauge group is the group of local internal symmetries. Its Lie algebra labels the gauge-boson species, while a matter field carries a representation of the group.

A gauge generator is the representation matrix of a basis element of the Lie algebra of a continuous gauge group on a specified set of fields. For Hermitian generators $T^a$, an infinitesimal transformation is $\delta \varphi =i{\alpha }^a{T}^a\varphi$, and $[T^a,T^b]=i{f}^{ab}{}_c{T}^c$. Gauge invariance of a superpotential consequently requires $W_i(T^a{)}^i{}_j{\varphi }^j=0$ for each generator.

A gauge group representation specifies how a matter field transforms under a gauge group. It fixes the gauge charges and whether gauge-invariant mass terms can pair a field with a field in the conjugate group representation.