invariant theory

Today, invariant theory is often understood as a branch of representation theory, algebraic geometry, commutative algebra, and algebraic combinatorics. Each of these four disciplines has roots in nineteenth-century invariant theory. […] In modern terms, the basic problem of invariant theory can be categorized as follows. Let V be a K-vector space on which a group G acts linearly. In the ring of polynomial functions K[V] consider the subring K[V]ᴳ consisting of all polynomial functions on V which are invariant under the action of the group G. The basic problem is to describe the invariant ring K[V]ᴳ. In particular, we would like to know whether K[V]ᴳ is finitely generated as a K-algebra and, if so, to give an algorithm for computing generators.

The point is that, to construct locally invariant theories, new fields have to be introduced which are referred to as the gauge fields.