Given a gauge theory with gauge group G, it is sometimes useful to find an equivalent formulation in terms of a non-trivial gauge subgroup H of G. This amounts to fixing the gauge partially from G down to H. We study this problem systematically, both from the algebraic and the path integral points of view. We find that the usual BRST cohomology must be replaced by an equivariant version and that the ghost Lagrangian must always include quartic ghost terms, even at tree level. Both the Cartan and Weil models for equivariant cohomology play a role and find natural interpretations within the physics framework. Applications include the construction of D-brane models of emergent space, the 't Hooft's Abelian projection scenario in quantum chromodynamics and the formulation of the low energy effective theories of grand unified models.

41 pages; v2: references added, typos corrected, comments added further emphasizing the signification of the tree-level quartic ghost terms, their crucial relation to gauge invariance and their sharp difference with quartic ghost terms used for renormalization purposes; v3: more types corrected