Equivalence classes of theories

Many physical theories have BRST-like structure. This means that there is a nilpotent fermionic
symmetry , and the space of physical states is the cohomology of .

Such theories come with an equivalence relation. Physically, the theory with the action
is equivalent to the theory with the action for any operator . Physically
meaningful quantities should be invariant, i.e. should not change under this
equivalence transformation. Important question is:

how can we obtain such invariants?

Naively it would seem that the only way to obtain an invariant is to take the path integral:

where are some BRST-closed operators.

But actually there are other possibilities, which we will now describe.