Let us study the whole equivalence class of theories. It is infinite-dimensional, because there where could be more or less arbitrary functional. There are infinitely many possible .

Let us choose a basis ; the space of BRST trivial deformations is parametrized by the coordinates :


Let us call this equivalence class . It turns out that the following differential form on is closed:


a pseudo-differential form on

We want to obtain BRST invariants by integrating

  • Usually the operator is only nilpotent on-shell; this requires more careful (than Eq. (1)) definition of a deformation

  • It appears that is not well-defined, because if is BRST-closed, it does not deform the action, and yet the form is nonzero

Solution: We should use the BV formalism.

We will now switch to BV language