Form Ω

Form

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

(1)

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

Problems:

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