What happens to 
?
In this section, we assume that the integrated vertex
satisfies the full
Master Equation:
(rather than Eq. (
2)).
Here
with
. We will also
postulate that
is diffeomorphism
invariant, which means
in our formalism that:
Eqs. (
4) and (
5) imply that
is
-closed:
but we will also
assume (or just postulate) that it is
-exact:
| | | | | |
| | | | | |
| | | ←  -invariance of  |
| | |
Under these assumptions we can deform:
In other words:
the space does deform, according to Eq. (9), but
the action of diffeomorphisms on this space remains the same (Eq. (8))
the BV Hamiltonian of diffeomorphisms 
stays undeformed
Is it true that the deformed
remain
in involution modulo -exact?
Notice that the deformed
is automatically
-closed
under already taken assumptions:
Opening the parentheses, we derive that the expression
is
-closed:
Let us
assume that it is also
-exact:
(the validity of this assumption depends on the cohomology of
).
Therefore, to the first order in the bosonic infinitesimal parameter
:
:
