Kalkman element
It seems that there is a general theme which plays a role here. It is basically Kalkman map between Cartan and Weyl models of equivariant cohomology.
Restriction of form to orbit
Slight generalization
Generalization
PDF on LAG
Restriction of form to orbit
with the action of a Lie group 
.
Consider differential forms of mixed degree on .
Pick a point 
and consider the orbit 
.
Then, for every 
we can consider the differential form on :
|
|
as functions on 
,
and forms on as functions on 
.)
The secret of the formula is:
Slight generalization
After the first step (which was 
), we could have
(instead of putting 
) integrated over a submanifold 
where
is some compact submanifold of .
(Choice of a point 
is a particular case, when is one point.)
If we take the whole group of diffeomorphisms of , then this prescription
cooks out of an inhomogeneous differential form on the space
of submanifolds 
.
Generalization
We have some representation 
of 
. Then Eq. (2) still
holds, where 
should be compatible with the differential of .
PDF on LAG
,
acts on half-densities on the BV phase space. Moreover, there is an action of , where
is multiplication of a half-density by the BV Hamiltonian of 
(which we denote 
):
|
is compatible with 
. (the string measure):
