Integration over the location of insertion points
1 Fixing the Lagrangian submanifold
Therefore we have to study the restriction of 
to
the subgroup 
.
We will for now assume that elements of 
are in involution
(in our notations 
).
We will make use of the fact that
(as opposed to full
) preserves
.
This implies that the integration measure can be transformed to a
fixed Lagrangian submanifold:
2 Modified de Rham complex of
2.1 Definition
We define the “modified de Rham complex” of
as the space of
-invariants:
where the action of
is induced by the right shift on
and the action of
on
;
in particular, any function of the form
is
-invariant. The differential
comes from the canonical odd vector field on
; we will denote it
.
The integrand in Eq. (
1)
belongs to this space:
On the next page
we will demonstrate that this is the same as the Lie algebra cohomology complex of 
with coefficients in 
.
This is a version of the well-known theorem saying that Serre-Hochschild complex of the Lie algebra with trivial coefficients
is the same as right-invariant differential forms on the Lie group. This is a general statement, true for any Lie group (not only )
acting on any manifold .
2.2 Notations and useful identities
Generally speaking, in this section the tilde over a letter will denote the composition with
:
Here are some identities that we will need:
Elements of the space
can be obtained
from letters
and
by operations of multiplication and computing the odd Poisson
bracket, or applying
.
3 Intertwiner between 
and
Consider any function
(not necesserily
-invariant). We have:
In particular, when
only depends on
and does not depend neither on
nor on
.
(
i.e. when
):
In other words, the operator of multiplication by
intertwines between
and
.
After we integrate over the Lagrangian submanifold,
becomes just
.
4 Integration
4.1 The one-form component
4.2 The two-form component
TODO: continue to higher powers of
.
is just the product of two functions,
and
