Integration over family of Lagrangian submanifolds
1 Choice of 
We will here make use of the standard choice of always applicable in the BRST case.
We will choose
so that
is
the algebra of diffeomorphisms of the worldsheet:
where
is a vector field on the worldsheet.
2 Equivariant form 
The resulting equivariant form is:
where the term
comes from
and
from
of Eq. (
5).
The pairing
is as in Eq. (
2).
3 Recovery of the standard approach
Usually in the literature, the integration cycle is chosen so that
, and
therefore the term
vanishes. Moreover, we do not even need to
do the horizonthal projection of
. This is a consequence of the following
general statement. Suppose that
and the integration cycle in the moduli space of
Lagrangian submanifolds is such that:
In this case the construction simplifies:
We can now use the
Baranov-Schwarz transform
and interpret the integration of
as the integration of
over some new Lagrangian
submanifold, which can be described as follows.
Consider {\em any}
-dimensional surface in the space of metrics
parametrized by
:
(no need to require any holomorphicity). Let us consider a submanifold
defined as follows: it is the bundle over
whose fiber at a point
consists of the subspace of
orthogonal to the tangent space to
at
that point. Notice that this submanifold is Lagrangian. We will promote
to
a Lagrangian submanifold
in the BV phase space of the bosonic string by
adding
and
(and keeping
and
). We have:
Notice that we can use
instead of
because Eq. (
6)
is satisfied in this case. Indeed,
and we choose the Lagrangian
submanifolds so that
; this proves Eq. (
6).
4 What is “base integral form”?
When some part of contains a gauge-trivial direction 
, then
the integral of 
over that part is automatically zero. Indeed, in this
case all the 
orthogonal to the tangent space to satisfy in
particular 
and therefore the integral of over will give
zero (because of the zero mode 
). In this sense, is a “base integral form”.
5 Standard integration cycle
We will now discuss the “usual” (in the bosonic string theory) integration cycle on the moduli space of Lagrangian submanifolds.
Remember that our Choice of is such that 
is the algebra of diffeomorphisms of the worldsheet.
This allows us to construct the base form on the space of Lagrangian submanifolds modulo diffeomorphisms.
Remember that on a Lagrangian submanifold from the standard family the
metric 
(same thing as 
) is fixed, and the path integral goes over 
.
This picture explains why we can identify metric with its pullback by a diffeomorphism:
, depending on some motivated choice of the subspace
. We will then implement the standard procedure to construct a closed form on
with some
.
only provides small diffeomorphisms. But then, why do we need this
? Technically it would probably be possible to factor out only large diffeomorphisms without including small ones. (For example, by restricting to constant curvature metrics. However, this would appear quite unnatural.