Reduction to integration over single Lagrangian submanifold
Here we will explain that integration over a family of Lagrangian submanifolds can be reduced to
integration over a single Lagrangian submanifold in a larger phase space.
A family of Lagrangian submanifolds
defines a single
Lagrangian submanifold in
. We will
describe the construction of
in two steps: first locally in a vicinity of some point
of
, and then globally.
1 Local construction
Let us pick some fixed Lagrangian submanifold
from our family
:
Let us define a “first try” Lagrangial submanifold
as follows:
Locally, in the vicinity of
, we can present
as a family of gauge fermions
(for each
, sufficiently close to
, the
is obtained from
as a deformation
corresponding to the gauge fermion
). Let us consider the following function:
We consider it as a gauge fermion deforming
to some new Lagrangian submanifold.
This new Lagrangian submanifold is what we need; we call it
.
2 Global definition of
As a first step, let us consider a submanifold
which is defined as follows:
This can be promoted to a subspace
quite trivially:
Finally, we will construct
as some section of the vector bundle
restricted to
. Which section?
The simplest guess would be the zero section. However
that would not be a Lagrangian submanifold, so zero section is a wrong guess.
There is, however, a natural nonzero section. It is constructed as follows:
where
computes for every tangent vector to
the value of its corresponding generating function on
.
This section defines our big Lagrangian submanifold:
3 Integration
There is a natural BV Hamiltonian
on
. It descirbes
the lift to
of the natural nilpotent vector field
on
.
We have:
| | | |
| where  |
| | |
4 Proof that 
is Lagrangian
First of all, we have to eliminate the ambiguity in the definition of
. We do this by lifting
to a subset of
. Now to every point
corresponds
, and:
We have to prove that the restriction of the symplectic form of
to
is zero.
We notice that it is equal to the pullback under the natural projection
(see Eq. (
48)) of the following 2-form on
:
Therefore we need to prove that
is zero.
Therefore
vanishes on any pair of vectors
when one of them is tangent to
.
The sum of Eq. (
52) and (
53) is zero, showing that
vanishes when both vectors are transverse to
.
Since is a Lagrangian submanifold, this completes the proof that is zero.
and the zero section
is a function on
. But since we lifted
to
, it defines for us a function on
, which is the same as a section of
. On the other hand, from any section:
of
in the following way: first construct
where
and
. This is what Eq. (50) does.
be a vector field on 
on 
be a vector in 
tangent to 
