Definition and main property of 
1 Definition of
For any function 
, consider the following pseudo-differential form on 
:
, consider the following pseudo-differential form on :
|
we will actually use just
2 is closed
Under the assumption that 
the form is closed.
2.1 Preparation for the proof of closedness
Notice that for any even 
:
:
We can interpret this formula using the notion of
Lie derivative of half-density as follows:
where 
but we prefer to work with
instead of
, because it is
2.2 Proof of closedness
where we have taken into account the
definition of the canonical odd Laplace operator
and the fact that 
.
.2.3 Maurer-Cartan equation
We used the equation:
The strange minus sign can be explained as follows. The Lie algebra of the group 
of canonical transformations
is actually the opposite with the Lie algebra of Hamiltonian vector fields.
of canonical transformations
is actually the opposite with the Lie algebra of Hamiltonian vector fields.3 Infinite-dimensional case
Our proof is rigorous for finite-dimensional odd supermanifolds. But in field theory models there is a complication:
is ill-defined
Indeed, implicitly contains 
which is ill-defined on local functionals
(what is 
?).
implicitly contains which is ill-defined on local functionals
(what is ?).We will resolve this difficulty as follows. First of all, let us define so that
just vanishes on local functionals —
.
This also affects the definition of the Lie derivative 
which we used in the proof of
closedness of . The question is, does the proof of closedness of
work with such “regularized” definition of ?
First of all, we must assume that we still have 
.
This means the absence of anomaly in BRST current, an absolutely essential requirement. This is usually
encoded in the formula:
.
This means the absence of anomaly in BRST current, an absolutely essential requirement. This is usually
encoded in the formula:
With such assumptions:
it is OK to act with
on
But this is not enough, because in our derivation we also act by on functions of 
. Therefore we also need:
on functions of . Therefore we also need:
We will now argue that we can always choose 
, at least locally, so that Eq. (8)
is satisfied. Indeed, let us consider a finite-dimensional family 
of Lagrangian submanifolds
in the vicinity of some fixed 
. For every 
, there is a corresponding Lagrangian
submanifold 
, and some canonical transformation 
such that 
.
Obviously, there is an ambiguity in the choice of , because they are defined up to the stabilizer of .
Let us choose Darboux coordinates 
so that is given by 
.
Consider a subgroup 
generated by the flows of those Hamiltonians which depend only on 
,
but do not depend on 
.
, at least locally, so that Eq. (8)
is satisfied. Indeed, let us consider a finite-dimensional family of Lagrangian submanifolds
in the vicinity of some fixed . For every , there is a corresponding Lagrangian
submanifold , and some canonical transformation such that .
Obviously, there is an ambiguity in the choice of , because they are defined up to the stabilizer of .
Let us choose Darboux coordinates so that is given by .
Consider a subgroup generated by the flows of those Hamiltonians which depend only on ,
but do not depend on .
In the vicinity of
In this case Eq. (8) is automatically satisfied. We also have:
This equation substantially simplifies many formulas. However, we will not assume Eq. (9)
for the following reasons:
such a choice of
definitely exists locally, but there could be global obstacleskeeping
in the formulas is a good consistency check on computations
With such a choice of the proof does go through —
the proof does go through —, the
closedness of the higher components is proven analogously.