Descent to the moduli space of Lagrangian submanifolds
Let us interpret
as a PDF on
using the formula:
where
means factor over the symmetry:
For the descend to work,
should be base,
i.e. both invariant and
horizonthal, with respect to (
14). Actually, it is invariant and
almost horizonthal. The invariance is straightforward. Let us study
the question of horizonthality, and understand why it is “almost horizonthal”
instead of “horizonthal”. Let
denote a Hamiltonian
whose flux preserves the Lagrangian submanifold
. The horizonthality would
be equivalent to the statement that
is zero. But in fact,
Eq. (
13) implies:
Notice that
implies
.
Apriori this only
implies that
is a constant (but not necessarily zero).
This is potentially a problem (we certainly do want to descend on ).
We will now outline possible ways of resolving this problem:
1 Use 
instead of
is naturally well-defined on the space of
Legendrian submanifolds in some contact submanifold 
which is a
bundle over 
2 Use ghost number symmetry
There is usually a
symmetry called
ghost number. Let us do the following:
This eliminates the possibility of adding a constant to
and therefore
renders
unambigously defined from the variation of
. Our form
is now horizonthal, and descends from
to
.
3 Use transverse Lagrangian submanifold
Suppose that we can find a Lagrangian submanifold
which is transverse to all
Lagrangian submanifolds from our family:
where
is a
marked point on every
.
This eliminates the ambiguity of a constant in
.
What happens if we change
to another transversal Lagrangian submanifold
?
Let us assume that we can choose:
Then we just have to change:
As
is invariant, this would not change the result of the integration.
4 Upgrade to 
The most elegant solution is to use, instead of the space of Lagrangian submanifolds
,
the space
of Lagrangian submanifolds with marked point. A point of
is a pair
where
and
. This defines the double fibration:
Given
, we can consider two projections
and
. Our
is a pseudo-differential form,
i.e. a function of
. We will define it so that it will depend on
only through
.
We can characterize
as a section of
modulo
. We then define
as
follows:
In order to make sense of
we must think of
as a section of
; the fact
that it is only defined up to tangent to
does not matter because
is isotropic. Eq. (
17)
eliminates the ambiguity, and we can now
safely define :
For every function
on
and for any PDF
on
, consider
the product
where:
We will now prove the following formula:
For small deformations of
we can always find canonical transformations
such that:
The introduction of such
is essentially a trick. It does not participate in any way
in the
definition of
; we will use it just to compute
.
We can now use the
moment map
.
We observe that:
—
the subtruction of
is needed to satisfy Eq. (
17).
Therefore:
Using Eq. (
3) with
:
and the fact that
we get:
But we also have to evaluate the exterior derivative of the prefactor:
This concludes the proof of Eq. (
18).
5 Classifying the possible ambiguities
Constant local or multilocal functionals are somewhat rare. As an example of such a functional,
consider 
in the Yang-Mills theory on a compact four-manifold.
In the context of string theory, there are no local constant functionals if the target space
has sufficiently trivial topology.
6 Conclusion
Possible anomalies which could prevent the descent of
from
to
should be analized on the case-by-case basis.
from
down to
