Summary
We want to define some pseudo-differential form 
which can serve as string measure.There are several closely related definitions:
1 PDF on the group of canonical transformations
Let 
denote the
central extension
of the algebra of Hamiltonian vector fields by constant Hamiltonians,
and 
the corresponding central extension of the group of canonical transformations.
Possible subtlety: it is not clear to us if this central extension exists globally; for now let us assume that we are working in the vicinity of the unit element of
. The definition actually
depends on the choice of a fixed Lagrangian submanifold 
. Strictly speaking, we
can characterize as a map of the following type:
to every Lagrangian submanifold).2 PDF on the space of Lagrangian submanifolds
|
on 
. A natural question is, does descend to a PDF on 
?
The answer is essentially “yes”, although some minor modifications are needed.
Essentially:
does not have interesting integration cycles. In order to get interesting
integration cycles, we have to also descend to
the factorspace of over some symmetries.3 PDF on an equivalence class of actions
In BV formalism the choice of a Lagrangian submanifold is closely related to the
choice of a quantization scheme. In other words, it is essentially the choice of a representative in
a class of physically equivalent theories.
Given 
and 
, the restriction 
gives a physical action functional which we use in the path integral. A different choice of 
gives a BRST equivalent action functional.
Therefore it would be natural to try to interpret as a PDF on such an equivalence class.
This, however, is not straightforward. The space of Lagrangian submanifolds is actually larger than the
space of action functionals; to descend to the space of action functionals one has to take
the factorspace over the symmetries of . Generally speaking,
does not descend to this factorspace.