Form 
in BV language
We will define form in BV formalism as a pseudo-differential form on the space
of all Lagrangian submanifolds:
|
To define , we need to explain how to calculate it on a point 
.
This point is actually a pair 
where 
and 
is a function on 
describing the tangent vector.
, we need to explain how to calculate it on a point .
This point is actually a pair where
and is a function on describing the tangent vector.
The tangent vector to
at a point
corresponds to an infinitesimal deformation of
is odd, even deformations correspond to odd functions and vice versa. In BV literature
is called “infinitesimal gauge fermion”.
In order to define , we need a half-density 
on 
. We will assume
that satisfies the Master Equation 
.
(The words “BV structure” usually imply both 
and .)
, we need a half-density on . We will assume
that satisfies the Master Equation .
(The words “BV structure” usually imply both and .)
this
Small problem. Suppose that the restriction of 
on is constant. This, unfortunately,
corresponds to zero tangent vector (i.e. the corresponding infinitesimal deformation
of is zero). However, the substitution into will be nonzero.
A differential form which is nonzero on zero vectors is, obviously, inconsistent.
on is constant. This, unfortunately,
corresponds to zero tangent vector (i.e. the corresponding infinitesimal deformation
of is zero). However, the substitution into will be nonzero.
A differential form which is nonzero on zero vectors is, obviously, inconsistent.The right way to deal with this problem is to consider Lagrangian submanifolds with marked points, and require
the vanishing of at the marked point.
Another possibility is to use ghost number symmetry, which is a 
symmetry
typically present in string theory. If we require Lagrangian submanifolds to be -invariant,
then must have ghost number 
and cannot be constant.