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.

The tangent vector to at a point corresponds to an infinitesimal deformation of . As in Classical Mechanics, such deformations are in one-to-one correspondence with functions on . Since 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 .)

this is closed:

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.

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.