B model
1 Definitions
We declare that
and
be scalars:
while
and
be components of a 1-form:
The action is:
The BV action is:
2 Action of diffeomorphisms
Let us consider the infinitesimal diffeomorphism (= vector field) of the worldsheet:
The corresponding BV generation function is
where:
Explicitly:
Identities useful in proving this:
3 Closedness of 
We need to prove:
The only nontrivial computation is the bracket of the last term in Eq. (
3) with the last line of (
4):
and the bracket of the term
in Eq. (
3) with
:
Eqs. (
5) and (
6) combine into
which
is the required term
in
requires explanation. As
is a scalar field, its antifield
is a 2-form. The expression
is the usual contraction of a vector field
with the 2-form
should be understood in the following way. We interpret
as a vector-valued two-form, i.e. a section of
. We contract its vector index with the single remaining covector index of
.
