Expansion in antifields
On this page we will investigate the structure of
from the point of view of the
antifield expansion. Suppose that we have the solution of the Master Equation in the form of the
series in the antifields:
1 Symmetry modulo BRST-exact expression
Suppose that the leading term is invariant under some Lie algebra
of infinitesimal transformations,
parametrized by
:
Moreover, we assume:
This defines the cohomology class:
Let us
assume that we can construct the full expansion for a ghost number
object:
The class (
42) is an obstacle to choosing
in such a way that:
2 Enters
For our applications, we need a special case when
is BRST-exact:
Moreover, there is a
special case when the expansion of
starts at the linear
order in antifields:
In this case
is
zero on-shell.
Generally speaking, the expansion of
starts with the term constant in antifields.
In this case, all we can say about
is:
3 Particular cases
3.1 Standard BRST, standard polarization, off-shell symmetry
Apriori may be unrelated to the gauge group. However, if
is the gauge group,
then we have:
3.2 Bosonic string, 
being diffeomorphisms
Notice that the term
(constant in the antifields)
is the effect of rotated (with respect to standard) polarization. (In the standard approach
would
be counted as an antifield.)
Moreover, we have:
where
.
3.3 RNS
TODO, but conjectured to be of the same type as bosonic.
3.4 Pure spinor string
TODO:
still 
, althought the expression for is rather complicated
the expansion (43) probably does not terminate on the term linear in the antifields
the possible obstacle (42) is potentially non-zero and should be computed
of something
