Construction of equivariant form
We will now describe an 
-equivariant analogue of 
1 Construction of 
Remember that we required
. This means that
for every
we can unambiguously find the corresponding
element of
, which we will denote
such that:
We can always write, without any further assumptions:
So defined
should satisfy (we abbreviate
to just
):
The first two terms in the
-expansion (which is the same as
-expansion) are:
The
consistency of Eq. (
28) requires that
at
every order in
. This can be proven in the following way:
The first term
is automatically zero because of the Jacobi identity.
The vanishing of
follows from Eq. (
29):
Now we have
is
-closed order by order in
. Let us
assume that
it is actually
-exact order by order in
:
and, moreover, exact in a way compatible with Eq. (
29),
i.e.:
Then we define
to be the next term of the
-expansion of
.
For example, up to the fourth order in
:
2 The equivariant version of 
| | | | |
| | | | |
| where |
| | | |
is a cocycle of the Cartan model of the
-equivariant cohomology of
.
3 Deformations of equivariant
Now let us study the moduli space of solutions 
of Eq. (28)
The tangent space at the point
is parametrized by
satisfying:
Remember that we understand
(and therefore
) as a series in
.
In particular, as we already assumed that the map
is
-invariant, we
should assume the same about
. (Just because
is a deformation of
.)
This imlies Eq. (
32), because
.
The term of the zeroth order in
in
can be interpreted as the deformation of the solution of Master Equation:
(where
is the infinitesimal deformation parameter).
Suppose that
starts with the term linear in
, or, equivalently, with the term linear in
.
Let
denote this leading linear term.
The
-invariance of the map
implies for the linear term:
Moreover, since
starts with the term linear in
, Eq. (
33) implies:
Therefore:
We can caracterize the linear term in the deformations of the equivariant form as
an integrated vertex operator
of ghost number
parametrically dependent on a diffeomorphism
satisfying the equivariance property described by Eq. (
34).
In principle it is possible that the linear term is also zero and
actually starts with quadratic or higher order
term in
. In this case a similar integrated vertex of ghost number
can be defined, but depending parametrically
on an element
.
We do not know nonzero examples of such vertices in actual string theories; we suspect that our construction
of equivariant form is rigid against small deformations.
4 Possible anomaly
We used a
potentially ill-defined
expression
. One obvious way to make it well-defined is to assume that
depends either only on fields,
or only on antifields. That would also imply
.
The assumption that
only depends on the antifields is valid for
bosonic and NSR strings. But for the
case of
pure spinor string depends on both fields
and antifields, so our assumption is
not valid. We feel that the validity of our
conclusions
in such cases depends on additional physical assumption. Roughly speaking:
the symmetry generated by the 
-ghost should be nonanomalous
The validity of this assumption is model-dependent; it is probably stronger than the vanishing of the BRST anomaly.
are:
are:
is
-invariant.
