Bosonic string quartet
We will now show how to integrate over metrics keeping the BRST operator nilpotent off-shell
1 Introducing additional fields
Let us return to our solution of Master Equation given by Eq. (1).
Let us add the “BRST quartet” 
and modify Eq. (1) by adding 
:
The standard choice of Lagrangian submanifold:
leads to degenerate action funcional.
Now let us shift it using the gauge fermion:
This gives the family of Lagrangian submanifolds:
2 BRST operator is nilpotent off-shell
It is straightforward to construct the new Darboux coordinates compatible with the new Lagrangian submanifold;
all we need to do is to shift:
In terms of these new variables:
We observe that:
Therefore, the BRST operator is nilpotent:
3 Form 
Since 
is nilpotent off-shell, we can use our procedure for constructing in BRST formalism.
3.1 For standard family of Lagrangian submanifolds
For the standard family, generated by the family of gauge fermions
parametrized by
as in Eq. (
14),
our formula gives:
where
is the first line of Eq. (
15).
3.2 Generic family of Lagrangian submanifolds
For the general family we have to use our general formula;
there is no further simplification.
3.3 Equivariant
We will define diffeomorphisms in the usual way:
Notice that there are two kinds of symmetries: diffeomorphisms parametrized by 
and 
-shifts parametrized
by arbitrary 
-independent functionals 
.
It seems that we now have a larger symmetry group: instead of just diffeomorphisms we have diffeomorphisms
plus -shifts. This is probably related to adding additional fields.
” labels on antifields.
is the Lagrange multiplier for 
plays the role of 
-ghost
of Eq. (16) is already base.
For the general case, we have to use
our general formula for equivariant 
. There is nothing particularly specific to bosonic string, except for:
should be
-independent follows from
