Bosonic string
Bosonic string belongs to the class of topological quantum field theories of Witten type.
By definition such theories are specified by an action functional which does depend on metric,
but the dependence is be BRST trivial.
In other words we have a family of physically equivalent action functionals labelled by metric.
1 Master Action
The fundamental fields of the worldsheet theory are: matter fields
,
complex structure
satisfying
and the diffeomorphism ghosts
.
The matter part of the action depends on matter fields and complex structure:
The Master Action follows the general scheme for BV structures coming from BRST procedure:
2 Form 
We have a family of Lagrangian submanifolds parametrized by
. The variation
is generated by
. Therefore we have:
We will see in a moment that this is the standard bosonic string measure.
The base analogue is:
3 Choice of Lagrangian submanifold
It would seem to be natural to choose the Lagrangian submanifold setting all the antifields to zero.
But the restriction of 
to this Lagrangian submanifold (i.e. 
)
turns out to be very complicated. (After integrating out , we get the Nambu-Goto string.)
The standard approach in bosonic string is to switch to a different Lagrangian submanifold so that
the restriction of to this new Lagrangian submanifold is quadratic.
Let us choose some reference complex structure 
, for example
, and parametrize the nearby complex structures
by their corresponding Dolbeault cocycles, which we denote 
.
And we rename 
as 
.
We have just changed the polarization; is now a field (called ) and an antifield (called ).
The action can be written in the new coordinates:
4 BRST operator
We expand
in powers of the antifields
and consider only the
linear term. The corresponding Hamiltonian vector field preserves the
Lagrangian submanifold and is the symmetry of the restriction of
on the
Lagrangian submanifold. There are also higher order terms,
because the dependence on
is nonlinear. In constructing the
we simply neglect those higher
order terms; this negligence leads to
being nilpotent only on-shell. Explicitly:
