Lagrangian submanifold
1 Motivation for changing polarization
Notice that the dependence of 
on the antifields (letters with 
)
is at most
linear. Indeed, this corresponds to “just the usual BRST operator”
of the form 
.
It would seem to be natural
to choose the Lagrangian submanifold setting all the antifields to zero. However,
the restriction of to this Lagrangian submanifold (i.e. 
)
turns
out to be very complicated (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. However there is some price to pay:
BRST operator is only nilpotent on-shell, resulting in a complicated 
structure.
2 Proceed with changing polarization
Let us choose some reference complex structure 
parametrize the nearby complex structures
by their corresponding Dolbeault cocycles, which we denote 
. Locally it is possible
to choose 
.
To summarize:
The other components are:
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:
3 Lagrangian submanifold
We now choose the Lagrangian submanifold in the following way:
On this Lagrangian submanifold the action is quadratic.
In particular:
4 BRST structure
The BRST operator
of the bosonic string can be understood as follows.
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;
is just the linear
approximation. In constructing the
we simply neglect those higher
order terms. This leads to
being nilpotent only on-shell. The explicit formula for
can be
read from Eq. (
3):
5 Ghost numbers
