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

#### 2` `Form

#### 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 ).