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:

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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:

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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:

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