Worldsheet diffeomorphisms

We believe that diffeomorphisms of the string worldsheet are crucial ingredient in string worldsheet theory.

How do they act in the BV phase space?

They should preserve . As they are gauge symmetries, it is natural to
conjecture that their BV hamiltonians should be exact. In other words, for every vector
field on the worldsheet we should get some function
on the BV phase space such that generates the
action of diffeomorphisms. We should definitely require:

where is the commutator of two vector fields on .

What else should we require? We need to turn into a base form. It is already invariant, but it is not horizonthal.

But it has some

Case of bosonic string

The BV phase space is:

Here we quotient by the action of where acts on from the right.
The comes from the canonical nilpotent vector field on .
We want to preserve the volume on . This is some
condition on the trace of the structure constant and the ’s of generators.

But the left action of on remains. It is generated by an exact Hamiltonian:

This means that:

Gauge symmetries act on the BV phase space

An equivariant form is given by: