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Case of bosonic string

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 special property (which we explain on next slide: 〚Special properties of 〛) which helps to turn it into a base form.

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: