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:
. 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:
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where 
is the commutator of two vector fields on 
.
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 〛)
which helps to turn it into a base form.
Case of bosonic string
The BV phase space is:
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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.
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:
on remains. It is generated by an exact Hamiltonian:
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This means that:
Gauge symmetries act on the BV phase space
An equivariant form is given by:
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