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
on the worldsheet we should get some function
on the BV phase space such that
action of diffeomorphisms. We should definitely require:
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
acts on from the right
comes from the canonical nilpotent vector field on
to preserve the volume on
. This is some
condition on the trace of the structure constant and the
’s of generators.
But the left
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: