Example: interpretation of Cartan complex
Let us consider the case:
The odd tangent bundle
has a natural volume form. This induces a half-density on
:
It satisfies the Quantum Master Equation.
Now consider a different half-density, which also satisfies the Quantum Master Equation:
where
is the BV Hamiltonian generating the lift to
of the canonical odd vector
field
on
.
Suppose that a Lie group
acts on
. This action can be lifted to
and
.
An infinitesimal action of
on
is generated by the BV Hamiltonian:
Therefore, in this case:
This formula can be generalized as follows. Suppose that
is a function on
.
For any
, we think of
as a function on
,
i.e. a
pseudo-differential form on
. With a slight abuse of notaions,
will also denote
the pullback of
along the projection, from
to
. Then we have:
Therefore
is an intertwiner from the
Cartan complex to the complex of equivariant half-densities on
.
with an arbitrary function of
. This will still satisfy the Quantum Master Equation.
