Example: Duistermaat-Heckman integration
Consider the case when our odd symplectic submanifold is the odd cotangent bundle of some
symplectic supermanifold
:
Let
be the projection.
Let
denote the Poisson bivector of
. It defines the following solution of the Quantum Master Equation:
| | | |
| where  |
| | |
| | | |
Let
be a subgroup of the group of canonical transformations
of
. A canonical transformation
of
can be lifted to a BV-canonical transformation
of
as follows:
An infinitesimal canonical transformation of
generated by a Hamiltonian
lifts to an infinitesimal
BV-canonical transformation of
generated by the BV-Hamiltonian
.
Every point 
defines a Lagrangian submanifold in : the fiber 
. Therefore,
we can think of as a family of Lagrangian submanifolds in .
In this case
is the
of the subspace of Hamiltonians generating
.
The equivariant form
becomes:
where
.
as a function on the space of bases (“tetrads”) in
, such that the value on two different bases differs by a multiplication by the super-determinant.
