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

where is a Darboux basis in and its dual

In this formula we define a half-density at 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.

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:

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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 .